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BELL'S MATHEMATICAL SERIES 
ADVANCED SECTION 

General Editor-, WILLIAM P. MILNE, M.A., D.Sc. 
Professor of Mathematics, University of Leeds 



A FIRST COURSE IN STATISTICS 



BELL'S MATHEMATICAL SERIES. 

ADVANCED SECTION. 

General Editor: WILLIAM P. MILNE, M.A., D.Sc, 

Professor of Mathematics, University of Leeds. 



AN ELEMENTARY TREATISE ON DIFFEREN- 
TIAL EQUATIONS AND THEIR APPLICA- 
TIONS. By H. T. H. PiAGGio, M.A., D.Sc, 
Professor of Mathematics, University College, 
Nottingham. Demy 8vo. I2s. 

A FIRST COURSE IN NOMOGRAPHY. By S. 
Brodetsky, M. a. , B. Sc. , Ph. D. , Reader in Applied 
Mathematics at Leeds University. Demy 8vo. lOs. 

AN INTRODUCTION TO THE STUDY OF 
VECTOR ANALYSIS. By C. E. Weatherburn, 
M. A., D.Sc, Professor of Mathematics and Natural 
Philosophy, Ormond College, University of Mel- 
bourne. Demy 8vo. I2s. net. 

A FIRST COURSE IN STATISTICS. By D. 
Caradog Jones, M.A., F.S.S., formerly Lecturer 
in Mathematics at Durham University. Demy 8vo. 
15s. net. 

THE ELEMENTS OF NON - EUCLIDEAN 
GEOMETRY. By D. M. Y. Sommerville, 
M.A., D.Sc, Professor of Mathematics, Victoria 
University College, Wellington, N.Z. Crown 8vo. 
7s. 6d. net. 



LONDON : G. BELL AND SONS, LTD. 



^^\ 



raV>s 

A 



FIRST COURSE 



IN 



STATISTICS 



rC^ 



) 



BY 



D? CARADOG JONES, M.A., F.S.S. 

FORMERLY LECTURER IN MATHEMATICS 
AT DURHAM UNIVERSITY 





LONDON 

G. BELL AND SONS, LTD. 

192 I 



PREFACE 

Fifty years ago a large section of the general public were not 
only uninterested in what we now call the social problem, but they 
scarcely gave a thought to the existence of such a problem. They 
felt vaguely perhaps, during periods of acute distress due to lack of 
employment, that all was not well and they thought the Govern- 
ment or possibly the big landowner was to blame, but only the 
more enlightened realized the complexity of the body poHtic and 
how fearfully and wonderfully it is made. To-day all this is changed, 
and comparatively few imagine that a single panacea — the pro- 
hibition of drink, the nationalization of land, or a levy on capital — 
will cure all evils. 

The very fact that nearly the whole civilized world has given 
itself up for over four years to the destruction of life and the dragging 
down of the social fabric in all countries on so vast a scale has 
led to a surfeit and a reaction in which thoughtful men are eager 
to take part in proclaiming again a common brotherhood and in 
building a better world. Those who have always been interested 
in this kind of architecture welcome the change of spirit, but they 
also recognize the difficulty of the task undertaken and the need 
for no little mental effort to second the good- will, which is the jQrst 
essential for success. To pull down no teacher is needed, but we 
must learn to build. 

This leads one to the subject of the present book. The man who 
wishes his work to stand must make sure of its foundations. He 
cannot afford to rest satisfied, as too often the poUtician and social 
worker do, with wild and ill-informed generalizations where more 
exact knowledge is possible, and there are few human problems in 
the discussion of which some acquaintance with the proper treat- 
ment of statistics is not in the highest degree necessary. 



Vi STATISTICS 

Most people, however, are suspicious of figures. They imagine 
that quantitative considerations must of necessity deaden all 
feeling for the purely aesthetic or qualitative spirit which is the 
very life of the phenomena observed or measured. But this surely 
need not be the case. Kepler, when he succeeded in translating 
the motions of the planets into the language of number was not, we 
believe, the less but rather the more enamoured of the beauty and 
order with which the whole of creation is clothed. 

A second reason for suspicion is that partisans of one school or 
another with more push than principle sometimes trade upon the 
general ignorance of statistics to ' prove ' their own pet theories, 
while others no less enthusiastic lead the credulous public into the 
ditch, not with malice intent, but because they are really blind 
themselves to the right interpretation of the figures they so glibly 
quote. 

Although a concern in social questions led the present writer in 
the first instance to study the theory of statistics, there is no reason 
why this bias should prevent the book being of service to those who 
wish to know something of its application in other directions, seeing 
that the general principles underlying the theory are the same in 
all cases, and illustrations have been taken from any field, biological, 
economic, medical, etc., just as they suited the immediate purpose 
in view. 

The author makes no claim to any originality : he is no more 
than a student seeking to put together, with some kind of system 
and as he understands them, the simpler and more important ideas 
he has gathered from other sources. The matter is entirely the 
work of others, the manner only is his own, and he will be happy 
to receive criticism if thereby he may learn more. His chief quali- 
fication for writing is that he has had to worry through most of 
his difficulties alone, and consequently he knows where another 
student is likely to be in trouble better perhaps than the kind of 
writer who is so quick as to be able to see through things at a glance 
or, failing that, so fortunate as to be able to borrow immediate 
light from others. 

The book is divided into two parts. Practically all the first part 
should be well within the understanding of the ordinary person. 



PREFACE vii 

Part II. is more mathematical, but an effort has been made through- 
out to explain results in such a way that the reader shall gain a 
general idea of the theory and be able to apply it without needing 
to master all the actual proofs. The whole is meant, not as an 
exhaustive treatise, but merely as a first course introducing the 
reader to more serious works, and, since real inspiration is to be 
found nowhere so surely as at the source, it is intended to encourage 
and fit him to pursue the subject further by consulting at least the 
most important original papers referred to in the text, only enough 
references being given to awaken curiosity. With the same inten- 
tion a short chapter is inserted after the Appendix by way of sug- 
gesting a few of the sources of statistics Hkely to be of interest to 
the social student. 

Some living writers, notably Professor Karl Pearson, have 
contributed so largely to the development and application of 
statistics that it is impossible to write upon the subject at all without 
incorporating large parts of their work, and the least one can do 
is gladly to record the benefit and pleasure one has received from 
them. The author's indebtedness to the two most important 
English text-books — Yule's Theory of Statistics and Bowley's 
Elements of Statistics — will be evident also to any one who knows 
these books, for they became so familiar through constant study 
that he fears he may have drawn upon them unconsciously even 
to the point of plagiarism in places. 

Finally, he wishes specially to acknowledge the kindness of four 
friends — Mr. Peter Fraser, Lecturer in Mathematics at Bristol Uni- 
versity, without whose encouragement in the early stages the work 
would never have been attempted ; Pjrofessor H. T. H. Piaggio, 
University College, Nottingham, and Mr. A. W. Young, sometime 
Lecturer at the Sir John Cass Technical Institute, London, whose 
criticisms and suggestions were most valuable ; and Professor 
W. P. Milne, of Leeds University, who, both as a practical teacher 
and as Editor of this series, ungrudgingly gave his help and advice. 

D. C. J. 



CONTENTS 



PART I 



II. 
III. 
IV. 

V. 

VI. 

VII. 

VIII. 
IX. 



- X. 
XL 



Introductory — Early Historical Beginnings : Logical 
Development ...... 

Measurement, Variables, and Frequency Distribution 

Classification and Tabulation 

Averages ...... 

Averages (continued) — Applications of Weighted Mean 

Dispersion or Variability . . . ' . 

Frequency Distribution : Examples to illustrate Calcu 
lating and Plotting : Skewness 

Graphs — Correlation suggested by Graphical means 

Graphs (continued) — Graphical Ideas as a Basis for Inter 
polation : Reasoning made Clear with the help of 
Graphs or Curves ..... 

Correlation ...... 

Correlation — Examples .... 



1 
5 

14 
22 
32 
42 

52 
68 



85 
102 
115 



PART II 

XII. Introduction TO Probability AND Sampling . . 132 

XIII. Sampling (continued) — Formul.*: for Probable Errors . 150 

:► XIV. Further Applications of Sampling Formulae . . 165 
XV. Curve Fitting— Pearson's Generalized Probability 

Curve ........ 178 

XVI. Curve Fitting (continued) — The Method of Moments for 

connecting Curve and Statistics .... 194 

XVII. Applications of Curve Fitting .... 206 

XVIII. The Normal Curve of Error . . . . .231 

XIX. Frequency Surface for Two Correlated Variables . 249 



APPENDIX 

Certain Current Sources of Social Statistics 
A Note on Tables to aid Calculation 



263 
279 
284 



ix 



I 



PART I 
CHAPTER I 

mTRODUCTORY 

Early Historical Beginnings. Statistics, more or less valuable, 
have been compiled in most civilized countries from very early 
times. One reason for doing this on a large scale has been to 
ascertain the man-power and material strength of the nation for 
miUtary or fiscal purposes, and we read in the Old Testament of 
such censuses being taken in the case of the Jews, while among the 
Romans also it was a common practice. 

In England, as economic terms began to be used and their mean-' 
ings analysed, and especially during the period when the mercantile 
system prevailed, and the Government endeavoured so far as was 
practicable to direct industry into channels such that it would add 
most to the power of the realm, men tried frequently to base argu- 
ments for social and political reform upon the results of figures 
collected. A distinct advance had been made in the seventeenth 
century when mortality tables were drawn up and discussed by Sir 
William Petty and Halley, the famous astronomer, among others, 
and their labours prepared the way for a more scientific treatment 
of statistical methods, especially at the hands of one, Siissmilch, a 
Prussian clergyman, who published an important work in 1761. 

It is almost true to say, however, that until the time of the great 
Belgian, Quetelet (1796-1874), no substantial theory of statistics 
existed. The justice of this claim will be recognized when we 
remark that it was he who really grasped the significance of one 
of the fundamental principles — sometimes spoken of as the constancy 
of great numbers — upon which the theory is based. A simple illus- 
tration will explain the nature of this important idea : Imagine 
100,000 EngUshmen, all of the same age and living under the same 
normal conditions — ruling out, that is, such abnormalities as are 
occasioned by wars, famines, pestilence, etc. Let us divide these 
men at random into ten groups, containing 10,000 each, and note 
the age of every man when he dies. Quetelet's principle lays 

A 



2 STATISTICS 

down that, although we cannot foretell how long any particular 
individual will live, the ages at death of the 10,000 added together, 
whichever group we consider, will be practically the same. De- 
pending upon this fact insurance companies calculate the premiums 
they must charge, by a process of averaging mortality results re- 
corded in the past, and so they are able to carry on business without 
serious ,risk of bankruptcy. 

As a distinguished statistician once said, ' By the use of statistics 
we obtain from milliards of facts the grand average of the world.' 
But if the average resulting from our observations were subject to 
violent fluctuation as we passed from one set of facts to another 
cognate set there would be little satisfaction in finding it. It is 
the comparative constancy of the average, if the number of our 
observations is large enough, which makes it so important, as 
Quetelet observed, for although the idea was not altogether new he 
first realized how wide an application it had and how fruitful of 
practical results it might prove. 

Quetelet was born in Ghent, and taught mathematics in the 
College there in his early youth. After graduating as Doctor of 
Science he became Professor of Mathematics in Brussels Athenaeum 
when only twenty-three years old, and later he was made Director 
of the Brussels Observatory, in the foundation of which he had 
taken a leading part. In 1841 he was appointed President of the 
Central Commission of Statistics, where he was in a position to 
render valuable assistance to the Belgian Government by his advice 
on important social questions. He initiated the International 
Statistical Congress, which has served to bring together the leading 
statisticians of all countries, and the first meeting was held in 1853 
at Brussels. His death occurred at the ripe age of seventy- eight. 

Some idea of the extent of Quetelet's statistical researches may 
be gathered from the titles of his chief works : (1) Sur Vhomme et 
le developpement de ses facuUes, ou essai de physique sociale (1835) ; 
(2) Lettres . . . sur la theorie des probabilites appliquee aux sciences 
morales et politiques (1846) ; (3) Du systeme social et des lois qui le 
regissent (1848) ; (4) U AnthropomHrie, ou mesure des differentes 
facultes de Vhomme (1871). 

In his writings he visuaUzes a man with qualities of average 
measurement, physical and mental (V%omme moyen), and shows 
how all other men, in respect of any particular organ or character, 
can be ranged about the mean or average man, just as in Physics 
a number of observations of the same thing are ranged about 
the mean of all the observations. Hence he concluded that the 



INTRODUCTORY 3 

methods of Probability, which are so effective in discussing errors 
of observation, could be used also in Statistics, and that deviations 
from the mean in both cases would be subject to the binomial law. 

Hain in Vienna put some of Quetelet's ideas to good service in 
1852, employing a superior method for the calculation of statistical 
variability. Knapp and Lexis in Germany, also following up 
Quetelet's principles, made an exhaustive investigation several years 
later of the statistics of mortality, and their work has been extended 
in many directions, and in our own time notably by Galton, Karl 
Pearson, and Edgeworth. 

The name of Sir Francis Galton (1822-1911), to whose work as 
a pioneer the science of Statistics owes so much, is deserving of 
even greater honour than it has yet received. Founder of the School 
of Eugenics, Galton himself came of famous stock, being grandson 
of Erasmus Darwin and a cousin to Charles Darwin. He studied 
medicine in early youth, but after graduating at Cambridge his 
attention was turned to exploration, and the Royal Geographical 
Society awarded him a gold medal on the results of his investiga- 
tions in South- West Africa. His first great work on heredity was 
not published till 1869, after he had already earned distinction in 
other directions, for he was elected a Fellow of the Royal Society 
in 1860. Alive with new ideas, marvellously patient and persistent 
in bringing them to the test of observation — qualities essential for 
real scientific research — he set himself to inquire into the laws 
governing the transmission of characteristics, physical and mental, 
from one generation to another. Large tracts of this ground have 
since been carefully explored and mapped out by the school of 
his great successor, Karl Pearson, who has originated formulae for 
testing the extensive anthropometrical and biological data col- 
lected. Largely as a result of their work it is now widely recognized 
that ' the whole problem of evolution,' as Professor Pearson himself 
has well said, ' is a problem in vital statistics — a problem of longevity, 
of fertility, of health, and of disease, and it is as impossible for the 
evolutionist to proceed without statistics as it would be for the 
Registrar-General to discuss the national mortality without an 
enumeration of the population, a classification of deaths, and a 
knowledge of statistical theory.' 

Logical Development. The best way to approach the study of 
any subject, if one had time, would be along the lines of its historical 
development, but these lines seem so often to diverge from the 
main theme, like branches from the parent stem of a tree, that 



4 STATISTICS 

when one tries to describe them the general effect is apt to be some- 
what confusing. It is therefore usually the custom to adopt a 
logical rather than a historical sequence, but it may assist the reader 
to see the connection between the two and the unity which embraces 
the whole if we now briefly trace the natural growth of the subject, 
suggesting the steps we might expect it logically to take. This we 
have tried to keep in view as nearly as possible in the succeeding 
chapters, except that the order may have been altered here and 
matter may have been omitted or inserted there as* reason and 
the elementary nature of the work dictated : — 

1. Owing to the difficulty which the mind experiences in grasping 
a large mass of figures, the necessity for an average arises to sum 
up shortly the character of the mass, and various kinds of averages 
are proposed. 

2. An average proves insufficient alone to define the whole scheme 
of observations, and other constants are invented to measure their 
spread or dispersion about the average. 

3. Considerations of space and the desire for some kind of system 
lead further to the formation of tables with the observations classi- 
fied in ordered groups. 

4. The formation of these tables suggests the possibility of a 
graphical representation of the numbers in the different groups to 
bring out the nature of their distribution. 

5. The impossibihty of dealing with a whole population results 
in the selection of samples, and the comparison of one sample with 
another introduces the subject of random errors. 

6. The closer examination of this subject leads us into the domain 
of mathematical probability and discovers the probabiHty curve, or 
normal curve of error, first formulated in connection with the study 
of errors of observation. 

7. This same curve serves in the sequel to describe a certain 
important type of statistical distribution, in which each observation 
is determined by a multitude of so-called chance causes puUing this 
way and that, so that it is impossible to foretell what the resultant 
effect will be. 

8. The failure of the normal curve to describe other common dis- 
tributions, especially those which are unsymmetrical in character, 
leads to the development of skew varieties of curves which will 
fit them. 

9. The extent of connection between one set of data and a pos- 
sibly related set is a natural subject for inquiry giving rise to the 
theory of correlation. 



CHAPTER II 

MEASUREMENT, VARIABLES, AND FREQUENCY DISTRIBUTION 

Measurement. There are two fundamental characteristics which 
pertain to nearly all measurement : it is (1) relative : it involves 
a comparison between one magnitude and another of the same kind, 
and (2) approximate : the comparison in practice cannot be made 
with absolute exactness. 

A man's height, for example, is stated to be 5 ft. 8| in., but this 
would convey Httle to one who did not know how long a foot was 
and how long an inch was. The first step in the measurement is 
made by comparing the man's length with a certain constant 
length previously agreed upon as a standard or unit, namely, a 
' foot ' ; he is placed to stand up against a scale which is divided 
up into feet, and the highest point of his head is seen to come 
somewhere between the 5 ft. line and the 6 ft. Hne : he is there- 
fore longer than five of these units, set end to end, but not so long 
as six of them. To carry the measurement a stage further a smaller 
unit has to be introduced ; each foot length of the scale is sub- 
divided into twelve equal parts called inches, and the top of the 
man's head is found to come somewhere between the 5 ft. 8 in. 
line and the 5 ft. 9 in. line : he is therefore over 5 ft. 8 in., but 
not quite 5 ft. 9 in. in height. For the next stage in the measure- 
ment each inch of the scale has to be further subdivided into quarter- 
inches, and the top of the man's head is found to come somewhere 
between the 5 ft. 8 in. 3 qu. in. line and the 5 ft. 9 in. line ; more- 
over it is nearer, let us suppose, to the former line than to the latter. 
In this case, then, we say that the man's height or length is 5 ft. 
8f in., measured to the nearest quarter inch. 

In measurement the decimal notation has very obvious advan- 
tages, because each unit is always divided into ten equal parts to 
get the next smaller unit. Thus a weight of 7 kilogr. 5 hectogr. 
3 decagr. 8 gr. 4 decigr. 3 centigr. can be expressed at once in 
grammes, namely 753843 gr. ; hence if we were measuring to the 
nearest decagramme, the result would be expressed as 764 decagr. ; 
to the nearest decigramme, it would be 75384 decigr., etc. 



6 STATISTICS 

Similarly, a length of 12 kilom. 7 metres 2 centim. can be written 
12007-02 metres, or, in kilometres, 12*00702 kilom., or, to the nearest 
decametre, 1201 decam., and so on. 

The mere act of counting things of a like kind is, in a sense, 
measurement of a primitive type, one thing being the linit, though 
the measurement may in many such cases be exact ; for example, 
we may count the number of persons in a room exactly. Even in 
this type of case, however, the counting or measuring cannot 
always be done accurately, but the inaccuracy arises from lack of 
precision and uniformity in definition rather than from want of 
power in the measuring instrument itself : e.g. in determining the 
population of a city, inaccuracies may arise because of failure to 
define exactly the boundaries of the city, or the time at which the 
census is to be taken, or how to deal with the migration of the in- 
habitants from or into the city, and with births and deaths during 
the actual time of numbering. 

Variables. By a variable is meant any organ or character which 
is capable of variation or difference in size or kind. The difference 
may be measurable as in the case of head-length, height, tempera- 
ture, etc., or not directly measurable as in the case of colour, intelli- 
gence, occupation, etc. Further, the variation, when measurable, may 
be continuous, or it may take place only by integral steps, omitting 
intermediate values : population, for example, can never go up or down 
by less than one, but if temperature is to change from 60 degrees to 
61 degrees it must pass continuously through every intermediate 
state of temperature between 60 degrees and 61 degrees. 

In dealing with a measurable variable sometimes we are inter- 
ested not so much in its actual value at a particular instant as in 
the change which has taken place in its value during some specified 
interval, but to gauge fairly the amount of this change it is necessary 
to measure it relative to the original value of the variable. For 
example, if we are told that the wages of a certain person have 
gone up during the year to the extent of 3d. an hour, we cannot 
say whether this is much or little to him until we know what his 
wages were originally. The addition would be relatively much less 
if he were a skilled patternmaker earning Is. 6d. an hour than it 
would be if he were a chainmaker earning only 6d. an hour.* This 
point can be met by stating, not simply the change in the value of 
the variable, but the ratio of the new value to the old. For instance, 
the patternmaker in the above instance has had his wages increased 

[* Wages to-day are, of course, much higher — the above figures are only hypothetical.] 



MEASUREMENT, VARIABLES, FREQUENCY DISTRIBUTION 7 

in the ratio of Is. 9d. to Is. 6d. It is important to notice that 
this form of measurement is quite independent of the particular 
units used ; if we take Id. as unit, the ratio=21/18=7/6, and if 
we take Is. as unit, the ratio=l|/IJ=7/6 just as before. 

There are other ways of measuring this change in the value of 
a variable. One of the commonest is to express it as a percentage 
of the original value ; thus the patternmaker's increase is at the 
rate of yVxlOO, or 16f per cent., which is simply the ratio of 
increase in wage to previous wage multipUed by 100. The multiplier, 
lOG, is quite an arbitrary factor, but it has obvious advantages : among 
others, it works well with the decimal notation and it often serves 
to put the result into a form which is greater than unity instead of 
leaving it as a fraction. Again, a man who gets a dividend of £25 
on an investment of £500 receives interest at the rate of -^ X 100, 
or 5 per cent. ; in other words, this is the rate at which his capital 
accumulates if the interest is added to it instead of being spent. 

Annual birth rates and death rates, on the other hand, are best 
expressed per thousand of the population, as estimated, say, at 
the middle of the year in question ; e.g. the birth rate of the United 
Kingdom in 1911 was 24-4 per thousand, and the death rate was 
14-8 per thousand, which is equivalent to 244 and 148 per 10,000 
of the population respectively. If we could assume the birth 
and death rates to remain constant from year to year, and if we 
could afford to leave migration out of account, the population 
would be subject to exactly the same law of increase as capital 
accumulating at compound interest [see Appendix, Note 1], thus : — 

1. If P be the original population, and if the annual net increase 
be at the rate of 25 per thousand, then 

the population in I year's time=Px (1*025) 

2 „ =Px (1-025)2 

3 „ =Px (1-025)3 
n „ =Px (1-025)". 

2. If £P be the original capital, and if the annual increase be at 
the rate of 2 J per cent., then 

the capital in 1 year's time==Px (1-025) 
„ 2 „ =Px (1-025)2 

„ 3 „ =Px (1-025)3 

„ „ n „ =Px (1-025)". 

Lest we may seem to have laboured to make plain what is really 
a simple idea, it may be remarked that quite frequently confusion 
arises with regard to percentage even in reputable quarters. As an 



'8 



STATISTICS 



illustration of the kind of mistake which, without thinking, is easily 
made, the following argument has been taken from a monthly 
circular sent out a little while ago to the members of the Boiler- 
makers' Society by their Secretary : Since July 1914, wages have 
risen 15 jper cent., the cost of living has gone wp 45 'per cent., therefore 
the workers' real wages have fallen 30 per cent. This same argument 
was quoted shortly after in one of the leading articles of The Man- 
chester Guardian under the heading ' Prices and Wages,' and again 
in The Labour Leader tersely as truth ' In a Nutshell,' but in 
neither instance did it seem to have occurred to the writer that it 
was inaccurate. It may be worth while for the sake of clearness to 
show what the statement should have been : — 





Wages. 


Cost of 
Living. 


Ratio of Wages to 
Cost of Living, 


Same Ratio 
multiplied by 100. 


July 1914 . 
October 1916 . 


100 
115 


100 
145 


1 


100 
79 



Since 



11 5 

14^ 



X 100 is roughly 79, this calculation shows that ' real 
wages ' had faUen only about 21 per cent. (100—79=21), and not 
30 per cent, as stated, between the two dates. 

Index Numbers. A very important case of variables changing 
with time appears in the discussion of changes in the value of 
money as measured by the movement of prices of commodities, 
introducing the notion of an index number. For example, supposing 
the wholesale price of beef was 6d. a lb. at one date, 8d. a lb. at 
another date, and 5Jd. a lb. at a third date, the change might be 
exhibited as in the following table : — 





1st Date. 


2nd Date. 


3rd Date. 


Price of beef 


6d. 
100 


Sd. 
133 


5K 
92 



Here 100, 133, and 92 are called index numbers, the price at the 
first date being taken as a standard and denoted by 100, while 
the prices at the other two dates are altered proportionally, so that 

6:8:5J=100:I33:92. 

Index numbers calculated on this principle have been published 
systematically for several years by Mr. A. Sauerbeck (in the Journal 



MEASUREMENT, VARIABLES, :E'REQUENCY DISTRIBUTION 9 

o/ the Royal Statistical Society up to January 1913, and continued 
afterwards in The Statist under the supervision of Sir George Paish) 
and in The Economist. 

In Sauerbeck's index numbers the average wholesale prices of 
forty- five commodities for the eleven years 1867-77 are taken as 
the standard, being denoted each by 100 as above, and the prices 
of the same commodities for any other year are then written as 
percentages of these standard prices. The commodities chosen are 
various — ^food of all kinds (cereals, meat, potatoes, rice, butter, 
sugar, coffee, tea), minerals (including coal), textiles, and sundries 
(including hides, leather, tallow, palm oil, olive oil. Unseed, 
petroleum, soda, soda nitrate, indigo, timber). Articles of similar 
character are grouped together ; naturally no class is exhaustive, 
but the selection is a fairly representative one. A sort of general 
average is then formed by combining all the results, and the move- 
ment of this average is taken to measure changes in the value of 
money. An example will make clear the way in which an index 
number for each group and the general average are obtained. 

The index number for each separate commodity may be first 
calculated thus : — 



Price of English Wheat. 


Years, 


QtX: ; Index Number. 


1867-77 
1912 . 


s. d. 

54 6 100 

34 9 64 



Now forming similar index numbers for each of the eight vegetable 
and cereal foods and combining them together, we have : — 

Index Numbers for Vegetable and Cereal Foods. 

























Years. 




1^ 




0) 

T 


1 




1 


i 


^1 


< 


1867-77 . 


100 


100 


100 


100 


100 


100 


100 


100 


800 


100 


1912. 


64 


68 


70 


79 


83 


85 


74 


101 


624 


78 



The figures in the last column but one are obtained by simply 
adding the figures in the eight previous columns, and, dividing these 



10 



STATISTICS 



results by eight, we get the average index number for the group 
in 1912 as a percentage of that in the standard years 1867-77. 

Treating all the other commodities in the same way we ultimately 
get index numbers for all the different groups and for all com- 
modities combined as follows : — 

Index Numbers for different Groups and 
FOR ALL Commodities. 



No. of CoTnmodities 


8 


7 


4 


19 


7 


8 


11 


45 


Years. 




< 


6 


1 


in 

1 


X 


w 

1 

3 

SO 


OS 

S 

6 


1867-77 . 

1912 .... 


100 

78 


100 
96 


100 
62 


100 

81 


100 

no 


100 
76 


100 

82 


100 

85 



The index number for ' All Food ' is obtained by summing the 
nineteen index numbers for the separate commodities which are 
included in this class and dividing the result by 19. Similarly the 
general index number for all commodities is obtained, not by 
adding the numbers for the different groups and dividing by the 
number of groups, but by adding the forty-five index numbers of 
ail the separate commodities and dividing the result by 45. 

In The Economist the average prices of twenty-two commodities 
for the years 1901-5 are taken as the standard, being denoted 
each by 100, and the prices of the same commodities for any other 
year are then written as percentages of these standard prices ; the 
sum of these percentages is taken as the index number, and it is 
a simple matter to divide by 22 if we wish to get the average per- 
centage change. The following table explains the method of 
calculation : — 

Index Numbers aHbESENTiNG Prices of Commodities 



Date. 


Cereals 

and 
Meat. 


^^L Textiles. 


Minerals. 


Miscel- 
laneous. 


Total. 


Index No. 


22. 


1901-5 . 

End of Dec. 1916 


500 1 300 500 
1294 553 1124-5 


400 
824-5 


500 
1112 


2200 
4908 


100 
223 



MEASUREMENT, VARIABLES, FREQUENCY DISTRIBUTION 11 

In this table five commodities are included under the head of 
* Cereals and Meat,' three under ' Other Foods,' and so on. The 
numbers in the last column are obtained by dividing those in the 
previous column by 22. 

It is clear that what is at bottom the same principle may be 
appHed in any case of a variable changing with time when we wish 
to measure the extent of the change, so that the use of index numbers 
is not confined to the problem of prices. We shall return again to 
discuss one or two further points in connection with the same 
subject in the Chapter on ' Averages.' 

Frequency Distribution. So far we have been thinking more 
particularly of the change which an individual variable, or a col- 
lection of such variables, may undergo in the course of time, or the 
difference between two values which the same variable may have 
at two different instants of time, and how to measure it. Now 
the science of Statistics is based upon the study of the crowd 
rather than of the individual, although observations on individuals 
have to be made before they can be combined together to produce 
the crowd, just as individual income-tax schedules have to be 
completed and combined before the balance-sheet of the State can 
be drawn up. As we pass from one individual to another there 
may be great differences in the organ or character observed — hence 
the word variable already introduced — but in the mass these differ- 
ences are merged together and lose their individual importance : 
it is rather their resultant effect we seek to measure. In order 
therefore to discover this effect it is necessary to make a collection 
of individual observations and to analyse the results. Now if our 
ultimate conclusions are to be safe the number of observations 
must be considerable, and in order to be able to cope with them 
and reduce them to some sort of system the first step in the analysis 
consists in arranging them in different classes according to the 
value of the variable under consideration. 

It is to be noted that now we are ""frtpr with changes in the 
value of a variable as we pass from onRy^ff0>midual to another at the 
same period of time and under the same ^ifteral conditions, and not 
with the change in a variable in the same individual occurring with 
the lapse of time. We wish, for example, to draw a distinction 
between (1) the change in wages as we pass from one man to another 
at the same time in the same trade, and (2) the change in wages of 
the same man, or class of men, in the same trade occurring in a 
given period of time ; in the fii'st case we want to find the amount 



12 STATISTICS 

of diversity within the trade at some stated time, and in the second 
our object is to discover whether an improvement has taken place 
in the wages of a particular individual or a particular trade with 
the passage of time. 

In picturing variation of the first type the conception arises of a 
frequency distribution where the observations are distributed in 
ordered groups, with a number corresponding to each showing 
how many, or how frequent, are the individuals possessing the type 
of variable or character which defines that group. More generally, 
if a series of measurements or observations of a variable y are 
made corresponding to a selected series of another variable x we 
get a distribution, which becomes a frequency distribution when y 
represents the frequency of events happening in a particular way, 
or of individuals corresponding to a particular value of some 
common variable or character, represented by x. Thus (1) the 
boys in a school might be grouped according to their intelligence : 
so many, dull ; so many, of ordinary intelligence ; and so many, 
bright or above the ordinary. Again (2) in an inquiry into the 
housing of the people in any town or district it would be necessary 
to draw up a table showing the number or frequency of existing 
tenements with one room, the frequency of tenements with two 
rooms, the frequency of tenements with three rooms, and so on. 
Once more (3) a zoologist, wishing to discover whether crabs of a 
certain species caught in one locality differ in any remarkable way 
from members of the same species caught in another locality, might 
start by making measurements of the length of carapace or upper 
shell for crabs of like sex in the two places and then proceed to 
form frequency tables for each, setting out the frequency of crabs 
for which the carapace length lies, say, between 5 and 6 millimetres, 
the frequency with length between 6 and 7 millimetres, the frequency 
with length between 7 and 8 millimetres, and so on. He would 
then have in these tables some basis for comparing the specimens 
caught in the two locaHties. 

The three illustrations just used give three different types of 
distribution corresponding to the three types of variable to which 
attention has been drawn before. In the first, where the variable 
or character observed is not measurable, doubt will sometimes 
arise as to the appropriate class in which individuals should be 
placed who seem to be on the border line between dulness and 
mediocrity or between mediocrity and brilliance, so that accurate 
classification will greatly depend upon what is called the ' personal 
equation ' of the observer. The second illustration corresponds 



MEASUREMENT, VARIABLES, FREQUENCY DISTRIBUTION 13 

to the case where the variable changes not continuously but by 
unit stages ; the choice of classes in such a case depends little 
upon the observer unless the unit is very small compared to the 
total range of variabiHty; for example, a tenement might either 
definitely have two rooms or it might have three rooms, but it 
clearly could not be put down as having 2J rooms or 2^ rooms : 
in other words, the only natural classification is so many tenements 
with two rooms, so many with three rooms, so many with four 
rooms, and so on, though here too some confusion might arise 
through failure to define clearly what is ' a room.' In the third 
tjrpe, where we can conceive of the continuous variation of the 
character under observation, there would be nothing surprising in 
the appearance of any value of the variable between the lowest 
and highest values observed ; the choice of suitable limits for the 
several groups becomes therefore in this case rather a delicate 
matter which requires careful judgment. 

We shall begin the next chapter with some general remarks 
upon the subject of classification and tabulation. 



CHAPTER III 

CLASSIFICATION AND TABULATION 

No part of Statistics is of more importance than that which deals 
with classification and tabulation, and it is the one part for which 
no very precise rules can be given. A neat arrangement of ideas 
in the mind, capacity to express them clearly, and patience are 
indispensable, but experience alone will convince one of the extreme 
care which must be exercised if blunders are to be avoided and 
time is to be saved in the long run. This has to be emphasized 
because most people, until they have tried and failed, imagine 
that to arrange things in classes and in tables is a straightforward 
proceeding involving no great thought or trouble. 

Abundant matter of a statistical character is published periodi- 
cally in Blue-books, Government Reports, Reports of Local Authori- 
ties, Directors of Education, Medical Officers of Health, Chief 
Constables, Employers' Associations, Trade Unions, Co-operative 
Societies, etc., but it needs a trained intelligence as a rule to assimi- 
late it and turn it to further advantage. The larger the scale upon 
which any inquiry is made, the more valuable should the results 
be, granted that equal accuracy is possible on the large as on the 
small scale, but it is fairly clear that mistakes of various kinds 
have also much more chance of creeping into a large work than into 
a small one. To appreciate the various and numerous' possibilities 
of error when the scope is wide it is enough to read the introduc- 
tions to the Registrar-General's Reports on the Census from decade 
to decade ; this should also impress the student with the care that 
is necessary if he proposes to use such material for the investigation 
of some other problem. It may seem a comparatively simple task 
to abstract two sets of figures from a Census Report, to establish 
a one-to-one correspondence between them, and to make deductions 
therefrom, but such figures when taken from their context will 
sometimes lead to absolutely unsafe,- if not false, conclusions. The 
exact meaning and limitations of any data can only be properly 
appreciated by one who has been closely in touch with the persons 
who have collected them, and it is therefore important, before 

14 



CLASSIFICATION AND TABULATION 15 

attempting to re- classify or re- tabulate any old statistics for a new 
purpose, to read very carefully through the notes made by the 
original compilers. 

Perhaps the best advice that can be given to any one in this 
connection is that he should embark upon some small inquiry 
which will necessitate the collection of statistics for himself ; the 
final result of his efforts may seem disappointing, but the experi- 
ence he will gain will be invaluable. Ideas for such an inquiry will 
occur to him if he reads through some authoritative work on social 
questions, e.g. Beveridge's Unemployment, the decennial Census 
Reports, or The Minority Report on the Poor Law (1905). But he 
must read with an open and critical mind, questioning particularly 
the foundation for all statements as to cause and effect which may 
be made. A few simple hints may be useful as to method of 
procedure. 

When he thinks he has discovered some subject of interest which 
would appear to deserve examination, it wiU be well to put it 
down on paper in order to get it clearly defined, because a precise 
written statement is likely to carry one further than a shadowy 
idea somewhere at the back of the mind which is hardly formu- 
lated at all. When the actual collection of statistics is begun 
it will almost certainly be found that it is impossible to solve the 
original problem contemplated ; but that need not prevent further 
progress — what is important is that the limitations should be 
exactly realized, and this will be impossible unless the original 
problem is clearly presented side by side with the nearest solution 
obtainable. 

The problem stated, the next thing is to set down categorically 
a number of questions, the answers to which are to be the raw 
material for the solution of the given problem. For the answers 
let us assume the inquirer is dependent upon the goodwill of others, 
either employers, or trade union secretaries, or public officials. 
The questions in that case must be clearly, concisely, and courteously 
phrased, and must not be capable of more than one interpretation. 
In number they should be few and in character not inquisitorial ; 
moreover, the replies should be obtainable without any great labour 
on the part of the persons approached. Here again it will be fou^nd 
that the questions first set down are not all satisfactory : one will 
be too vague ; another, though clear enough, may involve a con- 
siderable search through a mass of other matter before it can be 
properly answered ; while to another it might be impossible to give 
an exact reply in any case. Revision and amendment may there- 



16 STATISTICS 

fore be necessary in the light of the first replies received, and the 
inquirer will begin to see at this stage how far the solution to his 
original problem is reaUy possible. 

When the bulk of the returns have come in they should be critically 
examined one by one. A number will, for one reason or another, 
be worthless, and they must be discarded ; as for the remainder, 
if the questions were well chosen, the answers should not be difficult 
to interpret and classify ; the most successful questions are those 
to which a simple ' yes ' or ' no ' in reply gives all the information 
required ; numerical answers are less easy to deal with, especially 
if there is the least chance of misunderstanding on either side as 
there often is, for example, in the case of observations which are 
on the border line between two classes. 

Tables should then be drawn up and the headings to the different 
columns of the tables should state concisely and exactly what the 
figures below represent. So far as possible any one should be able 
readily to grasp their general meaning without being obliged to 
wade through a page or two of written explanation ; if any heading 
cannot be clearly expressed in a few words it may be helped out 
by a further note at the bottom of the page, but too many such 
notes are to be avoided. 

Finally, a summary should be made of the various conclusions 
suggested by a study of the tables. Some of the points raised in 
the course of the inquiry will perhaps be only incidental to the 
main problem under discussion, but may still deserve a passing 
reference. It will also be of advantage to foUow up the summary 
by any recommendations which can be fairly based on the con- 
clusions obtained, when the problem is such that recommendations 
are expedient, and, if ultimately the whole is of sufficient value to 
be printed, emphasis can be introduced where necessary by suitable 
variations in t3rpe. 

For this part of the work considerable judgment is necessary 
which can only be acquired by long training — a faculty to pick out 
the real from the false and an eye to distinguish the important from 
the trivial. A sense of numerical proportion too is desirable inci- 
dentally ; one of our leading exponents on finance in a book dealing 
with the meaning of money uses a very interesting illustration which 
is perhaps worth quoting here to show how even an acute mind 
may on occasion prove itself curiously lacking in such a sense. 
He is seeking to show how the credit system of the country is built 
upon a foundation composed of a little gold and a lot of paper ; 
for this purpose he amalgamates together the balance-sheets of half 



CLASSIFICATION AND TABULATION 



17 



a dozen big banks, and proves that their habilities on current and 
deposit account amounted at a certain date prior to 1914 to 249 
million pounds, while the cash in hand and at the Bank of England 
was 43 millions. Of the 43 millions he estimates that roughly 
20 millions would be cash in the Bank of England, and further 
that about two-thirds of this 20 millions would be represented really 
by securities and not by gold. Hence he concludes that to support 
this vast erection of credit there would only be £6,666,666 of actual 
gold. Thus after talking throughout in millions the author closes 
by giving his i-esult true apparently to a pound ! 

Much may be learnt as to methods of classification and the 
drawing up of tables by a careful study of those which appear in 
various official reports, and a few such tables are reproduced in 
the pages which follow. 



Table (1). Condition as to Cleanliness of 
School Children in Surrey. 



Cleanliness. 


5 years, 1908-12. 79,070 children inspected. 


Above the average . 
Average 
Below average 
Much below average 


15-4 per cent. 
76-5 

7-6 

0-5 „ 



Table (2). Condition as to Infectioijs Diseases of 
School Children at Different Ages in Surrey (1913). 



Age Groups inspected 


5-6 


8-9 


13-14 


Total at 
All Ages. 


Numbers inspected 


5,191 


5,151 


4,962 


15,304 


Proportion who before inspe 


c- 








tion had suffered from — 


per cent. 


per cent. 


per cent. 


per cent. 


Diphtheria . 


1-3 


3-5 


5-4 


3-4 


Scarlet fever 




2-7 


7-2 


10-9 


6-9 


Measles 




55-3 


79-3 


84-6 


72-9 


Whooping cough 




41-8 


56-4 


54-3 ! 


50-9 


German measles 




1 2-9 


51 


7-5 • 


51 


Chicken pox 




1 261 


401 


38-6 


34-9 


Mumps 




i 10-6 


220 


29-8 


20-7 


No infectious diseases 


18-9 


61 


4-7 


100 


No definite information 


3-3 


2-2 


0-9 


2-2 



18 



STATISTICS 



Table (3). Height op School Children according to 
District, Age, and Sex (1913). 



Age 
' Groups. 


Boys. 


Girls. 


1 

Nos. 
measured. 

1 " 


Average 

Height 

in inches. 


Average Height 
in cms. 


Nos. 
measured. 

2467 
2573 
2433 


Average 

Height 

in inches. 


Average Height 
in cms. 


Surrey. 


England 

and 
Wales. 


Surrey. 


England 

and 
Wales. 

102-6 
119-4 
144-2 


5-6 

8-9 

13-14 


2724 
2578 
2529 


41-4 

47-8 
57-0 


105-2 
121-4 
144-8 


103-4 
120-4 
142-4 


41-3 
47-5 
57-9 


104-9 
120-7 
1471 



The first four are taken from the Annual Report of the School 
Medical Officer for the County of Surrey, 1913. The first is an 
example of single tabulation showing the distribution according to 
cleanhness of children inspected in the elementary schools. The 
second is an example of double tabulation, showing the distribu- 
tion according to age of school children who at some period before 
the date of inspection had suffered from certain infectious diseases. 
The third is an example of quadruple tabulation, showing the dis- 
tribution of school children according to height, district, sex, and 
age. Thus in the first case we have one factor brought into relief, 
viz. cleanliness ; in the second case we have two factors, age and 
disease ; in the third case we have four factors, height, district, 
sex, and age. 

When we have two or more factors tabulated together as in cases 
(2) and (3), we may be sometimes led to discover a connection of 
some kind, possibly causal, between them, and the search for such 
a connection, or correlation as it is called, represents one very useful 
purpose to which tabulation may be put. Table (4) is an illustra- 
tion of this. It is the result of certain measurements carried out in 
order to discover the effect of employment out of school hours upon 
the physical condition of boys. The particular factor examined as 
the possible cause of evil in this connection is lack of sleep, and 
the figures given certainly seem to warrant a closer examination 
into the matter. 



CLASSIFICATION AND TABULATION 



19 



Table (4). Physical Condition of certain Boys according 
TO Hours op Sleep Obtained. 



No. of Hours 
Sleep obtained. 


No. of Boys 
examined. 


Average 

Height in 

inches. 


Average 

Weight in 

lbs. 


Nutrition. 


Percentage 

above 

average. 


Percentage 
average. 


Percentage 

below 

average. 


7 to 8 . 

8 to 9 . 

9 to 10 . 

10 to 11 . 

11 to 12 . 


14 

80 
296 
280 

50 


54-5 
55-4 
56-4 
57-9 
59.0 


71-3 
73-9 
79-3 

83-2 
87-0 


71 
101 
15-3 

22-8 
220 


35-8 
65-9 
64-5 
66-5 
680 


571 
240 
20-2 
10-7 
100 

















Tables (5) and (6) are two illustrations of neat tables, containing 
a large amount of information in a small space, set out in such a 
form that the eye can easily take it in — and that is the main purpose 
of tabulation. These examples are selected from the Sixteenth 
Abstract of Labour Statistics of the United Kingdom, Cd. 7131. 

In Table (6) note the classification of age groups : it is not ' 5 to 
10 years,' ' 10 to 15 years,' and so on, but ' 5 and under 10 years,' 
' 10 and under 15 years,' and so on. This removes difficulties at 
the border lines between two classes ; the difficulties are not com- 
pletely removed, however, unless there is some understanding as 
to what shall constitute under any particular age. Shall it be six 
months under, or one day under, or one hour under ? This sort 
of ambiguity has more importance in some cases than in others. 
Suppose, for example, we were classifying men according to their 
height : a group of the type ' 60 inches and under 62 inches,' 
assuming that measurements were made to the nearest half-inch, 
would really include all men who were ' 59J inches and under 
61 1 inches ' ; because one who measured anything from 59f in. 
to 60i in., being nearer to 60 in. than to 59 J in. measuring to 
the nearest half -inch, would be registered as 60 in. in height, while 
one who measured anything from 61f in. to 62J in., being nearer 
to 62 in. than to 61J in., would be registered as 62 in. in height. 

Another point to be noted is that in general people making 
returns seem to have a psychological weakness for round figures, 
so that a man in the neighbourhood of 40 years of age, for example, 
is apt to record himself as actually 40 although he may really 



20 



STATISTICS 



Table (5). Classification of Overcrowded Tenements — * 
England and Wales (1911). 





Urban Districts. 


Rural Districts. 


Total. 




Occupants 
thereof. 




Occupants 
thereof. 




Occupants 
thereof. 


Tenements 

WITH 


No. of 
Over- 
crowded 
Tene- 
ments. 




No. of 
Over- 
crowded 
Tene- 
ments. 




No. of 

Over- 

, crowded 

1 Tene- 

i ments. 




No. 


Per- 
cent- 
age of 
total 


No. 


Per- 
cent- 
age of 
total 


No. 


Per- 
cent- 
age of 
total 








popu- 
lation. 






popu- 
lation. 






popu- 
lation. 


1 room . 

2 rooms . 

3 rooms . 

4 rooms . 


56,290 
119,695 
107,892 

64,470 


206,022 
712,613 
847,937 
624,747 


0-7 
2-5 
3-0 

2-2 


1,545 
15,397 

22,380 
17,341 


5,748 

91,458 

175,988 

167,969 


01 
1-2 
2-2 

2-1 


57,835 
i 135,092 
i 130,272 
' 81,811 


211,770 

804,071 

1,023,925 

792,716 


0-6 

2-2 
2-8 
2-2 


5 or more 
rooms . 


21,200 


251,405 


0-9 


4,700 


55,585 


0-7 


25,900 


306,990 


0-8 



Table (6). Population grouped according to Age- 
England AND Wales (1911). 



males. 



Age Groups. 


Urban Districts. 


Rural Districts. 


] 
All Districts. 

] 


Number. 


Percentage. 


Number. 


Percentage. 


Number. 


Percentage. 


Under 5 years 


1,517,432 


11-3) 


418,681 


10-6^ 


1,936,113 


111^ 


5 and under 10 years 


1,431,900 


'nw^ 


415,395 


10-5 Li.o 
10-3 P^ -^ 


1,847,295 


^^•*H41-2 

100 p^ ^ 


10 „ 15 „ 


1,341,586 


406,045 


1,747,631 


15 ,, 20 „ 


1,267,500 


94 


387,395 


9-8j 


1,654,895 


9-5J 


20 „ .30 „ 


1 2,332,135 


17-3^ 


626,300 


15-9^ 


2,958,435 


17-0^ 


30 „ 40 „ 


2,094,934 


15-5 144-4 


542,370 


13-7 Uo-9 


2,637,304 


15-1 U3-6 -- 


40 ,, 50 „ 


! 1,556,818 


11-6 


444,360 


II.3J 


2,001,178 


11.5/ 


50 „ 60 „ 


i 1,042,868 


7-7^ 


333,368 


84) 


1,376,236 


7-9) 


60 „ 70 „ 


1 612,741 


4-5 V144 


230,306 


5-8 V17-9 


843,047 


4-8 V 15-2 


70 and upwards 


1 296,246 


2-2j 


147,228 


3-7j 


443,474 


2-5 J 


Total 


13,494,160 


100-0 


3,951,448 


100-0 


17,445,608 


100-0 



[* For the purpose of the Census Reports 'ordinary tenements which have more 
than two occupants per room, bedrooms and sittinjj-iooms included,' are considered 
overcrowded.] 



CLASSIFICATION AND TABULATION 21 

be 39 or 41 years old. To diminish the error arising from this fact 
it is usual, when not otherwise inconvenient, to fix the centres 
of the class-intervals at round figures : e.g. to take * 15 and under 
25 years,' ' 25 and under 35 years,' etc., in preference to ' 20 and 
under 30 years,' ' 30 and under 40 years,' etc. Where there is 
any known bias in the data, as, for instance, in the famihar case 
of certain women who consistently register themselves as younger 
than they really are, a correction can be made in the final figures. 

In any frequency distribution where we wish to group a number 
of observations according to the magnitude of some common 
variable, as in Table (6) a number of males grouped according to 
age, the question arises — ' How many groups should there be ? ' 
With this question is involved also the size of the corresponding 
class-interval, and this should be so large that, with possible excep- 
tions at either extremity of the table, there are a fair proportion of 
observations to each class or group ; and, contrariwise, it should 
be so small that all the observations in any one group may be 
treated practically as if they were located at the centre of the group 
so far as the variable in question is concerned, e.g. it should be 
possible to treat males recorded in class ' 50 and under 60 years,' 
where the interval is 10 years, as if they were all of age 55 years. It 
will be found in general that a number of groups somewhere in the 
neighbourhood of 20 is the most satisfactory, granted that the 
number of observations is reasonably large, although in some cases 
it is impossible to spHt up the unit of class-interval, and we are 
obliged to be satisfied with a smaller number of groups on this 
account : Table (5) is a case in point where we are tied down to 
one room as the class -interval. In Table (6) the class-interval 
varies, being only 5 years at first, and afterwards 10 years, but 
as a rule the labour of calculation of the different statistical constants 
we require is considerably simplified if it is possible to keep the 
size of the class-interval the same for each group. 



CHAPTER IV 

AVERAGES 

Common Average or Arithmetic Mean. Let us consider one of the 
commonest meanings of the term average. If a train travels a 
distance of 180 miles in 3 hours we say that it has been moving 
at 60 miles an hour. By this we do not mean that its speed is 
always 60 m/h, never more, never less, but that if it had moved 
always at that uniform speed it would have accomplished its 
journey in exactly the same time. As a matter of fact, during 
some instants it may have been moving at a much slower rate 
than 60 m/h, but, if so, it must have made up for this slackness 
by travelling at a much faster rate than 60 m/h during other 
instants, so that on the whole a balance was effected, and, as we 
say, the speed averaged out at 60 m/h. 

Again, suppose the wages of three men are : A, 27s. a week ; 
B, 18s. a week ; C, 30s. a week. We should say that the average 
wage of the three was equivalent to 

J(27+18+30)s.=25s. a week. 

In other words, if A, B, and C were all under the same employer, 
and if, instead of paying them different amounts, he wanted to 
pay them all equally, he would have to give each man 25s. a week, 
assuming that his total wages bill was to remain unaltered. This 
method of measurement gives what is known as the arithmetic 
mean, or, more simply, the mean. 

Once more, in discussing the state of the labour market as regards 
different trades, when we wish to compare one with another, it is 
not the actual numbers unemployed in each trade that are quoted, 
but these numbers expressed as percentages of the total numbers 
employable in each trade. 

In each of these three cases we reduce our observations or 
measurements to a sort of common denominator, so that they may be 
mentally compared or contrasted more readily with other observa- 
tions of a similar character. Thus we have in mind a certain mean 



AVERAGES 23 

train speed per hour, or mean wage per week, or moan percentage 
out of work; as the case may be. 

An average then in general we may regard as one of a class 
of statistical constants (others of which we are to meet later) which 
concisely label a set of observations or measurements pertaining 
to a common family. It is designed to describe the family type 
more nearly than is possible by observing any chance member, and in 
value it should therefore come somewhere near the middle of the 
family group, so that if the individual members of the family 
chance to be equal each to each in respect to the organ or character 
observed it should have the same value as they have. This consti- 
tutes a test for the validity of any formula giving the average of a 
set of observations : e.g. we might, if we wish, define the average 
of three numbers, p, q, r to be, not J(i?+g+r) but 

for (1) this formula, too, can be shown to give a number intermediate 
in value between the greatest and least of the numbers j), q, r ; 
also (2) if we put p=q=r=k (say), the formula reduces to 

l/i{J(^+k^-i-J(^)= X/k^=k. 

Clearly the range of choice for the definition of an average is 
infinite, though only a few definitions give averages which have 
proved their utility and come into general use. Of these the most 
important is the common mean already introduced, with its ex- 
tension, the weighted mean, but at least two others deserve special 
consideration, the median and the mode. 

Median. In any observed distribution if aU the individuals 
can be arranged in order of magnitude of the character or organ 
observed, which may be conveniently done when they are not very 
numerous, the median organ or character will be that pertaining to 
the individual half-way along the series, so that there are in general 
an equal number of individuals above and below the median. 
For instance, if seven boys of different heights be placed to stand in 
a row, the tallest first, the next tallest next, and so on, the median 
height is the height of the fourth boy from either end. If there 
are an even number of boys, say eight, it would be natural to take 
as median the height midway between that of the fourth and that 
of the fifth boy. 

When the items are numerous they are frequently grouped into 
classes, as we have seen, such that all in the same class are reckoned 



24 STATISTICS 

to have some value lying between the extreme limits of that class. 
We should then, as before, halve the total number of observations 
to fix the i^articular individual which defines the median organ or 
character. This would enable us to pick out the group in which 
the median lies, and on reference to the original record of observa- 
tions, assuming it was at hand, it would be a simple matter to 
identify the median. 

If the original record be not available, however, it will be neces- 
sary to proceed to get the best value we can for the median in some 
other way. Consider, for example. Table (7), showing the distribu- 
tion of marks obtained by 514 candidates in a certain examination. 
We begin by rearranging the data in the manner shown below. 
Table (7). Now in accordance with the definition the median in 
marks should, strictly speaking, be midway between the marks 
assigned to the 257th candidate and the marks assigned to the 
258th candidate : in fact, the marks corresponding to candidate 
number 257-5, if it were possible for such a candidate to exist. 
But we are ignorant so far as Table (7) goes of the marks gained 
by either the 257th or the 258th candidate, though it is possible, 
by the simple proportional process known as ' interpolation,' to 
calculate approximately the marks we require. We think of all 
the candidates as forming an ordered sequence, ranged one after 
the other according to their marks just like the boys of different 
heights, and the table shows that in this mental picture 

the 231st candidate gets approximately 30 marks, while 
„ 318th „ „ „ 35 

Hence candidate number 257-5, if one existed, ought to get a 
number of marks somewhere between 30 and 35. But, in this 
neighbourhood of the sequence, 

a difference of (318-231) candidates corresponds to a difference 

of 5 marks, therefore 
a difference of (257'5-23l) candidates corresponds to a difference 

of (mtX26-5) marks. 

Thus the marks obtained by candidate number 2575 are ap- 
proximately = 30+ ^T X 26-5 

=31-523, 

and this may be taken as the median. 

On examining the actual marks-sheet it was found that 252 
candidates obtained 31 marks or less, and 273 candidates obtained 



AVERAGES 



25 



32 marks or leas, so that the real median was 32, because this was 
the number of marks gained by both the 257th and the 258th 
candidates. The number 31-523 found above, however, would be 
a good approximation to take for the median when all the informa- 
tion at our disposal was that shown in Table (7). 



Table (7). Marks obtained by 514 Candidates in a 
CERTAIN Examination. 



Marks Obtained. 


No. of 
Candidates. 


Marks Obtained. 


No. of 
Candidates. 


lto5 
6 to 10 
11 to 15 
16 to 20 
21 to 25 
26 to 30 
31 to 35 


5 
9 

28- 

49 

58 

82 

87 


36 to 40 
41 to 45 
46 to 50 
51 to 55 
56 to 60 
61 to 65 


79 

50 

37 

21 

6 

3 

■ 


Total 


514 

1 



The table is to be read as follows : — 

5 candidates obtained 1, 2, 3, 4, or 5 marks, 
9 „ „ 6, 7, 8, 9, or 10 „ 



and so on. 



By straightforward addition it can evidently be rearranged so 
as to read thus : — 



5 candidates obtained not more than 5 marks. 



14 
42 
91 
149 
231 
31,8 
397 
447 
484 
505 
511 
514 



10 
15 
20 
25 
30 
35 
40 
45 
50 
65 
60 
65 



26 STATISTICS 

It will be noted that in calculating the median no use is made of 
the marks of any of the candidates except those in the two groups 
in the immediate neighbourhood of the median, and it is one of 
the great advantages of this average that it can be found when an 
exact knowledge of the characters of the more extreme individuals 
in the series is not in our possession, and even when their measure- 
ment is impossible : it is enough if they can be roughly located. 
The arithmetic mean on the other hand is often unduly influenced 
by abnormal individuals which are not really typical of the popula- 
tion in which they appear. 

Mode. If we measure or observe some organ or character for 
each individual in a given population, the mode, as its name sug- 
gests, is simply the organ or character of most fashionable or most 
frequent size. A large draper, for example, will have collars of 
several different shapes and sizes in his shop, but the fashionable 
shape and the predominant size correspond to the mode : it is the 
mode that sells most readily, and the intelligent draper will always 
have it in stock. Again, in Table (2), the disease mode or fashion- 
able disease among certain school children inspected in Surrey in 
1913 was measles, for a greater percentage of children had suffered 
from measles than from any other of the diseases recorded. 

Now when the variable in which we are interested is ' discrete,' 
that is, when it changes by unit steps, leading to classes like ' tene- 
ments with 1 room,' ' tenements with 2 rooms,' ' tenements with 
3 rooms,' and so on, it is an easy matter to pick out the class of 
greatest frequency : thus, in Table (5) there are more overcrowded 
tenements with 2 rooms than with any other number of rooms 
in the urban districts, so that 2 is the mode so far as this character 
(number of rooms) is concerned, whereas in the rural districts 3 is 
the mode, for there are more overcrowded tenements with 3 rooms 
than with any other number. There may be ambiguity, however, 
in determining the mode in this way for a grouped frequency dis- 
tribution when we are dealing with an organ or character subject 
to * continuous variation.' To cover such cases the modal value 
has been defined as that value for which the frequency per unit 
variation of the organ or character is a maximum. The precise 
significance of this wording will only be appreciated after discussing 
frequency curves : at present it must suffice to give a practical 
illustration of how the ambiguity arises and calls for some more 
refined treatment. 

For this purpose turn again to the examination marks in Table (7), 



AVERAGES 



27 



from which it appears that the mode, if it is to be the marks obtained 
by the greatest number of candidates, should lie in the group 
(31 to 35), since there are 87 candidates with marks between these 
limits, and this number exceeds that in any other group. But 
how are we to decide the exact point in the interval (31 to 35) which 
is to correspond to the mode ? Shall it be 33 ? We might say 
' yes ' if the distribution were perfectly symmetrical on either side 
of the (31 to 35) group, but if we examine the neighbouring groups 
we see that the balance leans rather more heavily to the (26 to 30) 
group with a frequency of 82 than to the (36 to 40) group with a 
frequency of 79, and we might allow for this by interpolating in 
some way — ignoring, of course, any errors which may occur in the 
frequencies themselves owing to the observations being generally 
limited in number. But the pull in the direction of lower marks 
becomes still more pronounced to our minds when we contrast 
also the frequencies in the next groups on either side, namely 
58 and 50. So we might go on until the influence of the whole 
field of observations comes into action. 

Now it so happened that in this particular case the original 
marks-sheet was to be seen, and a regrouping of the candidates as 
in Table (8) makes it clear that the value found in this way for the 
mode may be artificially displaced sometimes to a serious extent 
by the particular method of grouping adopted. Thus, according 
to this new arrangement, the mode would seem to lie in the interval 
(28 to 32), the mid-value of which differs materially from 33, the 
mid- value of the previous maximum frequency group. 



Table (8). Marks obtained by 514 Candidates in a 
CERTAIN Examination (Alternative Grouping). 



Marks Obtained. 


No. of 
Candidates. 


Marks Obtained. 

38 to 42 
43 to 47 
48 to 52 
53 to 57 
58 to 62 
63 to 67 


No. of 
Candidates. 



73 
45 
31 
12 

3 

3 


3 to 7 
8 to 12 
13 to 17 
18 to 22 
23 to 27 
28 to 32 
33 to 37 

1 


10 
17 
35 
56 

47 

108 

74 


1 

Total 


514 



28 



STATISTICS 



[It should be observed that while an alteration of the grouping 
may also affect the median, it does not affect it nearly to the same 
extent : e.g. the median determined from Table (8) is 31-3, which 
differs little from 31-5 the value obtained by the first grouping.] 

If, again, we combine the results of our two groupings to find 
the mode we might be tempted to conclude that it lies somewhere 
between the limits 31 and 32, but on examining the original records 
it was discovered that the real mode was 28. The frequency 
distribution of candidates in this neighbourhood was in fact very 
interesting ; it ran as follows : — 

Number of candidates who obtained 25 marks =14 

26 „ =10 



27 
28 
29 
30 



= 6 
= 33 
= 17 
= 16 



The explanation of this peculiar distribution seemed to be that 
28 marks were required for a candidate to pass, and apparently as 
many candidates as possible were pushed over the pass line : if, 
on the first marking, a candidate was found to want only one mark 
to pass, the examiner presumably looked through his paper again 
and did his best to find an answer which by kindly treatment 
might be granted an extra mark. The effect of this leniency was 
ultimately to leave only 6 candidates in the division immediately 
below the pass line, and to swell the number immediately above 
to 33, which thus made 28 easily the ' most fashionable ' mark of 
any, the next largest group of candidates being only 21. It will 
be observed that even a candidate who wanted 2 marks to pass 
was treated in the same tolerant fashion, although it is not so 
easy, of course, for a conscientious examiner to discover two extra 
marks as it is to discover one ; and if the candidate is 3 marks 
below the pass line it is still harder to give him the necessary lift 
to carry him over. Thus in the final list we find more condidates 
with 26 marks than with 27, and still more with 25 than with 26. 
If the above diagnosis is correct, and aU marks-sheets tell the same 
tale, who shall again say that examiners do not temper justice with 
mercy ? 

This example has illustrated fairly clearly the difficulty of fixing 
the mode with any great precision by mere inspection when the 
individuals are arranged in groups, the value of the variable under 
discussion lying between prescribed limits for each group. While 



AVERAGES 29 

it is possible to get a rough approximation to its value in this way, 
we conclude that for a really satisfactory determination we require 
some method which makes use of the whole distribution, as in the 
determination of the mean, and not merely of the portion in the 
supposed neighbourhood of the mode. This must be left to a later 
chapter ; we shall only point out before passing on that there 
may sometimes be more than one mode in a given frequency dis- 
tribution just as there may be more than one fashionable type of 
collar which it is expedient for the draper to stock in large quan- 
tities. The second grouping in the examination example suggests 
such a possibility, for it will be noticed that the frequencies of 
candidates do not rise steadily to a single maximum at 108 for 
class (28 to 32), and then fall steadily : there is a previous rise and 
fall in the neighbourhood of class (18 to 22). 

Weighted Mean. Let us suppose a farmer employs for the 
harvest 5 men, 3 women, and 4 boys. In estimating the amount 
of work they can do in a given time it is clear that in general a 
woman or boy cannot be reckoned as equal to a man. He must 
therefore decide what ' weight ' must be given to each in proportion 
to a man. If a woman's work be taken, for example, to be three- 
quarters as effective and a boy's work to be half as effective as 
that of a man, we have as the appropriate proportional weights 

1 :f a, or 4:3:2. 

Hence 5 men, 3 women, and 4 boys would on the average be equiva- 
lent in output to 

(5+3xi+4xJ) men 

4x5+3x3+2x4 
= men 



^ =91 men. 

An average of this type is called a weighted mean, 1, |, and 
I being the weights, because they tell us what weight to give to 
each separate worker in calculating the average. 

Let us consider the effect such weighting has in general upon a 
mean, and for this purpose we shall test it on a set of index numbers 
measuring rents in certain groups of towns in 1912, as given in a 
Report on the Cost of Living of the Working Classes issued by the 
Board of Trade (Cd. 6955). 



30 



STATISTICS 



Table (9). Mean Index Numbers of Rents for certain 

Geographical Groups of Towns in 1912 (with reference 

TO Middle Zone of London as standard = 100). 

(2) (8) 



(1) 



(4) 



(6) 



(6) 



Geographical Group. 


Rents. 


No. of 
Towns 
included 
in the 
Group. 


Each 

Group 

counting 

as 1. 


Arbitrary 
Weights. 


Approxi- 
mate sub- 
multiples 
of Noa. in 
previous 
column. 


Northern Counties and Cleve- 
land .... 
Yorkshire (except Cleveland) 
Lancashire and Cheshire 
Midlands .... 
Eastern and East Midland Cos. 
Southern Counties 
Wales and Monmouth . 
Scotland .... 
Ireland .... 


660 
58-5 
56-9 
52-3 
53-4 
63-7 
64-8 
620 
51-7 


9 
10 
17 
14 

7 
10 

4 
10 

6 


1 


27 
54 
45 

125 
63 
14 
22 

178 
55 


3 

6 
5 

14 
7 
2 
2 

20 
6 


Average . . 


•• 


58-4 


58-8 


57-6 


57-6 



The first mean in the above table, 58-4, is obtained by multiply- 
ing (or weighting) the mean rent of each geographical group by the 
number of towns in the group, given in col. (3), adding the numbers 
so obtained, and dividing the total by the total number of towns, 



thus : — 



9(66-0)+ 10(58-5)+ 



+ 6(51-7) 



9 + 10 + 



+ 6 



This is simply the arithmetic mean treating each town as unit. 

The second mean, 58-8, is obtained by adding the mean rents of 
all the groups and dividing by the total number of groups, thus : — 



66^0+58-5+ 

"r~+^rT 



+51-7 



+ 1 



This is the arithmetic mean treating each geographical group as 
unit. 

The third mean, 57-6, is obtained by multipljdng, or weighting, 
the mean rent of each group by a perfectly arbitrary number given 
in col. (5) ; the numbers selected were taken quite at random from 



AVERAGES 31 

another column of figures in another Blue-book, and had no con- 
nection whatever with the subject of rents ; this gives : — 

27(66-0)+ 54(5 8'5)+ . . . +55(51-7) 
27 + ~54 + . . . + 55~' 

The last mean, 57-6, is obtained by choosing as weights any 
numbers (and for simplicity we choose the smallest) as in col. (6) 
which are very roughly proportional to the arbitrary weights used 
in the last instance ; we thus get : — 

3(66-0)+6(58-5)+ . . . +6(51-7) 

3 + 6 + . . r+ 6~ * 

Now the first of these means is clearly the most satisfactory, since 
it is the result of very properly weighting the mean rent of each 
group of towns according to the number of towns the group con- 
tains. But the second result shows that if we are ignorant of the 
number of the towns in each group we shall not be very far out in 
our calculation if we treat them all as of equal importance, and find 
the simple arithmetic mean of the mean rents in the nine groups. 
We can even go further, for we find, from the third and fourth results, 
that by weighting the mean rents in the various groups on quite a 
random basis, the mean we get still does not differ very greatly from 
the best value first found. 

The important principle of which the above example is an illus- 
tration is perfectly general, and may be stated as follows : If the 
total number of measurements or observations be not very small, 
and if the resulting values of the organ or character measured 
(rent in our case) be not very unequal, any reasonable selection of 
multipliers or weights (as, for instance, the first two adopted above) 
will give means which differ from one another by but little ; and 
even an apparently unreasonable selection of multipHers (as, for 
instance, the third adopted above), assuming they are not so 
wildly chosen as to give any particular group a very unfair weight 
in comparison with the others, will not throw the mean out badly. 
Further, in place of a set of large multipHers we may substitute 
small numbers which are roughly proportional to them (as we have 
done in the fourth case above), and the mean wiU again be very 
little affected. [See Appendix, Note 2.] 



CHAPTER V 

AVERAGES {continued) 

Applications of Weighted Mean. In determining the weighted mean 
of a set of observations it is usual, of course, to weight each observa- 
tion according to its importance, though what number should be 
chosen as a measure of its importance may sometimes be a matter 
of doubt. It is not a very difficult matter to decide when we 
wish, for example, to compare birth, marriage, or death rates in 
two districts, if we know how the constitution of the population 
in the one district differs from that in the other, for the weighting 
in each of these cases must be in proportion to the population 
concerned, and it is too important to ignore. 

Death rate, crude and corrected. Imagine a city in which the 
total number of deaths in a certain year is N out of a population 
numbering P. 

The ordinary or crude death rate for that city will then be 

N 

-XlOOO, by deianition. 

Now this number N may be analysed according to the ages of 
the people who have died ; let us suppose it is made up of 

^1 people between limits and less than 5 years of age, 
^2 n ,, ,, 5 ,, 15 

^3 ' '' " 15 " 25 

and so on, where 

^l + ^2 + '^3+ • • • =N- 

Again the number P may be analysed according to the ages of 
the people who compose the total population, giving, say, 

p^ of the population between limits and less than 5 years of age, 

15 
25 



P2 „ 


5> 55 


5 


Vz »j " 


>5 55 


15 


and so on, where 


Pl^-V2-\-lh-\- 





AVERAGES 33 

Thus we may write for the crude death rate 

N 
D= xlOOO 
P 



:_JIL_ 2^ 3^ X 1000 



=''-ilOOO+'^1000+'^1000+ . . . 

^^/^-MoooV^^^-^ioooV^f'^ioooV . . . 

where d-^\^ the death rate between limits and less than 5 years of age, 
^2 " " .5 5 . ,, 15 ,, 

d^ „ „ „ 15 „ 25 

and so on. 

Now if we compare this expression with the corresponding one for 
another city, say, 

it is quite conceivable that the death rates in the various age groups 
might be equal — 

di^=d , d^==d^, d_=d^ . . . 

and yet D might exceed D' because in the first city there are a 
greater proportion of infants or old people, on which classes the 
hand of death falls heaviest, that is, because the ^'s or weights 
which multiply the biggest d's are greater in the first case than in 
the second. But so long as the d's in the two cities are equal, age 
group by age group, it would be reasonable to regard the cities as 
equally healthy, or unhealthy as the case might be, and therefore 
to insure a fair comparison it is usual in the Reports of the Registrar- 
General to give a corrected death rate in place of the crude death 
rate defined above. 

This is done by weighting the death rate for each age group, not 
in proportion to the actual number of persons in that group in 
the city itself, but in proportion to the corresponding number in 

C 



34 STATISTICS ' 

the country at large. Thus, if we denote the proportion of the^ 
population, Q, , 

between limits and less than 5 in the country at large by qJQ, ! 

» J5 15 ,, 25 „ ,, ,, QslHf i 

and so on, we get as the corrected death rate i 

i 

toA+g'2<^2+9^3^3+ • • O/Q, i 

a form wliich has the effect of making the results agree in two' 
cities which have equal d'8 throughout. 

A similar method of correction is clearly applicable in consider- • 
ing the incidence of the death rate when we are concerned not | 
with a difference of district but with a difference of sex, occupation, ; 
religious profession, wage-earning capacity, or any other well- j 
defined character. Further, it may be used also in comparing birth ! 
rates, marriage rates, heights, weights, chest measurements, or any . 
similar attributes, when it is necessary to refer the observations i 
or measurements to a standard population in order to avoid \ 
complications due to age variation. ' 

There is another method of correction, equally general in appUca- j 
tion, which is useful when the death rates in the various age groups \ 
are not known. In this case D, the crude death rate for the whole i 
population of the district is known, also pJ'P, 'P^l^, Psl^, • • • the \ 
proportions of the population between the various age limits, but 
di, ^25 ^3 . • • are supposed unknown. i 

Now if the population in the country as a whole were the same in ■ 
corresponding age groups as it is in the district under consideration, j 
we should get as the death rate for the whole country j 

where S^, Sg? ^a • • ^re the death rates in the various age groups in 
the country at large, and these would in practice as a rule be known. 
The actual death rate for the whole country is, however, ; 

{qA+q2^2+q2^z+ . . . )/Q, \ 

where g'l/Q, g'2/Q' S's/Q • • • denote, as before, the real proportions ! 

of the population in the various age groups in the country at large. 1 

We take as the corrected death rate required for the district a ] 

number bearing to the crude death rate the same ratio as ■ 

{qA+q2K+ • . O/Q bears to {PiS,+p,S,+ . . .)/P. \ 



AVERAGES 



35 



Hence we have 



corrected death rate_g'iSi -1-^282+ ... P 

iTidex Numbers to compare Household Budgets. Another highly 
important illustration of a weighted mean occurs in the search for a 
satisfactory measure of the change in the cost of Uving from year 
to year. We have already introduced the subject of variation in 
wholesale prices, and we have seen that Sauerbeck, in forming his 
index numbers, treats as one each of the forty-five commodities 
he uses to measure this variation : the observations, that is to 
say, are not weighted. 

But, confining our attention to food alone, supposing we have 
five items, such as bacon, bread, tea, sugar, milk, for which the 
index numbers of prices at two different dates are : — 





Bacon. 


Bread. 


Tea. 


Sugar. 


Milk. 


First date 
Second date 


100 
117 


100 
95 


100 

94 


100 
102 


100 
109 



Is it really right to treat each of these items as of equal importance 
with the rest, or ought we to regard bread and tea, say, as of more 
weight than bacon, and count bread perhaps five times and tea 
three times while counting bacon only once ? It is clear that, in 
order to select a reasonable set of multipUers in this case, we should 
need to know the standard of living of the class of people under 
consideration, and how much in the aggregate they spend upon 
bacon and how much upon bread, etc. 

A partial answer to these questions can be obtained by making 
a collection of household budgets as was done, for example, by two 
Government Committees which recently reported (1918-19) on the 
Cost of Living among the Urban and the Agricultural Worki'ng Glasses 
respectively. If the number of commodities employed is large, 
even an arbitrary set of multipHers, as we have indicated, will not 
displace the mean any great distance from the value when reason- 
able weights are chosen, but unfortunately in collecting such house- 
hold budgets we are confined to the comparatively limited variety 
of food-stuffs which are in general use. 

Different principles may be followed in making the comparison 



36 



STATISTICS 



between one year and another which may be illustrated by a few 
figures from the Urban Classes Report (1918) : — 

Table (10). Household Budgets showing Prices of bach Com- 
modity AND Quantities Purchased at Two Different 
Dates by Typical Family. 



Commodity. 


First year (1914). 


Second year (1918). 


Price (pence 
per lb). 


ni 
No. of lb. 

bought. 


a;2 

Price (pence 

per lb.) 


n2 
No. of lb. 
bought. 


Sugar. 
Tea . 
Potatoes 


2-2 

21-3 

0-7 


5-9 
0-68 
15-6 


7-07 
33-3 
1-26 


2-83 
0-57 
200 



Let Xi be the price, in pence per unit, of any one commodity 
at the first date, and let n^ be the number of units of this commodity 
bought per week by a typical family {n may be estimated in different 
ways, e.g. (1) by dividing the total number of units bought by 
all famihes by the total number of those famiHes, or (2) by ranging 
the different amounts bought by different families in order of 
magnitude and picking out the median amount, or (3) by choosing 
the mode, i.e. the amount most commonly purchased). Also let ajg 
be the price, in pence per unit, of the same commodity at the second 
date, and let Wg ^^ ^^® number of units of the commodity then 
bought per week by the typical family estimated in the same way 
as before. 

The actual expenditure, measured in pence, at the two dates 
will then be 

Z{Xini) and ^(x^n^ 

respectively, where E(x-^n-^ simply denotes the sum of expressions 
like (x-jji-^) for all the commodities recorded and ^{x^n^) denotes the 
sum of expressions like (x^n^ for aU the commodities recorded, 
Sy the old English S, being a well-known conventional abbreviation 
for ' Sum of expressions like.' Thus, with the numbers in Table (10), 
we should have 

2'(a;i7ii)=(2-2)(5'9)+(21-3)(0-68)+(0-7)(15-6)+ . . . 
^(^2^2)=('7-07)(2-83)-f(33-3)(0-57)+(l-26)(20-0)+ . . . 



AVERAGES 37 

Taking 100 as the index number to represent expenditure at the 
first date, the index number measuring expenditure at the second 
date may be formed in any of the following different ways,* which 
as a rule, of course, lead to different results : — 

(1) lOOZ{x^n^)IU{x^n^) ; 

(2) lOOi:(x^nj)IU{x^ni) or lOOZ (x^n^jjUix^n^) ; 

(3) l002J{Xjn2)IZ{Xjnj) or 1002'(a:2?i2)/^(^2%)- 

The first of these expressions compares the acttuil expenditure at 
the second date to that at the first date. 

The next two expressions take into account directly only the 
change in prices ; they compare, not actual expenditures but, the 
expenditures at the two dates as they would be if the amounts 
purchased at the two dates were the same : the first supposiug 
these amounts to equal those actually bought at the first date, 
and the second supposing them to equal those actually bought 
at the second date. 

The last two expressions, on the other hand, take into account 
directly only the change in amounts purchased ; they compare 
the expenditures at the two dates as they would be if the prices 
ruling at the two dates were the same : the first supposing these 
prices to equal those actually charged at the first date, and the 
second supposing them to equal those actually charged at the 
second date. 

The particular method of weighting adopted must naturally 
depend upon the circumstances of the period under discussion 
and the nature of the inquiry one is making ; it is a nice question 
to decide how far emphasis should be laid upon the old standard 
of life (measured by food, lighting, rent, recreation, etc.) with the 
expense required to maintain it, and upon the new standard of life 
and the cost necessary to reach it. 

It may be useful here to summarize a few of the questions of 
interest which present themselves in connection with the formation 
of index numbers of prices designed to measure changes in the 
value of money in general without reference to any particular class 
of the community : — 

1. What years should be selected in fixing our standard prices ? 

2. What commodities should be chosen as a basis for our 
average ? 

[* See also The Measurement of Changes in the Cost of Living, by A. L. Bowley, Sc.D., 
in the Jov/rnal of the Royal Statistical Society, May 1919, for a more complete dis- 
cussion of the subject.] 



38 STATISTICS 

3. What weight should be given to each commodity in relation 
to the rest ? 

4. How should the prices of the several commodities be deter- 
mined, bearing in mind that ' price ' itself frequently varies from 
place to place ? 

5. Finally, how should these prices be combined to give the 
average required ? Should we use the simple arithmetic mean, the 
geometric mean [see Appendix, Note 3], the median, or some other 
measure ? 

While we are not prepared to attempt to answer these questions 
fully, seeing that authorities are not altogether agreed as to what 
the answers should be, one or two points may be worth noting. 
Generally speaking we may say that : — 

1. The years selected in fixing our standard prices should be 
years in which economic conditions were normal rather than 
abnormal. 

2. The commodities chosen . should be articles of general con- 
sumption, and as wide a field as possible should be covered in their 
choice. 

3. Many consider that little is gained by weighting, but, if 
weights are introduced, the greater the importance of any com- 
modity in relation to the rest, judged for example by the relative 
quantity consumed, the greater should be the weight assigned 
to it. 

4. The practical difficulty of assessing retail prices when they 
are uncontrolled compels us in general to fall back upon whole- 
sale quotations, on which some light may be thrown by keeping 
under observation the important markets for the sale of each 
commodity. 

5. The average commonly used is the simple arithmetic or the 
weighted mean, though arguments can be adduced in favour of 
other averages such as the median. 

Leaving index numbers now on one side and returning to the 
general subject of averages, we may remark that the question 
which average is correct in any given case, the mean (weighted or 
otherwise), the median, or the mode, does not arise : no one average 
is more correct than another, because they are all entirely con- 
ventional and represent different ideas ; they correspond in fact 
to so many different ways of summing up a set of observations or 
measurements in a single numerical statement, and the real question 



AVERAGES 39 

to determine is which statement, which^ kind of average, brings the 
set of observations before us to the best focus. 

For this purpose one average will clearly be best in one case and 
another in another, but it may be stated without hesitation that 
the arithmetic mean is certainly the most useful of the three and 
it is the most frequently used. Other averages have been sug- 
gested, such as the geometric and the harmonic means [see Appendix, 
Note 3] familiar to students of Algebra, but they are only suitable 
in a comparatively small class of problems. 

In a reasonably symmetrical distribution of observations, one in 
which the variables of medium size are the most frequent and the 
frequency diminishes about equally on either side towards the 
largest and the least of the variables, the values of the mean, the 
median, and the mode will be found to lie all very close together ; 
and a useful practical rule to remember is that the median comes 
in general between the mean and the mode, the difference between the 
mean and the mode being about three times the difference between the 
mean and the median. This rule, for lack of a better, might be used 
to determine the mode in suitable cases, or it might be used to test 
the value found in some other way. 

The general term ' average ' is frequently used when the par- 
ticular denomination ' arithmetic mean ' is implied, but the context 
will usually prevent misunderstanding. 

In order to get a clear impression of the outstanding features 
presented by the three chief averages discussed, let us go over them 
once more in the case of marks awarded to a number of students 
in a class. All three may be regarded as in a sense measures of 
the standard reached by the class as a whole in the examination, 
but the measures are made in different ways : — 

1. The Arithmetic Mean is found by merely dividing the aggregate 
marks of the class by the number of the students, and it gives the 
marks earned by each student if we conceive them all to be of 
equal merit. 

2. The Median is found by rangmg the students in order of merit 
from top to bottom, and picking out the marks awarded to the one 
who comes half-way down the list. 

3. The Mode is the most fashionable number of marks, i.e. the 
marks obtained by the greatest number of candidates. 

The advantages and disadvantages of the three types may be 
set out broadly as follows, although the boundary lines must not 
be too strictly drawn : — 



40 



STATISTICS 



Mean. 


Median. 


Mode. 


Easy to calculate when 
the values of the vari- 
able can be summed 
and their number is 
known. 


Easy to pick out when 
the individuals can 
be ranged in order 
according to the 
value or degree of 
the variable ob- 
served. 


Not easy to determine 
with precision, when 
the observations fall 
into groups of differ- 
ent ranges, without 
fitting a frequency 
curve to the distribu- 
tion as a whole. 


Well designed for alge- 
braical manipulation, 
as, for example, when 
we wish to combine 
different sets of obser- 
vations [see Appendix, 
Note 4, for two illus- 
trations]. 


Unsuited for algebrai- 
cal work. 


Unsuited for algebrai- 
cal work. 


Affected sometimes too 
much by abnormal in- 
dividuals among the 
observations. 


Determined merely by 
its position in the 
distribution, and its 
actual value is thus 
quite unaffected by 
abnormal individuals. 


Unaffected by abnor- 
mal individuals, and 
owes its importance 
to the fact that it is 
located in the region 
where the frequency 
is most dense. 



The reader should test his grasp of the principles so far intro- 
duced by applying them himself to a concrete case. For example, 
he might use the data in Table (11), with regard to wages earned 
by certain women, taken from Tawney's Minimum Wages in the 
Tailoring Trade, and based upon the 1906 Wages Census. Let him 
begin by roughly estimating the mean, the median, and the mode 
from an inspection of the distribution. He might then proceed 
to calculate the mean wage : — 

(1) taking the actual frequencies given in the table ; 

(2) taking simple sub-multiples of these frequencies, roughly one- 

hundredth part of each : 2, 4, 6, 7, 9, 11, etc. ; 

(3) assuming unit frequency in place of that given in the table for 

each wage group. 

Finally, he might determine the median and the mode in the 
manner explained in the text, deducing the latter from the relation 
(mean— mode) = 3(mean— median) . 



AVERAGES 
The results obtained should be 



41 



(1) 13-08S. ; (2) 13-lQs. ; (3) 15-59s. ^ 
Median=12-53sKf Mode=ll'43s. 



Table (11). Distribution of Wages of certain 
Women Tailors. 





(1 




(2) 


(3) 


(4) 




No. of Women 




No. of Women 


Wages betAveen limits 


earning wages 
as shown in 


Wages between limits 


earning wages 
as shown in 




Column (1). 




Column (3). 


5s. and less than 6s. 


\m^ 


16s. and less than 17s. 


642^^ 


68. , 




7s. 


384- 


17s. „ „ 18s. 


453 


7s. , 




8s. 


553^ 


18s. „ „ 19s. 


401 


8s. , 




9s. 


690 


19s. „ „ 20s. 


272^^ 


9s. , 




10s. 


900 j:^ 


20s. „ „ 21s. 


251 


10s. , 




Us. 


1145 


21s. „ „ 22s. 


138 


lis. , 




, 12s. 


1201 


22s. „ „ 23s. 


124 


12s. , 




13s. 


1138 


23s. „ „ 24s. 


64 


13s. , 




14s. 


930 


24s. ,, ., 25s. 


5r~^ 


14s. , 




15s. 


885 


25s. „ „ 30s. 


122 


15s. , 




16s. 


790 - 


.. 


•• 



CHAPTER VI 

DISPERSION OR VARIABILITY 

Let us suppose that two men set out separately on walking tours 
and that they walk as follows : — 





First Man 
walks 


Second Man 
walks 


First day . 
Second „ . 
Third „ . 
Fourth „ . 
Fifth „ . 
Sixth „ . 


20 miles. 
20 „ 
25 „ 
25 „ 
30 „ 
30 „ 


15 miles. 
20 „ 
25 „ 
25 „ 
30 „ 
35' „ 


6 days 


150 miles. 


150 miles. 



The total distance covered in six days, namely 150 miles, and 
therefore also the mean rate of walking, 25 miles a day, are thus 
exactly the same in both cases, but the dispersion of the values of 
the variable (the variable being in this instance the number of 
miles walked per day) round about their mean value, the variability, 
is different in the two cases. The greatest deviation from the 
average in the first case is five and in the second case it is ten miles. 

Thus, besides knowing the average of a set of values of a variable 
it is important to measure the dispersion of the distribution. Are 
the observations crowded in a dense mass around the average, 
or do they tail off above and below it, and to what extent ? 
In other words, what is the variability from the average of the 
distribution ? 

Mean Deviation. Now we are not concerned here with the signs 
of the separate deviations, with the question, that is, whether any 
particular value of the variable lies above or below the average : 

42 



DISPERSION OR VARIABILITY 



43 



it is only of their amount we wish to take cognizance, and perhaps 
the most obvious way to measure the total variability and at the 
same time to ignore the signs of the separate deviations from the 
average is to add up these deviations, treating them all as signless, 
and to divide the result by their total number. This gives what 
is known as the mean deviation of the system of observations — it 
is the ordinary arithmetic mean of the separate deviations, treated 
as if they are aU in the same direction, and, in measuring them, we 
may use either the mean or the median as the average, but it 
would seem preferable to take the latter because the mean deviation 
is least when the median is chosen as the origin, or zero point, from 
which the differences are measured. The proof of this fact will 
be found in Note 6 in the Appendix, but we may readily test it in 
a given case. 

Let us adapt the ' walking ' illustration used above, sUghtly 
extending the figures and making them unsymmetrical, i.e. of 
unequal variability on either side of the average, so as to prevent 
the median coinciding with the mean. We then have an amended 
table setting out the number of miles walked by a certain man on 
successive days during, say, a fortnight's tour, as follows : — 



Table (12). Number of Miles walked on Successive Days. 

(1) (2) (3) (4) (5) (6) (7) (8) 



No. of 
days. 


Miles 
walked. 


X 

Deviation 
from 25. 


^1 

Deviation 

from 

24-64. 


Xo 

Deviation 
from 24. 


Xi 

Deviation 
from 2G. 


[No. in 

Col. (l)]x 

[No. in 

Col. m 


[No. in 

Col. (l)]x 

[No. in 

Col. (4)]. 


1 
2 
3 
3 
2 
2 
1 


10 
15 
20 
25 
30 
35 
40 


■ 

15 

10 

5 

5 
10 
15 


14-64 
9-64 
4-64 
0-36 
5-36 
10-36 
15-36 


14 
9 
4 
1 
6 
11 
16 


16 

11 

6 

1 

4 

9 

14 


15 
20 
15 

10 
20 
15 


14-64 
19-28 
13-92 
1-08 
10-72 
20-72 
15-36 


14 


•• 


•• 


•• 


•• 


••, 


95 


95-72 



The first two columns show that 10 miles was the distance walked 
on the first day, 15 miles on each of the next two days, 20 miles 
on each of the next three days, and so on until the last day, when 
40 miles was the distance walked. 



44 STATISTICS 

The median in this case, being the number of miles walked on 
the middle day when the days are ranged in order of mileage from 
the least to the greatest, is 25, for this is the distance covered on 
both the seventh and the eighth days which come half-way along 
the series. 

Col. (3) shows the deviations from the median, 25, of the distances 
covered each day as recorded in col. (2), and col. (7) enables us to 
sum these deviations when each is multiplied by the number of 
days to which it corresponds, since these numbers, given in col. (1), 
show how many times each deviation is repeated. Hence the mean 
deviation, regardless of sign, measured from the median 

=:[(lxl5)+(2xl0)+(3x5)-f(2x5)+(2xl0)+(lxl5)]/14 
= (15+20+15+10+20+ 15)/14 
=95/14 
=6-79 miles. 

We may compare this with the corresponding deviations measured 
from (1) the arithmetic mean, (2) the number 24, and (3) the 
number 26 as origin respectively. 

1. The arithmetic mean of the distribution is obtained at once 
by multiplying the corresponding numbers in cols. (1) and (2), 
adding the results, and dividing the total by 14, thus 

1 + 2 + 3+3 + 2 + 2+1 

10+30+60+75+60+70+40 



14 
=345/14 

=24-64 miles, 

and the deviations from 24-64 are shown in col. (4) ; the mean 
deviation from 24-64, obtained by combining cols. (1) and (4) and 
adding as shown in col. (8) 

= [l(14-64)+2(9-64)+ . . . ]/14 

=95-72/14 

=6-84 miles. 

2. Similarly, the mean deviation from 24, making use of col. (5). 

= [l(14)+2(9)+ . . . ]/14 
= 6-93 miles. 



DISPERSION OR VARIABILITY 45 

3. And the mean deviation from 26, making use of col. (6), 

=[1(16)+2(11)+ . . . ]/U 
=7-07 miles. 

The original determination gives a value which is less than any 
of these three results, as was anticipated. 

The mean deviation from the median is, however, difficult to 
calculate with exactness when the observations are recorded in 
groups between different limits : for this and other reasons we 
shall not spend much time upon it, and we shall as a rule choose 
the mean as origin of reference rather than the median. It 
may be as well to explain the source of the difficulty by a small 
hypothetical illustration. 

Let us suppose that in making measurements of some organ or 
character in 13 individuals we get a result lying between 4 and 6 
units on six occasions, between 6 and 8 units on four occasions, and 
between 8 and 10 units on three occasions. Here, assuming that all 
the individuals in any group have the mid- value measurement for 
that group, i.e. treating the distribution as one of 6 individuals 
with a variable measuring 5 units, 4 individuals with a variable 
measuring 7 units, and 3 individuals with a variable measuring 
9 units, we get § as the mean deviation with 7 as origin and ^^ 
for the mean deviation with 6-5 as origin, as the following table 
shows : — 



Measurement. 


Frequency. 


X 

Deviation 
from 7. 


y 

Deviation 
from 6-5. 


fx 


fy 


4 and less than 6 
6 „ „ 8 
8 „ „ 10 


6 
4 
3 


2 

2 


1-5 
0-5 
2-5 


12 

6 


9 
2 
7-5 




13 


•• 


•• 


18 


18-6 



Now the result obtained is in agreement with the minimum 
mean deviation theory, granted that 7 is the median measurement, 
as it might certainly be. But it is not so of necessity, and in that 
case the assumption italicized might lead, in the above calculation, 
to appreciable inaccuracy unless the number of observations is 
large and the class-interval is small. For example, the actual 



46 



STATISTICS 



distribution might, without contradicting the previous data, con- 
ceivably run : — 



Measurement. 


Frequency. 


x' 

Deviation 

from 7. 


Deviation 
from 6-5. 


fx' 


fy'. 


6 

6-5 
7-5 
9 


6 

2 
2 
3 


2 

0-5 

0-5 

2 


1-5 

i 

2-5 


12 
1 

1 
6 


9 

2 
7-5 


•• 


13 


•• 


•• 


20 


18-5 



But in this case the median, the measurement for the seventh indi- 
vidual from either end of the series, is 6-5, and according to the 
first calculation the mean deviation referred to 6*5 as origin appears 
to be greater than that referred to 7 as origin. If, however, we 
recalculate, using the more detailed table, we find that the mean 
deviation referred to 6*5 as origin (^) is really less than the mean 
deviation with reference to 7 as origin, as it should be, for the 
latter now turns out to be j^. 

Standard Deviation. An alternative method of avoiding the 
signs of the' deviations from the average in order to estimate the 
amount of variability of the distribution is to square each separate 
deviation, sum the squares, divide by their number, and take the 
square root of the result. This gives the root-mean-sqmire deviation, 
and it is least when the arithmetic mean of the variables is chosen 
as origin from which to measure the deviations, when it is known 
as the standard deviation. For proof of this minimum principle 
see Appendix, Note 5, but it is worth while testing it also with the 
data given in Table (12). 

The numbers in cols. (3) to (6) in Table (13) are obtained simply 
by squaring the corresponding numbers in the same cols. (3) to (6) 
in Table (12). Col. (7) is formed in order to enable us to calculate 
the mean-square deviation referred to 25 as origin ; the numbers 
in col. (3) show the squares of the deviations for each individual 
observation, and the numbers in col. (1), by which they are multi- 
plied, show how frequently the same values are repeated. Hence 
we get the mean- square deviation with reference to 25 

-^[l(225) + 2(100)+3(25)+2(25)+2(100)+l(225)]/14 

=975/14 

= 69-64. 



DISPERSION OR VARIABILITY 47 

Thus the root-mean-square deviation referred to 25 

=8-345. ^ 

Similarly, by means of col. (8), formed on exactly the same 
principle, we find that the root-mean-square deviation referred to 
24-64 as origin 

= V[(214-33+ 185-86+ . , . )/14] 
= V(973-22/14) 
= 8-338. 
But 24-64 is the mean of the distribution, hence 8-338 is the standard 
deviation. 

With the help of cols. (5) and (6) the student may himself calcu- 
late the root-mean-square deviation with regard to 24 and 26 
respectively as origin ; the results should be 8-36 and 8-45. Of 
the four values thus obtained for the root-mean-square deviation, 
the least is that referred to the mean as origin, the standard devia- 
tion, now proposed as a measure of variability or dispersion suitable 
for most general purposes. 

This measure possesses several decided advantages over the 
mean deviation ; among others it lends itself more easily to certain 
algebraical processes (see, for example, p. 158), a fact of importance 
when we wish, for instance, to discuss two sets of observations in 
combination, and it is in general less affected by ' fluctuations of 
sampling ' — errors which arise owing to the fact that we cannot as 
a rulev survey the whole field of operations, but have to be content 
with a sample. 

Table (13). Number of Miles walked on Successive Days. 

(1) (2) (3) (4) (6) (6) (7) (8) 



No. 

of 

days. 


Miles 
walked. 


a;2 
Square of 
Deviation 
from 25. 


x,^ 
Square of 
Deviation 
from 24 -64 


x,^ 
Sqviare of 
Deviatfion 
from 24. 


xs' 
Square of 
Deviation 
from 26. 


/X2 

[No. in Col (1)] 
[No.inCol.(3)J 


fxi" 
[No. in Col. (1)] 

X 

[No. in Col. (4)] 


1 


10 


225 


214-33 


196 


256 


225 


214-33 


2 


16 


100 


92-93 


81 


121 


200 


185-86 


3 


20 


25 


21-53 


16 


36 


75 


64-59 


3 


25 


. . 


0-13 


1 


1 




0-39 


2 


30 


25 


28-73 


36 


16 


50 


57-46 


2 


35 


100 


107-33 


121 


81 


200 


214-66 


1 


40 


225 


235-93 


256 


196 


225 


235-93 


14 


•• 




•- 


•• 


-• 


975 


973-22 



48 STATISTICS 

Quartile Deviation or Semi-interauartile Range. There is a third 
measure of dispersion, based upon the determination of the quartiles, 
and to introduce them we may refer again to Table (7) in order to 
show how the idea of the median may be extended. 

We define the individual occupying a position one- quarter the 
way along any series of observations, arranged in ascending order 
of magnitude of some organ or character common to all the indi- 
viduals of the series, as the lower quartile ; and we define the indi- 
vidual occupying a position three-quarters the way along the series 
as the upper quartile. 

When the distribution of observations is divided up into groups 

lying between different Umits of the variable under consideration 

the quartiles may, like the median, be calculated by interpolation. 

^ Thus, in the examination example, the total number of candidates 

^ is 514 and J(514)- 128-5. 

^ But the 9 1st candidate from the bottom gets approximately 20 

marks, and the 149th candidate from the bottom gets approxi- 
mately 25 marks. Hence the imaginary candidate. No. 128-5, 
should get a number of marks lying somewhere between 20 and 
25. But if, in this neighbourhood, a difference of 

(149-91) candidates corresponds to a difference of 5 marks, 

37-5 

(128-5-91) ,, should correspond ,, 5x marks. 

Do 

Thus, the marks assigned to the lower quartile candidate are 
approximately 

58 
^« =20+3-23. 

Hence the lower quartile=2S-23. 

^' Again |(514)=:385-5. 

But the 318th candidate from the bottom gets approximately 35 
marks, and the 397th candidate from the bottom gets approxi- 
mately 40 marks. Therefore, the imaginary candidate, No. 385-5, 
should get approximately a number of marks 

=35+5x^ 
79 

=39-27. 
Hence the upper quartile=^^'21 . 



DISPERSION OR VARIABILITY 49 

It is clear that the quartiles together with the median divide the 
whole series of observations into approximately f om* equal groups, so 
that the quartile marks 
give a rough idea of the 23'23 31 '52 39'27 

distribution on either q ^^^^ q, 

side of the average. For 

this reason half the difference between the quartiles provides a 
convenient measure of the dispersion, and it is called the quartile 
deviation or semi-interquartile range ; thus, if Q bfe the lower and 
Q' the upper quartUe, we have 

the quartile deviation=^{Q'—Q). 

In the above example, this measure 

=4(39-27-23-23) 
=4(16-04) 

= 8-02. 

If a more minute analysis of the distribution of variables is 
desired, we may range them in order of magnitude as before, and 
divide up the series into ten equal parts, recording every tenth along 
the line ; these tenths are called deciles. 

Thus, the deciles in the examination example correspond to the 
marks assigned to imaginary candidates numbered as follows : — 

51-4, 102-8, 154-2, 205-6, 2570, 308-4, 359-8, 411-2, 462-6, 
and they can be calculated by the interpolation method used in 
finding the median and quartiles. 

This way of representing the chief features of a distribution, by 
quartiles, etc., was much used by Galton in his researches and 
writings. 

The student may be perplexed as to which should be used of so 
many different measures of dispersion or variability, but there 
need be no real confusion. If a rough estimate only is wanted the 
quartile deviation is a convenient measure, assuming that the 
variables observed or measured can be ranged in order of magnitude 
so as to admit of the quartiles being readily picked out. Also the 
measure thus obtained is not unsatisfactory when the distribution 
of values of the variable is fairly symmetrical and uniform in its 
gradation from greatest frequency to least. If, however, it is 
conspicuously skew (unsymmetrical) and there are erratic differ- 
ences in frequency between successive values of the variable, it 
is better to choose a measure which gives the magnitude and 
the position of each recorded observation its due weight in the 
deviation sum. 

D 



50 STATISTICS 

Then again the choice as between the standard deviation and the 
mean deviation may be sometimes determined by the particular 
kind of average which suits the problem best. But as the arith- 
metic mean is the most important and the most commonly used 
average, so the standard deviation is certainly the most important 
measure of dispersion. 

It will be shown later that the following relations are approxi- 
mately true when the distribution of variables is not very far from 
being symmetrical : — 

(1) Quartile deviation= ^(Standard deviation). 

(2) Mean deviation =i{Standard deviation). 

In (2) the mean deviation should be measured from the mean. 

Also (3) a range of two or three times the standard deviation 
will be found to include the majority of the observations which 
make up the distribution. 

Coefficient of Variation. Before we pass on to illustrate the 
subject of averages and variability by means of a few examples 
it is necessary to introduce one more constant known as the co- 
efficient of variation. It is a measure of variabiUty but it differs 
from the chief measures already discussed in that they are absolute 
measures, whereas the coefficient of variation, written C. of V. for 
short, is a ratio or relative measure. The need for it arises when 
we reflect that in order to gauge fairly the amount of variability we 
ought to have in mind also the size of the mean from which the 
variation is measured ; just as a difference of 1 foot between the 
heights of two men is a conspicuous difference when the normal 
height is between 5 and 6 feet, whereas the same difference of 1 foot 
between two measured miles would be trifling because the standard 
mile contains over 5000 feet. 

The coefficient of variation has been defined by Karl Pearson 
(Phil. Trans., vol. 187a p. 277), who first suggested its use, as ' the 
percentage variation in the mean, the standard deviation (S.D.) 
being treated as the total variation in the mean,' so that 

C. of V. = 100 S.D./Mean. 

He pointed out that it would be idle, in dealing with the variation 
of men and women (or indeed very often of the two sexes of any 
animal), to compare the absolute variation of the larger male organ 
directly with that of the smaller female organ, because several of 
these organs, as well as the height, the weight, brain capacity, etc., 



DISPERSION OR VARIABILITY 



51 



are greater in man than in woman in the approximate proportion 
of 13 : 12. 

As an example of the use of the C. of V., figures may be quoted 
from a paper by R. Pearl and F. J. Dunbar {Biometrika, vol. ii. 
pp. 321 et seq.), On Variation and Correlation in Arcella. Measure- 
ments in mikrons were made of the outer and inner diameters of 
504 specimens of a shelled rhizopod belonging to the group Imper- 
forata, family ArcelUna, with the following results, to two decimal 
places : — 





Mean. 


S.D. 


C. of V. 


Outer diameter . 
Inner „ 


55-79 
15-91 


5-73 
2-17 


10-27 per cent. 
13-66 „ 



Thus, judging by the S.D. column, giving the absolute size of 
deviation, the outer diameter would appear to be more variable 
than the inner, but the C. of V. column shows that, if we take the 
sizes of the two diameters into account, the inner is reaUy the 
more variable of the two. To turn aside the edge of possible criti- 
cism it should be added that the authors also give the errors to 
which the above measures are subject, as unless these are known 
we cannot teU whether the differences observed in variation are 
significant or not of a real difference in fact, but that question 
must be left until the theory of errors due to sampling has been 
developed in a later chapter. 

The C. of V. varies considerably for different characters. W. R. 
Macdonell states that * 3 to 5-5 are representative values for varia- 
bility in man, while in plants it may run to 40,' and Pearson and others 
have shown that for stature in man it varies from about 3 to 4 
and for the length of long bones from 4 to 6. 



CHAPTER VII 

FREQUENCY DISTRIBUTION : EXAMPLES TO ILLUSTRATE 
CALCULATING AND PLOTTING : SKEWNESS 

Calculation of Mean and Standard Deviation. Example (1). — We 
return now to the examination example in order to show how the 
labour of calculation in finding the arithmetic mean and standard 
deviation of a frequency distribution may be somewhat lessened. 

The various steps in the process appear in Table (14). In the 
first column the marks at the middle of each class-interval have 
been written down, and we make the assumption that all the candi- 
dates in any one class have the same number of marks, namely, the 
marks at the middle of the class-interval. In any case where the 
number of observations is large, and where the class -intervals are 
reasonably small, the errors resulting from such an assumption will 
be insignificant, because the individuals in each class are just as 
likely to have values above as below the value at the middle of the 
class-interval, and they will therefore compensate for one another. 

We now seek to alter the scale of marking so as to produce a 
simpler set of marks than the original, wliich will make the work 
of finding the mean also simpler, but we must not forget at the 
end to change back again to the original scale. We choose a number 
from col. (1), somewhere near the required mean, to act as a kind 
of origin from which to measure the other numbers in the column. 
This choice is only a rough guess, and it is really immaterial which 
number is selected as origin, except that the nearer it is to the 
mean the lighter will be the calculation to follow ; the number 33 
has been selected in this instance. 

In col. (2) are written down the deviations of the marks in each 
class from 33, so that now some candidates appear as if they were 
5, 10, 15 . . . marks to the bad, and others as if they were 5, 10, 
15 ... to the good. So long as we remember to add 33 at the 
end we can content ourselves therefore by finding the mean of the 
marks as given in col. (2). But these again can be further simplified 
by dividing each candidate's marks by 5, and we then only need 

62 



FREQUENCY DISTRIBUTION 



53 



to find the mean of the marks as shown in col. (3), so long as we 
remember to multiply by 5 at the first step back to the old scale 
of marking. The addition of col. (5) makes it easy to calculate 
this mean, for it gives the result of multiplying each value of the 
variable (the number of marks in each class) by its appropriate 
weight (the number of candidates who obtained that number of 
marks). 



Table (14). Marks obtained by 514 Candidates in a certain 
Examination — (Analysis of Method for Calculating 
Mean and Standard Deviation). 



(1) 


(2) 


(3) 


(4) 


(6) 


(6) 


Marks on old 
scale. 


Deviation of 
Nos.inCol.(l) 


Marks on 
new scale. 


Frequency 
of 


Product of 
Nos. in 


Product of 
Nos. in 


from 33. 


Candidates. 


Cols. (3) & (4). 


Cols. (3) & (5). 






{X) 


(/) 


(/^) 


if^') 


3=33-30 


-30 


-6 


5 


- 30 


180 


8=33-25 


-25 


-5 


9 


- 45 


225 


13 = 33-20 


-20 


-4 


28 


-112 


448 


18 = 33-15 


-15 


-3 


49 


-147 


441 


23=33-10 


-10 


-2 


58 


-116 


232 


28=33- 5 


- 5 


-1 


82 


- 82 


82 


33 = 33 


. . 


. . 


87 


. . 




38=33+ 5 


+ 5 


+ 1 


79 


+ 79 


79 


43=33 + 10 


+ 10 


+ 2 


50 


+ 100 


200 


48=33 + 15 


+ 15 


+ 3 


37 


+ 111 


333 


53 = 33 + 20 


+20 


+4 


21 


+ 84 


336 


58=33+25 


. +25 


+ 5 


6 


+ 30 


150 


63 = 33+30 


+30 


+ 6 


3 


+ 18 


108 






•• 


514 


-110 


2814 



Thus, on this new scale, the mean marks obtained are 

5(_6)+9(-5)+28(-4)+ . . . +87(0)+ . . . +6(+5)+3(+6) 



514 



-532+422 
614 

-110 



514 
-0-214. 



54 STATISTICS ■ 

This, then, is the mean of the marks obtained by the candidates on ; 

the scale indicated in col. (3). If the marks are on the scale given ■ 
in col. (2), the mean is 5(— 0-214), i.e. —1-070. To bring them back 

to the original scale as in col. (1) we must add 33 to this result, so ^ 

that the required arithmetic mean ^ i 

-33+5(-0-214) i 

= 33-1-070 ^ 

=31-93. 1 

To find the Standard Deviation, or the root-mean-square deviation i 

from the arithmetic mean, it is convenient as before to work with ■ 

the simpUfied scale, to measure the deviations from the arbitrary | 

origin (33) associated with that scale, and to make the necessary i 

corrections at the end of the work. j 

Col. (5) in Table (14) gives the deviation multiplied by the ■ 
frequency in each class, the frequency denoting the number of 

times the particular deviation occurs. Hence, if these numbers be j 

multiplied again by the numbers in col. (3), we shaU have each • 

separate deviation squared and multiplied by its frequency. The '. 

results are shown in col. (6), and they must be added, and their ! 

sum divided by the sum of the frequencies (514), to give the mean- j 

square deviation, which we may represent by s^. ; 

Thus . 52=2814/514 f^' 

=5-475, \ 

and this is the mean-square deviation referred to 33 as origin. 
We require the corresponding expression referred to the mean, j 
31-93, as origin. If we denote this by s^^^ there is a simple relation ' 
connecting the two, namely, 



where x is the deviation of the mean itself from 33 [see Appendix, \ 

Note 5] ; of course s^^, s, and x are all to be measured on the same i 
scale, the simplified scale adopted with 5 marks as unit. 

Now we have already shown that the deviation of the mean from :, 
33^—0-214, and this is therefore the value of x. 

Hence s^2=5-475- (-0-214)2 I 

=5-475-0'046 ] 

=5-429 \ 

= (2-33)2. ] 



FREQUENCY DISTRIBUTION 55 

And; returning to the old scale, the standard deviation, usually 
denoted by a 

=5(2-33) 

= 11-65. 

We notice that 3cr= 34-95, and this range on either side of the 
mean amply takes in all the observations. 

The mean deviation is readily found from Table (14) by adding up 
the numbers in col. (5) regardless of sign and dividing by the sum 
of frequencies, 514. 

Thus, on the new scale, the mean deviation 

954 
= 5^TT 



1-856 



which, on the old scale, becomes 5(1-856) or 9-28. This, however, 
is the mean deviation measured from 33 as origin, and a correction 
has to be applied to get the mean deviation measured from the 
median or from the mean. 

To get the mean deviation from the mean we note that the 
difference between the mean, 31-93, and 33 is 1-07. Hence it 
should be clear from Table (14) that, by measuring from 33 instead 
of from 31-93, we have made the deviations of all the marks from 
33 upwards too little by 1-07, and we have made the deviations of 
all the marks from 28 downwards too much by 1-07. Hence, to 
get the deviation required we must add to 9-28 an amount 

= 6T4[l-07(87+79+ . . . +3)- 1-07(82+58+ . . . +5)] 

1-07 
=:^(283-231) 
514^ ' 

= — X52 
514 

=0-108. 

Therefore, the mean deviation measured from the mean=9-39. 
This may be compared with I (standard deviation) =9-32. 

Also the quartile deviation for this distribution has been shown 
to be=8-02, and it may be compared with §(standard deviation) 

= 7-77. 

Plotting of a Frequency Distribution. The data for the two 
examples which foUow are taken from the Quarterly Return of 
Marriages, Births, and Deaths, No. 261, issued by the Registrar- 
General, 



56 



STATISTICS 



The first shows the proportion to population of cases of infectious 
disease notified in 241 large towns of England and Wales for the 
thirteen weeks ended 4th April 1914. This proportion was given 
for each town separately in the Return, but, in order to bring out 
the distinctive features of the distribution, the several towns have 



Table (15). Proportion to Population of Cases of Infectious 
Disease notified in 241 Large Towns of England and 
Wales during the Thirteen Weeks ended 4th April 1914. 



Case Rate 
per 1000 
persons 
living. 


Each dot below represents One Town with Notified Rate of Infectious Disease 
between limits as given in previous column. 


Total No. 

of Towns 

with given 

Rate. 


0- 


....| 


5 


2— 


....|....|....|....!....|.. ..!... .!.... 


39 


4— 


....!....|....|....i....|....l....|....|....I....!....l...,l....!.... 


69 


6- 


....:....l....i....|.... !....!.... I. ...|. 


41 


8— 


....|....|.. ..!....!....!.... 


29 


10— 


....!....!....!....:.. 


22 


12— 


....'....I....1. 


16 


14— 




7 


16- 


— 


5 


18— 


... 


3 


20- 


— 


4 


22— 







24— 







26— 


• 


1 1 






241 



been, in Table (15), represented by dots and put into different classes 
according to the proportion of infectious cases notified in each, 
with a separate line for each class : e.g. ii the proportion for any 
town was 5-37 a dot was placed in the line corresponding to the 
class of towns for which the rate was ' 4 and less than 6.' Every 



FREQUENCY DISTRIBUTION 



57 



fifth dot in each line was ticked off, so as to make them easy to 
count up and also to keep the lines, down the paper as well as 
across, straight. The frequency, i.e. the number of dots in each 
class, was then recorded in a column at the extreme right-hand 
side of the paper. 



'^3 


Ifr 










.^ 












^ .- 












stub- 












's 




























































2 












(5? 












^^ 












<Occ " 
























§5 












OT -J 












•«, 












'^SO H 












05°° - 












§ 












o 












"•5 












O ii"? . 












vS**' : 












"t^ 
























^^ 












*t^An 
























c 


T 










o 


? 












*" 










"^•^s - 


T 


J 








o oo 


^ 












-T^ 










s 












Q. 


X ' 










Ort 


} 










s;^° " 














5 










^ = 


5 ! 










c:^ 












01 OK _J 


Xh 






















■k> 












-C _j 


±: 










Ho 


XI 






^ 




'^on 








X 




520 


-r 






A 










± : : 




^ 


X 




5 


j_ 






t 










^ i«^ 








xi - - 




15 








TT 






-1. 






^-^ 




v.. 






-A -II 




o' 


* 






•- •^ + 




•>^ irt 


* 










g»10 


h-j-- 
















5 




^ 


X- 






T T 




:5 








T 




ST «i ^ 


tx- 






I x 




^ 5 -^ 








Z_ A ? 




^ i 


5 






! ' 






:i: 


-^- 




5 i I ! 






xX- --j-^^-T-- 






10 



20 



25 



30 X 



Rate of Disease per 1000 persons living 

Fiu. (1). 

It will be at once seen that this procedure, without calculating 
any averages, etc., ultimately gives to the eye a very good picture 
of the distribution, and indeed it is the basis of the graphical method 
of studying statistics. In drawing a proper graph we use a specially 
ruled sheet of paper which is divided up into a large number^of 
equal small squares by * horizontal ' (cross) and ' vertical ' (up-and- 



58 



STATISTICS 



down) lines. This merely enables us to place our dots accurately 
in position, as shown in fig. (1), where the numbers 0, 5, 10 . . . 
have been marked off along the line Ox to correspond to ' case 



y -~ 
















Tn _ 






T 




_r 




T_ 


^ - 


it 


^fir: _ 


- Jt - -- - 


H^65 


t 


•♦j 


t 










^ en - 








a 




Ci 


-■ 






«0 EC _ 








Q) 


r 


^ 


r 


C5 




CO en - 




S 50 - 




.o 




•tj 


. 


S 


. . 4^ 


'^AR - 


J 


^45 


t - - 




■ t it 


o^ - 








s: An 




O 40 - 


L 


'•P 


1 HIT . 4 f T 


? 


\ ^ Modal Line 


Q. : 


- 1 ^^ 


P Qc; 


It ^^ 


^35 


t U^^ 




:: i^ 


s 




§ 






4__ ^/ 


OjOO - 


X^^ 


Q> 


-t^ 


^ 


A 






:S25 - 




■ti ^o 


t^^ 


S 


^\ 




^ V 




x 


ri ork 


\ __ ± 




i~ 


f2 


i 




t I 


"> 


V 




_r __ _ 


s^'5 ■ 


t 


o 


\ 


03 




S 




^ in 




Q^ 10 -j- 




u: 


\ ^ 








^ 


c 


i 




5^ i 




v^^ 




^"^ X 




x 


o X 


1 1 lllllll 1 1 1 1 1 1 1 1 M M 1 IMU^I 1 1 1 1 1 1 1 



5 10 15 20 25 30 

Rate of Disease per 1000 persons living 
Fig. (2). 

rates ' of these magnitudes : thus rates of ' 4 and less than 6 ' 
were recorded by 69 successive dots along a vertical line at a dis- 
tance 5 (the centre of the class-interval 4-6) from the axis Oy. 



FREQUENCY DISTRIBUTION 



59 



The final configuration in fig. (1), when turned half round, is 
exactly the same as that of Table (15). If desired the frequency 



y 


-j_-_ -_- ___ ___ 


















70 - - 












"B 












•g 












« eo 




% 60 - — 




^« i 




o 




- 




fO 




5S 55 




CO 












<* i^O - 




3 £>U 




.o 




s^ 




s : 




*K/it: 




^45 - 








o* - 






-r 11 )C[s;l U;rae 


=: /.n 




o 40 - 


r ^'^ 


t: — - 




'' Median" Li le 


CJ. 


--^ 




y 


V o5 




^- 


; /X r ;an Itme- 


s= 


' ^^ I 


§ 




•§.an 




Oi 30 - - 




<» 




:g 








5 OK 




.t; 25 - 




S 








^ 




2i nr\ 








^ 








*t> 








2h ^S " 




o 




§ 




^ 




5»" in 




CD lO -- 




li: 
















5 1- - 






■* 










OLJ-L 






5 10 '15 20 25 30 X 



Fig. (3). 



may be recorded, dot by dot, on a side piece of paper and then 
only the topmost dot in each class need be marked on the graph 
sheet. In order, however, to enable the eye to measure the height 



60 STATISTICS 

X)f each frequency in relation to the rest, it is advisable in that 
case to connect up adjacent dots as in fig. (2) or as in fig. (3). 

The last method of representation (fig. (3)), to which the name 
histogram has been given by Professor Karl Pearson, is particularly 
useful and should be carefully studied. It is formed in this case by 
erecting a succession of rectangles with the lines 02, 24, 46 . . . 
along Oa; as their bases, corresponding to the successive classes of 
the given distribution, and with heights proportional to the fre- 
quencies proper to those classes. It is not necessary to complete 
the sides of the rectangles, but, if they were completed, each would 
enclose a number of squares proportional to the frequency of towns 
with the rate of disease defined by its base : e.g. the first rectangle 
would enclose 10 squares, the second 78, the third 138, and so on, 
numbers respectively proportional to 5, 39, 69, and so on. It 
follows that the total area enclosed between the histogram and the 
axis Ox is proportional to the aggregate frequency of towns observed. 

Now we might conceive a step further taken and a smoothed 
curve drawn freehand so as to agree as closely as possible with 
fig. (2) or fig. (3), but with all the sharp corners smoothed out, and 
so nicely adjusted as to make the area enclosed between the curve, 
the axis Ox, and lines parallel to Oy defining the limits of any class, 
proportional to the frequency of towns in that class. To this 
fig. (2) and fig. (3) might be regarded as approximating if only a 
sufficient number of observations were recorded, and only in that 
case would it be possible to draw it with any accuracy. Such a 
curve is called a frequency curve, measuring as it does the frequency 
of the observations in different classes. 

[Assuming that corresponding to a given frequency distribution a curve 
of this kind does really exist — and the assumption turns upon the frequency 
being continuous — the reader who is acquainted with the notation of the 
Calculus will recognise that, if {x, y) represents any point on the curve, ybx 
measures the frequency of observations or measurements of an organ or 
character lying between the values x and {x-\-bx), when the total frequency 
comprises a large number of observations, say 500 to 1000. 

Further, it will appear later that the mean, the median, and the mode 
have a geometrical interpretation of no small importance associated with the 
curve. 

The mean x corresponds to the particular ordinate y which passes through 
the centroid or centre of gravity of the area between the frequency curve 
and axis (ix, because 

the mean= J ^ 2{x.y8x)/J^ My^x), 

where the summation extends throughout the distribution, 

=jxydx/jydx 
where the integral extends throughout the curve. 



FREQUENCY DISTRIBUTION 



61 



The median x corresponds to the ordinate y which bisects this same area ; 
e.g. in fig. (3), the number of small squares on either side of the median in the 
space bounded by the histogram and the axis represents half the total number 
of observations, two small squares corresponding to each observation. 

The mode « corresponds to the maximum ordinate of the curve, measuring 
the greatest frequentsy in the whole distribution.] 

Skewness. There is one feature of a frequency distribution which 
catches the eye sooner almost than any other, and that is its sym- 
metry or lack of symmetry. It is important therefore that we 
should have some means of measuring it. 

In a symmetrical distribution the mean, mode, and median 
coincide, and we have, as it were, a perfect balance between the 
frequency of observations on either side of the mode or ordinate of 
maximum frequency. In a skew distribution the centre of gravity 
is displaced and the balance thrown to one side : the amount of this 
displacement measures the skewness. But there is another factor 
to be taken into account, for when the variability of the distribu- 
tion is great the balance is more sensitive than when it is small, 
and the difference between mean and mode is consequently more 
pronounced though it may not be significant of any greater skew- 
ness. This will be clear in the light of the analogy of the swing 
of a pendulum. If OPP' denote the pendulum in the accompanying 
figure, OAA' its mean position, and OBB' an extreme position, the 
displacement in the position OPP' from the mean, if measured 
along the scale AB, is AP, 
and, if measured along the 
scale A'B', is AT'. But, 
since the amount of swing 
in either case is the same, 
it would be more appropri- 
ate to write the linear dis- 
placement as a fraction of 
the full swing so as to make 
these two measures also the 
same, thus 

AP/AB=A'P'/A'B'. 

So, in the case of a fre- 
quency distribution, Profes- 
sor Karl Pearson has suggested as a suitable measure for skewness, 
not the difference between mean and mode, but the ratio of this 
difference to the variability. Thus 

skewness— (mean— mode) I S.D. 




62 



STATISTICS 



or, approximately, 

=3(mean— median)/S.D. (see p. 39), 

a form which is sometimes useful. 

According to this convention the skewness is regarded as positive 



Skewness + 




Skewness — 



Mode Mean 




Mean Mode 



X increasing 



X increasing 



when the mean is greater than the mode, and as negative when 
the mode is greater than the mean. 

Illustrations of frequency curves, with the position of mode and 
mean marked, will be found in Chapter xvii. 

We proceed to the detailed calculations necessary in the infectious 
diseases example. 



Table (16). Proportion to Population or Cases of Infectious 
Disease notified in 241 Large Towns of England and 
Wales during the Thirteen Weeks ended 4th April 1914. 





(1) 


(2) 


(3) 


(4) 


(5) 


Case Rate per 
1000 persons living. 


Deviation 
from 7. 


Frequency of 
Towns with 
given Rate. 


Product of . 
Nos. in 
Cols. (2) & (3). 


Product of 

Nos. in 

Cols. (2) & (4). 


and less than 2 


[x) 
- 3 


(/) 
5 ^ 


-15 


{fx^) 
45 


2 


, „ 4 


- 2 


39 u. 


-78 


156 


4 


6 


- 1 


69 »■> 


-69 


69 


6 


8 




41 






8 


„ 10 


+ 1 


29 


+ 29 


29 


10 


» 12 


+ 2 


22 


+44 


88 


12 


„ 14 


+ 3 


16 


+48 


144 


14 


„ 16 


+ 4 


7 


+28 


112 


16 


„ 18 


+ 5 


5 


+25 


125 


18 


„ 20 


+ 6 


3 


+ 18 


108 


20 


„ 22 


+ 7 


4 


+ 28 


196 


26 


„ 28 


+ 10 


1 


+ 10 


100 


•• 


•• 


241 


+68 


1172 



FREQUENCY DISTRIBUTION 63 

Example (2). — ^The various averages and measures of variability 
of the distribution can be calculated just as in the case of the last 
example, and the data required to determine the mean and the 
standard deviation are set out in Table (16). We can afford now 
to miss out some of the more obvious steps in explanation. 

On the scale of col. (2), where a difference of 2 in the case rate, 
per 1000 persons living, is the unit and where a case rate of 7 is 
taken as origin, the mean, by the result of col. (4) 

68 
="2TT 

=0-282. 
Hence, on the original scale, the mean 

=7+2(0-282) 
=7-564. 

Again, the mean-square deviation, on the scale of col. (2), measured 
from 7 as origin is 



=4-863 ; 

and X, the deviation of the mean from 7 as origin, on the scale of 
col. (2) =0-282. Thus the mean-square deviation measured from 
the mean, 

=4-863- (0-282)2 
=4-783. 

Therefore, the standard deviation a, on the original scale 

=2V4^78^ 
=4-374. 

Since 3or= 13-122, the range ' (mean— 3o-) to (mean+3o-) ' includes 
all but one or two observations. 

To determine the median, we conceive the towns ranged in order 
according to the proportion of infectious cases notified in each, 
from the least to the greatest, and the town with the median rate 
is the 121st from either end. 

But the 113th town has a notified case rate of approximately 6 
per 1000, and the 154th town has a notified case rate of approxi- 
mately 8 per 1000. 

Thus a difference of 41 towns corresponds to a difference of 2 in 
the rate, hence a difference of 8 towns corresponds to a difference 
of 0-39 in the rate ; therefore the median ra<e= 6-39 approximately. 

By referring to the original records and writing down the rate 



64 STATISTICS 

for each town in the group ' rate 6 and less than 8 ' in which the 
median lay, the accurate value of the median turned out to be 6-30. 
The lower quartile or case rate of the imaginary town, No. J(241), 
or 60-25, one-quarter way along the ordered sequence of towns, is 
readily shown to be 447, and the upper quartile or case rate of 
town No. i(241), or 180-75, is 9-84. 
Hence the quartile deviation 

=i(9-84-4-47) 
=269. 

With this may be compared |(S.D.)=f(4-37)=2-92. 
Again, the mean deviation measured from 7 
=2(111) 
=3-253. 

Measured from the mean, it becomes 

=3-253+ -[(41+69+39+5)-(29+22+16+7+5+3+4-fl)] 
241 

= 3-253+ (0-564)(67)/241 

=3-41 

and this may be compared with i(S.D.)= 1(4-374) =3-50. 

If we estimate the mode by inspection of the frequency graphs in 

figs. (2) and (3), we should say it comes between 5 and 6 ; supposing 

we call it 5-5, very roughly. 

In this case, taking the values actually calculated for mean and 

median, 

(mean— mode)=7-56— 5-50 

=2-06, 

and 3(mean—median)= 3(7-56— 6-39) 

=3(1-17) 

= 3-51 ; 

so that the rule 

(mean— mode) = 3(mean— median) 

is far from being true according to these results ; this is partly due, 
of course, to the very unsymmetrical character of the distribution. 

The relative positions of the mean, median, and modal points 
as calculated are indicated in figs. (2) and (3) by three fines drawn 
paraUel to Oy through these points to meet the graph. 

Finally, 5A;ew;iie55=(mean— mode)/S.D.=2-06/4-37=0-47. 

Example 3. — The next example deals with the deaths of infants 
under one year, out of every thousand born, in 100 great towns in 
the United Kingdom during the thirteen weeks ended 4th April 1914. 



FREQUENCY DISTRIBUTION 



65 



The details of the calculation may be left in this case to the reader, 
who is recommended to follow the method shown in the last example 
so far as possible throughout, including the plotting of the distribu- 
tion in different ways. The statistics are as follows : — 

Table (17). Death Rate of Infants under 1 Year 
PER 1000 Births. 

(1) (2) (3) (4) 





No. of Towns 




No. of Towns 


Death Rate. 


with Death Rate 


Death Rate. 


with Death Rate 




as in Col. (1). 




as in Col. (3). 


30 and under 40 


1 


120 and under 130 


16 


50 „ 60 


3 


130 „ 140 


11 


60 


70 


2 


140 „ 150 


10 


70 


80 


6 


150 „ 160 


8 


80 


90 


7 


160 „ 170 


3 


90 


100 


6 


170 „ 180 


1 


100 


110 


11 


200 „ 210 


1 


110 


120 


13 


240 „ 250 


1 * 



The more important results are : — 
Arithmetic mean= 118-9; S.D. = 32-2 ; 

median= 120-9 ; quartile deviation ^ 



19-5. 



Example (4). — As another example corresponding details may be 
worked out for the following temperature records J^ken at noon 
at a certain spot in Chester week by week during a period o^me 
covering five years, the results in this case being : — 
mean=55-10; S.D.=10-33 ; 
median= 54-88 "f quartile deviation =7 -94 



Table (18). 257 Weekly Records of Temperature (Fahrenheit). 

(1) (2) (3) (4) 



Temperature 


No. of Records 


Temperature 


No. of Records 


Limits in 


between Limits 


Limits in 


between Limits 


Degrees. 


shown in Col. (1) 


Degrees. 


showninCol.(3) 


25-5-29-5 


I 


53-5-57-5 


30-5 


29-5-33-5 


1 


57-5-61-5 


31-5 


33-5-37-5 


9 


1 61-5-65-5 


30 


37-5-41-5 


11-5 


1 65-5-69-5 


26 


41-5-45-5 


28 


69-5-73-5 


13-6 


45-5-49-5 


31-5 


73-5-77-5 


4 


49-5-53-5 


36-5 


77-5-81-5 


3 




U i\^ 







G6 > STATISTICS 

Before closing the chapter a shghtly different manner of graphing 
the statistics is worth noticing, as it provides us with a fairly quick 
though rough alternative method of determining the mode and 
median. 

Take, for example, the examination marks data which for this 
purpose must first be thrown into the second form shown below 
Table (7). We mark off on some convenient scale along OX dis- 
tances 5, 10, 15, 20 ... 65 from O to represent these numbers 
of marks respectively, and at the points obtained we erect lines 
parallel to OY of lengths 5, 14, 42, 91 . . . 514 to represent the 
numbers of candidates who obtained not more than 5, 10, 15, 20 
... 65 marks respectively. A freehand curve is then drawn 
through the summits of these lines in the manner indicated in 
fig. (4), starting from a height 5 and rising to a height 514 above 
the axis OX. 

By means of this curve we can approximately state at once how 
many candidates obtained any given number of marks or less. 
Suppose, for example, we wish to know how many candidates 
obtained 22 marks or less, we have only to measure off a distance 
22 from 0, represented by ON, and erect a perpendicular NP to 
meet the curve at P. Since NP=110 we infer from the manner in 
which the curve has been formed that 110 candidafes obtained 
22 marks or less, so that, incidentally, the 110th candidate from 
the bottom must have obtained approximately 22 marks. This 
suggests that by working backwards we can also read off roughly 
the^umber of marks gained by any particular candidate when his 
order in the Hst is known. Thus, to find the median, i.e. the marks 
due to candidate No. 257-5, we merely draw a line parallel to OX 
at a height 257-5 above it and the portion of this line cut off between 
the curve and OY measures the median. The value given by this 
method is approximately 31-5. Similarly the quartiles are found 
by drawing lines parallel to OX at heights 128-5 and 385-5 above 
it with results about 23-3 and 39-2 respectively. 

Again, as we gradually increase the number of marks, the number 
of candidates getting that number of marks or less must increase 
also, but the rate of this second increase is variable. The reader 
will perceive that where the height above OX changes slowly the 
gradient of the curve is small, but where it changes by big steps 
the gradient is steep, and it is at its steepest just in the neighbour- 
hood where the greatest addition is being made to the height as 
the marks increase, i.e. where the frequency of additional candi- 
dates is at its greatest, so determining the mode : this should be 



FREQUENCY DISTRIBUTION 



G7 



clear on a comparison of the two arrangements of the data in and 
below Table (7). By sliding a straight-edge along the contour of 
the curve we can estimate approximately where the curve is 
steepest, for at this point the direction of turning of the ruler or 



Y """::- - 




















i '< 


^ . . . 


/ 






J 


^ _ .. _ .. 


T . z ":. : : ± 




K4i~ LpBer-Ouaitiilp-iUne. 


CO :::::::::::::"::::f ::::::: ::::::::::::: 


•S J 


Q oen - - __/_ .._ _. 


"S ^°° 


•S 


1 J :::;:: : 


« f .. _ . . 


<^300__ _ _ t - - . - 


v^^°° -,i± : . . 


^ t\ 


V. ,.f J 


55 \\l-\ . Wie^ian.J ii e^. 


1 250 = = = = = ^ = = = == = = =7=:: = = ==:= = ==:===- 


a - . - 


^ ..__.. 








T '■ - 


r 






150- -H- H-jf -- -- -- --- 


"Q/r ■ .(H ^Julu F glliai: 




W. \ - 








i ■ - ■■ ■■ - 








y _ . ._.._..- 


y ._..-..- 


^^ . __ . y^ . _ .. _ .. ^ 


o--i:::Lj::...--lL:u--. L-X,.- -.. 



to 



20 N 30 40 50 

Number of Marks 



60 



70 



Fig. (4). Graph showing the Number of Candidates who obtained not 
more than any given Number of Marks. 



straight-edge must change. This gives for the mode a value in the 
neignbourhood of 32. 

It might be advisable to treat the other examples by this method 
also, so as to compare results. 



CHAPTER VIII 



GRAPHS 



From the mathematical point of view graphs may be regarded as 
the alphabet of Algebraical Geometry. 

We can locate a point in a plane, relative to two perpendicular 
lines or axes as they are called, OX, OY, which serve as boundaries 

of measurement, when we know y and a;, 
its shortest distances from these boun- 
daries. This fact serves to connect up 
Geometry, in which points are elements, 

with Algebra, in which a:'s and t/'s, 

X standing always for numbers, are ele- 
ments. The names abscissa (ah — ^from, 
and scindo — I cut) and ordinate are given to x and y, or, when we 
refer to them together, they may be spoken of as the co-ordinates of P. 
The celebrated French philosopher, Descartes (1596-1650), was 
the founder of Cartesian Geometry, and if we may venture to com- 
press the essence of his system into a single statement, it is this — 
When a point P is free to take up any position in a given plane, 
its X and y are quite independent : they may be allotted any values 
irrespective of one another. Suppose, however, that P is constrained 
to lie somewhere on an assigned 
curve, such as APB in the figure, 
then X and y are no longer inde- 
pendent, for, so soon as x is fixed, 
y is fixed also ; it follows that in 
this case some relation, algebraical 
or otherwise, such as y=x^—2x-{-'l, 
must exist between x and y, and the relation may be called the 
equation of the curve which gives rise to it. 

Now, if to every curve there corresponds in this way some 
equation and to every equation some curve, it seems likely that the 
simpler the curve the simpler will be the corresponding equation, 
and vice versa. In fact, the student who does not know it already 




GRAPHS 



69 



need only refer to the most elementary treatise on graphs to find 
that every equation of the first degree in x and y, i.e. one which does 
not involve any x^, y^, xy, or higher powers, represents some straight 
line. Any such equation, e.g. 

x-3y-{-l2=0, 
can be at once thrown into 
either the form 



(1) 



12 4 




where — 12 and 4 are intercepts 
made by the line on the axes 
OX and OY ; or 

(2) 2/=ia:+4, 

where J, i.e. 1 in 3, is the measure of its gradient and 4 the height 
above the origin at which it cuts the axis OY. 

Further, every equation of the second degree in x and y, which 
may involve x^, y^, and xy, but no higher powers, represents geo- 
metrically some conic, a family of curves comprising the parabola, 
the ellipse, and the hyperbola, with the circle and two straight 
lines as particular cases. The earth and other planets, likewise 
comets, in their journeys through space travel along curves belonging 
to the same family, one of ancient and historical connections. 

These conies need not, however, detain us, and we pass on at 
once to an example of a cubic graph to show how a very little 

knowledge of the theory may be put 
to some practical use. Suppose a 
box manufacturer has a large number 
of rectangular sheets of cardboard, 
3 ft. long by 2 ft. broad, and he 
wishes to make open boxes with them 
by cutting a square piece of the same 
size out of each corner and turning 

[The shaded flaps are bent upwards up the flaps that are left. How big 
along the dotted lines.] , , , . , .|. i . . u. i. 

should the squares be if this is to be 
done with as little waste as possible ? Clearly this is commercially 
an important type of problem to solve. 

Let us denote a side of the square to be cut out of each corner 
by X feet. Then the bottom of the required box will have dimensions 

(3-2a;) ft. by (2-2x) ft. 

and its depth will be x ft. 




70 



STATISTICS 



Hence the capacity of the box when completed will be 

a:(3-2a;)(2— 2a;) cu. ft., 

and he makes best use of the material who produces the most 
capacious box. Call this expression y and let us find the values 
of y corresponding to different values of x so as to be able to draw 
roughly the curve of which the equation is 

y=^x(Z-2x)(2-2x) . . . (1) 



Table (19). Table of Corresponding Values of x and y 
IN the Curve y=x{3—2x){2—2x). 



X 


2x 


(3 -2a:) 


(2 -2a;) 


a;(3-2ic)(2-2x) 


y 


-1 


-2 


5 


4 


-20 


-20 


:| . 


-1 


4 


3 


- 6 


- 6 


-\ 


I 


f 


-M 


- 219 








3 


2 








+t 


+ h 


1 


f 


+ M 


+ 0-94 


+ * 


+ 1 


2 


1 


+ 1 


+ 1 


+ 1 


+ f 


1 


i 


+ A 


+ 0-56 


+ 1 


+ 2 


1 











+li 


+ # 


i 


-i 


-^ 


- 0-31 


+li 


+ 3 





-1 








+ 2 


+ 4 


-1 


-2 


+ 4 


+ 4 


+ 2i 


+ 6 


-2 


-3 


+ 15 


+ 15 


0-2 


0-4 


2-6 


1-6 


(0.2)(2.6)(1.6) 


0-83 


0-4 


0-8 


2-2 


1-2 


(0.4)(2.2)(l-2) 


1-06 


0-6 


12 


1-8 


0-8 


(0.6)(1.8)(0.8) 


0-86 


0-8 


1-6 


1-4 


0-4 


(0.8)(1.4)(0.4) 


0-45 


0-38 


0-76 


2-24 


1-24 


(0-38)(2-24)(1.24) 


1055 


0-39 


0-78 


2-22 


1-22 


(0.39)(2.22)(1.22) 


1056 


0-40 


0-80 


2.20 


1-20 


(0.40)(2.20)(1.20) 


1-056 


0-41 


0-82 


2-18 


M8 


(0-41)(2-18)(1.18) 


1-055 



We get a tolerably good idea of the shape of the curve by plotting 
the points (x, y) shown in Table (19) from x— — \ to x—-\-2 as in 
fig. (5). It is simply a matter of practice to be able to determine 
the whole curve from a few points in this way, and the greater the 
number of points plotted the more accurately will it be possible 
to draw the curve. It should be noticed that the points for which 
^=0 are in a sense key-points to the curve : they are readily 



GRAPHS 



71 




QfTtthf" 



mag-nifi^ 



0-25 0-50 0-75 100 X 

Length of Side of Square cut out 

Fig. (5). ^'*"! 



72 STATISTICS 

found by making the factors separately zero in the right-hand side 
of equation (1), namely x=0, 3— 2ic=0, and 2— 2ic=0, and by 
plotting them first they serve as a guide to the position of points 
subsequently plotted. 

We want to laiow for what value of x the capacity of the box, t/, 
is greatest and the preliminary plotting is enough to indicate a 
maximum value for y between x=0 and x=l, for the curve first 
rises and then falls between these two limits. In order to discover 
more exactly where the maximum is located we therefore plot 
in addition the points corresponding to x=0-2, 0-4, 0-6, 0-8 respec- 
tively, and this is done on a larger scale than that used in the 
first diagram because the accuracy is thereby increased (see fig. (5) 
inset). 

The calculations and figure suggest that the maximum required 
is very near the point for which a;=0-4, so we next work out values 
of y in this neighbourhood, corresponding, say, to a;=0-38, 0-39, 
0-40, 0-41, with the results shown at the foot of Table (19). From 
these we conclude that to a fair degree of accuracy the maximum 
value of y is given by taking a:=0-395. It would be possible in 
the same way to calculate more decimal places, but we have gone 
far enough to make the method clear. 

Hence the side of each square cut out should be of length 

0-395 ft., or 4| in. 

Whenever the value of one variable, y, depends upon that of 
another variable, x, in such a way that when x is given y is known, 
so that y may be termed a function of x, corresponding values of 
X and y can be plotted — as was done in the example just discussed — 
and a curve drawn by joining up the points obtained, the relation 
which connects x and y being the equation of this curve. More- 
over, it is possible, by calculating enough points from the equation 
and plotting them, to get the curve as accurately as we please. 

In Statistics, however, we usually have to start the other way 
round and reach the equation, if at all, last. We make observations 
of two sets of variables, a set of x's, and a set of y's, one of which 
is dependent in some way upon the other — e.g. y, the dependent 
variable, might denote the number of individuals observed to have 
a certain organ of length x, the independent variable — and thus 
we get pairs of corresponding values like {x^^, y^), {x^, y^), {x^, 2/3) •• • 
We met with examples of this method of recording results in the 
last chapter, and we need only repeat here that its chief virtue is 
suggested in the root of the word itself — it is more graphic than a 



GRAPHS 7S 

long table of figures and, by means of it, many of the essential 
features of a problem are immediately seized upon. 

Now for some purposes it may be necessary to go further and 
to find what curve would best fit the points plotted, assuming they 
were numerous enough, and what equation between x and y would 
best describe the curve. But the graphs we meet in Statistics, 
bearing, for instance, upon sociological or biological problems, are 
in general much more wayward than the mathematical kind we 
have referred to in the present chapter : it is impossible to set 
down simple equations to which they can be rigidly confined, and 
when we are unable to find any relation which accurately and 
uniquely defines 2/ as a function of x we must rest satisfied with the 
most manageable equation and the best fit we can get. 

In sciences such as Engineering and Physics it is often possible 
to fix upon two mutually dependent variables, x and «/, and to 
observe enough corresponding values of each to enable us to draw 
a graph which answers very closely to the true relationship between 
them, so that a connecting equation can be determined ; e.g. we 
may plot the amount of elastic stretch, y, in a wire when different 
weights, X, are hung from the end of it, and it is found that y is 
directly proportional to a:. If we deal in this way with some 
simple figures which are amenable to our purpose it may help to 
make clear the nature of the same problem in Statistics. 

The following corresponding values of x and y were given in a 
Board of Education Examination (1911) : — 

a;=l-00, 1-50, 200, 2-30, 2-50, 2-70, 2-80 ; 
2/=0-77, 105, 1-50, 1-77, 2-03, 2-25, 2-42. 

Allowing for errors of observation, it was desired to test if there 
was a relation between y and x of the type 

y^a^hx'^ . . . (1) 

In the first place, the shape of the curve obtained by plotting 
y against x, as in fig. (6), would, to the initiated, probably suggest 
a parabola, the equation of which is of type (1). In order to test 
its suitability we proceed to plot y against x^, or, putting x'^=^, we 
plot y against f . If equation (1) holds, then, in that case 

2/=a+6^ . . . (2) 

should also hold, and this, in (f , y) co-ordinates, represents a straight 
line. The result of plotting y against f should therefore be a 
number of points approximately in a straight line — we say * ap- 
proximately ' to allow for errors of observation in the original data. 



74 



STATISTICS 



Now from the given statistics corresponding values of ^ and y 
are, since f =a:2 : — 

^=1-00, 2-25, 4-00, 5-29, 6-25, 7-29, 7-84 ; 
2/=0-77, 1-05, 1-50, 1-77, 2-03, 2-25, 2-42 ; 



Y 






- 




"■ 






































~ 










~ 
































































































r- 






































































































































2-5 






























































































































i 


» 






































































/ 






































































) 


























































































20 


















































, 


/ 




































































/ 






































































y 






































































1 


' 




































































A 




























1-5 












































f 








































































































































/ 






































































/ 






































































/ 






































10 






























■ 


/ 


































































y 




































































^ 
































































































































































































0-5 






































































































































































































































































































































































r» 










_, 


J 


_ 


J 






_ 




_ 




_ 


_ 




_i 


_ 




_ 


_ 


_ 


_ 


_ 


_ 






_ 




_ 




_ 







0-5 10 



'1-5 20 

Fig. (6). 



2-5 30 



and the resulting graph, fig. (7), is very approximately a straight 
line. To determine its equation, choose two points (not too close 
together) on the line, which has been drawn so as to run as fairly 
as possible through the middle of the points plotted, and, in choosing, 
take points which lie at the intersections of horizontal and vertical 
cross lines (the printed lines of the graph paper) if such can be 

Y 






4 5 

Fig. (7). 



found, because their x's and 2/'s can be read off with ease and 
accuracy. Two such points are 

(2-8, 1-2) and (6*0, 2-0), 



GRAPHS 75 

and since each of these pomts lies on the line whose equation is 

we have 

l-2=:a+6(2-8) 
2-0=a+6(6-0). 

Subtracting, we get 

0-8=6(3-2). 
Therefore 6=i. 

Hence az=2 — 2 = J. 

Thus the equation of the line is 

t.g. 4t/=f+2, 

and the law connecting x and y is therefore 

4i/=a;2-|-2. 

The following statistics, the result of an experiment in Physics 
to verify Boyle's Law, may be treated in the same way. a; is a 
number proportional to the volume of a constant weight of gas in a 
closed space, and ?/ is a number proportional to its absolute pressure. 
Corresponding values of x and y observed were : — 

\x= 46-89 41-96 40-33 38-88 37-37 36-06 34-71 33-47 
[y= 76-32 85-38 88-93 92-36 96-09 99-61 103-51 107-51 

{x= 32-39 31-08 29-97 28-76 27-26 25-32 24-04 
\y=n\m 115-69 120-05 125-08 131-99 142-09 149-81. 

Boyle's Law states that the product xy is constant, and this may be 

tested by putting ^= - and plotting y against | ; the points obtained 
x 

should be approximately in a straight line. 

Now in Statistics, as we have already explained, the exact con- 
nection between the variables, x and y, is rarely so clear, though 
the absence of law is not so complete as it might seem at first sight. 
At this stage, however, we need not enter into the difficult question 
of curve fitting : if drawn with care and used with judgment much 
that is of value may be learnt by simple plotting and by connecting 
up the resulting points by straight lines or a freehand ciu-ve. We 
shall briefly explain or illustrate by examples how graphs and 



76 STATISTICS 

graphical ideas may be used to serve three distinct purposes, 
namely : — 

(1) to suggest correlation or connection between two different 

factors or events ; 

(2) to supply a basis for finding by interpolation some values of a 

variable when others are known ; 

(3) as pictorial arguments appealing to the reason through the eye. 

We reserve (2) and (3) for the next chapter and proceed at present 
with an example of (1). 

Correlation suggested by Graphical means. Consider the index 
numbers, col. (2) Table (20), showing the variation from year to 
year in wholesale prices between the years 1871 and 1912. It is 
not an easy matter to take in satisfactorily the meaning of such a 
mass of bare figures, but they are much easier to grasp when plotted 
in a graph. 

In this case the numbers x, representing years, and the numbers ?/, 
representing prices, are measures of things of quite a different char- 
acter, so that it is not necessary to take the x and y units of the 
same size. Moreover they need not, in a case of this kind, neces- 
sarily vanish at the origin, but it is convenient to draw the graph 
in such a way that it shall occupy the greater part of the space at 
our disposal. Thus, we have roughly 80 small squares across the 
breadth of our graph paper, and between 1871 and 1912 we have 
roughly 40 years ; we therefore take two sides of a square to 1 year 
and mark off the years 1870, 1875, 1880, . . ., along an axis or 
base Une parallel to the breadth of the paper, as shown in fig (8). 
Again we have roughly 70 small squares in the available space 
from this base line to the top of our graph paper, and the whole- 
sale price index numbers vary from 88-2 to 151-9, a range of 63*7 ; 
we therefore take one side of a square to correspond to a difference 
of 1 in the price index number, and mark off the prices 90, 100, 
110, ... , along an axis parallel to the length of the paper, as 
shown in the figure. 

We then plot points to represent the numbers in col. (2) of 
Table (20). Thus, in 1880 wholesale prices stood at 129 ; we there- 
fore travel along the width of the paper till we reach 1880 and 
then upwards until we are opposite the 129 level on the axis of 
prices, inserting a dot to mark the position. Similarly for all other 
points, and the required graph is given by joining them up in 
succession. 



GRAPHS 



77 



Table (20). Mabriage Rate and Wholesale Prices 
Index Numbers. 



(1) 



(2) 



(3) 



(4) 



(6) 



(6) 



(7) 







Nine Years' 


Difference be- 


Marriage 


Nine Years' 


Difference be- 


Year. 


Prices. 


Average 


tween Nos. in 


Average of 


tween Nos. in 






of Prices, j 


Cols. (2) & (3). 


rate. 


Marriage rate. 


Cols. (5) & (6). 


1871 


135-6 






167 






1872 


145-2 






174 






1873 


151-9 


. . 




176 






1874 


146-9 


. . 




170 




. . 


1875 


140-4 


139-3 


+ 11 


167 


164 


+ 3 


1876 


137-1 


138-6 


-1-5 


165 


162 i 


+ 3 


1877 


140-4 


136-5 


+3-9 


157 


159 1 


_ 2 


1878 


1311 


133-8 


-2-7 


152 


157 1 


- 5 


1879 


125-0 


131-5 


-6-5 


144 


155 ! 


-11 


1880 


129-0 


128-5 


+0-5 


149 


153 


- 4 


1881 


126-6 


125-2 


+ 1-4 


151 


151 


.. 


1882 


127-7 


120-8 


+ 6-9 


155 


149 


+ 6 


1883 


125-9 


117-2 


+8-7 


155 


148 


+ 7 


1884 


114-1 


114-7 


-0-6 


151 


148 


+ 3 


1885 


107-0 


111-8 


-4-8 


145 


149 


- 4 


1886 


1010 


109-2 


-8-2 


142 


149 


- 7 


1887 


98-8 


106-9 


-8-1 


144 


149 


- 5 


1888 


101-8 


104-2 


-2-4 


144 


149 


- 5 


1889 


103-4 


102-5 


+0-9 


150 


149 


+ 1 


1890 


103-3 


101-0 


+2-3 


155 


149 


+ 6 


1891 


106-9 


99-9 


+ 7-0 


156 


150 


+ 6 


1892 


1011 


98-7 


+2-4 


154 


151 


+ 3 


1893 


99-4 


97-4 


+2-0 


147 


153 


- 6 


1894 


93-5 


96-3 


-2-8 


150 


155 


- 5 


1895 


90-7 


950 


-4-3 


150 


156 


- 6 


1896 


88-2 


94-3 


-6-1 


157 


156 


+ 1 


1897 


90-1 


93-8 


-3-7 


160 


157 


+ 3 


1898 


93-2 


93-4 


-0-2 


162 


158 


+ 4 


1899 


92-2 


93-8 


-1-6 


165 


159 


+ 6 


1900 


100-0 


94-7 


+ 5-3 


160 


159 


+ 1 


1901 


96-7 


95-7 


+ 1-0 


159 


159 




1902 


96-4 


96-9 


-0-5 


159 


158 


+ 1 


1903 


96-9 


98-3 


-1-4 


157 


158 


- 1 


1904 


98-2 


99-5 


-1-3 


153 


156 


- 3 


1905 


97-6 


1000 


-2-4 


153 


155 


- 2 


1906 


100-8 


101-3 


-0-5 


157 


154 


+ 3 


1907 


1060 


102-8 


+ 3-2 


159 


153 


+ 6 


1908 


103-0 


104-8 


-1-8 


151 


153 


- 2 


1909 


104-1 


. . 




147 






1910 


108-8 






150 




• • 


1911 


109-4 


.. 


. , 


152 




' 


1912 


114-9 


•• 


•• 


155 


•• 





78 STATISTICS 

It is comparatively easy from this graph to trace the change 
in prices from year to year and from decade to decade : for example, 
we note that from 1873 to 1896 the tendency of prices was on the 
whole downward, and from 1896 to 1910 the tendency was upward. 
Also on the assumption — not necessarily valid — that prices have 
varied continuously, or at least consistently, during the intervals 
between the dates to which the records refer, it is possible to read 
off intermediate values from the graph : e.g. midway between 1883 
and 1884 we get the figure 120 as the index number for prices. 

On the same graph sheet we have also plotted the marriage rate 
from year to year during the same period. The numbers are given 
in col. (5) of Table (20). This rate varies from 142 to 176, a range 
of 34, and we have a range of 40 small squares at our disposal in 
plotting ; a difference of 1 in the marriage rate has therefore been 
taken to correspond to one side of a square, and the marriage rates 
140, 150, 160 . . . are accordingly marked along the axis perpen- 
dicular to the same base line as before, which is used again to 
measure the passage of years, but the second graph is drawn below 
the line whereas the first was drawn above it. In this way we 
are able to compare the two graphs, namely, the one registering 
the change in prices and the one registering the change in marriage 
rate from year to year. 

It is interesting to observe that the two seem to be not uncon- 
nected : they go up and down almost in the same time, and moun- 
tains and valleys in the one correspond roughly to mountains and 
valleys in the other ; in other words, there is some kind of correlation 
or reciprocal relation between them. Now these mountains and 
valleys are largely the result of what may be called short-time 
fluctuations^ and it is important to distinguish between these changes 
which are transient and the more permanent or long-time changes. 
In order to get rid of the former, which sometimes conceal the 
latter, the following device has been adopted : noticing that the 
wave period, the length of time taken for each complete up-and- 
down motion, is one of about nine years, nine-yearly averages have 
been taken of the figures for wholesale prices right down col. (2) 
of Table (20) ; thus 139-3 is the average of the index numbers from 

1871 to 1879 inclusive, 138-6 is the average of the numbers from 

1872 to 1880 inclusive, and so on, the results being recorded in 
col. (3). When the points corresponding to these numbers are 
plotted we get the broken line in fig. (8) passing through the body 
of the original graph of prices and indicating its general trend in 
the course of years as separated from the temporary fluctuations. 



GRAPHS 



79 




1870 1875 1880 1885 1890 1895 1900 1905 1910 

Fig. (8). Graph showing Variation in Wholesale Prices Index Numbers. 




1870 1875 1880 1885 1890 1895 1900 1905 1910 

Fio. (9). Graph showing Variation in Marriage Rate Index Numbers. 



80 



STATISTICS 



The same procedure has been followed with the marriage rate 
statistics ; the nine-yearly averages are shown in col. (6) of Table (20), 
and their graph appears as a broken Une passing through the body 
of the original marriage rate graph in fig. (9). 



«+io 



I- 
I 



-5 



-10 
+10 



+5 



Graph 1 showing Fluctuations from iheirlNinf ^ea'rly Averaeres 


1 ; pf t;he' Ind^x Numbers of Wholesale Pr ces 


L T 


J k Jl 


4 t IX- r- 


j I tx S 


-iU t J ^ 4% ^ 


jr "_/ A . , K /\ A 


it X 4 it Z \- t^ XV 


Lit ^ X- ^T L_ tJ\ t\ _ 


5\/ \ iJJol iiJ *5 / 1J90 \ 1*95 K itoo V 190S' \ '9?o 


X 3 T /^ \ n ^^ t 


vU 4 i \ t ^ 


vl -^\ -A- \ -^ 


Xt ^v- i vl 


jiiy- CI ^t 


X 4 ' -t 


1 


V-4 


1 






1 


/l '^ . 1 i 


\ iA A y 


it -i A a\ jX 7h 


4 Iff V t \^ -^^ X -tx 


x t Jr t z^ ^ t 


\ / \ i I /a i \ 


± ± I ^JL t ^ ^^%--t A 


5 1 \ 1880/ \iJ8s / 1S90 \ 1895/I 1900 \ i9<5 \1 1910 


• ^ 7 \ / / \ I 


c f j I 4 i IT V X 


\ / 1 / 1 / 


X^zL v _t ^^^ il 


iri w ^ 


ti V 




\ / 


\r 


y Graph showing Fluctua ions from their Nine- year y Averages 


[ of Marriag^-rktes | | | f i 



-10 



Fig. (10). 

Suppose we wish on the other hand to study the short-time 
fluctuations as distinct from the long-time changes, we may do so 
by forming the differences between the numbers for each year 
and the corresponding nine-yearly averages, and plotting these 
differences on convenient scales. 

The numbers obtained in this way are recorded, with their proper 
signs — positive if above the average, negative if below — in cols. (4) 
and (7) of Table (20), and the graphs of these differences are drawn, 



GRAPHS 81 

one below the other for comparison, on the same graph sheet 
(fig. 10). The agreement in fluctuation from the average between 
the two factors, marriage rate and prices, is more easily remarked 
now than it was in the original graphs. High prices go as a rule 
hand-in-hand with prosperous times, and such times lead to more 
frequent marriages. This statement must not be taken to imply 
that when prices are high the times are always necessarily pros- 
perous for the community as a whole : the lie direct would be given 
to such an implication by any one who had experienced abnormal 
war conditions. 

After about 1892, while the fluctuations continue to be similar, 
a tendency appears for the marriage rate graph to reach each 
extreme point about a year in advance of the other, as though an 
increase in marriages raised prices and a decrease lowered them. 
There is no doubt that any economic change, especially if it takes 
place on a large scale, will set up a system of corresponding forces, 
sometimes in unexpected directions, actions and reactions succeed- 
ing one another at intervals like tidal waves producing each a back- 
wash as it breaks, but such effects, even when anticipated in theory, 
are not always easy to unravel in practice. 

The comparison we have been discussing between changes in 
prices and marriages is suggested in Sir W. H. Beveridge's Unemploy- 
ment. The whole book will repay careful study, but it contains 
one particularly illuminating chapter on ' CycUcal Fluctuation ' with 
a chart labelled ' The Pulse of the Nation,' because of the remark- 
able picture it gives of the ebb and flow of the tide of national 
prosperity. It consists of a series of curves representing respec- 
tively :-^ 

(1) bank rate of discount per cent. ; 

(2) foreign trade as measured by imports and exports per head 

of the population ; 

(3) percentage of trade union members not returned as unem- 

ployed ; 

(4) number of marriages per 1000 of the population ; 

(5) number of indoor paupers per 1000 of the population ; 

(6) gallons of beer consumed per head of the population ; 

(7) nominal capital of new companies registered in pounds per 

head of the population. 

The interesting thing about these curves is to see the way m 
which they move in waves of varying size up and down almost 
together, showing a connection between such phenomena moro 

F 



82 



STATISTICS 



intimate than one might at first have suspected. A note of caution 
must be inserted here however : cavsal connection must not be too 
confidently inferred in discussing the correlation of characters 
changing simultaneously with time ; because two events happen 
together, one is nof necessarily caused by the other. 

An instructive article bearing on this point appeared recently in a 
periodical well known to students of social problems. It was there 
stated that high positive correlation exists between birth rate and 
infantile death rate : in general the two rise or fall together, whence 
Neo-Malthusians argue that the way to lower a death rate is to 
lower the birth rate. The writer then contrasts Bradford, the last 
word in the scientific care of infants, with Roscommon, where con- 
ditions as to wealth and child welfare are the very reverse, and 
points out that Bradford has a birth rate of 13 and an infant death 
rate of 135, while Roscommon has a birth rate of 45 and an infant 
death rate of 35. These figures, he suggests, prove instantaneously 
that the Neo-Malthusians are guilty of the commonest of all fallacies, 
they confound correlation with causation. 

As an exercise in plotting the reader may see whether he can 
discover any suggestion of correlation between crime and unem- 
ployment by comparing the following statistics, showing the number 
of indictable offences tried in the United Kingdom and the trade 
union unemj)loyed percentages respectively from 1861 to 1905 : — 

Table (21). Number of tried Indictable Offences and 
Trade Union Unemployed Percentages (1861-1905). 





No. of Indictable 


Trade Union 




No. of Indictable 


Trade Union 


Year. 


Offences tried 


Unemployed 


' Year. 


Offences tried 


Unemployed 




(in thousands). 


percentages. 


1 


(in thousands). 


percentages. 


1861 


560 


3-7 


1874 


53-5 


1-7 


1862 


61-3 


60 


1875 


500 


2-4 


1863 


ei-4' 


4-7 








1864 


58-4 


1-9 


1876 


51-9 


3-7 


1865 


69-9 


1-8 


1877 


53-8 


4-7 


1866 
1867 
1868 


57-6 
59-5 
62-4 


2-6 
6-3 
6-7 


1878 

1 1879 

1880 


560 
550 
60-7 


6-8 
11-4 

5-5 

1 


1869 


61-3 


5-9 


1881 


60-6 


35 


1870 


561 


3-7 


1 1882 


63-3 


2-3 


1871 


531 


1-6 


1883 


60-8 


2-6 


1872 


51-9 


0-9 


1884 


59-6 


81 


1873 


53-5 


1-2 


1885 


56-4 


9-3 



Graphs 



S3 



Table (21). Number of tried Indictable Offences and Trade 
Union Unemployed Percentages (1861-1905) — Continued. 





No. of Indictable 


Trade Union 


j 


No. of Indictable 


Trade Union 


Year. 


Offences tried 


Unemployed 


1 Year. 


Offences tried 


Unemployed 




(in thousands). 


percentages. 


1 
1 


(in thousands). 


percentages. 


1886 


56-2 


10-2 


^ 1896 


50-7 


3-3 


1887 


56-2 


7-6 


■ 1897 


50-7 


3-3 


1888 


58-5 


4-9 


1898 


52-5 


2-8 


1889 


57-6 


21 


1 1899 


50-5 


20 


1890 


550 


21 


1900 


53-6 


2-5 


1891 


541 


3-5 


1901 


55-5 


3-3 


1892 


58-3 


6-3 


1902 


571 


40 


1893 


57-4 


7-5 


1903 


58-4 


4-7 


1894 


56-3 


6-9 


1904 


600 


60 


1895 


50-8 / 


5.8 


j 1905 


61-5 


50 



The chief point of difficulty in plotting such graphs is the initial 
one of fixing upon the most convenient scales to use, and in this 
matter hints only can be given, facility will come by practice. An 
examination of Table (21) shows that the data cover a period of forty- 
five years which can be marked off horizontally along a base line so 
as just to fit comfortably into the available space across the graph 
paper. The unemployed percentages vary between 0-9 and 11-4, 
giving a range of 10-5. Similarly the indictable offences recorded 
(in thousands) present a range of 13-3. We might therefore very 
well choose the same. vertical scale for the measurement of indict- 
able offences and unemployment, but, in order that the graphs 
may run more or less together (without exactly overlapping) for 
the sake of comparison, only the unemployment zero need be taken 
actually on the base line, whereas the indictable offences may have, 
say, the number 50 (thousand) at that level ; also it will be con- 
venient to show the scale for unemployment on the right side 
and the scale for offences on the left side of the paper. 

An example deaHng with matters somewhat different is provided 
by a comparison of changes from week to week in — 

(1) the mean air temperature ; 

(2) the percentage of possible sunshine ; and 

(3) the rainfall. 

The following is a record of observations taken at Greenwich in 
1912 [data from London Statistics, vol. xxiii.] : — 



84 



STATISTICS 



Table (22). Weekly Meteorological Observations 
AT Greenwich (1912). 



Week 
ended— 

Jan. 6 


Mean Air 
Tempera- 
ture- 
Degrees 
Fahren- 
heit. 


Per- 
centage of 
possible 
Sunshine. 


Rainfall 

in 
inches. 


1 

Week 
ended— 


Mean Air 
Tempera- 
ture- 
Degrees 
Fahren- 
heit. 


Per- 
centage of 
possible 
Sunshine. 


Rainfall 

in 
inches. 


45-7 


7 


0-76 


July 6 


58-7 


15 


0-36 


13 


41-9 


15 


0-45 


13 


670 


46 


0-20 


20 


40-2 


1 


0-93 


20 


65-8 


44 


004 


27 


38-9 


8 


0-88 


27 


64-8 


31 


016 


Feb. 3 


300 


21 


002 


Aug. 3 


57-8 


33 


0-54 


10 


39-5 


15 


0-52 ■ 


10 


57-6 


28 


1-26 


17 


45-5 


11 


0-44 


17 


56-2 


14 


0-23 


24 


47-4 . 


6 


0-65 


24 


57-2 


24 


1-27 


Mar. 2 


49-8 


21 


0-52 


31 


56-9 


27 


1-33 


9 


44-6 


31 


0-79 


Sept. 7 


54-8 


36 


0-21 


16 


451 


16 


019 


14 


52-4 


14 


.002 


23 


42-7 


15 


108 


21 


53-6 


22 


000 


30 


510 


46 


005 


28 


51-5 


59 


002 


Apr. 6 


48-0 


43 


007 


Oct. 5 


48-8 


36 


2-30 


13 


45-6 


43 


002 


12 


460 


53 


000 


20 


500 


50 


0-00 


19 


49-8 


38 


013 


27 


52-6 


76 


000 


26 


45-4 


23 


0-g8 


May 4 


501 


32 


0-21 


Nov. 2 


491 


31 


0-55 


11 


59-7 


29 


006 


9 


47-2 


6 


018 


18 


55-2 


49 


0-69 


16 


43-3 


3 


017 


25 


541 


38 


019 


23 


46-2 


6 


0-31 


June 1 


57-0 


47 


017 


30 


40-4 


13 


106 


8 


54-2 


35 


0-99 


Dec. 7 


42-4 


9 


0-31 


15 


581 


48 


0-39 


14 


490 


2 


0-62 


22 


61-7 


56 


0-65 


21 


44-4 


19 


0-59 


29 


60-2 


45 


0-30 


28 


48-1 


8 


1-22 



The rainfall graph here should be drawn reversed {i.e. so that 
it goes up as the rainfall goes down in amount, and vice versa), 
because one would expect in general much rain to go with little 
sun and low temperature. 

The range of temperature during the year is 37 degrees, of sun- 
shine 75 per cent., and of rainfall 2-30 in. Hence the vertical 
scales for these three graphs might be chosen so that, roughly, 
40 units of temperature should correspond to 80 units of sunshine 
arid 2 units of rainfall. Also the zeros of the three variables should 
be so placed,, relative to the horizontal base line registering 'the 
weeks, that the three graphs may be conveniently compared without 
causing confusion by too closely overlapping. 



CHAPTER IX 

GRAPHS {continued) 

Graphical Ideas as a Basis for Interpolation. It frequently happens 
in statistical records that awkward gaps occur which require to be 
filled in ; this may be due to the fact that no record has been 
made, or that it has been made with insufficient detail, or that it 
has been lost or destroyed. Cases in point arise in connection with 
returns Uke that of the Census which can only be undertaken every 
few years, so that if figures are wanted for any intervening year, 
as they are in very many instances, an estimate has to be made 
from the known results of the years recorded. It is imperative, for 
example, for many purposes of local or national government, to 
be able to find with a fair degree of accuracy the population of 
county boroughs and urban or rural districts at any given time, 
to know the number of workers engaged in different occupations, 
the amount of land in pasture and under various crops, the con- 
dition of the people as to housing, of the children as to education, 
and so on indefinitely. 

Symbolically, with the same notation as we have used before, 
we conceive the statistics in tabular form, like 

•^l> '^2^ '^a * • • "^n 

VV 2/2» 2/3 • ' ' Vn 

each y denoting the frequency corresponding to the character 
measured by its companion x, e.g. the ic's may stand for successive 
dates and the 2/'s for the frequencies of the population of a certain 
district at those dates. If it happens that one or more of the y's, in 
between the first and the last recorded, are missing, the problem is 
to estimate the missing values by some method of interpolation, as 
it is called. Various methods of arriving at such estimates are used, 
but we shall only refer to the more elementary here. 

A" rough way of making the estimate, but one which is often as 
accurate as the data will allow, is to plot the observations, each 
{x, y) being represented by a point, and connect them up, if there 

85 



86 



STATISTICS 



be enough of them, by a smooth curve drawn freehand P^ Pg P3 . . . P„ 
[see fig. (11)] ; to find the y proper to any other x we have then 
only to draw the ordinate through the point {x, 0) and measure the 
y at the point where it cuts the curve. This is a not unreasonable 

principle to follow, for in effect it 
gives due weight to each of the 
observations actually recorded, 
and it assumes an even course 
from each one to the next — a 
justifiable assumption in the 
absence of any evidence that 
some sudden discontinuity of 
value has taken place. 

If only two observations are 
given, represented by the points 
Pi (a^i, 2/1) and Pg {x^, y^), the 
curve connecting them is a straight line, and the y corresponding to 
any other x is at once given geometrically, as fig. (12) shows, by 

PM PiM 




P2M2 



PiM, 



%,e. 



or 



y-yi _ ^-^i 

2/2 Vi ^2 ^1 



y=yi-\- 






{x-x^), 



the familiar proportional relation which is employed in this simple 
case. 

P. 




Example. — Given 
Required log 5-826736. 



Fio. (12). 

log 5-82673=0-7654249, 
log 5-82674=0-7654257. 



GRAPHS , 87 

Here a;i=5-826730 2/i=0-7654249 

:r2=5-826740 2/2=0-7654257 

a:=5-826736. 

Therefore, by means of the above relation, 

,=0.7654249+^:222222_^o.000006) 
^ 0-000010 ' 

=0-7654249+0-00000048 

=0-7654254. 

The logarithmic curye 2/=log x is, of course, not a straight line, 
and the value obtained for y only represents a first approximation 
to the true value. 

When more than two points are given there is bound to be a 
margin of inaccuracy, more or less according to the data, intro- 
duced in drawing the curve. For an example of this method the 
reader may refer back to the curve on p. 67, which was used to 
determine the median and quartiles. We may, as we saw, read 
off from it the number of candidates who obtained not more than 
any stated number of marks : e.g. 300 candidates obtained not 
more than 34 marks ; or we may use it the other way round and 
find the number of marks obtained by a stated number of candi- 
dates : e.g. 10 per cent, of the candidates got less than 17 marks. 
Such examples might be multiplied endlessly, and the method will 
be foxmd extremely useful when a high degree of accuracy is not 
looked for. But greater confidence will be felt perhaps in such 
results — though the foundation for it may be no more secure in many 
cases — if we can translate them from geometrical to algebraical 
form, if we can find, that is to say, some formula, like the simple 
proportional relation already introduced above, which will give 
one y when others are known. 

In order to make the argument as general as possible we shall 
speak of x and y as variables, and we shall think of the value of y 
as depending upon that of x in such a way that when x is given, 
y is known or it can be estimated * (in the sense that when the 
year is given the population is known or can be estimated). 

Suppose 

y=CQ-\-c^x-\-c^x^-^r . 

[* This is equivalent to assuming that y is some function of x, say y=/{x), and 
clearly some such assumption is necessary if any estimate from the known values 
to the unknown is to be possible. Further, for simplicity we assume f(x) can 
be expanded in a Maclaurin's converging series of ascending powers of ar, which 
simply means that we take the relation between x and y to be of the form 
adopted above. ] 



88 STATISTICS 

where the c's are constants to be determined, and their number 
can be made to depend upon the number of known values of y 
which are used in the estimate. 
Geometrically, the equation 

represents a curve called a parabola of the nth order, and such 
a curve could be employed (and uniquely found — there is only one 
parabola of the kind which will go through all the points) if we 
based our estimate upon a knowledge of (^+1) 2/'s corresponding 
to given a:'s, for we could readily make it pass through the (n-\-\) 
known points (Xq, y^), [x^, y^), (x^, y^), ... {Xn, yj by choosing 
the (n-\-l) c's so as to satisfy the (n-\-l) simple linear relations : — 

2/0^^0 I ^1*^0"!" ^2*^0 "T" • • • 'f'^nXo 

■yi=Co-]-CiX^+c.^Xi^-\- . . . +c„a;i" 

When the curve is determined, in other words when the c's are 
known, we can find any other y required by substituting the corre- 
sponding X in the equation 

y=Co-\-CiX-{-C2X^-\- . . . +c„x", 

i.e. by supposing this point [x, y) to lie on the same curve that goes 
through the known points. 

It is well to mention here that the parabola is by no means always 
the best curve for fitting any given statistics, and when the number 
of observations is adequate it is possible often to make a more 
satisfactory choice. Once the equation of a suitable curve has 
been determined the subsequent interpolation or calculation of y 
for any given x is not as a rule a very difficult matter. The larger 
question of curve fitting in general is reserved for a later chapter. 

Example of First Method (fitting with a parabolic curve). Let us 
illustrate this process of interpolation by fitting a parabolic curve 
to the following figures, extracted from Porter's The Progress of 
the Nation, giving the annual cost of Poor Relief (excluding insane 
and casual) at five-yearly intervals, but with the amount for the 
year 1845 omitted : — 

Year . . . 1835, 1840, 1845, 1850, 1855] 
Cost in £1000 , . , 5526, 4577, ? 5395, 5890J 



GRAPHS 89 

Assuming that no extraordinary conditions prevailed in 1845 to 
cause abnormality in expenditure, let us estimate what the figure 
would be for that year judging from the given records just before 
and after. Since there are four known points in this case, we take 
as the curve through them a parabola of the 3rd order, namely : — 

y=Co-{-CiX-{-CiX^+C32^ ; . . . (1) 

the four known points will then just suffice to determine uniquely 
the four arbitrary constants Cq, c^, Cg, Cg. Also, since the x class- 
intervals are equal, it will simplify the algebra if we measure from 
the year 1845 as origin, taking five years as unit for x and £1000 
as unit for y, so that we get 

x--=-2, -1, 0, +1, +2 \ 
y-5526, 4577, y^, 5395, 5890J 

where yQ is the number to be determined. 

Since all five points are to lie on the curve with equation as in 
(1), we have by substituting in that equation — 

5526=Co— 2ci+4c2— 8C3 
4577=Co— Ci4-C2— C3 

2/o=Co 
5395=Co+Ci+C2+C3. 
5890=Co+2ci+4c2+8c3. 

Adding the first and last of these equations, 

2co+8c2 =5526+5890 . . . . (2) 

Adding the second and last but one, 

2co+2c2=4577+5395 
or 8co+8c2=4(4577+5395) . . . (3) 

Subtracting (2) from (3), 

6co=4(4577+5395)- (5526+5890) . . (4) 
=4(9972)- (11416) 
= 39888-11416 
=28472. 
Therefore yo = Co = £4,745 ,000 . 

If we only wish to make use of the records for the years 1840 
and 1850, the appropriate fitting curve reduces to a straight line 

y=c^+c^x, 



90 STATISTICS 

on which we assume the points 

(-1,4577), (0,2/o), (+1,5395) 
to lie, so that 

4577=Co— Ci 

5395=-Co+Ci. 
Therefore, adding the first and last of these equations, 

' 2co=4577+5395, 

so that yo=Co=£4,986,000. 

* Second Method {using a formula connecting the ordinates). When, 
as above, the steps from each x to the next are equal, as commonly 
happens in practice, it is possible to write down a simple relation 
between the y's, known and unknown, without introducing the c's 
at all. At bottom the method is the same as the last, inasmuch as 
the elimination of the c constants by the first method really results 
in the same formula for the unknown y. 

Let us represent the given statistics in this case by 

Xq, XQ-{-h, XQ-{-2h . . . XQ-{-nh\ 

2/0. 2/l, 2/2 • • • 2/n J 

so that, if the fitting curve be 

y=CQ+c^x+C2X^-\- , . . +c„a;^, 

we have, by substituting the co-ordinates of the first two points 
in this equation, 

yi=Co+c^{Xo-\-h)+c.^{Xo-{-h)^-{- . . . -\-c^{xQ-i-h)'' 

and yo=CQ-f-Ci Xq -\-C2 Xq -\~ . . . -|-c„:^q . 

Hence 

yi-yo=Cih+C2{2xoh-\-h^)-\- . . . +c^{nxQ^-^h-\- . . .). 

Now this result, which we call the 1st difference between the y's, 
is of (ri— l)th degree in Xq, so that by subtracting two of the y's 
we have reduced the degree in a^o by 1. Similarly, 

y2-yi=cJi+C2{2xoh+3h^)+ .... +c^(nxo^-^h+ . . .)• 

Thus we get a series of 1st differences, each with the highest 
term of the {n—l)th degree in Xq. Treating them as a series of new 

[* The non-mathematical ceader will do well to omit the rest of this section on 
interpolation.] 



GRAPHS 



91 



ordinates and forming their differences in the same way, we get 
what may be called the 2nd differences between the y^s, a series 
of ordinates each with the highest term of degree {n—2) in Xq. 
Proceeding in this way the ^rd differences between the y's are a 
series of ordinates of degree (n— 3) in Xq, the Uh differences q^xq of 
degree (^—4), and so on, mi til ultimately we reach the nth differences, 
which are of zero degree in Xq, and consequently involve only h. 
It follows that the nth differences must all be equal in value and 
therefore, if we go one step further and write down the (n-\-\)ih 
differences, these must vanish altogether. 

If the reader finds any difficulty in following the argument 
he should test it step by step for himself in the simple case of a 
parabola of the third order when it should be perfectly clear. 

The formation of the successive differences is conveniently shown 
in Table (23). 





Table (23). 


Successive Differences of 


Ordinates. 




First 


Second 


Third 


Fourth 


Fifth 


y 


difference 


difference 


difference 


difference 


difference 




A 


A2. 


A3. 


A4. 


ab. 




^l-3/o) 
J/2 - 1/1 ) 


¥2-22/1+2/0^ 
2/3 -22/2+2/1 J 










2/3-32/2+33/1-2/0^ 
2/4-32/3+32/2-2/1. 






Vf. 




2/4- 42/3+61/2 -■* 2/1+2/0) 
2/5-42/4+61/3-42/2+2/1' 






3/3-2/2 




2/5 - 52/4+103/3 - 102/2+52/1 - J/o 


3/» 




2/4-22/3+2/2 








1/4 -1/S 




2/5-32/4+32/3-2/2 






V4 


y5-2/4 


¥5-22/4+2/3 








2/5 













The law of formation should be apparent from this table, for it 
is precisely that which we meet in the binomial expansion, e.g. the 
Tith difference is of type 



, n{n—\) n{n—\)in—2) 



+ (- 1^2/0, 



and by equating to zero the {n-{-\)th. difference we have the relation 
required between the i/'s. 

Example. — Let us apply this method to the ' Poor Relief ' example 
already considered. Since there are four knowTi points the relation 
between x and y must be of the form 

y=CQ+Cj,X-\-C2X^-\-C3X^ 

as before. Hence the 4th differences must vanish, and taking the 



92 STATISTICS 

. points in order from years 1835 to 1855 as {Xq, ^/o). (^i> ^i)* (^2» Vi)' 
(a?a, ya). (^4. 2/4). we get 

2/4-%3+62/2-42/i+2/o=0 

as the formula connecting five y's, four known and one (1/2) unknown. 

Therefore %2=4(2/i+2/3)— (1/0+2/4) 

=4(4577+5395)- (5526+5890), 

which is equivalent to equation (4) on p. 89. 

Thus y,=£4,745,000. 

Third Method {by means of advancing differences). In the last 
method we employed a relation connecting ?/„ with all the preceding 
y's, but it is possible also to express y^ in terms of 2/0 a-nd the suc- 
cessive differences, which may be written /\, /\^, /\^, . . . A** » 
we have, in fact, with the notation of Table (23) : — 

Ao=2/i-2/o. Ao^=2/2-22/i+2/o» Ao^=2/3-32/2+32/i-2/o, • • • 

Thus 

2/1=2/0+ Ao- 

2/2=22/1-2/0+ Ao'=2/o+2Ao+Ao'. 

2/3=%2-32/i+2/o+ Ao^ 

=3(2/0+2 Ao+ Ao')-3(2/o+ Ao)+2/o+ Ao' 
=2/o+3Ao+3Ao'+Ao'. 
^4=42/3— 61/2+42/1— 2/0+ Ao* 

=4(2/0+3 Ao+3Ao'+ Ao') - 6(2/0+2 Ao+ Ao')+4(2/o+ Ao) - 

2/0+ A 0* 
=2/o+4Ao+6Ao'+4Ao'+Ao*. 

Here again the law of formation is clear, and it is readily estab- 
lished by induction that, for all positive integral values of n, 

,„=,„+„Ao+^W+"^^ff^W+ (5) 

a series which automatically comes to an end at the term Ao"- 

An extension of this formula is obtained by writing 6 in place 
of w, where 0<^<1. We then get 

2/fl=2/o+c^Ao— -Y-2-Ao'^ + — f"2~'3 — ^0 - - , - - - (6) 

which enables us to interpolate for a 2/ in between any two of a series 
of y's corresponding to x's advancing by equal steps. This relation 



is no longer identically true as was (5), for the series on the right 
in (6) is unending, but its application in practice is justified when, 
as the differences advance, the numbers obtained tend to grow 
smaller and smaller, so that the remainder after a certain number 
of terms can be treated as negligible. Unless this tendency is 
reaUzed without carrying the differences far the formula is not 
very satisfactory. 

To illustrate the method of procedure the following figures may 
be used from Table (7), p. 25:— 



Table (24). Ma.rks obtained by certain Candidates 
IN AN Examination 





No. of 


First 


Second 


Third 


No. of Marks. 


Candidates. 


difference 


difference 


difference 




y 


A 


A2 


A3 


Not more than 45 


447 


37 






,, „ 50 


484 


21 


-16 


1 


» „ » 55 


505 


6 


-15 


12 


„ „ 60 


511 


3 


- 3 




,, ,, 65 


514 









Suppose now we wish to know the number of candidates who 
obtained a number of marks not more than 48. In that case, in 
applying formula (6), we have 

2/0=447, ^=(48-45)/(50-45)=3/5, 

Ao= 37, Ao'=-16, Ao'=l, 

and hence, up to this order of differences, the required number of 
candidates is given by 



(1) 



447+5 . 37-:^^(-16)+i*lliA^ 
1.2 1.2.3 

=447+22-2+l-92+006 

=471, approximately. 

Also, number of candidates obtaining more than 48 marks, but not 
more than 50 

=484-471 

= 13, approximately. 



94 STATISTICS 

Fourth Method {by means of Lagrange's Formula). We shall 
consider one more formula, due to the famous French mathematician 
Lagrange (1736-1813), which is useful when the recorded i/'s corre- 
spond to ic's which advance by unequal stages. 

Let the given statistics be represented as before by 

(a^o. 2/o)> (^1. 2/i). (^2. y-i)^ ' ' • i^n, 2/n)» 
and consider the equation 

{x-Xi){x-X2) . . . ix-x„) 



y=yo 

+2/ 



{Xq Xj){Xq X2) . . . {Xq XjJ 
{x-Xo){x-X2) . . . {X-Xj ^ 
(Xi SJo/l-^l •^2) ' • • l*^! **'«) 

{x-Xq){x-Xi) . . . (a;-a;„_i) 
[X„ Xq)(X„ Xj^) . . . [Xji Xn-i) 

It is of the nth degree in x, and it is identically satisfied by the 
(n-{-l) pairs of values 

{x=^Xo, y=yo), {x=xi, y=y^), . . . {x=x^, y=yn)- 

It will therefore clearly serve as the fitting curve 

y=Co+c^x+C2X^-{- . . . +c„a:«, 

being exactly of this type, and in order to get the y corresponding 
to any other x we have only to substitute that value of x in (7). 

Example. — The following figures, based upon data from Porter's 
The Progress of the Nation, show the age distribution of criminals 
in the j^ear 1842. 

Percentage of criminals up to age 25=52-0 (?/o). 

„ 30=67-3 (2/1). 

„ 40=84-1 (2/2). 

„ 50=92-4 (2/3). 

Let us employ Lagrange's formula to find the approximate 
percentage of criminals up to 35 years of age, making use of the 
four ordinates given, and taking a;=35. We have 

_ (3 5-30)(35-40)(35-50) ^^g(35-25)( 35-4 0)(35-50) 
^~ (25-30)(25-40)(25-50) '* (30^2^5) (30 -40) (30 -50) 
g^^ (35-25)(35-30)(35-50) ^^^^(35-25)(35-30)(35-40) 
(40-25)(40-30)(40-50) (50-25)(50-30)(50-40) 

^_ 10.4-1- 50-475+4205-4-62 
=77-5. 



GRAPHS 



95 




Number of cigarettes bought 
Fig. (13). 



Reasoning made Clear with the Help of Graphs or Curves. The 

graphical method not only produces an instructive picture of a 
scheme of observations, but it may also be used effectively on 
occasion to pilot one through the intricacies of economic or similar 
argument. The eye is a very ready pupil and is quick to pass on 
what it sees to the mind ; it acts, that is to say, as an ally to the 
understanding, which might get on without it, but which certainly 
gets on faster with it. 

To illustrate this we shall consider the first principles of an 
interesting class of curves relating 
to supply and demand.* 

Cur'ge of Demand. Conceive a 
smoker who buys cigarettes at 
the rate of x per day, and pays for 
them at the rate of y pence each. 
Altogether they cost him there- 
fore a sum of xy pence per day, 
which is conveniently measured 
by the rectangle OABC in fig. (13). 
Notice that the cost price of each single cigarette is here represented 
by the area (2/X 1), while the total expenditure is represented by the 
area (yxx). 

Now let us suppose his country is at war and that the smoker, 

to put himself in a position to discourage luxuries, decides to give 

Y up smoking. Let us try to 

D ' measure in terms of pence the 

cost of this great sacrifice to 
' him on the first day. 

The first cigarette is probably 
the hardest to do without, and 
the desire for it is so strong 
that, if it were a mere matter 
of money and not of patriotism, 
~X he would be willing to give as 
many pence as are represented, 
say, by the rectangle 1-1 in 
fig. (14) in order to have it to smoke. If he went on to bargain 



'^C 



12 34 

Number of cigarettes bought 
Fig. (14). 



[* A fuller account of these curves will be found in Cunynghame's Geometrical 
Political Economy, where a rather more accurate interpretation of "surplus 
ralue" is given, involving the introduction of subordinate curves. The 
simplified statement here adopted seemed sufficient in an introductory course. 
Marshall's Principles of Economics also contains many fascinating illustrations 
of the use of such curves, mainly in footnotes. ] 



96 



STATISTICS 



with himself in imagination, he would not be ready to offer quite 
so much for the satisfaction of a second smoke soon after the 
first : he would perhaps only give a number of pence represented 
by the rectangle 2-2 in the figure for this second cigarette. And 
if it came to a third he would offer less still, only ' 3-3 ' pence 
perhaps, for the fourth ' 4-4 ' pence, and so on. The rectangles 
here are of varying height, but each stands on a base of unit length. 
Thus we find that the total sum he would be prepared to offer, 
bargaining for cigarette after cigarette in this way, would be repre- 
sented by the sum of the rectangles 1-1, 2-2, 3-3 ... in fig. (14), 
where the addition of each unit length along OX means one more 
cigarette in imagination smoked, and a diminution of unit length 
in an ordinate parallel to OY means a reduction of Id. per cigarette 
in the price the smoker would be prepared to pay. 

But if he fell a prey to his persistent craving and actually bought 
a number of cigarettes represented by OA in the figure, each would 
cost him in the ordinary way only a number of pence represented 
by AB, say, i.e. area (ABx 1), and his total expenditure would thus 
be measured by the area of the rectangle OABC. He would get 
them, that is to say, for less than he would be prepared to give 
rather than go without them. The difference, the area of the 
rectangles making up the portion BODE of fig. (14), represents the 
measure in pence of surplus enjoyment which he would obtain free 

of charge, or it represents the 
measure of free sacrifice he 
makes if he is true to his 
patriotic principles. 

Let us now take an example 
on a larger scale. Imagine a 
small community of people, 
producers and consumers, buy- 
ing and selling among them- 
selves. Some of them are 
coalowners and sell coal to 
the others in the open market, 
where competition is supposed free and unrestricted in any way. This 
last condition is emphasized, because it is seldom perfectly satisfied 
in the real world of commerce. 

Just as in the previous case we may represent the number of 
cwts. of coal bought by a length OA measured along OX in fig. (15), 
and the price actually paid in shillings per cwt. by the area of a 
rectangle on unit base and of height 00 along OY. Thus the 




12 3 4 A 

Number of cwts. of coal bought 

Fig. (15). 



GRAPHS 



97 



total cost to the consumers in shillings is measured by the area of 
the rectangle OABC. 

But here again we may picture the consumers during a coal 
shortage, when, rather than go without the first cwt. of coal, some 
one among them would be ready to offer for it as many shillings as 
are represented by the rectangle 1-1 in fig. (15), and for the second 
cwt. some one would be ready to offer ' 2-2 ' shillings, for the third 
' 3-3 ' shillings, and so on. The demand for coal could thus be 
measured in shillings by the sum of the rectangles 1-1, 2-2, 3-3 
. . . and, if OA runs into thousands of units of coal, the lengths 
0-1, 1-2, 2-3 . . . along OX, corresponding to additions of 1 cwt. 
in the quantity bought, would in the limit be so small that the 
sum of the rectangles would become practically equivalent to the 
curvilinear area OAED in the figure, where DE is a curve drawn 
through the summits of the rectangles, namely the curve of demand. 

The consumers' surplus in this case would be measured in shillings 
by the area BCDE, this being the difference between the measures 
of the sum actually paid for the coal bought and the sum consumers 
would have been willing to pay rather than go without it. 

Curve of Supply. Now let us consider the question from the 
point of view of the coalowners. We shall assume that the average 
cost of production per cwt. of 
coal increases steadily as the 
number of cwts. produced in- 
creases ; this would not be an 
unreasonable assumption in most 
cases after passing a certain point, 
since the richer coal measures 
known are likely to be mined 
before the poorer ones, and the 
cost of mining near the surface 
is bound to be less than when 
deep shafts have to be bored. 

If, then, OA, fig. (16), represents the number of cwts. of coal 
sold, and if the price in shillings per cwt. at which it is sold is de- 
noted by the area of a rectangle on unit base and of height OC 
along OY,/the total payment received by the coalowners will be 
measured in shillings by the area of the rectangle OABC. 

But the cost of producing the first cwt. is perhaps measured 
by the rectangle 1-1, that of producing the second cwt. by the 
rectangle 2-2, the third by the rectangle 3-3, and so on, each rectangle 
being drawn on unit base representing an advance of 1 cwt. (The 




12 3 A 

Number of cwts. of coal sold 
Fia. (16). 



98 



STATISTICS 



advance in the cost of production would not in reality be measured 
by so much the cwt. of course, but the assumption is inaccurate 
in degree only, not in principle, and, by making it, the argument 
is rendered clearer.) Thus the actual cost of production is, in the 
limit when OA is very large and divided up into relatively very 
small parts, measured in shillings by the curvilinear area OAED, 
where DE is a curve drawn through the summits of the rectangles, 
namely, the curve of supply. 

The difference, BODE, between the areas OABC and OAED 
represents what is known as producers' surplus, for it measures the 
profit made by the owners in selling the coal at a higher price than 
the cost price of production. 

Now let us combine the curve of supply (S.C.) and the curve of 
demand (D.C.) in the same 'figure, fig. (17). Their meeting point 

P determines the number of cwts. 
of coal bought (x), and the selling 
price in shillings per cwt. (y). 
For it is clear that under normal 
conditions it would not be profit- 
able to coal producers to pass this 
point, because beyond it the de- 
mand on the part of coal consumers 
measured in money is less than 
the cost of production : they are 
not willing on the average to pay 
so much as ys. per cwt. for it, 
and it costs more than i/s. per cwt. 
on the average to produce. If, 
on the other hand, the amount of coal produced decreases below 
X cwts., the greater this decrease the higher does the profit become 
on the sale of it, because the greater is the difference between the 
cost price and the selling price ; hence, as profits become more 
pronounced, recruits will be attracted into the coal-producing 
business, and, if this goes on, deeper shafts will have to be bored 
and poorer fields worked until profits begin to decrease again and 
the supply once more approaches x cwts. Thus sooner or later 
the production of coal and its market price will tend to the level 
determined by the equilibrium point P where the supply and 
demand curves meet. 

Endless varieties of problems may be discussed by altering the 
conditions and observing the effect produced in the standard 
diagram. Three examples will suffice to illustrate the method. 




N X 

Number of cwts. of coal bought or sold 

S.C. = Supply curue 

D.C. =Demand curue 

Fig. (17). 



GRAPHS 



99 



1. Effect of a Change in Normal Demand. Here we suppose the 
normal conditions of supply are unaltered — it costs just as much 
as before to produce the same amount of the commodity in question ; 
but a more eager demand on the part of consumers shows itself in a 
readiness to purchase more at any given price than would have 
been purchased under the old conditions : this may conceivably 
be due to a general increase in the purchasing power of these con- 
sumers, or it may be the result of a shortage of some other com- 
modity which causes this one to be more widely used, just as 
margarine, for instance, has been known to take the place of butter ; 
whatever the reason may be, the effect is that the demand curve 
now occupies a higher level throughout its length, D'C in place of 
D.C. in the figures. 

When we turn to the supply side of the question, there are three 

Y 




N N' X 

Decreasing Return 

stages which, although they shade into one another in practice, it 
is well to separate clearly in theory : (1) the only supplies immedi- 
ately available are those actually in the hands of dealers ; (2) to 
meet the increased demand, and so earn for themselves increased 
profits, manufacturers wdll speed up production, by working over- 
time, etc., with the help possibly of any disengaged labour or 
capital they may be able to secure, and the resulting extra supphes 
will be available after a short time ; (3) if the demand continues 
unabated, manufacturers, by offering higher wages and interest, 
will seek to attract fresh labour and capital from other engagements 
into their business, and, by renewing their machinery and generally 
improving their organization, they will produce on a larger and 
relatively more economical scale. Moreover, other manufacturers, 
seeing the profits to be earned, will be attracted into the same line 
of business also, so that by this time the current available supplies 
of the commodity may exceed very appreciably their old figure. 



100 



STATISTICS 



But all this happens only in the long run, and the economist has 
always to bear this extremely important element of time carefully 
in mind when he seeks to estimate the effects of any proposed 
action. 

We assume then that the new demand remains long enough at 
its higher level to allow for the gradual adjustment in this way of 

supply to the changed 
conditions, and for the 
economic forces called into 
play once again to arrive 
at a balance between 
them, most likely at a 
new equilibrium point. 
3C. The two figures illustrate 
the difference in effect 
according as the produc- 
tion of the commodity is 
subject to a decreasing or 




Increasing Return 



an increasing return, i.e. according as the cost of production rises 
or falls when the amount produced is increased. In both cases it 
will be noted that more of the commodity is produced (ON' in place 
of ON) in answer to the keener demand, but the difference is much 
greater in the second case than in the first. Also the price has 
gone up in the first case, while in the second it has gone down, 
the difference being measured by the change in PN. 

2. Effect of a Tax. If the 
tax is at the rate of so much 
per unit (say Is. per unit, if 
the price is measured in shil- 
lings) of the commodity pro- 
duced, this will raise the 
supply curve, S.C., bodily up 
a distance of 1 unit into the 
position S'.C, fig. (18), be- 
cause the effect is the same 
as if Is. were added to the 
cost of each unit in produc- 
tion. The production will 
thus be diminished by N'N units, for P' is the new equilibrium 
point ; the selling price will be increased by P'Ms per unit — by 
less, it should be noted, than P'Q or K'K, the amount of the tax ; 
producers' surplus, which is analogous to what economists term 




N' N 

Fig. (18). 



GRAPHS 



101 



Y 










d" 


V 






L 


^v^^ 


i 


y 

3^''^ 




S 


""^v^F 






P' 


0^ 






^^^^^^"^^ 


* 


K 













^ 


\' N 


X 



Fig. (19). 



rent, is diminished by (area KPL— area KT'L')s ; consumers' 
surplus is diminished by (area PLL'P')s ; finally, the tax produces 
for the Treasury a number of shillings represented by a rectangle 
with sides of length ON' and KK'. 

3. Effect of a Monopoly. A monopolist has the power to stop 
production short of the true equilibrium point, so that ON' cwts., 
fig. (19), are produced in place of the ON cwts. which free competi- 
tion would demand. The selHng price is thus raised by Q'Ss. per 
cwt. ; producers' surplus is increased by (area KP'Q'M'— area 
KPL)s ; while consumers' surplus 
is diminished by (area PLD— area 
DM'Q')s. 

A word of explanation is neces- 
sary before leaving the subject of 
these supply and demand curves. 
It is probable that the reader will 
have questioned the possibiHty of 
drawing such curves for any com- 
modity with sufficient accuracy to 
be of any value, but it would be 
enough as a rule to be able to estimate what would happen 
if a slight variation occurred in price or in production, and such 
an estimate may sometimes be made by actual trial : e.g. a good 
practical farmer most likely knows nothing about supply and 
demand curves as such, yet from past experience he has a pretty 
shrewd notion as to how far it may be profitable to spend an extra 
pound here in rearing calves and a pound less there in cultivating 
crops, bearing in mind the prices which cattle and com might be 
expected to fetch. From his point of view the interest of the 
curves, if he knew anything of them, would be centred in those 
portions which correspond to normal conditions, i.e. somewhere in 
the neighbourhood of the equilibrium point under the free play of 
ordinary competition. 

Their real value, however, as suggested at the beginning, does 
not consist in the practical assistance which they afford to the pro- 
ducer or consumer, by way of foretelling the actual measure of 
consumption or production, so much as in the light they throw 
upon general tendencies which are rather apt to be obscured if they 
are ponderously presented with elaborate economic argument. 
They make plain in a moment to the eye what can only be stated 
in two or three pages of writing. 



CHAPTER X 

COBRELATION 

One of the most important questions which can be discussed by 
statistical methods is that of possible connection, or correlation, as 
it is called, between two sets of phenomena. If some factor in 
each can be isolated and measured numerically, our object is to 
discover if the size of either is sympathetically affected when a 
change occurs in the size of the other ; or, to put the matter in 
another way, do large values of the one factor go with large values 
of the other factor and small with small, or vice versa ? And, if 
some mutual dependence of this kind exists, can an estimate of 
its extent be made ? 

Consider, for example, the factor or character of height in husband 
and wife. Is there any connection between stature of husband (x) 
and stature of wife (y) ? Do tall men tend on the average to wed 
tall women, or do we find tall men choosing short women for wives 
just about as often as they choose tall women ? When correla- 
tion exists we shall want some measure for it which wiU tell us 

the amount of change or devia- 
tion from the average in either 
character associated with a given 
change or deviation from the 
average in the other. 

In studying graphs we saw how 
some hint of the existence of 
correlation might be discovered, 
but we wish now to go a little 
more deeply into the subject. 
The first step is to measure an 
adequate number of pairs of values, x and y, of the characters 
concerned in order to find what values are associated together, 
and how frequently the same values are repeated. When this is 
done we can draw up a table of double entry, see fig. (20), setting 
out in rows and columns the frequencies observed. An examina- 
tion of Table (25), showing the variation of braiii weight with age 

102 





x^ 


AT, 


^3 




xp 


y. 












y. 
























y^ 













Fig. (20). 



CORRELATION 



103 



in the case of 197 Bohemian women, will make clear what is meant. 
The x's from x^ to x^ and the y's from y^ to y^ are supposed to 
ascend in magnitude, and when, for example, the pair of values 
(Xg, yz) is observed to be repeated nine times, the number 9 is placed 
in the second column and third row of the table, so that the frequency 
of each class is found recorded in the square proper to it : thus, 
out of the sample in Table (25), there are 10 women between the 
ages of 40 and 50 with brams weighing between 1300 and 1400 
grams. 



Table (25). Variation of Brain Weight with Age in the 
Case of certain Bohemian Women. 

[Data from Biometrika, vol. iv. pp. 13 et seq.. Variation and Correlation 
in Brain Weight, by Raymond Pearl.] 







Age in years 








^1 

20-30 


^2 

30-40 


^3 
40-50 


50-60 


60-70 


70-80 


Totals 


CO 

i 

.^ 

-c: 

s 
1 


y. 

1000-1100 


1 


_ 


1 


1 


- 




3 


y^ 

1100-1200 


2 


2 


4 


2 


5 


4 


19 


^3 

1200-1300 


28 


9 


8 


14 


10 


4 


73 


1300-1400 


26 


14 


10 


6 


5 


4 


65 


1400-1500 


13 


7 


7 


2 




2 


31 


1500-1600 


2 


3 


■ - 


1 


- 


- 


6 




Totals 


72 


35 


30 


26 


20 


14 


197 




Mean y 


1325 


1350 


1310 


1285 


1250 


1279 





When each class interval, as in this table, includes a small range 
of values, the x and y may, as an approximation, be taken as the 
mid values of their class intervals : 2/3 would be taken, for instance, 
as 1250, though it really includes all values between 1200 and 



^x 



104 STATISTICS 

1300 grams. Strictly in such cases each single observation is not, 
geometrically speaking, located at a definite point, but lies some- 
where within a small area, though it is treated as if it had the values 
X and y which apply to the centre point of the area. It is some- 
times possible to correct for this assumption by what is known as 
Sheppard's adjustment, but we shall not concern ourselves with 
the correction in the present discussion, so as to avoid complications, 
because the difference made is not generally large. 

The table, when drawn up, may immediately suggest some 
intimate connection between x and y. It may indicate that as 
X increases y also in general increases, or that y tends to fall in 
value as x grows bigger. But a more refined analysis is neccHsary. 
It would be instructive perhaps to travel along the row of x'a, find- 
ing what mean value of y is associated with x^, what mean value 
of y is associated with X2, and so on. This would give a sounder 
basis for judging whether, as x increased, y in general increased or 
decreased as the case might be : for example, in Table (25) the 
mean values of y associated with the several types of x are shown 
in their proper columns at the foot of the table and clearly, as 
X increases, y tends to decrease, apart from conflicting readings at 
the beginning and end of the table, and the latter of these may not 
be significant of any real difference in brain weight at the end of 
life, for it is only based on fourteen observations ; generally, the 
inference from this table would be that the weight of the brain 
decreases as the age increases after maturity is once reached, 
although, of course, it would be rash to make more than a tentative 
statement with so small a sample at our disposal. 

Let us suppose y^ to be the mean value of y associated with x^, 
y^ the mean value of y associated with 0^2, y^ with Xq, and so on. 
If these values {x^, y^), [x^, y^), {^3, §3), etc., are plotted, it is very 
often found that they cluster more or less closely about a straight 
line, see fig. (21), so that we are led to ask whether there is not 
some line which will very fairly describe the run of the points ; 
the equation of such a line would be 

y=.mx-\-c, 

and if m and c were known we could find from this equation the best 
average value of y corresponding to any given '^. 

But, on reflection, ^1, §2, ^3 • . • are themselves only the best 
2/'s corresponding to the particular values Xi, Xg, Xq . . . oi x, so 
that the problem is really the same as that of finding the relation 

y=mx-\-c, 



CORRELATION 



105 



based on all the observations, which will enable us to estimate the 
best y corresponding to any given x. 

Now for any value x^ of x the value of y given by this relation 
is (mx^-i-c), while by observation we may find more than one value 
of y corresponding to the value Xj^ of x. If y^ be one such value 
the dijfference between it and the value given by the above rela- 
tion is 

(ma;i+c)— 2/1. 

This difference we may regard as the error made in estimating _y A 
from the relation instead of taking the value given by observatibn // 



Y Pr'a^A ky^A-a^ i^fl,*.' lU^ritl R ^inJTxU iml^ifL 




> - asso :i a ;ed w 1 1 va 1 )vis A p b f es 








s 


s 


V 




1325 ■• " :S^ " 1" " " 


S ^ - - 


Q • V ' 


1. -t ^t 






^ , _^s, ± 


^ ± ^,. 


o> X *S 




1 1275 - - __vj_ _ 


. : ~ -. : ^ : : 


s ^ - 


5 ^s^ 






\ 




^ 















O20 



40 



50 60 

Age in years 
Fio. (21). 



70 



80 



which for the moment we think of as the true value. The best 
relation will then clearly be the one which makes all such errors of 
estimate as small as possible. But, algebraically, some of these 
errors are positive, i.e. the value of y given by the relation is greater 
than that given by observation, and some are negative, and it is 
only their magnitudes that we wish to take into accomit. Accord- 
ingly we follow the method used in finding the standard deviation 
in order to get rid of the ambiguities of sign : we form, that is to 
say, the sum of the squares of the errors, because the expression so 
formed will clearly be least when each separate error is as small as 
possible in absolute magnitude. 



106 STATISTICS J 

To find, then, the values of m and c which will make 

(mXi-f-c-2/i)2_|_(,^^^c-2/2)^-f . . . +(^a^n+c--yn)^ j 

a minimum (see Note 7 in the Appendix), where n is the total ' 

number of pairs of observations. 

The required values are given by differentiating, first with regard i 

to c treating m as constant, and then with regard to m treating c 
as constant, putting each result equal to zero. Thus 

(ma;i+c-2/i)+ . . . +(ma;„+c-2/J=0 J^ .| 

Therefore m(x^-{- . . , -\-Xn)-\-nc—{y^^ . . . +2/n)=0 

m{x^^^ . . . +a:„2)+c(xi+ . . . ■^x^)-^(x^y^-\- . . . x^y^)=0. 

The first of these equations gives i 

m(nx)-\-nc—{ny)=0, '■■ 

I 
i.e. mx-{-c—y=0, j 

i 
where x is the mean of all the x's and y is the mean of all the y's, j 

and it expresses the fact that the line y=mx-\-c passes through ' 
the point {x, y). \ 

This might have been expected, for, graphically, each pair of ; 
observations (:tj^, 2/1), (a:2J 2/2)' (^3' 2/3) • • • corresponds to some point, < 
and if we look for the line y=mx-{-c passing through the region 
where they cluster most thickly together we should certainly expect 
it to pass through their mean or centre of gravity [x^ y). This j 
suggests how the values of m and c may be considerably simplified. 
If we measure all the cc's from x, their mean, and all the ^/'s from y, j 
their mean, which is equivalent to taking the point (x, y) as origin 
and replacing every x by its deviation ^ from x and every y by \ 
its deviation "n from y, the first of the above relations is reduced 
to c=0. and therefore the second becomes ! 



© 


m(^^^+ . . . 


+L')-(^x\+ ■ ■ ■ 


+^»%)=o. 


Hence 




■ ■ +L-nn)l(^x'+ ■ ^ 


. • +L') 



where p is the mean of all the product pairs f^, and a^ is the standard 
deviation of all the a:'s. 



CORRELATION 107 

Thus the required equation for estimating the best v correspond- 
ing to any particular f is 

p ' ' > 

whence y—y=-^A^—^) • • . (1)"" i. 

The coefficient p/crj^ in this equation evidently gives the deviation 
in y from the mean y con*esponding to unit deviation in x from 
the mean x, for when {x—x)=l, {y—y)=p!(Tj^. Hence the greater 
this coefficient is, the greater will be the change in y resulting from, 
or at all events coexistent with, unit change in x. 

Thus p/aa:^ would seem to supply a not unreasonable measure of 
the correlation between x and y. But there is something very 
unsymmetrical about this result. Why should the correlation be 
measured by pla^s^ any more than by pJGy^ ? In fact, we might 
repeat the whole of the previous argument, interchanging x and y 
throughout wherever they appear. In that case we should first 
travel down the column of i/'s and calculate the mean values of x 
associated with 2/is 2/i» 2/3 • • • respectively. This would give a set 
of points {xj^, 2/1), {x^, yo), (Xq, 2/3), • • . , which, when plotted, would 
perhaps lie approximately in a straight line. We should thus be 
led to look for some relation 

x=m'y-{-c' 

which would enable us to estimate the best average x corresponding 
to a J/ of given type, and, proceeding just as before, we should 
ultimately obtain the equation 

or (x-x)=^^Ay-y). . . - (2) 

^/ 

in which the coefficient pjuy^ givQQ now the deviation in x from the 

mean x corresponding to unit deviation in y from the mean y. 

Hence p/ffy^ has, seemingly, just as much claim asp/cja-^ ^o measure 

the correlation between x and y. The one gives the change in x 

corresponding to unit change in y : the other gives the change in y 

corresponding to unit change in x ; and the only reason why they 

differ is because unit change in x does not mean the same thing as 

unit change in y : their standards of changeableness or variability 

are not equal. If then we could alter the scales of measurement 

so that unit change in each were of the same magnitude, the two 

coefficient!^ obtained ought to become identical, and we should then 

have a really satisfactory measure for the correlation required. 



108 STATISTICS 

With this object let us examine the variability of the x's and 
compare it mth the variability of the t/'s. Now the total dispersion 
of the di£ferent x's on either side of x, the mean x, is conveniently 
measured by g^, their standard deviation. And similarly the 
dispersion of the y's on either side of y, the mean y, is measured 
by Gy. The bigger cja- is, the greater is the variability of the oj's, 
and the bigger Cy is, the greater is the variability of the y's. Hence, 
in equations (1) and (2), (x—x) should be divided by o-a. and [y—y) 
by Gy if we want to work with the same unit of change or variability 
in each case. The equations then become 



and 



\ Gy / Gg,Gy\ Gj. 

x-x\ p iy-y 



^x^y\ ^y 



Write r=plGjPy ; then r is taken to be the coefficient of correla- 
tioUy for it measures the change in either character corresponding to 
unit change in the other when the units are made comparable. 

The lines giving the best y for a given x and the best x for a 
given y may now be written 



y—y^r-^(x—x) 



and x—xz^r—iy—y), 

G„ 



and they are called lines of regression. The term regression was 
first used by Sir Francis Galton in a paper entitled Regression 
towards Mediocrity in Hereditary Stature, though the root idea 
is not by any means confined to characters affected by heredity : 
it holds for any pair of correlated variables. Galton found that 
if a number of tall fathers are selected and their heights measured, 
the mean height being calculated, and if, further, the heights of the 
sons of these fathers are measured, their mean height being like- 
wise calculated, the latter is not equal to the mean height of the 
selected fathers, but is rather nearer the mean height of the popula- 
tion as a whole. There is, that is to say, a regression or stepping 
back of the variable towards the general average. Professor Karl 
Pearson has remarked that ' in the existing state of our knowledge 
the recognition that the true method of approaching the problem 
of heredity is from the statistical side, and that the most we can 
hope at present to do is to give the probable character of t^ e offspring 



CORRELATION 109 

of a given ancestry, is one of the great services of Francis Galton to 
Biometry.' 

The expressions r— and r— are called coefficients of regression, 

and they register in the above particular case the amount of abnor- 
mality to be expected in the height of the sons when the amount of 
abnormality in the height of the fathers is known, and vice versa. 
The regression of the sons' height, y^ on the fathers' height, x, is, 
in fact, defined as the ratio of the average deviation of the heights 
of the sons from the mean height of all sons to the deviation of the 
heights of the fathers from the mean height of all fathers, and hence 
it may be written 

To make the definition more general, instead of speaking merely in 
terms of height, we refer to any row or column — ^for there is no 
intrinsic difference between row and column — in a table like 
Table (25) as an array of y's or of x's, and selecting a particular 
type J say a particular value of x (like fathers of height x), we define 
the regression of the corresponding array of y's (like heights of sons 
of these fathers) on the type x to be the ratio of the average devia- 
tion of the array of y's from the mean y to the deviation of the 
selected type x from the mean x. 

Example. To illustrate, let us take some figures due to Professor 
Pearson and Dr. Alice Lee [Biometrika, vol. ii. pp. 357 et seq., On 
the Laws of Inheritance in Man]. Suppose the mean stature of all 
observed fathers, based on a sample of over 1000 observations 
=67-68 in., with S.D.=2-70 in. 

Also suppose the mean stature of all sons= 68-65 in., with S.D. 
=2-71 in., and that the correlation r between stature of father 
and stature of son= 0-514. 

The regression of son on father as regards stature is then given by 

(.V-68-65)=: (0-514)— (x-67-68) 

where x is the height of selected fathers and y the mean height of 

their sons. 

Hence 2/=0-516x+33-73, 

so that if we selected fathers of height 70 in., for example, the 
mean height of their sons would not be 70 in., but 

(0-516)(70)+33-73=69-85 in., 



110 STATISTICS 

i.e. there is a regression towards the general mean, 68-65 in., of 
all sons. 
Also the coefficient of regression 

^ =(0-514)(2-71)/(2-70) 

=0-516. 

It is not difficult to show that the greatest numerical value r 
can in general take is unity, for consider the expression for the 
sum of the squares of the differences between the observed devia- 
tions of the y characters from their mean and the corresponding 
deviations as deduced from the best fitting regression line, 

y—y=r^(x—x). 

If, with our previous notation, 'n denote the observed deviation of 
the one character y, associated with a particular deviation, ^, of 
the other character, x, then, since (rajay,)^ denotes the best value 
given by the line, the sum of the squares of the differences between 
these values 



o-„ . a 



2 



=na^\\-r\ 

Since the sum of a number of squared quantities must be positive, 
it follows that r^ must be less than 1 and hence r lies between —1 
and +1. 

Further, n^y^{l—r^) can only vanish if every one of the squared 
quantities on the other side vanishes independently of the rest, 
so that we onJxget r=:^l, when 

In this case the deviation of the one character from its mean is 
always exactly proportional to the deviation of the other character 
from its mean, and the correlation is then said to be perfect, for 
it is equivalent to causation. In perfect correlation a one-to-one 
correspondence thus exists between the values of the two char- 
acters, for to one value of either there corresponds one and only 
one value of the other, and the standard deviation of the array 



CORRELATION 



111 



(measuring its variability) corresponding to any selected type 
vanishes. 

Zero correlation is at the opposite extreme where, no matter 
what the type selected in the one character may be, the mean 
value of the array in the second character i^ unaffected, because 
the two characters are quite independent or uncorrelated ; the 
deviation of y from its mean bears no relation at all to the deviation 
of X from its mean, and unit change in either is associated with no 
particular change in the other, so that r must in this case be zero. 

When r is negative, since (y—y)l{x—x)=ra^x a-nd the o-'s are 
necessarily positive, corresponding to any value of x above the 
mean of all the x's the best value of (y—y) is negative, that is, the 
best value of y is below the mean of all the y's, and vice versa. 
This means that in general high values of x would be associated 
with low values of y, and vice versa. 

If we take the mean as origin so that the regression lines become 

y=rayl(T^ . x, 
x=rajay . y, 

these Hues coincide with the axes when the correlation is zero, 
and with one another when r=±l and the correlation is perfect, 
fig. (22). Given two equally 
variable characters (cra.=<7j,) and 
perfect correlation, the regres- 
sion lines coincide with one of 
the bisectors of the angle formed 
by the axes. 

It may be helpful to look back 
again now at the graphical view 
of the argument leading up to 
the determination of the co- 
efficient of correlation. For 
successive values of x we calculated the means of the several 
2/'s observed, these being presumably the best available y's corre- 
sponding to the particular x's selected, and we assumed that, 
when plotted, the points so obtained, {x^, y^), (ajg, ^2)' G'^^a? ^3)' • • •' 
lay roughly in a straight line. In the same way we calculated the 
means of the several x's observed to correspond to particular y's 
selected, and again we assumed that the resulting points, (Xj, y^), 
(^2> 2/2)' (^3> 2/3) •• • lay roughly in a straight line. These assump- 
tions are justified in very many cases, but when they fail recourse 
must be had to other methods beyond the scope of this book. [See, 




^^^ (Mean) 



Regression Lines when 
Correlation is perfect (r'=-¥\} 

Fig. (22). 



112 



STATISTICS 



for example, Pearson's paper in Drapers' Company Research Memoirs 
Biometric Series ii., On the Theory of Skew Correlation and Non- 
linear Regression, introducing the correlation ratio, v, which is 
equal to r in the particular case when the regression is linear.] 
Sometimes, again, although the observations are so scattered that 
the assumption of a straight line to describe the best fit seems 
somewhat wide of the mark, it may be justified on the ground that 
no better graphical result would be given by using any other curve 
in place of the line. Moreover the linear expression, y=mx-\-c, 
is simple and may serve to give at all events the first two terms of 
some more complex relation supplying an estimate for the most 
probable y corresponding to a given x. 

If we had plotted all the original pairs of observations, instead 
of plotting certain ic's and the mean t/'s associated with them, or 

certain i/'s and the associated mean 
ic's, the two lines of regression would 
not have stood out so clearly : they 
would have lacked definition, like an 
optical image which is not strictly in 
focus, but there would have been a 
concentration of observations, as of 
light, in the neighbourhood where the 
lines of regression intersect, namely 
at {x, y), the mean of all the a;'s and 
all the 2/'s. When, however, the lines of regression lie close together 
they become more clearly defined, all the observations being centred 
then more nearly in one line, and the correlation tends towards 
perfection. Such cases are frequent in Physics but rare, if found at 
all, in that class of Statistics into which the element of human 
impulse enters. When r is less than 1 the lines of regression, if the 
regression is of linear type, will be inclined to one another at some 
angle between and 90 degrees. 

If only a rough value of r, the correlation coefficient, is required, 
that may be obtained by merely estimating the gradient of each 
regression line and multiplying the results together, one measured 
relative to the axis of x and the other relative to the axis of y, 
for this product 

= (regression of y on x) (regression of x on y) 







CORRELATION 



113 



Such an estimate may also be useful, though it may not be very 
dependable, when the complete distribution of characters is not 
known, for either regression line can be drawn when any two points 
on it are known and a single array of values of either character 
corresponding to a given type of the other is sufficient to fix one 
such point ; also the mean {x, y), if it were known, would at once 
give a point common to both regression lines. When all the facts 
are available, however, the method of calculation is to be preferred 
to that of simply graphing the observations and their means, as there 
is bound to be a certain amount of guesswork and consequent error 
in deciding from a graph how the best regression lines run. 

It is frequently convenient to refer the deviations of the given 
variables to some point other than the mean (x,y) as origin, and, 
when this is done, a correction 
must be applied to the resulting 
value of r. We have already 
explained how, in such a case, 
to correct for standard devia- 
tions, and, as r—pja^dy, it only 
remains to explain how to cor- 
rect for p. 

Now p is given by 

where the |'s and ^'s denote deviations from x and y respectively. 
Fig. (23) indicates the changes necessary in transferring from some 
origin to the mean G. The co-ordinates of P (representing a 
typical observation) referred to O are {x, y) and referred to G are 
(f, 7/). Also the point G itself referred to is (x, y). Thus 

i=X—X, •n = y—y^ 

and np becomes 

(x,-x)(y^-y)-\- . . . -^(Xn-x){yn-y) 

={^iyi-xyi-yxi+xy)+ . . . +{Xr,yn-^yn-y^n+^y) 

= K2/i+ . . . +^n2/«)-^(2/i+ • • • +yn)-y(Xi+ • • • +^n)+nxy 

= (^i2/i-|- . . . -\-x^7j^)-x.ny-y .nx+nxy 

=Z(xy)~nxy, 

where S(xy) denotes the sum of expressions of the type xy. 
Hence the corrected value of p 

=(^(xy)ln)-xy, i >. 

H 



Y 


y' 














^ 


F 


3 
t 


x' 










•T 






X \ 




i 







— -- 


-X- - 


- ->- 




X 



Fig. (23). 



114 STATISTICS 

We proceed to a few applications of these results in the next 
chapter. 

[As early as 1846 a French physicist, Auguste Bravais, had conceived the 
surface of error as a means of describing in space the path of a point whose x 
and y co-ordinates are subject to errors which are not independent. It is 
astonishing that although his work really embraces the fundamentals of the 
theory of correlation as afterwards developed, it lay dormant for nearly forty 
years until Sir Francis Galton introduced on graphical lines an improved nota- 
tion (Galton's function, or the coefficient of correlation) and gave practical 
examples of its use. 

A little later Edgeworth (1892), using Galton's function, independently 
reached some of Bravais' results for the correlation of three variables, and 
showed how they could be extended. Karl Pearson, in 1896, contributed 
to the Royal Society Transactions a fundamental paper on the subject, with 
special reference to the problem of heredity, drawing attention to the best 
value of the correlation coefficient, and how it should be calculated. (See 
Appendix, Note 11.) Yule, returning in the following year to Bravais' for- 
mulae, showed their significance also in the case of skew correlation. 

Pearson afterwards developed a method of determining the correlation of 
characters not quantitatively measurable, and in a discussion of the general 
th3ory of skew correlation in another paper he proposed a new function, the 
correlation ratio, applicable to the case of non-linear regression.] 



CHAPTER XI 



CORRELATION — EXAMPLES 



Example (1). — To find the correlation between Differences in Whole- 
sale Price Index Numbers and in the Marriage Rate from their corre- 
sponding Nine-yearly Averages during the twenty years, 1889-1908. 
using the data given on p. 77. 

Table (26). Correlation between Differences in Wholesale 
Prices and Marriage Rate from their respective Nine- 
yearly Averages. 

(2) (3) (4) (5) 



(1) 



(6) 



Year. 


Difference in 

Prices from 

9-yearly Average, 


Square of 
No. in 

Col. (2). 


Difference in 

Marriage-rate 

from 9-yearly 

Average. 


Square of 
No. in 
Col. (4). 


Product of No8. 

in Col. (2) and 

Col. (4). 


{X) 


(x^) 


(y) 


{y') 


1889 


+ 0-9 


0-81 


+ 1 


1 


+ 0-9 


1890 


+ 2-3 


5-29 


+ 6 


36 


+ 13-8 


1891 


+ 7-0 


49-00 


+ 6 


36 


+ 42-0 


1892 


+ 2-4 


5-76 


+ 3 


9 


+ 7-2 


1893 


+ 2-0 


4-00 


- 6 


36 


-12-0 


1894 


- 2-8 


7-84 


- 5 


25 


+ 14-0 


1895 


- 4-3 


18-49 


- 6 


36 


+ 25-8 


1896 


- 61 


37-21 


+ 1 


1 


- 6-1 


1897 


- 3-7 


13-69 


+ 3 


9 


-11-1 


1898 


- 0-2 


0-04 


+ 4 


16 


- 0-8 


1899 


- 1-6 


2-56 


+ 6 


36 


- 9-6 


1900 


'-\- 5-3 


28-09 


+ 1 


1 


+ 5-3 


1901 


+ 10 


100 


. . 


. , 


. . 


1902 


- 0-5 


0-25 


+ 1 


1 


- 0-5 


1903 


- 1-4 


1-96 


- 1 


1 


+ 1-4 


1904 


- 1-3 


1-69 


- 3 


9 


+ 3-9 


1905 


- 2-4 


5-76 


- 2 


4 


+ 4-8 


1906 


- 0-5 


0-25 


+ 3 


9 


- 1-5 


1907 


+ 3-2 


10-24 


+ 6 


36 


+ 19-2 


1908 


- 1-8 


3-24 


- 2 


4 


+ 3-6 


+241-26-6 


197-17 


+41-25 


306 


+ 141-9-41-6 



136 



116 STATISTICS 

The arithmetic is comparatively simple in this case because 
there is only one value of each variable corresponding to each year, 
so that there is no weighting or grouping to complicate the analysis. 
The variables x and y, between which we wish to find the correlation, 
appear in col. (2) and col. (4) in Table (26), and the positive and 
negative differences are separated from one another in each case 
so as to make their summation easier. 

Thus for the arithmetic mean of the numbers in col. (2), we have 

^=(+24-l-26-6)/20=-0'125 ; 

and for the mean of the numbers in col. (4), we have 

j^=(+41-25)/20=+0-8. 

The straightforward procedure would now be to get the twenty 
corresponding values of ^ and v, the deviations of the twenty aj's 
in col. (2) and of the twenty y's in col. (4) from x and y respectively, 
and, having found 0-3. and ay, we could immediately deduce r from 
the formula 

r~pla,^(Ty 

But it is simpler to measure the deviations from (0, 0) as origin 
rather than from the mean (—0-125, +0-8), because x^, y^, and xy 
involve fewer significant figures than would ^^ i/2^ ^nd ^^/, and, 
of course, it will be necessary to correct for this at the end in the 
usual way. 

The mean square deviation of x referred to zero as origin 
= 197-17/20, by col. (3). 

Therefore, cr^^^ 197-17/20- (0-125)2=9-843 

(7,-314. 

Again, the mean square deviation of y referred to zero as origin 
=306/20, by col. (5). 
Therefore, (j/=306/20- (0-8)2= 14.66 

c7^=3-83. 
Also the corrected p 

= {Exy)ln-xy 

= 100-3/20- (-0-125)(+0-8), by col. (6) 

=5-015+0-100 

=5115. 

Hence 'f=vl<^x'^y 

=5-115/(3-14)(3-83) 
=0-43. 



CORRELATION — EXAMPLES 117 

It is necessary to be careful with the signs in forming the numbers 
in col. (6), but otherwise the actual calculation should present no 
difficulty. 

The regression equation giving the best marriage rate difference, 
Y, for a given wholesale price difference, X, from their respective 
nine-yearly averages is 

{Y-0'8)=r^ . (X+0-125) 

= (043)||g-j(X+0.i25) 

i.e. Y=:0-52X+0-86. 

The regression equation giving the best wholesale price difference, 
X, for a given marriage rate difference, Y, from their respective 
nine-yearly averages is 

(X+0-125)=r^ . (Y-0-8) 

=0-35(Y-0-8) 
i.e. X=0-35Y-0-40. 

We noted that fig. (10), p. 80, suggested a closer correlation 
between the two factors we have been considering during the 
earlier years of the period 1875-1908 than during the later years. 
It might be worth while as an exercise to see if this is borne out 
by calculating r for the years 1875-1889, and comparing it with 
the value found for the years 1889-1908. 

Example (2). — To find the correlation between Overcrowding and 
Infant Mortality in London Districts. [Data taken from London 
Statistics, vol. 23, published by the London County Council.] 

The figures are apparently based upon the Census Report of 
1911. The numbers in col. (2), Table (27), show what percentage of 
the total population occupying private houses in each district were 
living in overcrowded conditions, any ordinary tenement which 
has more than two occupants to a room, including bedrooms and 
sitting-rooms, being defined as overcrowded. The numbers in 
col. (5) show the infantile mortality in each district, that is, the 
number of infants who died under one year out of every 1000 
born, including both sexes. 

For the sake of comparison these numbers have been plotted 
together on the same graph sheet. The districts, arranged in 
alphabetical order, were numbered from 1 to 29 so as to form a hori- 
zontal scale corresponding to the scale of years in discussing prices 
and marriages. The scale in this case is, of course, purely artificial, 



118 



STATISTICS 



and the only reason for joining up neighbouring points is that we are 
better able by so doing to see whether or not high values of the one 
variable go with high values of the other variable, and low with low. 
In calculating the mean and standard deviation for overcrowding 
we have measured deviations from 17-0 as origin, and in making the 
same calculations for infant mortality we have measured devia- 
tions from 125 as origin. It is convenient, therefore, to use the 
point (17-0, 125) as origin in working out also the product deviation 
sum, col. (8) of Table (27), instead of using the mean (17-86, 126). 



Table (27). Correlation between Overcrowding and 
Infant Mortality in London Districts (1911). 

(1) (2) ' (3) ■ (4) (5) (6) (7) (8) 





Per- 


















centage 
of 


Deviation of | 


Square 


Infant 


Deviation of 


^square 

of No. 

in 


Product of Nos. 


District. 


Popula- 


No. 


in Col. (2) 


of No. in 


Mor- ] 


^To. in Col. (5) 


in Col. (3) and 




tion 
Over- 


from 17-0. 1 


Col. (3). 


tality. 


from 125. 


2o\.(6). 


Col. (6). 




crowded 






















{x) 






(y) 






1) Battersea . 


13-3 




- 3-7 


13-69 


124 


- 1 


1 


+ 3-7 


(2) Bermondsey . 


23-4 


+ 


6-4 


40-96 


156 


+ 31 


961 


+ 198-4 


(3) Bethnal Green . 


33-2 


+ 


16-2 


2G2-44 


151 


+ 26 


676 


+ 421-2 


(4) Camberwell . 


13-5 




- 3-5 


12-25 


109 


- 16 


256 


+ 56-0 


(5) Chelsea .• . 


14-9 




- 2-1 


4-41 


109 


- 16 


256 


+ 33-0 


(6) City of London 


12-3 




- 4-7 


22-09 


124 


- 1 


1 


+ 4-7 


(7) Deptford . 


12-2 




- 4-8 


23-04 


142 


+ 17 


289 


- 81-6 


(8) Finsbury . 


39-8 


+ 


22-8 


519-84 


156 


+ 31 


961 


+ 706-8 


(9)Fulham . 


14-6 




- 2-4 


5-76 


125 








(10) Greenwich 


124 




- 4-9 


24-01 


128 


+ 3" 


" "9 


■■_ 14.7 


(11) Hackney . 


12-4 




- 4-6 


21-16 


119 


- 6 


36 


+ 27-6 


(12) Hammersmith. 


14-2 




- 2-8 


7-84 


146 


+ 21 


441 


- 58-8 


(13) Hampstead . 


71 




- 9-9 


98-01 


78 


- 47 


2209 


+ 465-3 


(14)Holborn . 


25-6 


+ 


8-6 


73-96 


115 


- 10 


100 


- 86-0 


(15) Islington . 


20-0 


+ 


3-0 


9-00 


127 


+ 2 


4 


+ 6-0 


(16) Kensington 


17-1 


+ 


0-1 


0-01 


133 


+ 8 


64 


+ 0-8 


(17) Lambeth . 


13-6 




- .3-4 


11-56 


123 


- 2 


4 


+ 6-8 


(18) Lewisham. 


3-9 




-13-1 


171-61 


104 


- 21 


441 


+ 275-1 


(19) Paddington 


16-2 




- 0-8 


0-64 


127 


+ 2 


4 


- 1-6 


(20) Poplar . 


20-6 


+ 


3-6 


12-96 


157 


+ 32 


1024 


+ 115-2 


(21) St. Marylebone 


20-7 


+ 


3-7 


13-69 


108 


- 17 


289 


- 62-9 


(22) St. Pancras . 


25-5 


+ 


8-5 


72-25 


112 


- 13 


169 


-110-5 


(23) Shoreditch 


36-6 


+ 


19-6 


384-16 


170 


+ 45 


2025 


+ 882-0 


24) Southwark 


25-8 


+ 


8-8 


77-44 


144 


+ 19 


361 


+ 167-2 


(25) Stepney 


35-0 


+ 


18-0 


324-00 


144 


+ 19 


361 


+ 342-0 


(26) Stoke Newington 


8-8 




- 8-2 


67-24 


102 


- 23 


529 


+ 188.6 


(27) Wandsworth . 


6-3 




-10-7 


114-49 


122 


- 3 


9 


+ 32.1 


(28) Westminster . 


12-9 




- 4-1 


16.81 


103 


- 22 


484 


+ 90-2 


(29) Woolwich. 


6.3 




-10.7 


114-49 


97 


- 28 


784 


+ 299-6 






+ 119-3-94-4 


2519-81 




+ 2.56-226 


12748 


+ 4322-9-416-1 



CORRELATION — EXAMPLES 



119 



For overcrowding, 

mean= 17+24-9/29=: 17-86 ; 

G,= V[(2519-81/29)- (0-86)2]= V(86-15)=9-3. 
For infant mortality, 

mean= 125+30/29= 126-03 ; 

(T,= V'[(12748/29)- (1-03)2]= V438:5=20-9. 

Also^, referred to (17-0, 125)=(4322-9-416-l)/29=3907/29, and, 
referred to the mean (17-86, 126-03), tliis becomes 
=3907/29-(0-86)(l-03) 
= 133-8. 
Hence r=133-8/(9-3)(20-9)=0-69, 

so that the correlation between overcrowding and infant mortality 
is fairly marked. 



§•0 



5 P20 



^o 



%4 




. . _ , _ „ . _ „ 10 11 12 13 14- 15 16 17 18 19 20 21 22 23 24- 23 26 27 28 29 

Numbers representing various London Districts 
Fig. (24). 

The regression equation giving the average infant mortality, Y, 
for districts in which the extent of overcrowding, X, is known is 

Y- 126-03=r^^(X- 17-86) 

^ ^(0-69)(20.9) 

9-3 ^ ^ 

i.e. Y=l-55X+98-4. 

Similarly, the regression equation giving the average percentage 
of overcrowding, X, for districts with a known amount of infant 
mortality, Y, is 

X- 17-86=r^^(Y- 126-03) 

=0-31(Y- 126-03) 
*.e, X=0-31Y-81-0, 



120 



STATISTICS 



Example (3). — The reader might apply the same method to the 
determination of the correlation between Ratio of Indoor Paupers 
and Ratio of Outdoor Paupers, each measured per 1000 of the esti- 
mated Population in England and Wales, excluding casuals and 
insane, during the years 1900-1914. The following are the statistics 
required for the purpose : — 



Table (28). Correlation between Ratio of Indoor and Ratio 
or Outdoor Paupers, each measured per 1000 or the 
Population. 





Indoor 


Outdoor 




Indoor 


Outdoor 


Year. 


Paupers- 


Paupers- 


Year. 


Paupers—" 


Paupers — 




Rate per 1000. 


Rate per 1000. 




Rate per 1000. 


Rate per 1000. 


1900 


5-9 


15-8 


1908 


6-8 


16-4 


1901 


5-8 


15-3 


1909 


7-1 


15-6 


1902 


6-0 


15-3 


1910 


7-2 


151 


1903 


6-2 


15-4 


1911 


7-2 


14-1 


1904 


6-3 


15-4 


1912 


6-9 


11-2 


1905 


6-6 


161 


1913 


6-7 


111 


1906 


6-8 


160 


1914 


6-4 


10-4 


1907 


6-8 


15-6 









The coefficient of correlation in this case comes out negative 
and = — -15, but it is very small and probably not significant. 
If it were, it would imply that as indoor pauperism diminishes 
outdoor pauperism increases, and vice versa. 

Example (4). — To find the correlation between the Number of 
Cattle and the Number of Acres of Permanent Grass-land in the Coal- 
Producing Counties of England (1915). 

A Government Report was consulted giving the acreage under 
crops and grass and the number of live stock in each petty sessional 
division in the country, as returned on 4th June 1915, and the 
counties included were those which appear in the coal-mining 
reports published monthly in the Labour Gazette. 

In each county the petty sessional divisions with the greatest 
and the least numbers of cattle and of acres of grass-land were 
noted, the numbers being written down to the nearest 1000, and, 
after a rough examination of the range of these variables from 
county to county, suitable class intervals were chosen and a table 
of double eutry was drawTi up, Table (29), with an empty square 
ready for each possible pair of variables. 



CORRELATION — EXAMPLES 



121 



Table (29). Correlation between the Number of Cattle 
AND THE Number of Acres of Permanent Grass-land in 
THE Coal-Producing Counties of England (1915). 







Total Head of Cattle (expressed to nearest thousand) 










^1 
0-5 


^2 
5-10 


^3 
10-15 


15-20 


20-25 


25-30 


30-35 


Xp 
35-40 


Totals 


Meanx 


o 

:S 

i 

s: 

1 

£ 
1 

1 

1 

i 

»2 


^1 

0-5 


: lo 
1 : 15 
i : 150 
















15 


2-50 


5-10 


: : 9>. 
W :27 
: : :2i6 


4 
3 

: " 














30 


300 


^3 
10-15 


:: : 6 

jjjso 

: : :i8o 


•: 3 
:: 18 
:: 54 














48 


4-37 


15-20 


4 

3 

: 12 


III- 

:::6o 














33 


704 


20-25 








i ^ 

; 












30 


8-33 


25-30 




1 



: 

I: ^* 
: : 


. 

: 9 
: 




2 

J 










26 


9-81 


30-35 




-t 
: 6 
: -6 


: : 
: j 22 
:::o 


z 

3 

• 3 










31 


1202 


35-40 




-2 

1 


: 

i 12 

: : 


2 
: 6 
: " 


4 
. 4 

: 16 








23 


15-33 


40-45 








3 
: 


3 
. 4 
• 12 




9 

1 

. 9 






8 


16-87 


45-50 








3 

: 


4 

3 

: 12 


8 
3 

: M 




16 
. 16 




10 


19-00 


50-55 








s 

: * 
: 20 


10 

. 4 

: 40 


15 

1 
. »5 






9 


20-83 


55-60 








6 

1 

. 6 


12 

2 

: 24 




24 

1 

. 24 


30 

1 

. 30 


5 


26-50 


60-65 












21 

1 

. 21 






1 


27-50 


65-70 












24 

1 
. 24 






1 


27-50 


70-75 












27 

1 
. 27 






1 


27-50 


75-80 














40 
3 

; 120 




3 


325 


80-85 








n 

1 

. zx 


22 

1 








2 


200 


' 


Totals 


76 


97 


54 


24 


14 


5 


5 


1 


276 






Mean y 


9 14 


20-13 


33-24 


43-33 


5000 


59 50 


67-50 


57-5 







122 STATISTICS 

Each petty sessional division was then considered in turn and a 
dot was inserted in the particular square applicable to it : e.g. a 
petty sessional division with 42,000 acres of grass-land and feeding 
19,000 cattle would be represented by a dot in the square defined 
by row (40-45) and col. (15-20) in Table (29) ; x was used to repre- 
sent the number of cattle and y the number of acres of grass-land 
in any division, each expressed to the nearest 1000 units. All the 
dots were ultimately added in each square giving the frequency 
for each corresponding pair of variables, and these frequencies were 
recorded in the centres of the squares to which they applied : e.g. 
the frequency of petty sessional divisions stocking 10 to 15 thousand 
cattle and with 30 to 35 thousand acres under permanent grass 
was 22. The total frequency for each row, i.e. each array of 
selected y ty^e, was also noted, in the column at the end of the 
rows : e.g. altogether 31 petty sessional divisions were observed of 
the type having 30 to 35 thousand acres of land under permanent 
grass. Likewise the total frequency for each column, i.e. each 
array of selected x tjrpe, was noted in the row at the foot of the 
columns : e.g. altogether 54 divisions were observed of the type 
stocking 10 to 15 thousand head of cattle. 

It was possible now to treat each column separately and to 
calculate the mean y^s associated with different types of x, namely 
^i> ^2j ^3j • • • > ^'id the frequencies so obtained were inserted in 
the bottom row of Table (29) : e.g. when x lies between 20 and 25 
thousand, the mean value of y is 50 thousand. The resulting 
points— (a?!, y^, (x^, y^, (x^, Vs) - - ■ in the notation of Chapter x. — 
are plotted together in fig. (25), and they are seen to lie approxi- 
mately in a straight line. The successive rows were treated in 
precisely the same way and the mean cc's calculated corresponding 
to 2/'s of different types, namely y^, y^, 2/3, • • • ? the frequencies 
obtained being recorded in the extreme right-hand column of 
Table (29) : e.g. when y lies between 45 and 50 thousand, the mean 
value of X is 19 thousand. The resulting points (x^, y^), (rcg, 2/2)? 
{Xq, 2/3), • • • , are also plotted in fig. (25), and, excepting for values 
which depend upon only one or two records, they too lie roughly 
in a straight line which is not far from coinciding with the previous 
one, so that we shall expect on calculation to get a high value for 
the coefficient of correlation. 

In order to calculate r we need first to find the mean and standard 
deviation for each variable. For this let us take as origin the 
point (12-5, 27-5). The essential details are shown immediately 
below the relative Tables (30) and (31). 



CORRELATION — EXAMPLES 



123 



Table (30). Distbibution of Petty Sessional DmsioNS ac- 

COEDING TO THE HeAD OF CaTTLE (EXPRESSED TO NEAREST 
1000) STOCKED. 

(1) (2) (3) (4) (6) 



No. of Cattle 


Devia- 


No. of Petty 


Product of 


Product of 


stocked (in 


tion from 


Sessional 


Nos. in 


No9. in 


thousands). 


12-5. 


Divisions. 


Cols. (2) & (3). 


Cols. (2) & (4). 




(x) 


i 


(-•■ 


(.n- 


0-5 


-2 


76 


-152 


304 


5-10 


-1 


97 


- 97 


97 


10-15 





54 


. . 


.. 


15-20 


+ 1 


24 


+ 24 


24 


20-25 


+ 2 


14 


+ 28 


56 


25-30 


+3 


5 


+ 15 


45 


30-35 


+4 


5 


+ 20 


80 


35-40 


+5 


1 


+ 5 


25 






276 


-157 


631 



27 6X5: 



Mean number of cattle=12-5 
units referred to 12-5 as origin ; 



=9-66, since x=—^ji class 

___ and a,=5V[in-ann 

= 5Vl-963=7-00. 

[The numbers in col. (4) may be spoken of as the first moments 
of the totals of x arrays and the numbers in col. (5) as the second 
moments.] 

In order to calculate easily the product deviation with reference 
to (12-5, 27-5) as origin, the value proper to each square was inserted 
just above the frequency and the product of the deviation by the 
frequency was inserted just below the frequency in different type of 
print to prevent confusion : e.g. the row (50-55) is +5 class intervals 
distant from the row (25-30) containing the origin, and the column 
(20-25) is +2 class intervals distant from the column (10-15) con- 
taining the origin ; hence, for the particular square defined by this 
row and this column, the product deviation=5x2=10 ; also 
the frequency recorded in this square =4, so that it supplies a 
term 10 X 4 to the product deviation ; the numbers 10, 4, and 40 
are therefore the numbers which appear in the square. It is neces- 
sary to be careful with the signs ; if the product deviation is to 
be positive, the separate deviations must be of like sign, both 
positive or both negative : hence they must either be both above 
or both below the numbers 12-5 and 27-5 respectively from which 



124 



STATISTICS 



they are measured. In this instance there are only two negative 
terms among the product deviations in the whole table. 

Table (31). Distribution of Petty Sessional Divisions ac- 
cording TO the Number of Acres of Land (expressed to 

NEAREST 1000) UNDER PERMANENT GrASS. 



(1) 


(2) 


(3) 


(4) 


(5) 


No. of Acres 

under Grass 

(in thousands). 


Deviation 
from 27-5. 


No. of Petty 
Sessional 
Divisions. 


Product of 

Nos. in 

Cols. (2) & (3). 


Product of 

Nos. in 

Cols. (2) & (4). 


0- 5 


iy) 

- 5 


15 


- 75 


375 


5-10 


- 4 


30 


-120 


480 


10-15 


- 3 


48 


-144 


432 


15-20 


- 2 


33 


- 66 


132 


20-25 


- 1 


30 


- 30 


30 


25-30 


. . 


26 


. , 


. . 


30-35 


+ 1 


31 


+ 31 


31 


35-40 


+ 2 


23 


+ 46 


92 


40-45 


+ 3 


8 


+ 24 


72 


45-50 


+ 4 


10 


+ 40 


160 


50-55 


+ 5 


9 


+ 45 


225 


55-60 


+ 6 


5 


+ 30 


180 • 


60-65 


+ 7 


1 


+ 7 


49 


65-70 


+ 8 


1 


+ 8 


64 


70-75 


+ 9 


1 


+ 9 


81 


75-80 


+ 10 


3 


+ 30 


300 


80-85 


+ 11 


2 


+ 22 


242 






276 


-143 


2945 



Mean number of acres =27-5 - 



iifx 5-24-91, since y=-\^l 



class units ; and a^=5V[W/-(4Tf)^]=5VlO-402= 16-12. 

[The numbers in col. (4) are the first moments of the totals of y 
arrays, and the numbers in col. (5) are the second moments.'] 

It is now a simple matter to sum the product deviation terms, 
taking each column (or each row) in turn : e.g. the first column 
gives 

150+216+180+12=558; 

the second column gives 

12+54+60+25-6-2-143, 
and so on ; and, summing these results together, we get 
558+143+76+126+96+160+30=1189. 



CORRELATION — EXAMPLES 125 

But this is the sum of all the product deviations referred to 
(12-5, 27-5) as origin. Transferring now to the mean, we have 

=¥A'-(-ifi)(-ifJ) 

=4-013, expressed in class units. 
Hence, ^=vl(^x^y^ 

where u^ and Oy are also to be expressed in class units, 

=4-013/V(l'963)V(10-402) 

=0-89, 

a result not far from unity, so that the correlation is high. 

The regression of ' acreage of grassland ' (Y) on ' head of cattle ' 
(X) is given by 

(Y-24-91)=r^(X-9-66) 

= (0-89)^i5^(X-9-66), 
(7-00) ' 

i.e. Y=205X+5-ll. 

The points representing the mean 2/'s for a;'s of different types 
should lie close to this line which is shown in fig. (25). This equation 
enables us to predict the acreage under permanent grass to be 
found on the average in petty sessional divisions with a given total 
head of cattle in each. The words ' on the average,' to be tacitly 
understood even if not stated in all such cases, are emphasised 
because the prediction relates to the whole array of divisions of a 
particular type, and as it only professes to give the mean or most 
likely result it is not to be pronounced worthless if it fails in an 
individual trial with a selected division. 

Again, the regression of X on Y is given by 

(X-9-66)=r^-^(Y-24-91) 

(Jy 

i.e.. X=0-39Y+005, 

which tells us the total head of cattle (X) to be found on the average 
in petty sessional divisions when the acreage under permanent 
grass (Y) is known. This line is also drawn in fig. (25). 

Example (5). — The data for this example are taken from an 
exceedingly interesting Government Report on the Cost of Living 
of the Working Classes {Report of an Inquiry by the Board of Trade 
into Working Class Reyits and Retail Prices, together with the Rates 



126 



STATISTICS 



of Wages in certain Occupations in Industrial Towns of the United 
Kingdom in 1912 in continuation of a similar Inquiry in 1905, 



70 



60 



50 



40 



30 



'20 



10 



i 





m. 



10 20 30 40 X 

Total Head of Cattle (expressed to nearest thousand) 
Fig. (25). 



Cd. 6955). Some further particulars concerning this Report will 
be found on p. 281. 



CORRELATION — EXAMPLES 127 

The towns included in the inquiry numbered 93, but in five 
instances it was found desirable to consider closely adjacent muni- 
cipalities as single towns thus reducing the number of town-units 
to 88, namely 72 in England, 10 in Scotland, and 6 in Ireland. In 
the example which foUows the three zones of London, middle, 
inner, and outer, have been treated as separate towns, so making 
the net number of town-units 90. This number is too small to 
allow any real value to be attached to our results, but the fewness 
of the observations makes them easier to deal with as an illustration 
of method. 

We begin as before by choosing ^convenient class intervals for 
the two factors we propose to consider, namely, Increment of Un- 
skilled Wages and Increment of Bents — by increment in each case 
is meant the percentage increase (+) or decrease (— ) between 
1905 and 1912 — and then form a correlation table. In the last 
example separate tables were drawn up to find means and S.D.'s, 
but that was only done in order to keep the argument clear at its 
first presentment : generally we may dispense with these additional 
tables and show all the worldng in one (see Table (32)). 

The increment of wages runs from (—2-5) per cent, to (+11-5) 
per cent., so that, if we take (—0-5) as origin and a difference of 
2 per cent, as unit, the classes run from (—1) to (+6), these numbers 
being shown in different type in the table, but in the same com- 
partments as the others. In the fourth row from the bottom 
are shown the total frequencies for x arrays from class (—1) to 
class (+6), and in the row just below it these several frequencies 
are shown multiplied by their corresponding deviations measured 
from (—0-5) as origin in terms of the class unit — the resulting 
numbers give the first moments of the totals of x arrays. These 
numbers, multiplied again by their corresponding deviations, give 
the second moments of the totals of x arrays, and appear in the 
last row but one of the table. 

We deal in exactly the same way with increment of rents : a 
percentage increment of (—1) is taken as origin from which devia- 
tions are measured, a difference of 3 per cent, is taken as unit, 
and the different classes then have deviations running from (—3) 
to (+6). The totals of y arrays, the first moments, and the 
second moments of these totals appear in the last three columns 
on the right-hand side of Table (32). 

To calculate the deviation products, numbers were inserted in 
each square on the same principle as in the last example, and the 
sums of these products for each x array, that is for each column, 



128 



STATISTICS 



are given in the bottom row of the table— 1, 0, 14, 6, etc., making 
in all a total of 126. 



Table (32). Correlation between Increment of Unskilled 
Wages and Increment of Rents in certain Industrial 
Towns of the United Kingdom. 





X 


= Percentage Increment of Wages 






























1st. 


2nd. 






-I 





fi 


+ 2, 


+ 3 


+ 4 


+5 


+ 6 


Totals 
of y 


mo- 
ments 


mo- 
ments 






-2-5 


-0-5 


1-5 


3-5 


5-5 


7-5 


9.5 


IV 5 


arrays 


ofy 


ofy 
























arrays 


arrays 




J9 


-3 


-10 




o 

1 














1 


-3 


9 












o 






















^ 






2 


o 






















Qc 


-2 


-7 


1 


3 














4 


-8 


16 




CO 






2 

























^ 








o 


-I 


-2 




-4 


-5 


-6 










O 


-I 


-4 




4 




2 




1 


1 


1 


10 


-10 


10 




Gl 








o 


-I 


-4 




-4 


-5 


-6 


















o 




o 


o 

















t: 




-1 




15 




6 


6 




2 




30 


- 


- 




i 








o 




o 


o 





















-I 


o 




2 




4 


5 












*K 


+ 1 


2 


1 


9 




3 




3 


1 




18 


18 


18 










-I 


o 




6' 




12 


5 












? 








o 




4 




8 


lO 












E: 


+ 2 


5 




6 




1 




1 


2 




11 


22 


44 




.N 








o 




4 




8 


20 












c^ 








o 


3 




9 


12 


IS 












c 


+ 3 


8 




3 


4 




1 


1 


2 




11 


33 


99 




03 








o 


12 




9 


12 


30 












&. 








o 






















^ 


+ 4 


11 




3 














3 


12 


48 




^ 








o 






















1 








o 






















4S 


14 




1 

o 














1 


5 


25 




Si 


+ 6 


17 












24 
1 
24 






1 


6 


36 




Totals of X arrays 


2 


45 


8 


12 


7 


7 


8 


1 


90 


75 


305 




1st. moments of 


-2 




8 


24 


21 


2B 


40 


6 


125 








X arrays 


























2nd. moments of 
X arrays 


2 


- 


8 


48 


63 


112 


200 


36 


469 






!, 


Product Sums of 


1 





14 


6 


9 


52 


50 


-6 


126 


Total Product Sum ' 






















' 



The necessary calculations are as follows : — 

1. Mean a:=-0-5+2(125)/90=2'28, 

(7^=2V[-VV--(W)']=2V(26585)/90. 

2. Mean 2/=-l + 3(75)/90-:l-50, 

^t/^SVftV— (U)2]=3V(21825)/90. 



^ 120 /12 6.\/7 5\ 



1965 
(90)2' 



expressed in class units» 



CORRELATION EXAMPLES 



129 



Hence 



r=pl(j^<iy 
1965 



(90)2 
:0-08. 



X 



90 



X 



90 



V(26585) V(21825) 



In substituting for Gg. and cry to find r we have omitted the factors 
2 and 3 respectively, because the S.D.'s have to be expressed in 
the same units as p. Alternatively, if we worked with a difference 
of 1 per cent, as unit, instead of taking a difference of 2 per cent, 
as xmit for x deviations, and a difference of 3 per cent, as unit for 
y deviations, each individual product of x and y deviations would 



Y 




''1T\ ' 










(2) 
























































*3 




















S r 










QC 5 










«K 










? 






1 














Sa 




















s 










V. 










o 










-^n 




















Oi 










2 










c- 










So 








w 


^ 2 








»••-"""■' 


.? 






— - — """" 


c"" 


Ci. 




_ » ■ "1 1 " 


" 




^ 




'"" M(2 


-28,1^5)- 














1 



















































2| 3 4 5 6 X 

Percentage Increment of Wages 
Fig. (26). 

have to be multiplied by 2 x 3. Thus p would then be 6 X 1965/(90)2, 
and we should get the same result for r as before by taking g^ 
and Gy as in (1) and (2) above. In this case r is so small as to be 
quite insignificant of any correlation between the two factors dis- 
cussed, and the regression lines should therefore be not far from 
perpendicular to one another. 

The regression of y on x, or the equation giving the most probable 
y for a given type a; is 

(2/-l-50)=r^(a;-2-28), 



I.e. 



y=0'Ux+l'26. 
I 



130 



STATISTICS 



Similarly, the regression oi x on y is 

x=0'06y+2'2. 
To draw the first line we note that it passes through the points 
(0, 1*25) and (5, 1*8) ; also the second line goes through the points 
(2-2, 0) and (2-5, 5). The two lines intersect at M (2-28, 1-5), the 
mean of the distribution. They are drawn together in fig. (26). 



Table (33). Correlation between Unskilled Wages 
AND Rents in certain Industrial Towns of the 
United Kingdom. 





X = Index Number for Wages of Unskilled Labour 






45-5 


51-5 


57-5 


63-5 


69-5 


75-5 


81-5 


875 


93-5 


99-5 


§. 
•^ 


40-5 


2 








3 






1 






"to 

1 


48-5 






2 


1 


3 


1 


4 


3 


1 






56-5 




1. 




1 




2 


7 


15 


6 


2 


1 


64-5 










2 


1 


3 


9 


4 


1 




72-5 








1 






3 


3 


2 




1 


80-5 












1 




1 




1 


^ 

V 


88-5 




















1 


1 


96-5 




















1 


104-5 






















:si 


112-5 




















1 



Example (6). — Instead of discussing the Changes in Wages and 
Rents between 1905 and 1912, it might be of interest to find the 
correlation between index numbers representing Actual Wages and 
Rents in October 1912, taken from the same Report. The necessary 
data for this purpose appear in Table (33) showing the distribution 
of frequency between the different classes : e.g. seven towns were 
observed in which the index number for wages was between the 
limits (79-84) and the index number for rents was between the 
limits (53-60). The wages figures quoted in Table (33) refer only 
to unskilled labour in the building trade ; the inquiry actually 
embraced certain occupations in the building, engineering, and 



CORRELATION — EXAMPLES 



131 



printing trades, these having been selected as industries which are 
found in most industrial towns, and in which the time rates of 
wages are largely standardised. 

Table (34). Correlation between Increment of Working 
Class Prices and Increment of Working Class Rents 
IN certain Industrial Towns of the United Kingdom. 







X = Percentage Increment of Prices 






7-5 


9-5 


115 


13-5 


15-5 


17-5 


19-5 




-10 










1 








-7 


1 




1 






2 






-4 


1 


2 


2 


2 


1 


2 






-1 


1 


4 


6 


10 


8 


1 






















i 


2 




1 


2 


5 


8 


1 


1 


1 


5 


2 




4 


2 


3 






1 


8 




1 


2 


3 


1 


4 




aj 


11 








1 


1 




* 1 




14 








1 










T7 








1 









The coefficient of correlation turns out to be 0-46, distinctly larger 
than in the previous case. Also the lines of regression are : — 
(1) y=0'4nx-\-2l. (2) a;=0-452/+56. 

Example (7). — The Report also furnishes data for evaluating the 
correlation between the Increment of Working Class Prices and 
Increment of Working Class Rents, again meaning by increment the 
percentage increase (+) or decrease (— ) between 1905 and 1912 
(see Table (34)). 

The correlation in this case is very small, being only 0-13. The 
regression equations are : — 

(1) y=0-22x-l-5, (2) x=0'01y-{-l3. 



PART II 
CHAPTEK XII 

INTRODUCTION TO PROBABILITY AND SAMPLING 

Sfppose we wish to know the average measurement of some organ 
or character, e.g. length of forearm or weight or anything similar, 
in a large population containing several thousand individuals. The 
mean obtained by actual measurement if it were practicable to 
carry it out on so large a scale, would evidently depend to some 
extent upon the sex, the race, the age, the social class, and so on, 
of the individuals selected, and we shall accordingly assume our 
population to be composed of individuals of the same race and sex, 
at about the same age, taken from the same class, etc. ; it would be 
impossible in practice no doubt to secure that all conditions should 
be identically the same for all the individuals observed, but the 
population may be as homogeneous as we care to make it in theory. 

Now suppose* that, instead of attempting to measure every single 
individual, a random sample of 1000 from among the population 
be taken and that the mean and variabiUty of the measurements 
for this sample be calculated, giving results m-^ and a^. With 
these may be compared mg and g^, the results of measuriug a second 
sample of 1000 individuals, m^ and o-g, the results of a third sample, 
and so on. It is extremely unlikely that the values obtained for 
the m's in this way will equal one another, neither will the o-'s 
be equal ; but, if we have succeeded at the beginning in avoiding 
aU 411-balanced influences when we tried to make the field of 
observation as homogeneous as possible, the resulting m's and cr's 
will only differ from the values of the mean and variability for the 
whole population, assuming they could be measured, within a 
comparatively small range. 

Differences of this kind, which arise merely owing to the fact 
that we are often obliged in practice, for lack of time or means, to 
deal with a comparatively small sample instead of with the whole 
population of which it forms a part, are said to be due to random 

132 



INTRODUCTION TO PROBABILITY AND SAMPLING 133 

sampling. Granted that the samples themselves are adequate in 
size (containing, say, from 500 to 1000 individuals each) an esti- 
mate of differences to be expected between one and another can be 
made, and unless the observed differences fall outside recognized 
Umits it is said that they are not significant of any difference other 
than such as might quite weU be accounted for by random sampling 
alone. 

In theory, then, we can imagine a large number of such random 
samples selected, and by determining the S.D. of their means, 
m^, mg, mg, . . . , we should have a fair measxire of the deviation 
which might quite well occur from the true value, that is, from the 
mean of the population as a whole, through working only with a 
sample. Further, a range of two or three times the S.D. on either 
side of the true mean ought to take in the majority of the sample 
means observed. 

Exactly the same principle holds good in dealing with the pro- 
portion of individuals in a given population which can be assigned 
to a particular class, or in discussing the S.D. of the distribution, or 
the C. of v., or a coefficient of correlation, or any other statistical 
constant, no matter what the nature of the character may be which 
is measured or observed, or whether it relates to animate or inani- 
mate objects. Take, for instance, the variabiUty — by selecting 
several samples from a given population we get a series of values 
a I, o-g, 0-3 . . ., and in the S.D. of this distribution of variabilities 
we have a measure to which we can compare the deviation of any 
sample variability, Gj., from the true variabiHty of the whole popu- 
lation, while a range two or three times the S.D. might be expected 
to include the majority of the different variabilities met with in 
the samples. 

Although the S.D., as we have explained, provides quite a suit- 
able measure of the extent of deviation of a sample constant from 
its true value in the population as a whole, in practice, owing to 
the historical development of the theory having followed the track 
of the normal curve of error [see Chapter xviii.] a measure known 
as the probable error and equal roughly to two-thirds of the S.D. 
is not seldom employed in its place. The main, if not the sole, 
purification for retaining this measure is that it has estabHshed its 
position by long usage, and in any case it is very easily deduced 
from the S.D. by the relation 

p.e.=0-6745 S.D., 

which follows at once from the normal curve and is only strictly 



134 STATISTICS '' 

justified when the distribution is normal (see p. 246). Let it suffice 
here that instead of simply using the S.D., as might now seem 
the obvious course, some writers prefer to multiply the S.D. by a 
certain fraction, in which there is no particular virtue except that 
which arises through honourable descent, and to work with the 
* probable error.' 

Since we do not know how much weight to assign to any result 
unless the magnitude of its p.e. is ailso given, results are frequently 
stated in the following manner : in a study of the Variation and 
Correlation in the Earthworm, by R. Pearl and W. N. Fuller [Bio- 
metrika, vol. iv. pp. 213-229] :-- 

Mean length of worm= 19-171 ±0-094 cms., 

S.D.=3-077±0-067 cms., 

C. of V.=16-049di0-356 per cent., 

meaning that the mean length of the worms measured was 19-171 
cms., subject to a probable error of 0-094 cms. which might be in 
excess or defect, in other words the mean length lay probably some- 
where between 

19-077 cms. and 19-265 cms. ; 

similar remarks apply to the variabiUty, absolute (S.D.) or relative 
(C. of v.). 

When the standard deviation (p.e./0-6745) is used as the measure 
of error due to simple sampling, the fact is generally recorded, and 
it is sometimes spoken of as the standard error in that connection, 
but, as it seems unnecessary to multiply names for ideas which are 
not really new, only that they appear in a new setting, we shall 
not employ the term. 

It must be clearly understood that no outstanding and predict- 
able cause exists, by our hypothesis, for such differences as occur 
in the statistical constants between one sample and another : they 
are the resultant effect of a complex of forces which cannot be 
properly traced, still less measured, apart from one another, and 
which have been happily described as that ' mass of floating causes 
generally known as chance.' Since therefore the forces coming 
into play, under the ideal conditions formulated, are of the same 
chance nature as those affecting the spin of a well-balanced coin 
or the selection of a card from a smooth and weU- shuffled pack, 
it may be expected that the resulting distribution of means, 
m^, mg, mg, . . . , of S.D.'s, cti, o-g, o-g, . . . , and of all the other 
constants will likewise be subject to the same laws of probabiUty 



iNTRODtJCTION TO PROBABILITY AND SAMPLING 135 

which serve to describe within limits what happens in the case of 
coin or card. It follows that some acquaintance with the first 
elements of mathematical probability is essential if one is to under- 
stand the theory of sampling, and a short digression must here 
be made in order to introduce that subject. This will be found 
to lead directly to a solution, under certain prescribed conditions, 
in the simple case when the character observed is an attribute like 
complexion, fair or dark, or like birth, male or female, which can 
only fall into one of two definite classes and when every one observa- 
tion in the sample is independent of every other. In the more 
general case where the character observed is capable of direct 
measurement and may lie in magnitude anywhere along a scale 
of values divided up into a number of different classes, it is not 
so easy to determine the effect of random sampling, because it is 
not possible, as it is in the previous case, actually to draw up a 
frequency table describing in detail the character of the distribu- 
tion to be expected from theory in any given sample. 

The idea contained in the word probability is one familiar to us 
in our everyday talk, but if we seek to analyse it as used we find 
it as elusive as the personality of the user. A remarks : * Wars 
will probably be stamped out, like duelling, in the course of time.' 
B repHes : ' No ! fighting will probably go on as long as the world 
lasts — you can't change human nature.' Now the amount of 
credence we are prepared to give to each of these statements is 
vague and uncertain until we know something about A and B 
themselves and the value of their judgment, quite apart from the 
influence of our own opinion upon the matter ; perhaps A is an 
optimist or B is a pessimist, and in estimating the ' probably ' 
used by each we must allow for these facts. ProbabiHty, then, in 
ordinary conversation, is something largely subjective : it has a 
varying significance according to the person who uses the word 
and, unless we could get rid of this personal element, it would be 
hopeless to try and approach it along scientific lines. 

Mathematical probability is unlike colloquial probability in that 
all the uncertainty is taken out of it, or at least the uncertainty is 
confined within defined limits. We shall only touch the fringe of 
the subject in this book, and what we have to say may be best 
introduced by considering some examples which may appear trivial, 
but they possess the merit that no personal bias can enter into 
their discussion to distort the results. The reader must not be 
impatient at their artificial character : in many, if not in all, 
branches of science, before tackling any particular problem as it 



1S6 Statistics 

actually exists, it is helpful to examine what can be deduced in a 
simple case free from all complication, and, having settled that, 
we try to see how the results are affected when we come to allow 
one by one for the various compHcating factors which exist. For 
example, in Astronomy, the track of a planet in space may first be 
found on the hypothesis that the sun alone is the compelling influence. 
Then we may proceed to discuss how it is deflected from its path 
when the gravitational influence of neighbouring planets also is 
taken into account. 

Let us start with an ordinary pack of playing cards, and, after 
shufifling, turn up one card. Can we measure the probability that 
this card shall be (1) the 7 of spades ? (2) some spade ? 

Altogether there are 52 cards, and we will suppose that the 
cards are so cut and so smooth that each of the 52 has an equal 
chance of being turned up : for instance, there is to be no sticki- 
ness or anything to help any particular card to evade us by sticking 
fast to its neighbour. Now we are certain to turn up some card 
and there are 52 different possibilities, each of them by hypothesis 
equally probable. If, then, we agree to denote certainty by unity, 
we must divide 1 into 52 equal parts and assign one part to each 
card as the probabiHty of its appearance. 

1. The probability (or chance as it is sometimes caUed) of turning 
up any stated card, such as the 7 of spades, is therefore 1 out of 52, 
i.e. 1/52. 

2. Again, since there are 13 spades in all, the chance of turning 
up some spade is 13 out of 52, i.e. 13/52=1/4. 

These results may be put in another way which is often useful. 
If the experiment is repeated a great number of times, a return to 
the initial conditions of the problem being made after each trial 
by replacing the card drawn and reshuffling the pack, we should 
expect to turn up the 7 of spades on the average about once in 
every 52 experiments, and we should expect to turn up some spade 
on the average about once in every 4 experiments. This must 
not be taken to mean that in 4 experiments we are sure to turn 
up just one spade — a trial wiU readily prove such a statement to 
be untrue — but that, if we went on performing experiment after 
experiment, we should in the long run get a proportion of about 
1 spade to every 4 experiments and a trial will likewise prove the 
truth of this statement. 

GeneraUy, when an event can happen in n different ways alto- 
gether, and among these different ways there are a which .give 
what might be caUed successful events, the probability of success 



INTRODUCTION TO PROBABILITY AND SAMPLING 137 

at any single happening is a out oin^i.e. ajn, and is usually denoted 
by the letter p, and the probability of failure is (n—a) out of n, 
i.e. {n—a)ln, and is usually denoted by the letter q. 

Clearly {p-\-q)=l, and this is reasonable because we are certain 
to get either a success or a failure at a single trial and unity was 
fixed as the measure of certainty. In k trials, the probable number 
of successes would be kp and of failures kq, because in n trials, on 
the average, there are a, or np, successes and (n—a), or nq, failures. 

Example (1). — In the second case considered above, the pro- 
bability of success (turning up a spade) is a out of n 

=a/7i= 13/52- 1/4=^, 

and the probabiUty of failure (not turning up a spade, i.e. turning 
up one of 39 other cards) is (n—a) out of n 

= (n-a)ln=39l52=3l4:=q. 
And (^+g)=l/4+3/4=l. 

Example (2). — What is the chance of drawing either a picture 
card or an ace from the pack at a single trial ? 

Altogether there are 12 picture cards, and the chance of drawing 
any one of them is thus 12 out of 52 

= 12/52=3/13; 

and the chance of drawing any one of the 4 aces is 4 out of 52 

=4/52=1/13. 

Hence the total probability required 

=3/13+1/13=4/13. 

Generally, if the probability of one type of event is p^, and the 
probability of a second t3rpe of event is ^2» ^^^ if either type is 
reckoned a success, then the total probabiUty of success is (Pi+Pz)- 
This evidently holds good however many different types there 
may be, and even if there is only one event of each type. 

Consider now the simultaneous happening of two events, one of 
which can happen in n different ways, a among which are to be 
regarded as successful, and the second can happen in n' different 
ways, a' among which are to be regarded as successful. Further, 
the two events are to be absolutely independent of one another 
in the sense that neither is to influence the success or failure of 
the other. What is the probability of a double success occurring ? 

The total number of different combinations of the two events 



138 STATISTICS 

possible is nn' , for any one of the n possible happenings for the 
first event can be combined with any one of the n' possible happen- 
ings for the second event. Also the total number of different 
combinations of two successes possible is aa\ for any one of the 
a possible successes for the first event can be combined with any 
one of the a' possible successes for the second event. Hence, 
according to our definition of probability, the probabiUty of a double 
success is aa' out of rin' =aa' jnn' ={aln){a' jn'). 

Thus to get the probabiUty of a double success for a combination 
of two independent events we must multiply together the separate 
probabilities for the success of each event taken by itself. 

Similarly, in the above catee, the probability of a double failure 
= (n—a)(n' —a')lnn' ; and the probability of one success and one 
failure 

_a n'—a'n—a a' • 

— . - -\- . — - 

n n n n 

for the first event can be a success and the second a failure or the 
first a failure and the second a success. 

Here, again, if we take all the different possibilities into account, 
and add the probabilities corresponding to each case, we arrive 
at certainty, the measure of which is unity, thus : — 

probability of 2 successes =aa'lnn\ 

„ 1 success and 1 ia,iluTe=a{n'—a')lnn'-}-a'{n—a)lnn' 

„ 2 failures ={n—a)(n'—a')lnn\ 

Therefore total probability, all cases, 

_aa' a(n'—a') , a'(n—a) {n—a)(n'—a') 

,-T — T- -, + ; — 

nn nn nn nn 

= {aa' -\-an' —aa' -\-a'n—a'a-\-nn' —na' —an' -{-aa')lnn' 
=nn'lnn' 
= 1. 

Example. — Take two packs of cards. What is the probability 
of drawing an ace from the first pack and a king, queen, or knave 
from the second pack ? 

Here a=4, n=62, a' =12, n'=52 ; hence the required probability 

=aa7^7i'=4/52x 12/52-3/169= l/56i. 

Thus we might expect to succeed on the average about once in 
56 trials. 



INTRODUCTION TO PROBABILITY AND SAMPLING 139 

We proceed to discuss the case of a coin spun a number of times 
in succession, and we shall find the probabilities of the appearance 
of so many heads (H) and so many tails (T) in so many spins on the 
hypothesis that the coin is perfectly balanced and equally likely 
to fall on either side. 

In 1 spin there are 2 possible events, namely H or T, which 
we shall write simply as 

(H, T). 

In 2 spins there are 4 possible events, because we can combine 
the H or T of the first with an H or T at the second spin, and we 
may express the result thus 

(H, T)(H, T)=(HH, HT, TH, TT) ; 

the interpretation of which is that we may get either head followed 
by head, or head followed by tail, or tail followed by head, or tail 
followed by tail. 

In 3 spins there are 8 possible events, because we can combine 
the 4 events previously possible with an H or T at the third spin, 
thus getting 

(H, T)(H, T)(H, T) 

= (H, T)(HH, HT, TH, TT) 

= (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT) ; 

the interpretation of which is that we may get either 3 heads in 
succession, or 2 heads followed by 1 tail, or head followed by tail 
followed by head, and so on. 

In 4 spins there are 16 possible events, because we can combine 
the 8 events previously possible with an H or T at the fourth spia, 
thus 

(H, T)(HHH, HHT, HTH, HTT, THH, THT, TTH, TTT) 
= (HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, 
HTTH, HTTT, THHH, THHT, THTH, THTT, 
TTHH, TTHT, TTTH, TTTT). 

But the method here adopted to get the possible events at each 
stage is precisely the same as that which gives the successive terms 
in the ordinary algebraical expansions of 

(H+T), (H+T)(H+T), (H+T)(H+T)(HH-T), etc. 
Also each new spin has the effect of doubling the number of possible 



140 STATISTICS 

events obtained at the previous spin, and we conclude that in 
n spins, there are 

(2 X 2 X 2 X . . . to ^ factors), • 
or 2"*, possible events, and these events are given by the successive 
terms in the expansion of 

[(H+T)(H+T)(H+T) ... to 7^ factors.] 

Let us now consider the probabilities of the different events 
obtainable. The important point to notice is that at any stage 
each possible event has exactly the same probability, for there is 
no reason why any particular spin should give H rather than T, 
or T rather than H : for example, in 3 spins there are 8 possible 
events, each by itself equally probable, and we therefore divide 
the unity of certainty into 8 equal parts and assign one part to each 
event, thus 

probability of 3 heads— HHH=J 
probability of 2 heads and 1 tail— HHT=Jl 

HTH=i I 
THH^iJ 
probability of 1 head and 2 tails— HTT=Jj 

THT=iU 
TTH^iJ 
probability of 3 taUs-TTT=J. 

It is clear from this arrangement that, if the order of the appear- 
ance of H and T is indifferent, some events are of the same type 
and some types are likely to appear oftener than others, e.g. the 
probability of getting ' 2 heads and 1 tail ' (or ' 1 head and 2 tails ') 
is three times as great as the probability of getting ' 3 heads ' 
or ' 3 tails.' Hence for conciseness it is convenient to adopt the 
ordinary index notation and write 

HHH=H3, HHT=H2T, HTH-H^T, etc., 
so that the possible events in 3 spins are 

H3, 3H2T, 3HT2, T^ ; 
in 4 spins they are 

H*, 4H3T, 6H2T2, 4HT3, T^ ; 
and so on. 

The probability of any particular type is now readily written 
down : e.g. in 4: spins, the probability of getting 2 heads and 2 tails 

= (number of successful events possible)/(total number of events 

possible) 
=6/2*=6/16=i. 



INTRODUCTION TO PROBABILITY AND SAMPLING Ul 

But the binomial expansion always sums together terms of the 
same type for us in just the manner wanted, and we have the 
possible events in n spins given by the successive terms in the 
expansion of 

(H+T)(H+T)(H+T) ... to n factors, 
i.e. (H+T)«, 

i.e. H«+"Ci . H'»-iTi+«C2H"-2T2+ . . . +T", 

and therefore again the probability of any particular combination 
is readily written down : e.g. probability of ' (n—2) heads, 2 tails ' 
= (number of successful events possible)/(total number of events 
possible) 

Another way of stating the result obtained is to say that we 
might expect to get 

n heads appearing on the average about once in every 2** trials, 
(n—1) heads, 1 tail ,, ,, ,, "0^ times ,, „ 

(?i— 2) heads, 2 tails ,, „ ,, ^^Cg times ,, „ 

and so on. 

If, in accord with our previous notation, we call the appearance 
of, say, H at any spin a ' success,' and label its probability J by the 
letter ^, and if consequently the appearance of T at any spin is a 
* failure,' its probability, J, to be labelled by the letter q, we have the 
probabilities of the different combinations of events in (H+T)", or 

H«+«CiH«-iTi+«C2H«-2T2+ . . . +T«, 
given by the corresponding terms in {jp-\-qY, or 

where p=gr=:|. 

After each spin of the coin in the case considered the distribution 
of probabilities was symmetrical, e.g. after the fourth spin the pro- 
babilities were 

14 6 4 1 

T^J TFJ T^» T^' T^- 

We pass on now to a case where the distribution is not symmetrical, 
owing to the fact that p and q are no longer equal for any isolated 
event. 

Consider the throw of an ordinary die in which each of the six 
faces is assumed to have an equal chance of appearing uppermost. 
The probability of throwing, say, a 3 is 1/6, since we are certain 
to throw either 1, 2, 3, 4, 5, or 6 ; and the probability of failing to 
throw a 3 is 5/6, since we are certain either to throw a 3 or not 
to throw a 3. 



142 



STATISTICS 



If we represent the probability of success (say, in this case, 
throwing a 3) by ^ {i.e. 1/6), and failure {i.e. in this case, failing 
to throw a 3) by g {i.e. 5/6), we have 

iJ+g= 1/6+5/6= I. 
Bearing in mind then that the probability for a combination of two 
independent events is determined by multiplying together the 
separate probabHities for each, we have the following table showing 
what might be expected when 1, 2, or 3 dice are thrown up together, 
where 5 stands for success and / for failure : — 



No. of 

Dice 

thrown. 


Different 
Possibilities. 


Different 
Probabilities. 


1 
2 

3 


ss, sf. 

p,ff- 

sss, ssf, sfs, sff, 
fss,fsf,ffs,fff. 


QP, qq- 
pppy ppq^ pqp, pqq, 

qpp, qpg, qqp, qqq. 



The table is easily extended on the same principle, and at each 
step, it will be noticed, a fresh pair of possibiUties, s or/, is intro- 
duced, with corresponding p or q, to be combined with what has 
gone before. 

If the order of appearance of s and / is a matter of indifference, 
e.g. if it does not matter whether the first die shows s and the 
second /, or vice versa, so that results of the type sff and fsf may 
be regarded as equivalent, we may use the index notation, as in 
the coin case, to render the table more concise, thus : — 



No. of 

Dice 

thrown. 


Different 
Possibilities. 


Corresponding 
Probabilities. 


1 

2 
3 


s,f. 

s\2sf,p. 

^,3s%3sf^,f^ 


p>q- 

P\ 2pq, q'- 
p\ 3p% 3pq\ q\ 



When, therefore, n dice are thrown we again recognize the 
different possibiUties as given by the successive terms in the ex- 
pansion of (-5+/)", namely 

5«4_nC^5«-l/l+'^C2.S«-2/2+ . , . +/«^ 

and the corresponding probabiUties by the successive terms in the 
expansion of (33+g')", namely 



INTRODUCTION TO PROBABILITY AND SAMPLING 143 



Hence the probability of throwing n threes =j9"= 1/6" ; 



(n-l) 



(n-2) 



_ 1 5 
* 6"-i * (5 
=5n/6" ; 

n(n—\) 



62 



1-2 

■25n(n- 



l)/2 . 6« ; 



and so on. 

The result we have just obtained is of perfectly general appUca- 
tion. Whether we spin n coins, in which the probabihty, p, of 
success (say ' heads ') for each is 1/2, or throw n dice, in which the 
probability, p, of success (say ' to get a 3 ') for each is 1/6, or have 
any n similar but independent events happening in which the 
probability of success for each is p, the different resulting possi- 
bihties as to success are given by the successive terms in the expan- 
sion of («+/)", and their corresponding probabilities are given by 
the successive terms in the expansion of (p-\-q)^. 

We are thus in a position to form a frequency table, like that on 
p. 53, showiQg the probabilities of getting 0, 1, 2 ... ti successes 
(in other words, the proportional frequencies of these different 
numbers of successes) at the occurrence of n similar independent 
events, where p is the probability of success for each and q is the 
probability of failure : — 

Table (35). Binomial Distribution. 

(1) (2) (3) (4) 



Number of 
Successes. 



Frequency. 



{X) 


1 



(/) 



n{n-l) n-2^ 
1-2 ^ ^ 

^(^-l)(^-2)^-3p3 



1-2-3 



Product of Nos. in 
Cols. (1) & (2). 



(^) 


7i(?i-l)3"-V 
i{n-l){n-2)^_,^. 



1-2 



?ip" 



np 



Product of Nos. in 
Cols. (1) & (3). 





nq^~^p^ 



»-2*»2 



271(71- 1)2«-V 

Mn-l){n-2) , . 
U2 ^ ^ 



TlV 



np[l+p(n-l)] 



144 STATISTICS I 

Col. (1) gives the deviations from the origin of measurement, I 
which in this case is taken as ' no successes,' the class interval j 
being equal to a difference of 1 in the number of successes. \ 

The summations of the last three columns are effected as ^ 
follows : — 

Col. (2). grn_|_^^n-lpl_|_^H^^:ZO^n-2p2_|_ , ^ , _|_^n \ 

1*2 ' 

'( 

because jp+g=l. \ 

i 

CoZ. (3). \ 



fiifi lU^i 2) 

1 * Zi 



■up 



gn-l^(^_l)^n-2^1_|_(^ 1)(^ ^^g^-3j92-f . . . +^«-l1 



= ^29. 

Coi. (4). ' ;, 

wg"-V+2TO(ro- 1)^"-^+ ^"^""^ ^>( -"ZJ-V-3j)3+ . . . +to2^« ,j 

=^np[l + (n-l)p(q+p)n-^-\ 

= 7l^[lH-^(7l— 1)]. I 

The arithmetic mean of the distribution | 

=sum of terms in col. (3)/sum of terms in col. (2) \ 

=Z(fx)IS(f) \ 

=np. : 



INTRODUCTION TO PROBABILITY AND SAMPLING 145 

The mean-square deviation referred to zero as origin, zero in this 
case corresponding to ' no successes ' 

=sum of terms in col. (4)/sum of terms in col. (2) 

=Z{fx^)IZ{f) 

=njp[\+'p(n—\)']. 

Thus the standard deviation, a, is given by 

(j^=njp\\-\-p{n—\)]—x^, 

where x is the deviation of the mean from the origin of measure- 
ment, so that x=np. 

Therefore G^=np[l-\-p{n—l)]—n^p^ 

='np{l—p)-{- n^p^— n^p^ 

=npq. 
Hence cr= \/(npq), 

and p.e.= 0*6745 V'(npq). 

These two results are exceedingly important, and it is essential 
to understand what it is they measure. An example may help 
to make this clear. 

If we spin 300 coins, counting * head ' for each a success, the 
number of heads we shall get will be unlikely to differ very greatly 
from the average or mean number of successes, np, i.e. 150 if p=ll2 
for each coin, and in the long run, if we repeat the experiment a 
great number of times, we shall get a proportion of about 150 heads 
to every one experiment. Again, if we throw 300 dice, counting 
every throw of the number 5, say, for each die a success, so that 
p in this case =1/6, the number of fives we shall get will be unlikely 
to differ much from np, i.e. 50, and in the long run, if we repeat the 
experiment a great number of times, we shall get on the average 
a proportion of about 50 fives to every experiment ; we should 
find, for example, something like 5000 fives if we threw 300 dice 
IPO times in succession. The arithmetic mean of the distribution 
tells us therefore about what number of successes to expect in one 
experiment with n events if n is fairly large, though we should be 
unlikely to get exactly this number if we confined ourselves to the 
one experiment. 

The second result, the S.D., supplies us with a measure of the 
unlikelihood of getting the exact number of successes expected at 
any single experiment, for it defines the dispersion of the different 
numbers of possible successes about their average. Clearly the 
greater the dispersion, the greater is the likeUhood of missing the 

K 



146 STATISTICS 

average. The mean number of successes when an experiment is 
repeated a great number of times is n^, but at any single experi- 
ment it is not unlikely that the number of successes obtained may 
differ from np by as much as 0-6745 '\/(njpq) in excess or in defect ; 
it is, however, unlikely, as we shall see later (p. 244), that the 
number will differ from np by more than ^^/{npq) in excess or 
defect when the distribution is not very skew, or unsymmetrical, 
especially if n be large. The probable error in the case above when 
we throw a sample of 300 dice is 

=0-6745 V(300 X 1/6 x 5/6)=0-6745 V(41-67)=:4-4, 

and it is therefore quite likely that the number of fives obtained 
at one experiment will differ from the expected number, 50, by as 
much as 4 or 5 in excess or defect, but it is unlikely that the number 
will fall outside the limits 50±3V(41-67), say 30 to 70. 

It is sometimes more convenient to refer to the proportion of 
successes, etc., expected at any experiment rather than to the 
actual number expected. In that case, since with n events the 
expected number of successes is pn, but the number obtained may 
quite likely differ from this by ■±:Q^&14:^^/{npq), therefore with 
n events the expected proportion of successes is pnjn, i.e. p, with 
quite possibly an error =i 0-6745 y'(n25g)/7i, i.e. i 0-6745 v^(2)g'/^). 

Thus, with the 300 dice, the expected proportion of successes at 
one experiment lies between 

[1/6-0-6745 V(l/6x5/6-^- 300)] and [1/6+0-6745^(1/6x5/6-^300)] 

i.e. (1/6-0-6745/46-5) and (1/6+0-6745/46-5) 

i.e. 1/5-5 and 1/6-6 ; 

and it is unlikely that the proportion will differ from 1/6 by more 
than 3/46-5, i.e. 1/15-5. 

To illustrate how the binomial distribution might be directly 
applied, an experiment was made with 900 digits selected at random 
by taking in succession the digits in the seventh decimal place in 
the logarithms of the following numbers : — 

10054, 10154, 10254, . . . 99954, 

as given in Chambers's Mathematical Tables. In this way each of 
the 10 digits, 0, 1, 2, 3 ... 9, may be supposed to have stood an 
equal chance of selection each time one was written down. Gaps 
of 100 were left between the numbers selected so as to avoid runs 



INTRODUCTION TO PROBABILITY AND SAMPLING 147 

of the same figure which sometimes occm- even in the seventh 
decimal place owing to lack of independence. 

The digits were arranged in 36 columns, each column containing 
25 digits, and in this way we obtained what was equivalent to 
36 separate but like experiments with 25 events each. If we agree 
to regard the appearance of a 7 or an 8 as a successful event, and 
the appearance of any other digit as a failure, the chance of success 
at any appearance is 2/10, and the chance of failure is 8/10. The 
case is thus of exactly the same kind as that of throwing 25 dice 
36 times in succession, and if the probability of success, namely 1/5, 
for each independent event, be denoted by ^, and the probability 
of failure, namely 4/5, by q, the distribution of successes and failures 
should approximately conform to that given by the expansion of 

for any particular experiment, and since the experiment was re- 
peated 36 times, the total numbers of successes and failures of 
different orders obtained should approximately conform to 

m(p+qY\ 

for if the probability of an event is jp the number of events to be 
expected in N trials is Np. 

The actual distribution observed is compared with that given 
by the binomial expansion in Table (36). Col. (2) is obtained by 
picking out the appropriate terms in the expansion of 36Q3+g)25, 
where p=l/5, g=4/5 ; this expansion is 

/OK OK . OA \ 

36U'^+^.i)2Y+-^P^¥+ . . • +g'H 

Thus, 5 successes occur 

•^ 25 • 24 ... 6 5 20 
1 • 2 • 3 . . . 20^ "^ 

times, and this equals 7-06, or approximately 7. 

The mean number of successes by theory=rip=25/5=5. The 
mean by trial, since it is measured from zero as origin, the numbers 
in col. (1) being the deviations, 

=i;(/x)/2'(/)= 162/36=4-5. 
The standard deviation by theory 

= V(^M)=v'(25xix|)=2. 



m^ 



148 



STATISTICS 



Table (36). Distribution of Successes (getting a 7 or 8) m 
THE Random Choice of 25 digits 36 times in succession. 

(1) (2) (3) (4) (6) 



No. of 

Successes. 


Frequency 
Calculation. 


Frequency 
Experiment. 


Product of 

Nos. in 
Cols.(l)&(3). 


Product of 

Nos. in 

Cols.(l)&(4). 


{X) 

1 


1 


(/) 

1 


1 


1 


2 


3 


5 


10 


20 


3 


5 


5 


15 


45 


4 


7 


7 


28 


112 


5 


7 


9 


45 


225 


6 


6 


4 


24 


144 


7 


4 


3 


21 


147 


8 


2 











9 


1 


2 


18 


162 




36 


36 


162 


856 



By trial, the mean square deviation, measured from zero as origin 

=856/36. 
Thus the S.D. by trial= VftV—^'), 
where x is the deviation of the mean from the origin, 

= V[856/36- (4-5)2 
= 1-88. 

It wiU be seen that not one of the 36 experiments gave a number 
of successes differing from 5, the theoretical mean, by more than 
twice the S.D., for the number ranges only between 1 and 9. 

If we treat the 900 digits as 900 separate experiments with one 
event each, instead of treating them as 36 experiments containing 
25 events each, we have 1/10 as the chance for the appearance of 
any particular digit, and hence the number of times any digit may 
be expected to appear 

=^i'ibfV(^PQ')> approximately 
= (900)TVd=IV(900XTVxT'^) 
=90±6. 
The actual number of occurrences of each digit was as follows : — 



Digit .... 
No. of Occurrences 



95 


1 
96 


2 
93 


3 
105 


4 
91 


5 
80 


6 

82 


7 
72 


8 
90 


9 
96 



INTRODUCTION TO PROBABILITY AND SAMPLING 149 

so that the digit 7 showed the greatest divergence from 90 of any, 
and this was only just three times the probable error. 

[The Theory of Probability is older than that of Statistics. Todhunter, in 
his History, states that ' writers on the subject have shown a justifiable pride 
in connecting its true origin with the great name of Pascal.' The well-known 
story of the latter being found, as a lad of twelve, tracing out on the hall floor 
geometrical propositions which he had evolved in his own head is not to be 
wondered at, nor yet that at sixteen he wrote a small work on Conic Sections, 
when one reflects upon the fame he was to win as a philosopher and writer, 
as well as a mathematician, in his too brief life of thirty-nine years. He was 
born in 1623 of a distinguished French family, and for the last half of his 
life he suffered from the effects of a serious disease which contributed to turn 
his attention from mathematics to religion and philosophy. 

We learn from Todhunter how a certain gentleman of repute at the gaming 
tables set Pascal pondering on a question of probability concerning the fair 
division of stakes between two players who give up their game before its con- 
clusion — an old problem cited in a work by Luca Pacioli as early as 1494. A 
correspondence followed between him and Fermat, then probably the two most 
distinguished mathematicians in Europe, and so began a science which has 
fascinated at one time or another all great mathematicians from that day to 
this. 

The illustrious family of the BemouUis, friends of Leibnitz, who championed 
his claim against that made by English mathematicians on behalf of Newton 
to the invention of the Calculus ; De Moivre, an exile in England, owing to 
the revocation of the Edict of Nantes ; Euler, Lagrange, and Laplace, who 
worked out in algebraical form Newton's theory of gravitation for the motion 
of the planets — all these had a share in building up the science of Probabihty, 
often by investigating problems in games of chance, where the conditions can 
be made mathematically perfect, so by careful analysis preparing the way for 
the use later of the same principles in matters of greater importance. 

It has been said that the development of the subject owes more to Laplace 
(1749-1827) than to any other mathematician ; nor did he confine himself to 
its theory : he would have earned fame by his astronomical apphcations alone. 
His method was to take certain observations, and to determine by means of 
probability whether the abnormalities present were merely the results of chance 
or whether there was some as yet undiscovered but constantly acting cause 
behind the phenomena observed. In this way he was led to highly interesting 
and important results such as those relating to the theory of the tides, the 
effect of the spheroidal shape of the earth on the motion of the moon, the 
irregularities of Jupiter and Saturn, and the laws which govern the motion 
of Jupiter's moons. It needs but a step in thought to pass from the dis- 
cussion of such physical data to the statistics of social phenomena and the 
causes which determine abnormalities met with in that field. Professor Edge- 
worth, in making reference to books that have been written on Probability at 
the end of his excellent article under that heading in the Encycl&p^ia 
Britannica, remarks that ' as a comprehensive and masterly treatment of 
the subject as a whole, in its philosophical as well as mathematical character, 
there is nothing similar or second to Laplace's TMorie analytique des 
probabilites.'] 



CHAPTER XIII 
SAMPLING {continued) — formula for probable errors 



GENERAL POPULATION. 



So far we have only considered the most simple case of random 
sampling when we take a sample of n independent events each of 
which falls into one of two classes according to its natm*e, the 
chance of entering either class being the same for every event : 
we have dealt, that is to say, more particularly with non-measurable 

characters. We pass on now to measur- 
able characters which are distributed 
among several classes according to their 
size, so that a frequency distribution 
table can be set up for each sample ; and 
assuming that the population from which 
the samples are drawn is homogeneous, 
the samples themselves containing each 
an adequate number of individuals, there 
should not be greater differences between 
one table and another than can be ac- 
counted for by random sampling. It is 
our object to discover how great such 
differences may be. 

Given a homogeneous population of N 
individuals which we will suppose could 
be distributed into a number of groups, 
Yi individuals in the first group, Yg in the 
second group, Yg in the third, and so 
on, according to the size of the organ or 
character under observation. Suppose a 
random sample of n individuals be taken 
from this population, and when they are 
assigned to their several groups let the 
frequency table now take the form shown, 
with 2/1 individuals in the first group, y^ 
in the second, and so on. To find the "probable error of ?/&, the 
frequency observed in the kth group. 

160 



Class. 


Frequency. 


1st Group 
2nd Group 

Tcth. Group 






N 



SAMPLE. 



Class. 


Frequencj-. 


1st Group 
2nd Group 

Tcth. Group 


2/2 




n 



SAMPLING — FORMULA FOR PROBABLE ERRORS 151 



Consider the selection of the n individuals, one by one in succession, 
to form the sample. When the first choice is made the probability 
that we shall get an individual falling into the Jcth. group is, by defini- 
tion, Yj./N, and the probabiHty will remain practically the same for 
each successive choice granted that N is considerable. We have thus 
n independent events, the chance of success (falling into the kth. 
group) for each being ^(=Yj./N) and the chance of failure being 



/=!-: 



_ ^ The case is therefore analogous to the one pre- 

viously considered to which the binomial distribution is applic- 
able, so that the frequency to be expected in the kth group is np 

i.e. y]c=np with a p.e.=0'Q14:5Vnpq. 

Yg, Yg . . . would not be known, 
and hence the true value of p would also be unknown, but since 
yjg=np, approximately, when the sample is of adequate size, we 
shall get a fair idea of the probable error involved by taking 
p=yjcln, where 2/a; is the actual frequency observed in the Jcth group. 

y>, (1) 



with S.D., Gy =Vnpq 

Now in practice the numbers Y^, j- 2? ^ 3 



Hence, o^y^=npq=yj,(l—p)=yJl— 



and the frequency in the A;th group 



yj,±0-6745 



M^' 



yu\ 



(2) 



/. 



The size of the S.D. is under ordinary conditions a test of the 
-adequacy of the sample, for the frequency in the kth group, if due 



yvA simply to random sampling, 
/i J^ should not differ from its 
{ V expected value by more than 



(Z/±3cT, 



and a, J should therefore j 



A 



be small compared with ?/& 
itself. 

To find the correlation between 
the frequencies in any two 
groups of a sample distribution. 

Let the expected frequencies 
in the various groups of the 
sample be denoted by y^, y^, 
. . .^ 2/fcj • • •» ^^d suppose an 
error 82/ & "^ Vk is associated 
with errors Sy^, Sy^, . . • , S?/,, . • • in y^, y^, 
require then the correlation between yj^ and y^. 



Class. 


Expected 
Frequency. 


Observed 
Frequency. 


1st Group 
2nd Group 

kth Group 
5th Group 


yi 

y^ 

yk 
y» 


y^+^yz 

yk+^yk 

y,-^^y, 
i * * 




n 


n 



y., 



We 



152 



STATISTICS 



Now although the group frequencies may change relative to one 
another, the total sum of frequencies in all groups is not affected, 
because the n individuals of the sample make up its composition in 
each case : to keep n constant the group frequencies must adjust 
themselves accordingly, which explains the correlation between 
them. Hence to compensate for an excess, Si/^ (assuming hy^-\-'"^), 
of frequency in any one group there must be a defect {—Syj.) shared 
among the other groups, and the fairest way of sharing will be in 
proportion to the expected frequencies in the several groups. 

But the total frequency divided between groups other than the 
A;th is (w— 2/fc)» so that the proportion of {—Syjc) due to the 5th group 
is VsKn—Vk), thus 



S2/.= -^^(-82/,). 



n-Vic 



Therefore, 



^Vk'^Vs^ 



'Vk 
n 



Vs ^y^ 



Vki 1- 



Vs §2/^ 



n a' 



•Vk 



. (3) 



by (1). 



FIRST SAMPLE. 



Size of Organ 

or Character 

observed. 


Frequency of 
Observations. 


First Moment. 


Second Moment. 


X2 


2/2 

Vk 


XkVk 


Ay2 
x\yk 




n 


1{xyy 


2ix^y) 



This gives the product moment of the deviations from yj^ and yg 
in one particular sample ; summing for all such samples, remem- 
bering that by definitign the coefficient of correlation between ?/^ 



SAMPLING FORMULAE FOR PROBABLE ERRORS 153 

and 2/s is ry^y^=I!{Syji. • ^ys)lv(JyGy^, where v is the total number 
of samples, also cr^^ ^ZSy^j^lu, we have 



Therefore, 



r =-l,Ml. 



" ^^.^y. 



(4) 



gives the correlation required. 

To find the p.e. of the mean of a sample of n observations. Let a 
frequency table be drawn up in the usual manner showing the 
number of observations y^, y^ . . . corresponding to organs of 
different sizes x^, x^ . . . 

The mean referred to some fixed point as origin is then given by 

also the mean square deviation of the sample referred to the same 
fixed point is /^^2' ^^7^ given by 



and 



m22_M2=(t2 



where a is the S.D. of the sample. 

For another sample of the same size the frequency distribution 



SECOND SAMPLE. 



Size of Organ 

or Character 

observed. 


Frequency of 
Observations. 


First Moment. 




2/1 + %i 

2/2+^2/2 

Vk+^Vk 


^liVi + ^Vi) 
^2(2/2 + %2) 

XkiVk + ^k) 




■ 
n 


My+^y) 



may be slightly different, say, 2/1+82/1, 2/2+^2/2, • • •, and conse- 
quently the mean will also be different, say, 

U+m=[x,{y,+8y{l+x^{y^+Sy^)+ . . . ]/n, 



154 STATISTICS 

and, by subtraction, 

SM=(a;i82/i+a;282/2+ . . .)M • . . (5) 

Now we want to determine the S.D. of the different values of M 
found among the different samples, and that is given by 

where U denotes summation for all samples and v is the number of 
samples. This suggests that we should square both sides of 
equation (5), getting 

Therefore, n^ . vG\=x\va^yi+ . . . +2x^xJ -^ . v]-{- . . ., 
by (3). Hence, making use also of (1), 

\ n J n 

= (^2/1+ . . . )--(A2/S+ . . . +2^12/1.^22/2+ . . .) 
n 

=n^\-l(xiy^^ . . 0^ 
n 

Thus G\=(H'\-W)ln=G''ln, 

and the probable error of the mean == 0*6745(7/ \/y» . . • (6) 

The p.e. in the arithmetic mean found by taking a random sample 
of n events is a measure, so to speak, of the failure to hit the absolute 
mean, and it follows that the precision of the sample, the accuracy 
of aim at the mean, would be not unfairly measured by some 
quantity proportional to the reciprocal of the above expression, 
namely, ^/n/0'614:5G. With such a measure the precision would 
evidently be increased if the number of observations in the sample 
were increased, being proportional to the square root of their 
number, 

[It is desirable to draw a distinction here between what have been 
termed biassed errors and unbiassed errors ; errors due to random 
sampling are of the second class for there is, by hypothesis, no 

[* We do not know the true mean for the population as a whole, but we take 
in place of it M, the value given by the sample, which we may do with little 
error if n is large. Similarly c is the S.D. of the sample. ] 



SAMPLING — FORMULA FOR PROBABLE ERRORS 155 

reason why they should be in one direction rather than in another. 
Biassed errors, however, all tend to be in the same direction and 
they may arise in different ways, e.g. they may be due to faults of 
omission or commission on the part of the observer himself : he 
observes either carelessly or badly, omitting certain factors which 
ought to be taken into account, or so measuring or classifying his 
results that they appear always larger or less than they really are 
in fact. 

Sometimes, although the bias is known to exist, it may be im- 
possible to correct it : the most one can do is to bear it in mind 
and allow for it in using the results. A familiar example of this 
occurs in the collection of household budgets from the poor to find 
their standard of living, where it is only possible to get particulars 
from the more intelligent and thrifty class among them. 

Whereas in the case of unbiassed errors due to random sampling 
we can diminish the probable error of the average by increasing 
the number of observations, the same is not true of errors which 
are biassed, for suppose an error e in excess be made in each of 
n observations x^^ x^, - . - x^, the effect upon the average is to 
increase it from 

a?i+^2+ • • • +^n ^ (^i+e)+(^2+e)-f • • • +(^n+6) 

to — ' 

n n 

i.e. from 

n n 

so that the average is over-estimated by precisely the same amount. 
If, therefore, we know that bias exists, it is well, if possible, to 
correct it in each observation, for by so doing we change biassed 
into unbiassed errors, and though our corrections may be somewhat 
wide of the mark, the resultant error will then be diminished by 
increasing the number of observations : e.g. a farmer offers 400 
sheep for sale and, being anxious to make a good bargain, he asks 
a higher figure for them than he is in reality prepared to take ; 
let us suppose that this excess is 2s. 6d. for each sheep, then clearly 
the average price per sheep at which he is prepared to sell will be 
less than the amount he asks by 2s. 6d. also. But now suppose the 
buyer, a simple person knowing little of the prices of sheep and 
less of the ways of men, goes through the flock one by one and 
makes the error of offering a price either much above or much below 
what the seller is prepared to take ; even if his unbiassed offers 



156 STATISTICS 

differ by as much as 10s. for each sheep from the seller's reserve 
price, so long as they are random in direction, i.e. sometimes too 
much and sometimes too little, the resultant difference in the 
average from what the seller is prepared to take will probably not 
greatly exceed f 10s./\/400, or 4d. per sheep. 

We can sometimes diminish the effect of bias, even when its 
extent is unknown, by working with the ratios of the quantities 
affected instead of with the quantities themselves : e.g. suppose 
biassed errors, 61 and eg, enter into the measurement of the variables 
Xj^ and X2, both in excess, the ratio of the variables then 

= (^i+ei)/(a;2+e2) 

=xJl+'^)/xJ 1 + . 



X-i I \ Xi 



x^X ; 



-( 1+— )( 1——+ higher powers of eg 



:*' l + il-! 



if we omit higher powers of ej and eg than the first on the under- 
standing that they are both comparatively small. Suppose, for 
example, there was an error of 5 per cent, made in measuring ic^ 
and an error of 3 per cent, of like sign in measuring X2 then the 
resulting error in xjx2 would be 5 per cent. — 3 per cent. =2 per cent. 
Clearly the same holds good also if the errors are both in defect. 
This explains why a comparison of results arranged, say, on the 
index number principle may be trustworthy, although the method 
of formation of the numbers themselves may be in some respects 
faulty, granted that the same faults are repeated each year so as 
to produce Uke errors, i.e. the bias is to be unchanged in character. 
To correct the faults in one case and not in the other would prejudice 
the success of the method, since it depends upon the errors counter- 
acting one another.] 

Example (1). — To illustrate the important result we have obtained 
for the p.e. of the mean of n observations let us return to the experi- 
ment of selecting 900 random digits. The distribution actually 
obtained, and the theoretical distribution to be expected in the 



SAMPLING — FORMULJE FOR PROBABLE ERRORS 157 

long run if the experiment were repeated several hundred times and 
the average taken, are shown in the following table : — 



Table (37). Disteibution op 900 Random Digits. 


•Tk- •* 


Frequency 


Theoretical 


Digit. 


Frequency 


Theoretical 


Digit. 


Observed. 


Frequency. 


Observed. 


Frequency. 





95 


90 


5 


80 


90 


1 


96 


90 


6 


82 


90 


i 2 


93 


90 


7 


72 


90 


1 3 


105 


90 


8 


90 


90 


1 ' 
1 


91 


90 


9 


96 


90 



It is a simple matter to calculate the mean and S.D. for the dis- 
tribution from this table in the usual way ; the results are : — 



Observed mean =4-38 
Theoretical mean=4-50 



S.D.=2-911 

S.I).=2-872. 



Thus the p.e. of the mean based on the sample 

= ±0-6745 x2-911/\/900 
= ±0065, 

and 4-38 differs from 4-50 by less than three times the p.e. 

The 36 averages of samples of 25 events apiece were also calcu- 
lated, and the following were the results obtained : — 

2-76, 3-32, 3-68, 3-72, 3-72, 3-72, 3-76, 3-80, 3-92, 3-92, 408, 412, 
4-16, 4-16, 4-16, 4-28, 4-36, 4-40, 4-40, 4-40, 4-44, 4-60, 4-64, 4-68, 
4-72, 4-72, 4-76, 4-88, 4-96, 500, 5-00, 5-00, 5-08, 5-28, 5-40, 5-72. 

The mean of this distribution=157-72/36==4-381, and the 
S.D.=0-612. But the S.D. of the whole distribution of 900 digits 
=2-911, and therefore the S.D. of the distribution of averages of 
samples of 25 digits should be 2'911/V25=0-582, differing from 
0-612 by about 5 per cent. 

To find the p.e. of the sum or difference of two variables. Let the 
mean values of the two variables be denoted by y and z, so that 
deviations from these values found in a particular sample may be 
denoted by Sy and Sz. If then we write 



u=y-\-z 



we have 



Su=By+Bz 



(7) 



158 STATISTICS 

To find the S.D. of u we therefore require E{hu^)jv, where the 
Z denotes summation for all samples and v is the number of samples. 
But, squaring both sides of equation (7), we have 

Thus . Shu^=Ehy^+Zhz^-\-2E(hyhz), 

where the summation extends to all samples. Hence 

vg\= va^y+ VG%+ "IvOya^Ty^ 
or (7\=a^y+c72,+2r,,(7y(7, 

where r^^ ^^ ^^ correlation between the variables. And the 
p.e.=0-6745(7,,. 

The p.e. of the difference of two variables follows at once by- 
changing the sign of z throughout ; for, if 

v=y—z, 

we have hv'^=hy'^-\-'hz^—1hy^z, 

and o-\=o-''y+o-'^,— 2ry,o-yO-,. 

Generally, if x-^, x^, . . . x^ be the mean values of n variables, 
and if ^x-^, SiCg, . . . 8a;„ denote deviations from these values in 
a particular sample, we may write 

It — X-^ ~\~ X2 ~i • . . ~\~Xji 

and Su=Sxj^-\-Sx2-\- . . . +S^n. 

Thus 2:Su^=2:Sxi^-i- . . . +2I(SxiSx2)-i- . . . 

whence cr\=G\+ . . . -\-2i,^,a^a,^-i- . . . 

Important Corollary. If y and z are quite independent so that 
Vy^ is zero, the p.e. of their sum and the p.e. of their difference 
have the same value, namely, the square root of the sum of the 
squares of the p.e.'s of y and z themselves, which 

=0-6745-v/(o-%+or\) . . . (8) 

This result is exceedingly important, because it can be directly 
used to test whether a difference between two samples is accidental, 
i.e. whether it is such as might arise through sampling, or whether 
it imphes a real difference between the two populations from which 
the samples are selected. An example will illustrate the pro- 
cedure : — 

Example (2). In a study of Minimum Rates in the Tailoring 
Industry i by R. H. Tawney, a table is given (p. 114) which suggests 



SAMPLING ^FORMUL^ FOR PROBABLE ERRORS 159 

that * in the north of England women work in the tailoring trade 
when they are young ... in London and Colchester they have 
to work when they are older.' Taking some figures from that 
table we find : — 



District. 


AVorkers over 
35 years old. 


Workers at 
all ages. 


Proportion ' 
over 35. , 


London and Essex 
Manchester and Leeds . 


11,718 
4,029 


35,316 
21,822 


0-332 
0185 



-r ' >'n 



The difference between the proportions over 35 years of age 
= (0-332-0-185)=0-147. 

Let us suppose for the moment that this difference is not significant 
of any real difference in conditions between the two districts, but 
is merely due to random sampling. In that case the most natural 
value to assign to the true proportion of women workers over 35 
for the trade as a whole, as given by these figures, would be 

n^^718+4,029^15J47^^.2^^ 
35,316+21,822 57,138 

The S.D. for the first sample (London and Essex) would then be 

0-1= V(PQM= \/[0-276 X 0-724/35,316], 

and for the second sample (Manchester and Leeds) would be 

(72= a/[0-276 X 0-724/21,822]. 

Hence the p.e. for the difference between the proportions in the 
two samples would be roughly 

=Wi^\+^%), by (8), 

= f V[0-276 X 0-724(1/35,316+ 1/21,822)] 

=f VLO-276 X 0-724/13500] 

=0-0026. 

The actual difference between the proportions, 0-147, being much 
more than 3(0-0026), is certainly significant of a greater difference 
between the two populations ihan can be explained by random 
sampling alone. 



><U 



160 STATISTICS 

Another method of attack would be to assume a real difference 
between the two populations, if other considerations led us to 
suspect such a difference, and to find whether such a difference could 
be disguised by random sampUng. In that case the proper pro- 
portion to assume for the first sample would be 0-332, giving 

ai= V[0-332 X 0-668/35,316]= V628/10^ 

and for the second sample the proportion would be 0-185, giving 

(72= ^[0-185 X 0-815/21,822]= V^Ol/lO^. 

Hence the p.e. for the difference between these two proportions 
due to random sampling would be 

= IVK'+^2'), by(8), 
= |^^y(628+691) 
=0-0024. 

The actual difference is 0-147, which certainly could not be out- 
balanced by an error in the opposite direction due to random 
sampling, because it is much more than three times the probable 
error due to sampling. 

Sometimes we have to test the difference, not between two 
simple proportions, but between two sample distributions. In 
that case the mean of each sample may be calculated so that the 
difference (M^— Mg) between the means is known ; to find out 
whether or not it is significant of some real difference between the 
two populations from which the samples are drawn, (Mj— Mg) 
is compared with its p.e., namely 

0-6745V(ct2mi+^Im2), 
or 0-6745 V(o-\/r^i+c7\/^2) • • - (9) 

where Ui and n^ are the numbers of observations in the two samples 
respectively, and g^, g^ are the S.D.'s of the samples. Unless 
(Mj— Mg) is definitely greater than some two or three times this 
expression we cannot be very sure that the difference between M^ 
and Mg may not have arisen merely through random sampling, 
and it may quite Ukely not be significant * of any real difference 
between the two populations as regards the organ or character 
which is under consideration. 

[* It should be observed that the S.D. provides a wider margin for significance 
than the p.e., because a range of approximately 3 p.e. =3'§(r = 2o- onl3^ It is 
quite safe therefore to attach no great significance to a difierence which does 
not exceed two or three times the p.e.] 



SAMPLING FORMULA FOR PROBABLE ERRORS 161 

Example (3). — Statistics have been collected to test whether there 
is any significant difference between the eggs laid in general by- 
cuckoos and those laid by them in the nests of particular species 
of foster parents. Results of the following kind were obtained 
[see Biometrika, vol. iv., pp. 363-373, The Egg of Cuckulus Canorus 
(2nd Memoir), by 0. H. Latter] : — 





Number 


Mean 


S.D. 

(mms.) 


Signi- 




Group. 


of 


Length 


ficance 


Remarks. 




Eggs. 


(mms.) 


Test. 




Eggs of the Cuckoo 












race in general 


1572 


22-3 


0-9642 


. . 


, . 


Eggs laid in nests of — 












Garden Warbler . 


91 


21-9 


0-7860 


7-0 


Significant. 


White Wagtail . 


115 


22-4 


0-7606 


1-6 


Not significant. 


Hedge Sparrow 


58 


22-6 


0-8759 


3-75 


Probably significant 



The diJfference between the mean lengths of eggs laid in the nests 
of garden warblers and those laid by cuckoos in general 

=22-3— 21-9-:0-4 mms. 
The p.e. of this difference 

=0-6745 V[ (0-7860)2/91+ (0-9642)2/1572], by (9), 

=0-6745^(0-007380) 

=0-058. 

Hence the significance test 
=0-4/0-058=7-0, 

and we conclude that the difference in length between the two 
classes of eggs is certainly significant. Similarly the other cases 
may be tested. 

In the example just given, to find out whether one population 
differed from another, the arithmetic means have been compared ; 
but the mean alone will scarcely serve to establish the identity of 
any population. For example, we can conceive of two distinct 
races of men, both of the same mean height, but one race embracing 
a number of giants and dwarfs. Of course if we agreed to define 
two races as identical when they have the same mean heights, there 
would be nothing more to be said, but that would certainly only 
be a very rough-and-ready attempt at classification. 

Taking into consideration only the character of height, a further 
step in definition would be to measure the mode or most fashionable 

L 



162 STATISTICS 

height, and the dispersion or variabiHty — absolute : the standard 
deviation, and relative : the coefficient of variation — of the two 
races. Then, after comparing heights with sufficient detail, the 
attention could be turned to innumerable other characters, skull 
and body measurements, physical, mental, and even moral 
attributes. 

Clearly the difficulty of definition and of establishment of identity 
grows as we pass along the scale from physical to moral. Moreover, 
other statistical constants must be requisitioned when the question 
of the existence and degree of relationship between two organs or 
characters is to be determined. As the S.D. and the C. of V. serve 
to measure the amount of variability, so the coefficient of correlation 
comes in to measure the amount of likeness or association. Further, 
and especially in problems of inheritance, the coefficient of regres- 
sion must be measured. It might seem at first sight hopeless to 
try and measure the correlation between two such characters as 
athletic capacity and health in the same boy, or between the 
truthfulness of one boy and that of his brother ; but the genius of 
Karl Pearson has gone some way to solve even this difficult problem 
by means of a system of adjectival instead of numerical classifica- 
tion [see Phil. Trans., vol. 195a, pp. 1-47, On the Correlation of 
Characters not Quantitatively Measurable, and, as an exceptionally 
interesting application of the method, see Pearson, On the Laws of 
Inheritance in Man, ii. ; On the Inheritance of the Mental and Moral 
Characters in Man and its Comparison with the Inheritance of the 
Physical Characters; Biometrika, vol. iii. pp. 131-190]. In short, 
for a full and exact definition of a population of any kind, human 
or otherwise, it is necessary to measure not only the means, but aU 
the more important statistical constants, modes, medians, S.D.'s, 
C.'s of v., coefficients of correlation and regression, and so on, and 
it is no less necessary to calculate also their probable errors if we 
are to test the real significance of such differences as are observed 
in these constants between two samples from the same or from 
different populations. 

The probable errors for the more important constants, some of 
which are only introduced later in the book, are collected together 
in Table (38) for reference. The proofs in general are a little intricate 
and would be lacking in interest to the ordinary person, who is 
satisfied to take algebraical analysis on trust so long as he under- 
stands the nature of the results he uses, but the more mathematical 
reader who is anxious to see proofs may refer for some of them to 
Biometrika, vol. ii., pp. 273-281, Editorial, On the Probable Errors 



SAMPLING — FORMULA FOR PROBABLE ERRORS 163 

0/ Frequency Constants, which has been freely consulted on the 
subject here. 

The usual notation is adopted, n being the total number of 
observations in the given distribution, supposed normal in general, 
o- the S.D., etc. 



Table (38). Probable Errors of Statistical Constants. 



statistical Constant. 


Probable Error (=0-6745 S.D.). 


1 

Any observed group frequency, y 
; The mean of a distribution of any type 
' The S.D. of a normal distribution, o- . 
[The second moment about the mean, n^ 
. „ third „ „ „ Ms • 
[ „ fourth „ „ „ ^4 . 

The coefficient of variation, v . 

The coefficient of correlation, r 

The correlation ratio, »/.... 

f X, as determined from (X-X)=r^(Y- Y), 

ay 

1 when Y is given 

Y, as determined from {Y-Y)=r^(X-X), 

^ when X is given 

Distance between mode and mean in a skew 
distribution . . . . 

Skewness 

^2 (which should = 3 for a normal distribution) 

' ^i( »» »» =0 „ „ ) 

VW^ . 


0-6745j< V[y{l-y/n)] 

cr/Vn 

a/V2n 

aW2fn 

aW9Q/n 

„ {l-r^)/Vn 

{l-r]^)/Vn, nearly 

(rV(3/2n) 

„ V(3/27i) 
„ V(24/7i) 
» 
» V(6/n) 



Example (4). — Li the example which follows are given data 
necessary for testing the significance of differences in variability 
as well as in mean values. They represent an attempt made to 
find whether members of a particular species of crab caught in 
shallow water differed with regard to certain characteristics from 
those caught in comparatively deep water [see Biometrika, vol. ii., 
pp. 191 et seq., Variation in Eupagurus Prideauxi, by E. H. J. 
Schuster]. Only a few of the results are recorded here, to two 
decimal places ; the reader wiU find it a valuable exercise to verify 
for himself the p.e.'s given in each case. 



164 



STATISTICS 



Measurement Made. 


Sex. 


Locality. 


Mean (mm.). 


S.D. (mm.). 


C. ofV. 
per cent. 


Carapace length 


Male 
Female 

55 


Deep water 
Shallow ,, 
Deep 
Shallow „ 


8.59±0-05 
841 ±0-04 
7-54±0-03 
7-12±0-02 


l-67±0.04 
149±0-03 
0-94db0-02 
0.86±0.02 


1945db044 
17-75±0-37 
1249±0-28 
1212±0-25 


Difference of Means (mm,). 


Difference of S.D.'s (mm.). 


Difference of C.'s of V. 
per cent. 


Sex. 


0-18±0-07(poss. sig.) 
042iO-04(sig.) 


0-18db0-05{prob.sig.) 
0-08±0-03(poss.sig.) 


1.70±0.58(poss. sig.) 
0-37±0-37(not8ig.) 


Male 
Female 



The significance or otherwise of differences between variabiUties 
in the case of cuckoos' eggs (p. 161) might be tested in the same way. 



CHAPTER XIV 



FURTHER APPLICATIONS OF SAMPLING FORMULA 



We have been discussing in the last chapter how to test two samples, 
supposed each to contain homogeneous material, to find out whether 
they belong to the same or to different types of population, but 
the further question often arises as to whether a sample is or is not 
homogeneous. t 

Example (1). — To this we may obtain a partial answer by working 
out the statistical constants of the sample and their p.e.'s in order 
to compare them with the corresponding constants for a sample or 
series of samples believed to be homogeneous and of the same 
type. For example, Professor Karl Pearson has measured the 
skulls of skeletons of the Naqada race, excavated in Upper Egypt 
by Professor Flinders Petrie and presumed to be some 8000 years 
old, and he places his results for comparison alongside those 
for certain other races admittedly homogeneous [see Biometrika, 
vol. ii., p. 345, Homogeneity and Heterogeneity in Collections of 
Crania] : — 







Variability (mm.). 


Series. 


Number of 






Observations. 










Skull Length. 


Skull Breadth. 


! /"Ainos 


76 


5-936 


3-897 




Bavarians . 


100 


6-088 


6-849 


Skulls J 


Parisians 


77 


5-942 


5-214 




Naqadas 


139 


5-722 


4-612 




lEngUsh 


136 


6085 


4-976 


Living r ^^"i^ridge undergrad'tes 
heads ^^gl^^^ criminals 

tOraons of Chota Nagpur 


1000 


6-161 


6-055 


3000 


6-046 


6-014 


100 


5-916 


4-397 


Mean Variability 




5-987 


4-877 



166 



166 STATISTICS 

The S.D. of the variabihty of skull length calculated from this 
series=0-129 mm. and of the variabihty of skull breadth=0-545 mm., 
and these supply standards for valuing the differences between the 
Naqada and the mean variabilities. 

Another method of procedure is to take a random sample out of 
the sample itself, assuming the latter is large enough to admit of 
an adequate sub-sample, and to compare the constants of the 
whole and jjart. When they do not differ beyond the Hmits allowed 
by random sampling the inference is that the whole may be treated 
as a homogeneous class if judged by this test alone. 

Example (2). — In an interesting and important memoir, On 
Criminal Anthropometry and the Identification of Criminals, by W. R. 
Macdonell [Biometrika, vol. i., pp. 177 et seq.], the author uses this 
method to test the homogeneity of a class of 3000 criminals by 
measuring also a random sample of 1306 ciiminals out of the 3000. 
He obtained, for example, 

S.D. of head length-- 6-04593±0-05265 mm., for the 3000 criminals ; 
= 600247 ±007922 „ „ 1306 

The difference between the variabilities in the sample and sub- 
sample, by result (8) on p. 158, 

=0-04346±V [(0-05265)2+ (0-07922)2] 
= 004346+009512 

which is certainly not significant. If the same holds good with 
regard to the means and other constants, then the whole may be 
said to be homogeneous so far as this test goes. 

Example (3). — ^Another example may be given from the memoir 
on Variation and Correlation in Brain Weight, by Raymond Pearl, 
[Biometrika, vol. iv., pp. 13 e^ seq.]. The author wished particularly 
to investigate the change of brain weight with age ; on the hypo- 
thesis that the weight of the brain reaches a maximum between 
the ages of 15 and 20, remains unchanged from 20 to 50, and then 
begins to decline and so continues till death, the material was 
divided into a * Young ' series, ages 20 to 50, and a ' Total ' series 
including all between 20 and 80. The ' Young ' series thus formed 
a selection from the ' Total ' series, but a selection based on age 
and not on brain weight. If there were no correlation between 
age and brain weight, this selection, based as it is on age, would, 
of course, be random as regards brain weight. Now correlation 
does exist between the two, but it is so slight that, within the hmits 



l^URTHER APPLICATIONS OF SAMPLING FORMULA 16? 

of error, the ' Young ' series does form practically a random sample 
of the ' Total ' series, as is shown by the following figures : — 

Difference in Variation Constants between Young and 
Total Series (written with a positive sign when the 
Young Series gives the greater value). 





Male. 


Female. 


Swedes 
Bavarians 


S.D. 
+2-851+4-066 
-1-888+3-556 


C. of V. 
+0-122+0-291 
-0-173+0-234 


S.D. 
+ 4-786+5-465 
-10-357 + 3-909 


C. of V. 
+0-271 + 0-435 
-0-941+0-320 



Thus in only one case, that of the Bavarian females, is the differ- 
ence between the variabilities, S.D. or C. of V., of the two series as 
great as its probable error, and even in that case the differences, 
10-357 and 0-941, are not three times as large as their respective 
p.e.'s, 3-909 and 0-320. Dr. Pearl concludes from these and similar 
results that ' the series are reasonably homogeneous in other respects 
than age.' 

The reader is recommended to test his knowledge of the formulae 
for probable errors by applying them to the following examples. 
Dr. Alice Lee, in a note on Dr. Ludwig on Variation and Correlation 
in Plants [Biometrika, vol. i., p. 316] makes use of the statistics 
relating to Ficaria Vema in Example (4). Those in Example (5) 
are taken from among a large number of others in the highly 
interesting memoir, On the Laws of Inheritance in Man, by Professor 
Karl Pearson and Dr. Alice Lee [Biometrika, vol. ii., pp. 357 et seq.] 
cited once before. 



Example (4). — Variation and Correlation in Ficaria Verna. 



• No. of Observations. 


Mean No. of 
Petals; S.D. 


Mean No. of 
Sepals; S.D. 


Correlation between 

No. of Sepals and 

No. of Petals. 


1000 (Greiz A) 
1000 (Greiz G) 


8-286; 1-3382 
8-232; 0-9954 


3-695; 0-8524 
3-437; 0-7033 


0-2439+0-0201 
0-2480+0-0200 



We have here all the data necessary to find the p.e.'s of the 
means, variabilities, and correlations, and we wish to know whether 



168 



STATISTICS 



the differences between the means and variabilities of the A and G 
plants can be accounted for by random sampling alone. 
For example, the difference between the petal means 

= (8.286-8.232)±i /[ (1:3382)^ (0;9954n 
\j[_ 1000 1000 J 

=0054±0035. 

Clearly this difference, being not so great as twice its p.e., is not 
significant and may quite well be due to random sampling. 
Again, the difference between the petal variabilities 



-(l-3382-0-9954)±f 
=0-3428±0025 



\ 



(1-3382)2 , (0-9954)2 



2000 



2000 



which is certainly much too great to be explained away by random 
sampUng merely. 

Similarly the differences between the sepal means, between the 
sepal variabilities, and between the correlations, may be tested for 
significance by comparison with their p.e.'s. 

Example (5). — Size and Variability of Stature in the 
Two Generations. 





Father. 


Mother. 


Son. 


Daughter. 


Mean height (in.) 

S.D. (in.) . 

C. of V. (percent.) 


67-68±0-06 
2-70 ±0-04 
3-99±0-06 


62-48 ±0-05 
2-39±004 
3-83±0-06 


68-65 ±005 
2-71 ±004 
3-95±006 


63-87 ±005 
2-61 ±0-03 
4-09 ±005 



The student in this case might use one of the formulae for the 
p.e.'s to find the number of fathers, mothers, sons, or daughters 
observed when the p.e.'s are known, and then the remaining p.e.'s 
might be verified when the numbers of observations are found. 

As evidence of ' assortative mating,' the tendency of like to 
mate with -like, the following particulars are given, based on 1000 
to 1050 cases of husband and wife : — 



Correlation between stature of husband and stature of wife=0-2804±0-0189 

„ span „ „ „ span „ ,, =0-1989±0-0204 

,, ,, forearm ,, „ ,, forearm ,, „ =0-1977±0-0205 



To measure the average intensity of inheritance, the extent of 



FURTHER APPLICATIONS OF SAMPLING FORMULiE 169 



resemblance between parents and children in any character, 
efficients of correlation are calculated such as the following : — 



co- 



Coefficient of Correlation 

between stature of father and stature of son =0-514db0-015 

,, ,, ,, „ „ daughter =0-510±0-016 

,, mother ,, ,, „ son =0494±0-016 

,, ,, ,, ,, ,, daughter =0-507±0-016 



[In verifying the p.e.'s for this case take the number of observa- 
tions to be 1024.] 

One more extract may be quoted, a prediction table, giving the 
probable mean stature of sons of fathers of given stature, and 
so on : — 



Son's probable stature = 33-73 + 0"516 (father's stature) ± 1 '56 

Daughter's „ „ = 30-50 + 0*493 ( „ „ )±1-51 

Soq's „ „ =33-65 + 0-560 (mother's stature) ±1-59 

Daughter's „ „ =29-28 + 0-554 ( „ „ )±l-52. 



All values given in this example for the p.e.'s should be 
verified. 

Before we consider further applications of these principles to 
questions of a somewhat different kind, let us imagine a very 
simple though artificial illustration. Suppose we have 999 sheep, 
each one ticketed, the numbers on the tickets running from 1 to 
999. Also suppose 666 of these sheep are white and 333 are black, 
so that, if we pick out any one at random, the chance of it being 
black is 333/999 or 1/3. Let us call picking a black sheep a ' success,' 
then :p= 1/3, g=2/3. 

We proceed now to select 99 sheep in succession at random 
from the flock with the understanding that each sheep is returned 
into the flock before the next is picked out. This insures that 
the chance of a success at each selection remains equal to 1/3 and, 
of course, there is nothing to prevent the same sheep being picked 
more than once. The selection might practically be made by 
placing in a box 999 tickets, numbered from 1 to 999, one to corre- 
spond to each sheep, then picking out 99 of them in succession, 
being careful to replace each and to shake up the box before picking 
out the next ; if there were absolutely no difference between the 
tickets, such as would cause one to be picked more easily than 
another, the selection made in this way would be random in the 



170 STATISTICS 

sense required, and the tickets so chosen would determine which 
sheep were to be taken and which left. 

The proportion of black sheep to be expected in such a random 
selection of 99 is 1/3, but, if we only perform the experiment once, 
it is quite Ukely that the proportion we actually get will differ from 
1/3 by an amount 

=0-6745V(2??/w) 

=0-6745V(J . § . A) 

= 1/31, about, 

while it is unUkely that the proportion will differ from 1/3 by much 
more than 3/31, or 1/10. 

Conversely — and it is really the converse which is useful in prac- 
tice — if we do not know the proportion of black sheep in the whole 
flock, we may get a fair estimate of it by taking a random sample 
of 99 sheep (any other number wiU serve the purpose, but the 
larger the better for accuracy), and if we find that in this sample 
there are 33 black sheep, i.e. ^=33/99=1/3, it will appear that 
the value of jp for the whole flock is 1/3, subject to a probable error 
0-6745\/(2?9'/^) in excess or defect, i.e. the true proportion for the 
whole flock may quite likely differ from 1/3 by as much as 1/31, 
but it is unlikely to differ by much more than 1/10. It should be 
noticed that the calculation of the probable error in this converse 
case is based upon the value of p given by the sample taken, for 
that is the only value of which we have knowledge. 

Too much stress can scarcely be laid on the fact that the samples 
chosen must be absolutely unbiassed, otherwise the use of the 
formulae Tfp and ^/(npq), or the corresponding proportional formulae, 
cannot be justifled : each sheep in our illustration must have the 
same chance of being picked, and no one selection is to have any 
influence on another. The failure to appreciate this essential 
point has led to no little waste of time and effort in the collection 
of valueless statistics. 

The method of sampling has been employed in a way at once 
interesting and useful by Dr. A. L. Bowley, and, as some of this 
work has barely received the attention it deserves, it may be well 
to explain two of his experiments in some detail. 

The first was of interest because its results could be tested by 
an examination of the original record from which the sample was 
taken. The details concerning it are abstracted from the Journal 
of the Royal Statistical Society, September 1906. 

Example (6). — Bowley sampled the dividends paid by 3878 



FURTHER APPLICATIONS OF SAMPLING FORMULA 171 

companies as quoted in the Investors' Record. His sample con- 
sisted of 400 of these companies, i.e. about 10 per cent., selected in 
a purely arbitrary fashion thus : the investigator took a Nautical 
Almanac and noted down the last digits of one of the tables, record- 
ing them in groups of four, but if any particular group gave a 
number bigger than 3878 he rejected it. In this way each of the 
numbers between 1 and 3878 had an equal chance of selection (for 
numbers under four figures would appear like 0327, 0042, 0009, 
which would be taken to represent 327, 42, 9 respectively), and the 
selection of one had no influence on that of any other. The com- 
panies in the Investors' Record were numbered consecutively, and 
the dividends corresponding to the 400 arbitrary numbers obtained 
formed the sample with which Bowley worked. 

After making some interesting deductions with regard to the 
average for the whole distribution, to which we shall return pre- 
sently, he proceeded to forecast the grouping of the original com- 
panies as to their dividends by setting out the grouping discovered 
in the sample 400, as follows, using the standard deviation in place 
of the probable error as the error due to randorii sampling : — 

Table (39). Distribution of Dividends paid by a 
Sample of 400 Companies. 



(1) 


(2) 


(8) 


(4) 


Dividend. 


Sample of 

400 
Companies. 


Percentage of Sample 
Companies in each Class. 


Percentage of 
all Companies 
in each Class. 


Nil 

£1 to £2, 19s. 9d. 
£3 to £3, 9s. 9d. 
£3, 10s. to £3, 19s. 9d. 
£4 to £4, 9s. 9d. 
£4, 10s. to £4, 19s, 9d. 
£5 to £5, 19s. 9d. 
£6 to £7, 19s. 9d. 
£8 to £10, 19s. 9d. 
Ab6ve£ll 


28 
6 
37 
71 
64 
53 
60 
48 
29 
4 


7 with S.D. = l-27 

n 

9i „ =1-46 
171 „ =1-90 
16 „ =1-83 
13J „ =1-68 
15 „ =1-78 
12 „ =1-63 
7i „ =1-29 
1 


6 
1-5 

8-4 
18-8 
17-3 
13-8 
17-7 
10-8 
3-8 
1-9 



In col. (3) the S.D. for each group was calculated as follows : — 
for the first group : out of 400 possible events we have 28 successful 
events, meaning by ' successful ' here * a company paying no 
dividend,' thus 

^=28/400, 3=372/400. 



172 STATISTICS 

Hence the S.D. of the frequency in the first group 

= V(28x372)/20 
=5-l. 

Since this is for a sample of 400, the S.D. of the ^percentage * frequency 
in the first group 

-J(5-l)--l-27. 

The other S.D.'s are calculated in the same way, but when the 
number in a class is very small the forecast can scarcely be refied 
upon and consequently the S.D. is not inserted. 

It will be noted, by comparing with the numbers in col. (4), 
showing the corresponding percentages for aU the 3878 companies, 
that every forecast- was remarkably good except one, class £8 to 
£10, 19s. 9d., where the error approaches three times the S.D., and 
the exception will serve as a warning that, in working with samples, 
the unexpected sometimes happens. Professor Edge worth, in his 
Presidential Address to the Royal Statistical Society (1912), points 
out that the method appears to be a permanent institution in 
the Statistical Bureau at Christiania, where it has given very good 
results. These can be checked or ' controlled ' for safety if complete 
statistics are obtainable under some heads. He fairly sums up the 
utility of sampling when he says that ' we may obtain from samples 
a general outline of the facts — often sufficient for the initiation of 
a project like that of insurance — ^rather than the features in detail.' 

Bowley also divided up his 400 random samples into 40 groups 
of 10 companies each, and calculated the average for each group. 
The S.D. for these 40 averages was found in the usual way, giving 
0-775. But since this was the S.D. for averages of 10, we conclude 
that 

(theS.D.forthedistributionofthe400companies)/\/10=0-775 
i.e. the S.D. for the distribution of the 400 companies =0-7 75-^^10. 

Hence, appl.ying the same principle again, 

the S.D. of the average of the 400 sample companies 

-0-775V10/\/400 
=£0122. 

[* It would not be correct to take \/[7(l -iItt)] as the S.D. of the percentage 
frequency in the first group ; this value would be double the true value, namely, 
J v''[28(l -TW)] = i v''[7(l -T^ir)], because the accuracy is increased by increasing 
the number of events in a sample, and the sample here is really 400 and not 100. J 



i^ 



FURTHER APPLICATIONS OF SAMPLING FORMULA 173 

Now the average of the 400 samples turned out to be £4-7435. 
Hence it was judged that, if this was a fair selection (and the random 
method adopted was such as to make it fair in all reasonable likeli- 
hood), the average for the 3878 companies should certainly lie 
between 

£[4-7435±3(0-122)]. 

The true average was found by actual calculation to be £4*779, 
well within the above limits, although the original items varied from 
nil to £103, being grouped according to the nature of the security 
— Government, Railways, Mines, etc., etc., and the averages and 
S.D.'s on successive pages di£fered materially. This aggregation, 
Bowley remarks, is very similar to that found in wages in different 
occupations and localities, and in many other practical examples. 

The value of the second experiment due to Dr. Bowley lies in the 
suggestion that similar means can be applied with good results to 
the investigation of many social phenomena. 

If out of a large group a comparatively small sample of statistics 
is collected in the purely random manner already described, we are 
able by such means to estimate what is the average, and even to 
obtain limits between which the average wiU almost certainly lie, 
in the large group based upon values found for the average and 
S.D. in the small sample. 

Example (7). — With the collaboration of Mr. Burnett-Hurst and 
a number of other workers. Dr. Bowley conducted an inquiry into 
the conditions of working-class households in four representative 
towns — ^Northampton, Warrington, Stanley, and Reading — ^the 
results of which are published by Messrs. Bell and Sons under the 
title of Livelihood and Poverty. They are similar in character to 
those obtained by Rowntree in his study of conditions in York, 
but what is peculiar to Bowley's inquiry is that only a sample, 
about 1 in 20, of the working-class houses in each town was 
examined, and the conditions in the towns as a whole were deduced 
from these samples. 

We are not concerned here with the actual facts disclosed by the 
investigation, striking as they are, but with the explanation of the 
sampling method adopted, and as to that it may be remarked that 
the foundation on which it rests is precisely the same as that which 
underlay the example of the 999 black and white sheep. The 
main point to notice here again is that Bowley was careful to select 
his samples in unbiassed fashion as follows : ' For each town a list 
of all -houses . . . was obtained, and without reference to anything 



174 STATISTICS ; 

except the accidental order (alphabetical by streets or otherwise^ 
in the list, one entry in twenty was ticked. The buildings sa 
marked, other than shops, institutions, factories, etc., formed th^ 
sample.' It will be evident that this method of choice is not quit© 
on the same level of randomness as that followed, for example, m 
drawing cards from a well-shuffled pack, each card to be replaced 
and the pack reshuffled before the next is drawn ; but, for that 
very reason, the results of the experiment are all the more likely 
to be well within the limits of error provided by the formulae o| 
the ideal case. The deliberate selection of every twentieth hous^ 
in each street is likely, that is to say, to give a more representative 
picture of the town as a whole than would be obtained by selecting 
the same number of houses in a purely random fashion which might 
by chance give too much emphasis to some street or district. 

A practical test of the goodness of the sample was possible by 
comparing the results in a few instances with information available 
from other sources. In order to make the method of working 
quite clear, let the guiding principle first be recalled : — 

* If , in a random sample of n items, the proportion of successes 
is p, then the proportion of successes in the universe from which the 
sample is selected will not be likely to fall outside the limits 

p±3(0-6745)VtoM), 

and, if that universe contains altogether N items, the number of 
successes will not be likely to fall outside the limits j 

Njp±3(0-6745)NV(i?^M).' T: 

In Reading the total number of all inhabited houses in th^ 
borough was 18,000 at the time of the inquiry, i.e. N= 18,000. 
The total number of houses visited was 840, i.e. 71—840. If we^ 
call a house assessed at £8 or less a ' success,' the number of suchi 
houses found in the sample was 206. { 

Thus ;p=206/840, ^=634/840, ; 

and the number of houses rented at £8 or less in the whole borough 
should be ^ i 

N2? with a ly.e.=0•Q14:5N^/ (pq/n) 
i.e. 4414±180. : 

The actual number of houses so rented was known from other sources 
to be 4380, weU within the limits forecasted. 

The value used for p in the above is that given by the sampleJ 
but when we know the actual number of successes in the universe! 



FURTHER APPLICATIONS OF SAMPLING FORMULA 175 

as a whole, as in this case we do, we might use the true value of 
p, i.e. the value for the universe in place of that for the sample. 
The argument might also be put in another way without affecting 
the principle employed, thus : — 

The number of houses rented at £8 or less in the whole borough 
was 4380. 

But the proportion of houses sampled in the whole borough was 
840/18000, i.e. 1/21-43. 

Hence the number of houses at the above rental to be expected 
in the sample=4380/21-43=204. 

The number actually found in the sample was 206, with a probable 

^^^^^ =0-6745 V(^M) 

=0-6745 V(840 X ^\% X il^SS) 
= 8, approximately. 

AgaiQ, the number of persons engaged in a certain occupation at 
Reading was known to be 761 in the borough as a whole. Hence 
the number of persons so engaged to be expected in the sample 
was 761/21-43, i.e. 35. 

The number actually found in the sample was 29 with a probable 

^^^^ = 0-6745 V(wi55) 

=0-6745 V(840 X tIwt^ X kl^^l) 
=4, approximately. 

Further examples of the method are here given, in each of which 
the total number of events is small so that the number in each 
sample is also small, and since, as we have seen, the accuracy or 
precision of the proportion of successes discovered in any sample 
varies directly as the square root of the number of events the sample 
contains, the results cannot be expected to be so good when this 
number is small. 

Example (8). — 514 candidates sat a certain examination paper ; 
their marks ranged from 3 to 64. The candidates were numbered 
consecutively from 1 to 514, and a random sample of 90 (17 J per 
cent.) was selected from among them by writing down the 90 
numbers formed by the digits in the seventh decimal place, taken 
in groups of three, in the logs of the numbers 10104, 10204, 
10304, . . . , as given in Chambers's Tables, neglecting all numbers 
greater than 514 and calling such numbers as 005, 037, etc. — 5, 
37, etc. In this way each of the numbers between 1 and 514 stood 
an equal chance of inclusion. 



176 



STATISTICS 



The distribution of candidates in the sample is compared with 
that for all 514 together in the following table : — 





Percentage of All 


Percentage of Candidates in 


No. of Marks Obtained. 


Candidates who obtained 


Sample who obtained 




these Marks. 


these Marks. 


Less than 15 


8 


p.e. 
8d=l-9 


15 but less than 25 


19 


17±2-6 


25 „ „ 30 


16 


18±2-7 


30 „ „ 35 


18 


13±2-4 


35 „ „ 40 


15 


17±2-6 


40 „ „ 50 


19 


18±2-7 


50 and over. 


^o^ 


10±21 






^l) 



The reader might verify the p.e.'s given in the last column : 
e.g. proportion in the sample obtaining less than 15 marks— 7/90 ; 
therefore ^j=-7/90, g=83/90. 

Hence the S.D. for this group 

=.V[7(1-9V)] 
=2-54, 

and the S.D. for the percentage 

=VTrx2-54=2-8. 
Thus the p.e. for the percentage 

= |cr— 1-9, approximately. 

Example (9) deals in a similar way with the data concerning 
infectious diseases in 241 towns in England and Wales previously 
recorded on p. 62. 

A sample of 60 towns, i.e. about 25 per cent., was chosen in a 
random fashion as in the last example, and the sample distribution 
is compared below with that of the 241 towns as a whole. 

The verification of the probable errors in this and the next case 
is left to the reader. 



Case Rate per 1000 
of the Population. 


Actual No. of 
Towns so rated. 


No. as suggested by 
the Sample. 


1 and under 5 
5 „ 9 
9 „ 13 
13 and over. 


85 
86 
42 
28 


p.e. 
92 ±10 
96±10 

28± 7 
24± 6 



FURTHER APPLICATIONS OF SAMPLING FORMULA 177 

Example (10) is concerned with the annual output per head in 
142 different types of employment as given in 1907 by the Censiis 
of Production [data from Sixteenth Abstract of Labour Statistics of 
the United Kingdom, Cd. 7131]. The distribution suggested by a 
random sample of 50 different occupations is compared with that of 
the complete list of 142 occupations. 





No. of Occupations 


No. in Complete 


Actual No. 


Output per head. 


in Sample with 


List as deduced 


found in 




this Output. 


from Sample. 


Complete List. 


Under £60 .. 


4 


p.e. 
ll±3-6 


12 


£60 and under £80 


16 


45 ±6-2 


42 


£80 „ £100 


6 


17d=4-3 


25 


£100 „ £120 


10 


28±5-3 


20 


£120 „ £190 


8 


23±4-9 


27 


£190 and over 


6 


17±4-3 


16 



The S.D. in each of the last three examples has been calculated 
by using the value for p given by the sample, which is the value 
one must fall back upon in practice when the true p for the whole 
distribution is unknown. In any case where we are able to test 
our sample by comparison with the whole distribution, however, 
it is possible to use the true value of p, e.g. in Example (10) 
output £100-120, p==20/142 as opposed to 10/50. 



r 



CHAPTER XV 

CURVE FITTES^G PEARSON' S GENERALIZED 

PROBABILITY CURVE 

It may be recalled that in the introductory chapter an outline was 
given of the manner in which the theory of Statistics might be 
conceived to develop. It was shown how the desire for simpUfica- 
tion and the need for compression leads to the division of a large 
mass of figures dealing with any given matter into groups ; indeed, 
it may well be that the statistics have been so arranged at the 
source in the act of collecting : e.g. we may have to deal with 
so many males of height 54 in. and less than 55 in., so many of 
height 55 in. and less than 56 in., so many of height 56 in. and less 
than 57 in., and so on. Here corresponding to each given height, 
which we may label x, or each range of height, such as x^ to x^, 
we have a certain frequency of males of that height or range, 
which frequency we may label y, and hence a frequency table can 
be formed showing the variation of y with x. Further we have 
seen how such pairs of corresponding values of x and y can be 
plotted so as to picture the complete observed frequency distribution 
to the eye. 

Now the representation thus made, though helpful up to a point, 
is not entirely satisfactory. Whether we simply join up successive 
points (Xy y), or set up rectangles of varying height y on bases 
spanning the successive ranges of x, or erect ordinates (y's) at the 
mid-points of these bases, joining the summits in the manner 
previously described, the connection so established between each 
observation and the next is too superficial, depending merely on 
the fact of casual neighbourship, and may sometimes give a false 
impression of frequency and changes in frequency in the population 
of which the observations are but a sample. And this is neces- 
sarily so if we confine ourselves strictly to the data observed. 

One difiiculty which has to be faced is that only within certain 
broad limits can we trust our observations to give us information 
which is truly representative of the population in which we are 

178 



CURVE FITTING 179 

interested. We seldom if ever deal with the whole population : 
in fact it may be so large that it is impracticable even to reckon it ; 
instead we make a random or unbiassed selection of a smaller but 
adequate number of individuals belonging to the population, and 
classify them according to the size or nature of the character which 
concerns us. But, granted that our sample is adequate in size 
and unbiassed, the numbers obtained in the different groups- of the 
frequency distribution will still be subject to the errors of random 
sampling, and it is only after these errors have been calculated that 
we can lay down the probable Umits within which our sample may 
be regarded as really representative of the population as a whole. 

Another difficulty arises owing to the fact that our observations 
in general do not cover the whole field of values of the variables x 
and y ; we may quite likely want to know the percentage frequency, 
2/, of individuals with a character (height or whatever it may be) x 
which does not chance to be any one of the a;'s observed, if the 
observations are only recorded according to discrete (separately 
distinct, Hke 5 ft., 6 ft., 7 ft.) values of x ; on the other hand, if 
the observations have been classed in groups, the frequency in 
which we are interested may refer to an x which does not coincide 
with the centre of any group or which is even outside the range 
altogether. We have therefore further to inquire whether such 
information can be deduced in any way from the statistics collected. 
Now it so happens that both these difficulties disappear if we 
can only attain the ideal already outlined in discussing graphs, 
and find a suitable curve to ' fit ' the statistics observed. Such a 
curve would not necessarily pass through all or any of the points 
(ic, y) representing the observations, for these, as we have remarked, 
are subject to errors of random sampHng and the observed frequency 
y of any ic may be greater or less than the corresponding y in the 
population at large to which the curve is presumed to approximate. 
The curve in short must remove the roughnesses which are in- 
separable from ordinary observation. Moreover, given any x, not 
merely one of the x's observed, it must be possible to read off from 
it the corresponding y, the frequency appropriate to that x. 

It is not always accurate enough for our purpose to draw a curve 
by eye, passing as evenly as possible through the middle of the 
points observed in the manner conceived in an earHer chapter. It 
is necessary in some way to find an algebraical formula, possibly 
even a trigonometrical, exponential, or more complex expression, 
which will give the y corresponding to any x desired. This formula 
or equation must depend upon the statistics collected : i.e. the 



180 STATISTICS 

constants involved in it must be directly and fairly easily computed 
from the 2/'s observed, and the results of all the observations should 
enter into the equations which determine the constants in order to 
make use of the full information at our disposal. In addition, the 
method of determining the equation and its constants should be as 
general as possible, so relieving us of the trouble of discovering a 
new method owing to the failure of the original one at nearly every 
trial. Finally, the equation should not be so intricate as to make 
the labour of calculating y for any given x too heavy to be attempted 
with the ordinary equipment at the statistician's disposal. Once 
such an equation is found it is a fairly straightforward proceeding 
to trace the curve for which it stands, and it wHl remain afterwards 
to test the goodness of fit in some more refined way than by seeing 
how closely it passes through the observed points by eye. 

When we come to review the shapes of the frequency polygons 
or histograms most commonly met, we find that the majority 

of them start from low fre- 
quency, rise to a maximum 
as X, the character observed, 
increases, then fall again to- 
wards zero very likely at a 
different rate. In fact the 
statistics suggest a shape something like that shown in fig. (27) 
for the corresponding frequency curve, though we cannot be sure 
that it would coincide with the axis at either extremity. [Cases 
do occur where the curve has two or even more humps (maxima), 
but we purposely restrict ourselves to the simpler and more frequent 
tyipe described.] 

Now the simplest shape to deal with from the algebraical point 
of view would certainly be symmetrical in character, corresponding 
to statistics which rise and fall at the same rate, though this would 
not necessarily be the most common shape among the records of 
actual Hfe. In order to simplify our problem, therefore, we might 
start by making up for ourselves an ideally simple set of statistics 
which are perfectly symmetrical, and see whether we can discover 
a process for fitting a curve in a case of that kind. If this prove 
successful it might be possible afterwards to adapt the same process 
to an unsymmetrical or ' skew ' set of statistics made up in a similar 
way. Then finally we should inquire whether actual observations 
conform to any of the types of curve discovered, and, if so, how 
they can be fitted together. 

Now in manufacturing our statistics we must keep before us the 




CURVE FITTING 181 

object at which we are aiming. Given the statistics, what we 
want is a formula, algebraical or of some other kind, to fit them. 
This raises the possibility of choosing the statistics themselves in 
some algebraical form, and such a form is at hand in the binomial 
expansion, which is, in fact, one of the first examples of a general 
symmetrical expression one meets. Thus 

(a+6)i=a+6 
(a+6)2=a2+2a6+62 

(a+6)4=a4+4a36+6a262+4a63-fM. 
(a+6)5=a5+5a*6+ I0a^b^+ l0a^b^+5ab*-\-b^ 



1-2 

Clearly all these expressions become perfectly symmetrical if we 
put a=6, for they read the same whether we run from left to right 
or from right to left. 

We have already seen what an important part the binomial 
expansion plays in the early stages of the theory of probability : 
e.g. (i+i)^^, when expanded, tells us at once the proportion of times 
on the average we may expect 10 heads, 9 heads and 1 tail, 8 heads 
and 2 tails, and so on, when we toss an evenly-balanced coin ten 
times in succession ;" or again, if p is the probability that a certain 
event will happen, and q the probability that it will fail to happen 
at one trial, then the probabilities that it will happen p times, 
ip—l) times, {p—2) times, . . .inn trials are given by the succes- 
sive terms in the expansion of {p-\-Q)^- However, we make no 
assumption for the moment as to the values of a and 6, except 
that in the symmetrical case with which we begin they are equal, 
and we have as the successive terms of {a-\-a)^ : — 

an, ^a^ ^(^-^V, . . . , ^<^-^U ruin, an, - 
1-2 12 

Let us suppose that our observed statistics take the above form 
so that these terms may be plotted as a succession of ordinates, 
2/i> 2/2' 2/3» • • • . Vn+v associated with abscissae, x^, x^y Xq, . . . , x^+i, 
at equal distances apart measured, say, by c ; for convenience we 
may place the origin as in fig. (28), so that 

:2c, x^=Sc, . . . , XJ^+^=(n-\-l)c, 



182 STATISTICS 

and we can then form a frequency polygon, where 



x^=rc, y^- 



n(n—\)(n—2) 



(n-r+\. 



1-2-3 . . . (r-1) 

are typical values of a pair of the variables x and y, each such 
pair defining a vertex of the polygon. 

Now in this case, since the statistics have been artificially built 
up by ourselves and are not in reaHty a random selection, they are 



Y 




















/I 


p^ 


-> 


[\ 




^A 










K 




^ 


y. 


yr 










yn^>^^ 


O 


-e — rt 


n+2 


c- - 


( 

— *- 


y 









Fig. (28). 

not subject to errors of samphng and the fitting curve should, 
therefore, pass through the summits of all the 2/'s, or, perhaps 
better, touch each of the fines joining adjacent summits. The 
curve only differs from the neighbouring outline of the polygon in 
that the latter is discontinuous, it alters its direction relative to the 
axis of X by jerks at equal intervals c measured along OX, whereas 
the former must rise gradually and continuously and then fall in 
the same way. This is one sense in which we mean that the fitting 
curve removes the roughness of the observation statistics — ^it gets 
rid of jerks besides fiUing gaps in the observations. 

It will be clear that as n increases and c diminishes (and this is 
what we aim at in collecting statistics, though it has not been assumed 
in what immediately follows) the discontinuity in the polygon 
becomes less and less pronounced and the outline of the figure 
approximates more and more closely to the 
curve. Moreover this approximation gains in 
intensity if we make the slope of the curve at 
each appropriate point the same as the slope 
obtained by joining up the summits of adjacent 
ordinates of the polygon. 



^^•^r+r-^J 



yr%i 



Now the expression 



{yr+i-yr)lG 



CURVE FITTING 183 

is the measure of the gradient from the rth ordinate to the (r+l)th, 
and 

yr+i-yr _«"r M^-l) • • ♦ {n-r+l) _ n{n-l) . . . (n-r+2y \ 
c c\_ 1-2 ... r 1-2 .. . (r-1) J 

_a" n(n—l) . . . {n—r-{-2)rn—r-^l 1 
~~c 1-2 .. . (r-1) L r ~ J 

n—2r+l 

=yr . 

re 

If this be also taken as the gradient of the tangent to the curve at 
the point midway between (a;^, y^) and (aj^+i, yr+i), calling this point 
(a;, y) we have, since, in the notation of the differential calculus, 

— is the measure of the gradient of the curve at this point, 
dx 

dy^ yr+i-Vr 
dx c 

n-2r+l 

==yr 

re 
And 

x=i{Xr+x,^i)=Krc+{r+l)c]=l{2r+l) 

2 1-2 .. . (r— 1) L ^ J 2r 

Hence 

^^ n-2r+l _ 2ry ^ (n+2)-(2r+l) _ 2y L^2-^'' 
re n-\-l re {n-\-l)c\ c 



%_ 22/ /„^2_2. 



Thus 



dx {n-\-l)c\ c 

But if we had started with any other two adjacent ordinates 
instead of i/r and y^+i we should have been led to exactly the same 
relation connecting the corresponding x and y of the required 
curve, for r, which serves to particularize the ordinates, does not 
appear in the relation at all — their individuality has been eliminated. 
The above equation may thus, if we please, be taken as holding 
good for, and therefore defining, all points {x, y) of the fitting curve : 
it is, in short, the differential equation of that curve. 

The equation may be slightly simplified by transferring the 

origin to the point {n-{-2)- , , evidently the point O' in fig. (28) 



184 STATISTICS 

corresponding to the maximum ordinate of the polygon or curve. 
Algebraically, this merely means that for x we must write 

x-\-- L in the equation, which then becomes 

dy_ 2y [ 2a;\_ 4:xy 



dx (n-\-l)c\ cj (n+l)c2 

We may pass to the equation proper of the curve by integration. 
Thus, separating the variables, 



. J y {n-\-l)cV 



2x^ 
Therefore, log y-\- + A=0, 

where A is a constant. 
Hence ' 2/=2/o6''''^''^"+'^ 

where y^ is a new constant. 
This may be written 

y^Yoe-^'/''^', . . . (1) 

where a'^=(n-\-\)c^l4:, and it is called the probability curve or normal 
curve of error.* 

Let us now see whether the procedure so far followed is applicable 
in the case of an unsymmetrical or skew distribution of statistics. 
With this object we will suppose the frequencies of observations in 
successive groups to be represented by the corresponding terms in 
the expansion 

vyivh 1 \ 

and as before we can form a frequency polygon by joining the 
summits of the ordinates 

n(n—l) ^ „ o 

[* Karl Pearson's method of getting the normal curve equation has been 
adopted as the basis of the above discussion, in preference to that usually 
followed, which develops the curve also from the binomial expression but some- 
what on the lines of Laplace and Poisson. They showed that the sum of all the 
terms lying within a range t on either side of the maximum term in the expan- 
sion of {p + g)" is approximately 



V2ir-(T[_J_t 



where (r= ij{npq), whence the equation of the curve is derived. (See Historical 
Note at the end of Chapter xviii. )] 



CURVE FITTING 185 

erected on the axis of x at distances from the origin given by 
^i=c, a?2=^^c, x^=^6c^ . . . , x^j^y=\n-\-\)c, 

the figure being very similar to that in the symmetrical case. 

The gradient of the fitting curve where it touches the join of 
{x^, 2/r) to (a;^+i, i/r+i) is given by 

dy_ yr+i-yT ^ 

dx c 

and we must try and express the right-hand side as before in 
terms of {x, y), the co-ordinates of the mid-point of the hne joining 
(^r> 2/r) to (a:^+i, i/r+l)- 

We have 



dy 
dx 



_ir r.(^.-l). . . {n-r-\-\) _ n(n-l) . . . {n-r+2^ 1 

cL 1-2 ... r ^ ^ 1-2 .. . (r-1) ^ ^ J 

-1) . . . (7i-r+2)p 
1-2 .. . {r-l) L 



j3"-y-^ n{n-l) . . . {n-r+2)\'n-r+l _ 1 



c 
Also 

2x=Xj.-\-x^+-^=rc+{r-\-l)c={2r-{-l)c 



1-2 .. . (r-l) . L ^ J 



Thus 



2y/n-r+l \ //n-r+l , \ 



dy_2y/n—r+l 
dx 



2v 
=J-[{n^\)q-r(p-\-q)\l[(n+\)ci+r{p-q)\ 
c 

2?/ 
=A^(n+l)qc-{jp+q)(2x-c)]l[2(n+\)qc-^{p-q)(2x~c)\, 
c 

This, being true for all such pairs of values of x and 2/, is now in a 
form independent of any particular point on the curve we seek ; 
in other words, it may be taken as the differential equation of the 
curve, and it is evidently of the type 

dx iP+yx) 

where a, jS, y involve only p, q, w, etc., the constants of the distri- 
bution we set out to fit. 



186 STATISTICS 

The equation is simplified if we transfer the origin to the point 
(a, 0), when it becomes 

dx yx-\-h 
where 8=jS+ya. 

To integrate, separate the variables as before : 

'dy [ X 



/-+/■ 



-dx=0. 



y Jyx-\-h 

Therefore, log y+ ^ /•(y^+3)-3^^^Q 

yj yx-\-h 

X 8 

log 2/-h-— - log (ya;+S)+A=0, 

y 7 
where A is a constant, 

or y=Ee-^''^(yx+hfly'', 

where B is a constant. 
It may be written 



y=y.«-"(i+a)'' • • • 



(2) 



where k=^l/y;a=S/y, and 2/0 is a new constant. 

This, then, may prove a suitable type of curve to fit a set of 
statistics forming a skew frequency distribution, but the question 
now arises whether equations (1) and (2) are the most general 
types possible. Clearly (1) is only a particular case of (2) obtained 
by making p=q, and, this being so, (2) may itself be a particular 
case of some still more general type. 

Light may be thrown on this if we consider the geometrical 
bearing of the differential equation obtained in the last case : 

dy y{a—x) 



dx P-\-yx 



(3) 



The presence of y and (a—x) in the numerator of the right-hand 

dij 
side of (3) shows that — vanishes when y=0 and when x=a, i.e. the 
dx 

curve touches the axis of x where the two meet and there is a 

maximum point on the curve at x=a. (Since a is the particular 

value of the organ or character x for which the frequency is a 

maximum, a is of course the mode.) Now these two characteristics 

are the very ones to which we wished to give symboUcal expression 

since they serve to describe in broad outline what was agreed to 



CURVE FITTING 187 

be the trend of the majority of frequency distributions — the rise 
from zero to a maximum, at first gradually, then faster, and, after 
passing through the maximum, the fall to zero again, generally at 
a different rate. 

As to the denominator of equation (3), the corresponding equation 
for type (1), before the origin was changed, was similar to equation (3), 
except that it contained no x term in the denominator, and that is 
readily understood when we note that y is a multiple of {p—q) 
and thus vanishes when p=q. Now, if from (3) we get a less 
general tjrpe of curve by dropping the x term in the denominator, 
we may perhaps get a more general type by adding an x'^ term, and 
even an x^ term, an x^ term, and so on. In fact there seems no 
reason why the denominator should not be any function of x, say 
f{x), which, however, we shall suppose for simplicity capable of 
expansion in a Maclaurin's series of ascending powers of x which 
converges quickly. 

We are led to propose, therefore, as more general than (3), the 
differential equation 

dy y(x-\-b) 



dx px^-\-qx-{-r 



(4) 



We stop at a;2 in the denominator because it has been found, if we 
may anticipate results to save needless labour, that beyond this 
point the heaviness of the calculation involved and the decreasing 
accuracy of the higher moments that have to be introduced out- 
weigh any other advantage gained. The curve or set of curves 
resulting from the integration of equation (4) is known as Karl 
Pearson's Generalized Probability Curve, and their author has 
stated that, while it comprises the two other types as special cases, 
it practically covers all homogeneous statistics he has had to deal 
with. 

Just as the differential equations in the first two cases considered 
were related respectively to the symmetrical and the skew binomial 
expansions, so is equation (4) related to the hypergeometrical 
expansion 

the successive terms of which express the probabiUty that r black 
balls, (r— 1) black balls and 1 white ball, (r— 2) black balls and 
2 white balls, . . ., r white balls, will be drawn from a bag contain- 
ing pn black balls and qn white ones, where ip-\-q) = l, when r balls 
are drawn in all, each being replaced before the next is drawn. 



188 



STATISTICS 



If the terms of this expansion are represented by ordinates of 
which the summits determine a polygon as in the binomial cases, 
the corresponding expression for the gradient of the curve at any 
point is given by an equation of type (4). We need not go over 
the detailed proof of this statement since it follows precisely the 
same lines as in the previous cases. 

The method of integration of the equation 

dy_ y(x-\-b) 
dx ' px^-j-qx-j-r 

depends upon the nature of the roots of the quadratic in the 
denominator which may be written 



x+, 



px^-\-qx-]-r=p\ ^ . 

4 



W pJJ 



^+^ - 



2p 



4^2 



4:pr 



4pr /J 



x-h— 



4r2 , 
— — -/cU- 
g2 



.,], 



where k =q^l4:pr, and it is evident that the quadratic splits up into 
real factors if k{k—1) is positive. This is the case when k has any 

negative value, or when it is positive 
and greater than 1, the truth of which 
may be seen more effectively if the 
curve 

2/=/c(/c— 1), 

K + (>\) 

a parabola symmetrical about the line 
K=i, be drawn, fig (29), by plotting 
y against k. 
Further, the product of the roots of the quadratic 

px'^-^qx-\-r=0 
4:pr 




Fig. (29). 



IS 



4 



p q" ^pr g^ 

so that the roots when real will be of the same sign if k is positive 
and of opposite signs if /c is negative. The boundary lines 

K=0 and /c=l 

thus divide the whole field into three parts, as shown in fig. (30), in 
one of which the roots are real and of opposite sign, in the next 



CURVE FITTING 



189 



the roots are imaginary, and in the third the roots are real and of 
the same sign. At the boundaries we get particular cases as 
follows : — 

K=0 : this requires q=0, since K=q^/4:pr, which makes the 
roots of the quadratic equal but of opposite sign, unless p=0 also, 
and in that case both roots are 
infinite ; 

K=l : the roots are real and equal 
and of the same sign ; 

K=cc: this requires p =0 or r=0; 
in the former case one root of the 
quadratic is infinite, and in the 
latter one root is zero. 

Thus, returning to the differential 
equation, the curves which result 
from the integration 

'dy f (x-]-b)dx 





y 












'^ \ 


•fe- 


c 


? 

« 






/ 


i \ 


1 


.51 




09 




h 


«.?>^ 




u 


cs 






/« c 


y II 


^ 


"U 




/— 2" 


c °' 


\ ** 


03 


'■" 


c 




? "* 


II 


\i 


1 


■2 

1 


II 

1 


/ 


2l 




.o 







/ 


F 


K 



Fig. (30). 



J y J: 



px^-{-qx-\-r 

are of different types according to the value of k, which is therefore 
called the criterion. 

Type I. — /c—^^. Roots of px^-\-qx-\-r=0 real and of opposite sign. 
In this case we may write 



and so get 



px'^-^qx-{-r=p{x-{-a)(x—P') 
{x-]-b)dx 



J y J via 



=0, 



p{a'+x){p'-x) 
or, transferring the origin to the point (—6, 0), the mode, we have 

'dy . f xdx 



or 



j y^]p^a'-b+x){p'-^b-x) 

[dy [ xdx 

J y J p(a-\-x) 



=0, 



where 

Therefore, 

where A is a constant. 



a=a'-6, j8=iS'+6. 



, Ifa dx I f B dx ^ . ^ 

log y—- -+- -J- -+A=0, 

^^ pJa-\-xa+P pJp-xa+p 



190 STATISTICS 

Thus log y= ^ [a log (a+a;)+^ log (iS-a;)]+log B, 

where B is a constant, 



whence y='B(a-\-x)v^'>-+P')[Q—xY^'^'^^^ 

where v=l/p{a-^P) and i/o is a new constant. 

This is a skew curve of limited range, bounded by the lines x=—a 
and a;=+iS, with the mode at the origin. 

Type II. — K=0. q=0, but not p=0. Roots of px'^-{-qx-^r=0 
equal and of opposite sign. 

This curve is just a particular case of type I., which reduces to 

y=y.(i-„-.) , . . • (6) 

symmetrical about the axis of y (because for any value of y there 
are two values of x, equal and of opposite sign) and of limited 
range bounded by a;=— a and x=-^a, with the mode at the origin. 

Type III. — K = oz.* p=0, but notr — 0. One root ofpx'^-\-qx-{-r=0 
infinite. 

This is the skew binomial case over again. It may be also de- 
duced from type I. by making one root, say ^' , tend to infinity. 
The curve then takes the form 

because j8=jS'+6, so that j8 tends to infinity with ^'. Hence 

where A=— jS/ic. 

1+^) e , . . . (7) 

a skew curve limited in one direction by the Hne x=—a, with the 
mode at the origin. 

[* Although theoretically this type corresponds to an infinite value for /c, in 
practice it will as a rule give a reasonable fit provided k is numerically greater 
than 4. (See W. P. Elderton's Frequency Curves and CorrdcUion, p. 50)]. 



CURVE FITTING 191 

Type IV. — /<:+^^ and <1. Roots of px^-]-qx+r=0 imaginary. 
Put k{k—1) =—X^, and the differential equation then leads to 

:-{-b)dx 






2p) "^ q^ ^ J 
Transfer the origin to the point ( — -±-, 

2pl 






1. f ^■■r^\Jb_q_\_g_ ^_^^ 



log y=A+- log k2+4 — + — _^ J^ tan-i , 
^2p ^\ ^ q^ y\p 2p^j2rX 2rA 



where A is a constant. 



-1 

p2^ 



Therefore, y=yjl+^J e"^*'" '* . . . (8) 

where a= — , m=— — , v =— — b—-±- 

q 2p ap\ 2pj 

and 2/o is a constant. 

This is a skew curve of unlimited range in both directions. The 

position of the mode is found by putting — =0 in (8) after differ- 

dx 

entiation, or, what comes to the same thing, is seen by direct refer- 
ence to the differential equation itself. Thus the distance of the 
mode from the origin 

= — ib—^\=vpa 



2p 

■—va/2m. 



Type V. — K = l. Roots of px^-\-qx-]-r=0 real and 
The equation to integrate becomes 

"dy f {x-\-b)dx 



/?=/■ 



' '''h: 



192 STATISTICS 

Transfer the origin to the point ( — _, ), and this becomes 



J y J 'px^ 



dx 



log 2/=A+- log x— . ( 6-^ )-, '\ 

p p \ 2pjx ; 

where A is a constant. i 

-l(b-l)l \ 
Therefore, y=yQX^'Pe ^^ ^p'^~ 

y=yoX-e-7/x, ... (9) I 

where s = — l/p, y=-i b—— ), and 2/0 is a constant. 

P\ 2pl 

Here x cannot become negative, so that the curve is skew and \ 

limited in one direction. The distance of the mode from the origin j 

Type VI. — K-\-^^ and >\. Roots of px^-\-qx-\-r=0 real and of the i 

same sign. i 

Equation becomes ] 

rdy^ r {x-^h)dx j 

J y J p(x-\-a)(x^P) i 

iog,^fr±i^.-i-+i^.-ij.. ' 

]\jp{^-a) x^ra p{a-p) x+^j i 

=A+_J— [(6-a) log {x-\-a)-(b-P) log (a^+jS)], i 
p(P-a) 

where A is a constant ; ♦• 

or, transferring the origin to (— j8, 0), j 

y=yo\x-(p-a)f^^h^(^ . \ 

y=yo(x-a)^^-'^^ . . . (10) I 

where a=jS— a, q2 = {b—a)/p(P—a), qi = (b—^)/p{P—a), and yo iB a, : 

constant. j 

This is a skew curve bounded by x=a in one direction.- The j 

distance of the mode from the ongm=— {b—^)=aqj{qi—q2). j 



CURVE FITTING 193 

Type VII. — /c=0, ^=0,^=0. Boots of the quadratic px^-\-qx-]-r=0 
both infinite. 

This is the symmetrical binomial case over again and the integra- 
tion reduces to 







J y~ 


'!'>■ 


or, transferring the origin to (— 


■b, 0), 






fdy^ 
} y~ 


=/;- 


. 




\ogy-- 


=A+,>. 


where A is 


a constant. 








Therefore 


y- 


=y,e-""'^ 



• . (11) 

where i/o is a constant and a^=z—r. 

This curve, the normal curve of error, is symmetrical about the 
axis of y, where mean and mode coincide, and it is of unlimited 
range on either side of it. 



CHAPTER XVI 

CURVE FITTING {continued) — the method of moments 

FOR CONNECTING CURVE AND STATISTICS 

We have now completed the first stage of the discussion upon which 
we embarked : we have found by the application of general prin- 
ciples various types of curve, represented by different equations, 
which are said to fit more or less satisfactorily a considerable number 
at all events of frequency distributions composed of homogeneous 
material. 

Our next task is to pass from the general to the particular, to 
see how to set up a connection between an actually observed fre- 
quency distribution and the appropriate theoretical curve. This 
again seems to break up into two parts — (1) to find a way of deciding 
which type of curve to adopt in a particular case ; (2) to determine 
the constants of the curve in terms of the observed statistics ; but 
since the criterion, k, which distinguishes one type of curve from 
another is itself a function of the constants of the curve before 
integration, it follows that the solution of the first part is incidental 
to that of the second. 

The general method proposed for determination of the constants 
of the curve in terms of the observed statistics is the now well-known 
method of moments due to Karl Pearson, whereby the area and 
moments of the fitting curve are equated to the area and moments, 
calculated from the statistics, of the observation curve. 

If a frequency table be drawn up (see Table (40)) showing the 
number / of observations corresponding, to the deviation x of each 
value, or group mid- value, X of the character observed from some 
fixed value, the expression 

^1/1+^2/2+ • • • +^rfr+ ' • • 
is called the first moment of the distribution with reference to the 
fixed value, which may be termed the origin. Similarly, 

is called the second moment, Zx^f, the third moment, Ux^f, the 

191 



CURVE FITTING 



195 



fourth moment, and so on. The following notation will be found 
convenient for working purposes : — 



N\ Uxf 



= -=r^, V 



W^_Zx^f 



Undashed letters are reserved for use when the distribution is re- 
ferred to its mean as origin, in other words when the deviations of 
the X*s are measured from the mean X. 



Table (40). 



Deviation. 


Frequency. 


First 
Moment. 


Second 
Moment. 


Third 
Moment. 


Fourth 
Moment. 




fr 




^V2 


^V2 




Totals . 


N 


N', 


N'2 


N'3 


N'4 



Now each N in the frequency, table is the sum of a number of 
discrete quantities which only tend to form a continuous series as 
the class intervals are made very small and the number of observa- 
tions is made very large. The corresponding frequency polygon 
or histogram, if we drew it, would at the same time tend to become 
a continuous curve, the observation curve. If that Hmiting stage 
were attainable, if we could actually get an infinitely large sample 
of observations in which the character observed changed by infinitesi- 
mal amounts, we could then replace the isolated /'s of observation 
by the corresponding y^s, the ordinates of this observation curve, 
and to get the moments we could write instead of the discrete 
sums 

2"/, Uxf, Ex^ , . ., 

the continuous integral expressions 

\y'dx, jxy'dx, jx^dx, . . ., 

taking in the whole sweep of the curve by integrating throughout 



196 STATISTICS 

the range of deviation x. We should then have, if areas and 
moments are equated according to Pearson's method, 

jydx=jydx, \xydx=jxy'dx, jx^ydx=jx^y'dx, . . .,jx^ydx==jx^y'dx, 

where y is the ordinate of the fitting curve corresponding to the 
ordinate y' of the observation curve. 

In practice, however, it is impossible to go to this limit : we 
cannot deal with an infinitely large sample, so we take as large a 
sample as is convenient, calculate the rough moments, N, N'^, N'2 . • ., 
and find approximately what corrections or adjustments are neces- 
sary to obtain the moments of the observation curve, a procedure 
which is really equivalent to the determination of the area of a 
curve when only a number of isolated points thereon are known. 

For the full analytical justification of the method of moments 
the reader is referred to Professor Pearson's original paper, On 
the Systematic Fitting of Curves to Observations and Measurements 
[Biometrika, vol. i., pp. 265 et seq. ; also vol. ii., pp. 1-23], where 
it is shown that ' with due precautions as to quadrature, it 
gives, when one can make a comparison, sensibly as good results 
as the method of least squares.' The latter, which is the traditional 
way of approaching all such problems, is shown to be impracticable 
in a large number of cases, either because the resulting equations 
cannot be solved, or, when they are capable of solution, because 
the labour involved would be colossal. 

Let us consider next how to deduce the area and moments of the 
observation curve from the statistics, in other words how to get 

jy'dx, jxy'dx, jx^dx, . . ., 

the integrals being taken throughout the range of the curve, when 
we Imow the frequencies corresponding to only a certain number 
of values or elementary ranges of the deviation x. 

Now the character observed may be capable of the deviations 
actually recorded and of no values in between, e.g. measuring 
deviations from ' no rooms ' as origin, we might have /^ one-roomed 
tenements, /g two-roomed tenements, /g three-roomed tenements, but 
there could be no such thing as a two-and-a-half or a three-and-a- 
quarter-roomed tenement ; on the other hand, any recorded devia- 
tion, x^, may be only the mid- value (used as a convenient and 
concise approximation) of a group of observations including all in 
the continuous range from (a;^— J) to (a^^+J), where unit deviation 
is the class interval : thus we might have f^ males deviating by 
-j-6 in. from 5 ft. (comprising all the males observed between 5 ft. 



CURVE FITTING 



197 



5J in. and 5 ft. 6J in.), /g males deviating by +5 in. from 5 ft. (com- 
prising all males between 5 ft. 4J in. and 5 ft. 5J in.), and so on. 
These two cases must be discussed separately. 

(1) When the observations are centred at definite but isolated values 
of X. 

The problem is to find 

^x'^y'dx 

(the Tith moment) when we have no definite curve given but we 
know the values of x and y' at a number of isolated points, say 

This is equivalent to discovering a suitable ' quadrature formula,' 
i.e. a good approximation to 

\zdx 



O /f Wz 2 /z 3 

Fig. (31). 



h\ hlhZ 



Fig. (32). 



Ph 



in terms of known points 

(♦^0> '^0)' V**^!' -^l)' ('^'2' ^2/' • • • \^p> ^p)' 

where we have written z in place of x^y' , and we may generally 
take the ordinates to be at equal distances, h, apart. Several 
such formulae have been suggested and they vary according as the 
2's are situated at the ends (fig. (31)) or at the centres (fig. (32)) 
of the h intervals. The second type is perhaps the more useful of 
the two, and we shall work out one formula in illustration of it. . 
Consider the first five of the given points, namely, 

(^0> '^0/' ('^1' ^l)j • • • (•^4j ^4)-. 

As a simple * curve of closest contact ' let us find the parabola of 
type 

z=CQ-j-CiX/h-\-C2X^/Ji^-\-c^x^/h^-^CiXyh* . . (1) 

which goes through these five points, where the c's are constants to 
be determined. We may without loss of generality take the axis 



198 



STATISTICS 



of z to coincide with the middle one of the five ordinates, so that 
the known points on the curve become 

(-2h, zo), (-h 2i), (0, zg), (+/i, zs), (+2^. «4), 
and on substitution in (1) we get 



2=0=^0— 2C1+4C2—8C3+I6C4. 



Za Cn 



Z4=Co+2Ci + 4C2 + 8C3+16C4. 




-2h -h O +A +2>^ 

Fig. (33). 



^1 — ^0 ^11^2 ^3 I ^4* 
2^3 = Cq 4" C^ + C2 + C3 -j" C4 . 

These equations are just sufficient 
uniquely to determine the c's, and 
hence the parabohc curve of closest 
contact, in terms of the five given 
points, but for our purpose it is not 
necessary to find all the c's. Suppose 
our object is to find the area of the 
shaded portion of fig. (33) in terms 
of the co-ordinates of the five given 
points. This area 



+hl2 



zdx 



{CQ-\-Cix/h-i'C2X^/h^-\-CQX^/h^-\'C^x*/h^)dx 



=Co^+ cJb/U-\- cJi/SO. 
But the equations between the z's and c's at once give 

Z2=Co> ZQ-\-Z^=2{CQ+4:C2+lQCi), 2i + Z3=2(Co+C2+C4). 



Thus 



Therefore 



2C2+2C4 = (Zi+Z3)-22;2 



24c2=16(2;i+Z3)-(Zo+Z4)-3022 

24C4 = (Zo+Z4)-4(Zi+23) + 6Z2. 

Hence, by substitution, the shaded area becomes 

2£?a;=^[z2+^F«-|l6(Zi+Z3)-(2o + Z4)-30«2| 

+TT^\(Zo+z^)-Hzi+z^)+Qz^\] 
= A[5178z2-17(.o+^4)+308(Zi+.3)], 



£ 



h/2 



(2) 



/: 



CITRVE FITTING 199 

these particular ordinates being appropriate when the axis of z 
coincides with the z<^ ordinate. 
Similarly, it can be shown that 

•+3A/2 }i 

■m 2^^=2i^272;o+172i+52;2-Z3]. (3) 

by finding the parabolic curve of closest contact through (0, Zq), 
(A, Zi), (2^, Zg)' (3^» 2:3), the axis of z coinciding now with Zq. 

cHv+m 

Now we require / zdx 

(see fig. (32)), and this may be obtained by spHtting up the integral 
thus 

/•+3A/2 /•6/1/2 nhl2 r(p-m f(P+i)h 

+ + +...+ + 

J~h/2 J3h/2 Jbhll kv-i)^ \v-\)h 

and applying the formulae (2) and (3) to evaluate these sub -integrals. 
The first and last come under head (3), while all the rest come 
under (2). In fact, we fit together portions of curves of parabolic 
type based on the successive groups of points 

(0, 1, 2, 3), (0, 1, 2, 3, 4), (1, 2, 3, 4, 5), (2, 3, 4, 5, 6), . . . 
(p— 4, p-3, p—2, p—\, p), (p— 3, p—2, p—l, p), 

and as the points overlap, in the sense that neighbouring groups 
have points in common, the curves dovetail into one another and 
so provide a fairly good approximation to what we want in the way 
of integral expressions giving areas based upon the positions of 
certain known points. 
We have, then : — 



8A/2 h 

zdx=—[27zQ+llZii-5z2-z^'] 

-hl2 24 



5/1/2 }i 

r =2:^^=^-;:^[5178z2-17(2:o+2J4)+308(2;i+2:3)] 
3A/2 57 bO 



6A/2 57 dO 



i: 



"'''=«i'^'^~[5n»z,-n{z^+Zt)+S08(z,+z,)] 

hji 57 dO 



^^=^^5n8zp.^-n{Zp.,-{-z^)+S0S(Zj^^+Zp.,)] 
(j)-i)h 5760 



zdx=~[21zp-\-llZp.;i^+5zp.2—Zp-^]. 
J(j>-i)h 24 



200 STATISTICS 



f 



2(^a;=^^^^[6463;:o+4371Zi+666922+5537z3+6463z^ 



Hence, by addition, 

f(p+m , h 

zdx= ^ 

•A/2 5760 

+4371z^_i4-6669v2+55372;^_3] 
=A[M220(Zo+s)+0-7588(Zi+Vi)+M578(z2+V2) 

+ 0-9613(2;3+V3)+ (2=4+2^5+ . . . +V4)]- 

In effect, since z—x^y', this means that to calculate the moments 
from the given statistics we may work simply with the observed 
ordinates or frequencies, as drawn up in Table (40), so long as we 
modify the first four and the last four by multiplying them by 
suitable factors. In particular, when the frequencies at the be- 
ginning and end of the distribution are very small, that is to say, 
when there is high contact at each end of the frequency curve, 
we may dispense even with the modifying factors also since we 
may assume that before the first and after the last ordinate observed 
there are others which are so small as to be negligible. 

Thus, given high contact at each extremity of the observation 
curve, we may write 



/: 



zdx=h2Jz, 

-hii 



or, if we take the class interval as unit in measuring x so that h=l, 
this gives 

jyx^dx=Zfx^, 

where the integral may now be taken as referring to the fitted 
curve, since the moments of the theoretical and of the observa- 
tional curves are to be equal, and the integration traverses the 
extent of the curve. When, however, there is not high contact at 
the extremities the same equation holds good if we multiply the 
first and last of the observed /'s by 1-1220, the second and the last 
but one by 0-7588, the third and last but two by 1-1578, and the 
fourth and last but three by 0-9613. 

In particular, when 7i=0, integrating throughout the curve, 

\ydx=i:f=^, . . . (4) 

which, being interpreted, means that the area contained between 
the fitting curve and the axis of x measures the total frequency of 
observations, modified if necessary. 

Also, when the observation moments have been adjusted, if we 



CURVE FITTING 



201 



write /Lt and fju' in place of v and v in the notation previously pro- 
posed (see Table (40)), integrating again throughout the curve, 

\xydx/jydx=UxffN=fjL\, . . • (5) 

and the geometrical interpretation of this is that the foot of the 
ordinate passing through the centre of gravity of the area between 
the fitting curve and the axis registers the deviation of the mean X 
from the fixed origin. 

If deviations are measured from the mean of the distribution 
as origin i7(a:/) vanishes (see also Appendix, Note (5)) so that/>ti=0. 

Generally, we have, with the same limits of integration, 

jx^^ydx/jt/dx ^Z'x^Z/N =:/x' „ , 

and when the distribution is referred to its mean as origin the 
right-hand side is written /x„. 
We now pass to the second case. 

(2) When the observations appear in groups ranging between 
definite values of x, the range of each group as a rule being the same 
in extent. 

Since the usual procedure here is to treat each member of a group 
as though it were centred at the x at the middle of that group — 
e.g. a group of school girls 
each of some weight be- 
tween 7 stone and 7 stone 
5 lbs. would be treated as 
if all its members were of 
weight 7 stone 2-5 lbs. — 
this case evidently reduces 
to that already considered. 
It is necessary, however, to 
examine what correction 
must be made for assum- 
ing that all the members 
of the same group have 
the same x. 

Consider again the expression 

jx^y'dx. 

The contribution to the nth. moment coming from the Zj. group of 
observations (see fig. (34)) may be taken as the portion of the 

above integral between limits ' 



A 



Fig. (34). 



a^o-fr^--) and [x^j^rh-^-] where 



20^ STATISTICS 

Xq is the distance of the centre of the first group from the 
origin 0. 

But, since all the observations in the same group are treated as 
if they had the same x, by (2) this integral may be written 

Mr 



5760' 



[5llS{x^-{-rhr-l'7{{x^-\-r-2hr-\-{Xo+r-}-2h)- 



H-308{(a;o+r-U)"+(a:o+r+Ur}], 
where /^ is the frequency of observations in the group, and this, on 
expansion in powers of {xQ-]-rh) and h, 

^hMxo+rh)-+^[2^0n{n-lW(Xo+rhr-'^ 
57 bO 

+3n(n-l)(n-2){n-3)h^{xQ-\-rh)»-*+ . . .]. 
When we sum for all groups, the expression 

X^"'hMxo+rh)- 

r«=0 

gives evidently the nth moment of a set of isolated variables, 
/o, fi, /g, . . . fp, and by Case (1) it may therefore be taken as 
being practically equivalent to the required nth moment of the 
observation curve, assuming that there is high contact at each end oj 
the curve. 

The remaining terms, 

^^r^o 5760 

+Sn(n-l){n-2)(n-Z)hSxo-\-rh)^-^i- . . .], 

may accordingly be taken as the correction required. 

When n=0, these terms vanish, so we infer, just as in Case (1), 
that, when the integration is taken throughout the curve, 

j2/(Za:=27=N, . . . (4) bis, 

or, the area between the fitting curve and the axis of x measures 
the total frequency of observations when the class interval h is 
treated as the unit in measuring x. 

Again, when n—l, the corrective terms vanish, so we likewise 
infer, as in Case (1), that, with the same limits of integration, 

jxydxljydx=I!xf/N=fjL\, . . • (5) bis, 

and that jLti=0. 

When n—2, the reduction of the corrective terms gives 

h'^ 
second unadjusted moment = second adjusted moment-}- — ^M* 

1^ 



CURVE FITTING 203 

or, dividing throughout by Ehf and bearing in mind the notation 
adopted with the mean as origin, 

when A=l as before. 
When n=3, 

third unadjusted moment =third adjusted moment -\- —Zf^(xQ-{-rh) ; 

4 

but, if we refer the deviations to the mean of the distribution as 

origin, Zf^{xQ-\-hr) vanishes. 

Therefore, i"'3=^3 • • • (^) 

When w=4, 
fourth unadjusted moment 

=fourth adjusted moment-j--— i^/r(a^o+^^)^H — ^M- 

2 80 

Hence, dividing through as before by Zhf and taking A as 1, 

Therefore, /^4=^4-ii^2+2ib • • ' (^) 

To sum up, the general procedure in Case (2) is to calculate 
N, N'l, N'2, N'g, N'4 directly from the statistics and so deduce 
v'l, v\y v\, v\. Then, transferring the origin to the mean, the v"b 
become vi, V2, v^y v^ (see Appendix, Note 5), and finally the cor- 
rected /x's are given by 

These adjustments, originally due to Dr. W. F. Sheppard * [Pro- 
ceedings of the Lond. Mathl. Socy., vol. xxix., pp. 353 et seq.], are 
applicable only when the 
curve of distribution has 
high contact at each ex- 
tremity as very frequently 
happens. To this case 
we shall confine ourselves, 
and when it does not hold 
the unadjusted moments 
may be used as a rough approximation failing a more refined but 
also a more intricate adjustment. 

The way in which the three chief kinds of average are related to 

[* To obtain Sheppard's adjustments we have followed the method indicated 
in Elderton's Frequency Curves and Correlation, pp. 28, 29. ] 




204 STATISTICS 

the fitting curve is of interest and deserves recapitulation. Whether 
the observations are classed as in Case (1) or as in Case (2) : — 

(1) the ordinate drawn through the highest point of the curve, 

since the frequency there is a maximum, fixes the modal 
value of X ; 

(2) the median X is determined by the ordinate bisecting the 

area between the curve and axis, since there are an equal 
number of observations on either side of it ; and 

(3) the mean is determined by the ordinate through the centre 

of gravity of the area between the curve and axis. 

We have still to show how to express the constants of the fitting 
curve in terms of the moments calculated from the given statistics, and 
it will be convenient now to make our approach from the other end. 

Take the general equation of the fitting curve, express its con- 
stants in terms of its moments, and substitute for the latter the 
values determined from the statistics, since the basis of the fitting 
is the equalization of the moments of the observational curve and 
of the theoretical curve. This will enable us to determine k, the 
criterion for fixing the type of curve suitable to the given distribu- 
tion. When the type has been fixed it is, as a rule, not a very 
difiicult matter to express the constants of the particular type 
again in terms of the observational moments. 

Now the general differential equation of the fitting curve was 

dy y{x+b) 



dx px^-\-qx-\-r 
hence 

j{px'^-{-qx-]-r)dy=jy(x-]-b)dx, 

where the integration is to traverse the complete curve. 

Therefore, multiplying both sides by x^, 

j{px''+^-]-qx''+^-\-rx'')dy=j{yx''+^-\-byx'')dx; 

or, if we integrate the left-hand side by parts 



[{px''+^-}-qx''+'^+rx'')y']—jy{n-\-2px''+^-\-n-\-lqx''-{-nrx'>'-^)dx 
=j(yx^+^-{-byx^)dx. 

But the expression in square brackets vanishes at both limits if 
we suppose y to be zero at each end of the curve, so that the equa- 
tion reduces to 



{l-{-pn-^2)jyx''+^dxi-{b-\-qn-i-l)jyx''dx-\-rnjyx'^-'^dx=0, ... (9) 



CURVE FITTING 205 

Now if deviations are measured from the mean of the distribution, 
we have 

jyxdx—'NfjLi=0, jyx^dx=l^fi2y jyx^dx='NfjL^, etc., 

and therefore, putting n=3 in the above relation, 

put 71=2, (l+4i>)N/x3+(6+3g')N//,2=0 

put 71=1, (1+3^)N/X2+^N=0 

put 71=0, (6+g)N=0. 

Thus b =—q, and, on substitution in the other three equations, we get 
S/iiP + 3jLt3g+ 3/^2^ +/X4 =0, 

3/>t2P + r-\-fjio^=0, 

three simple linear equations to find p, q, r, the solution of which 
leads to 

^ = — (2^2i^4— 3/x23— 6/a32)/(10/X2/X4— IS/A^g— I2/X23), 

^=-6 = -/i3K+3ia'^2)/(10ia2/X4-lV2-12/x23), 

We have thus expressed p, q, r, and 6, the constants of the fitting 
curve in terms of the moments of the observed distribution, but the 
results may be rendered more concise by writing 

Pi=i^yi^\ P'z^i^ji^h, • • • (10) 

whence 

p=-(2^2-3i3i-6)/2(5ft-6ft-9), .... (11) 

g=-6=-V(/x2ft).(i32+3)/2(5^2-6ft-9), . . (12) 

r=-^2(4i82-3ft)/2(5ft-6^i-9) . . , . (13) 

And Ky the criterion for fixing the type of curve suitable to the 
statistics given, is immediately deduced from 
K =q^l^pr 

=A(iS2+3)V4(4ft-3A)(2^2-3ft-6) . . . (14) 

Also, since ~ vanishes when x = — b, this fixes the mode relative 
dx 

to the origin. ' But the origin is now at the mean, so that 
mode-mean=-6=- V(/^2iSi) • {p2+^)m^p2-^Pi-^) (15) 

And 

skewness = (mean— mode)/S.D. 

=6/V(M2) 

=Vi3i(i38+3)/2(5;8j-6;8i-9) . . . (16) 



CHAPTER XVII ' 1 

APPLICATIONS OF CURVE FITTING [ 

We are in a position now to test the application of these principles ' 
to given frequency distributions and we shall start by trying to ] 
find a curve to fit the record of marks obtained by 514 candidates 
in a certain examination (see p. 25). 

Example (1). — This example is chosen because it turns out, ' 
when we come to evaluate k, that it is well fitted by the normal ^ 
curve, Type VII , which is one of the simplest and at the same time ; 
the most important of all the types discussed. Before we start ' 
the numerical part of the work it will be well to express the [ 
constants 2/0 ^^d a of this curve in terms of the moments of the ! 
distribution. ! 

The equation of the normal curve is J 

If N be the total frequency, we have by equation (4) bis, p. 202, j 

f+co \ 

N= ydx 

J-co \ 

/'+03 '■ 

=2/0 e-*'/2a'^a., j 

dx — 

Put x72(j2=^^ so that —=(J^/2 and when a;=oo, f =00 also. j 

di J 

\ 
Thus N=2/o<^V'2f^%-^'(Zf j 

J -co j 

I 

=yocrV2V'7T (see Appendix, Note 8) .' 

=V(27r)cryo ... (1) \ 

206 ' . 1 



APPLICATIONS OF CURVE FITTING 



207 



Again 



r+oo / f+co 

^2= yxHx yds 

J -co I J -co 



2 V2 . (j^i/o 






N 



[Mr-f-"<^] 



2\/2 . ggyp Vtt 
' ' 2 ' 



N 



(«•-): 



vanishes at both Limits. 

fi^ = V2. (72/0 Vi • ctVN-ct^, by (1). 



since 

Therefore 

In fact, a is simply the S.D. of the distribution. 
And yo=N/\/(2^).cr. 

Table (41). Distribution of Marks obtained by 514 Candi- 
dates IN A CERTAIN EXAMINATION. 



Mean No. 

of 

Marks. 


Deviation 


Frequency 

of 
Candidates. 


First 


Second 


Third 


Fourth 


from 33. 


Moment. 


Moment. 


Moment. 


Moment. 




(^) 


.(/) 


ifx) 


ifx') 


if^) 


ifx') 


3 


-6 


5 


- 30 


180 


-1080 


6480 


8 


-5 


9 


- 45 


225 


-1125 


5625 


13 


-4 


28 


-112 


448 


-1792 


7168 


18 


-3 


49 


-147 


441 


-1323 


3969 


23 


-2 


58 


-116 


232 


- 464 


928 


28 


-1 


82 


- 82 


82 


- 82 


82 


33 




87 










38 


+ 1 


79 


+ 79 


79 


+ 79 


79 


43 


+2 


50 


+ 100 


200 


+ 400 


800 


48 


+3 


37 


+ 111 


333 


+ 999 


2997 


63 


+4 


21 


+ 84 


336 


+ 1344 


5376 


58 


+5 


6 


+ 30 


150 


+ 750 


3750 


63 


+6 


3 


+ 18 


108 


+ 648 


3888 


— 


— 


614 


-110 


2814 


-1646 


41,142 



208 STATISTICS i 

The first 4 moments referred to 33 as oriein and with the class ' 

interval, 5 marks, as unit of deviation, are i 

-110/514, 2814/514, -1646/514, 41142/514. I 

The arithmetic mean of the distribution j 

=:33H-5(-^if) ! 

=33-5(0-214008) j 

=31-92996. I 

The second, third, and fourth moments referred to the mean as . 

origin, and retaining five marks as unit of deviation, are given | 

(see Appendix, Note 5) by i 

1/2=2814/514-^2^5-42891 
j,3=_1646/514-3^i/2-^3_0-29296 
z/4=41142/514-4:ri/3-6:c2i;2-^*^78-79964. 

After making Sheppard's adjustments i 



/^2 — ^2~T2j /^3 — ^3' /^4 — ^4~4''2+ 2^4 5 



these become 



/x2=5-34558, /x3=0-29296, jLt4 =76- 11436. ] 

Thus j8i=/i23/^3^ =0-00056, jSg^/x^/^^^ =2-66365. ] 

Hence «=ft(iS2+ 3)^/4(4^2- %)(2i82-%- 6) | 

= (0-00056)(5-66365)2/4(10-65292)(-0-67438) \ 

=-0-00063. 

Since k and p^ are small and jSg does not differ greatly from 3, making ; 

p and q small, we may fit a normal curve to this distribution. 

The appropriate normal curve is . ^ 

2/=2/oe-^/2«.2, . 

where (t2=/x2 =5-34558 (5 marks as unit), j 

2/o=N/V(27rf>t2)=514/\/2^(5-34558)^=88-6903. I 

Hence the required curve has for its equation, writing results to 
three significant figures, 

j 
Now the mean of the distribution is at 31-92996, where the 

central ordinate of the normal curve is erected, and the distance i 

of any x, say x^^, from this point J 

=(33-31-92996)/5 (expressed with 5 marks as unit) ; 

=0-214008. . 



APPLICATIOlsrS OF CURVE FITTING 209 

Vny other x may be found in the same way and y can then be 
deduced from the equation of the curve by taking logs, thus 

log>o2/=log.o88-6903-^-^^bg,„e 

=1-9478762- (0-0406218)a;2. 

This enables us to calculate the ordinates of the normal curve and 
thence we could evaluate the areas by successive applications of a 
suitable quadrature formula. 

We can, however, get the areas direct by using a table of the 
probability integral, such as that due to Dr. W. F. Sheppard (see 
pp. 284, 285). In that case the corresponding abscissae have first 
to be expressed in terms of the standard deviation as unit, e.g. 

a;4o.5=40-5-31-92996=8-57004, 
and (7=5^/(5-34558) =11-56025, 

where the factor 5 is introduced because 5 marks was the unit in 
the calculation of /Xg (a process equivalent in effect to that previously 
adopted). 

Thus a;4o.5/cT=0-741336 
=$, say. 

The area of the normal curve up to the abscissa x/a or $ 
= 1 ydx 

J -co 

= r yoe-''''''^''''dx 

J -co 

J -CO 

=nP zdi 

J -co 

=N . i(l+<x), 

where - represents the area of the curve z= — =e~^^- between 
2 "^ V2,7 

and £. 



210 



STATISTICS 



Sheppard's Tables give the values of J(l+a) for different values 
of f , and when 

^=0-74, 1(1 +a) =0-7703500 
^=0-75, i(l+a)==0-7733726. 

Therefore, by interpolation, when 

^=0-741336, 1(1 4-a) =0-7707538. 

Thus the frequency of candidates with marks lying between and 
40-5 

=514(0-7707538) =396-17. 

Similarly the frequency of candidates with marks l3dng between 
and 45-5=452-20. 



othlGO 



lii 



80 



I 



s 




20 



ffii 



10 20 30 40 

Marks obtained 
Fig. (35). 



50 



60 70 



Hence the normal frequency for the group with 43 as mean 
number of marks =56-0, and the same method gives the area for 
any other group. 

The histogram of the observations and the curve plotted from the 
ordinates are shown together in fig. (35). 

In Table (42) are set out the calculated normal frequency (col. (4)) 
for each group alongside the corresponding observed frequency 
(col. (2)), and the differences between the two are shown in col. (5). 
We want to know whether the fit is a good one. 



APPLICATIONS OF CURVE FITTING 



211 



(1) 



Table (42). Comparison of Observed and Normal 
Frequencies in Examination Example. 

(3) (4) (5) (6) (7) 



(2) 



Mean No. 




Normal Frequency. 






Ratio of No. 


of 
Marks. 


Observed 
Frequency. 






Deviation. 


Sq. of 
Deviation. 


in Col. (6) to 
No. in Col. (4). 


Ordinates. 


Areas. 


3 


5 


3-9 


5-7 


+0-7 


0-49 


009 


8 


9 


10-4 


10-7 


+ 1-7 


2-89 


0-27 


13 


28 


23-2 


23-5 


-4-5 


20-25 


0-86 


18 


49 


429 


431 


-5-9 


34-81 


0-81 


23 


58 


65-8 


65-6 


+ 7-6 


57-76 


0-88 


28 


82 


83-7 


83-1 


+ 11 


1-21 


0-01 


33 


87 


88-3 


87-6 


+0-6 


0-36 


0-00 


38 


79 


77-3 


76-8 


-2-2 


4-84 


0-06 


43 


50 


561 


560 


+ 6-0 


36-00 


0-64 


48 


37 


33-7 


340 


-3-0 


900 


0-26 


53 


21 


16-8 


171 


-3-9 


15-21 


0-89 


58 


6 


7-0 


7-2 


+ 1-2 


1-44 


0-20 


63 


3 


2-4 


3-5 


+0-5 


0-25 


0-07 


•• 


514 


511-5 


513-9 


•• 


184-51 


X2=5.04 



Now with this object we might square each difference as in 
col. (6), sum the squares, and find the mean square deviation by- 
dividing by the total frequency ; this, after extracting the square 
root, would give what might be called the root-mean-square error, 
regarding the theoretical values as the true ones. In the above 
example it 

=V(184-51/514) =0-599. 

But this form of result, while it may be useful in some cases, 
e.g. in comparing two distributions of the same kind to some 
theoretical series, is open to objection ; for one thing it treats all 
the differences as if they were of equal importance in absolute 
magnitude, but a difference of 2, say, in a normal frequency of 10 
is clearly more serious than a like difference in a frequency of 60. 
The objection, however, goes deeper than that ; even when the 
root-mean-square deviation is found we are at a loss to estimate 
its precise relationship to the quality of fit, as there seems to be no 
definite connection between one distribution and another of a 
different kind : there is no standard case, so to speak, to which we 
can always appeal, where the fit is agreed to be good and supplying 
therefore a suitable root-mean- square deviation for comparison. 



212 Sl^ATlSTICS 

This leads us to the question : What constitutes goodness of 
fit ? Suppose by some means we have selected a theoretical or 
empirical formula to describe a certain frequency distribution in a 
given population ; if the frequency values observed do not differ 
from the theoretical frequencies by more than the deviations we 
might expect owing to random sampling, then clearly the fit may be 
regarded as a good one. And we have a measure of the fit if we 
can find the proportion of random samples, of the same size as the 
given distribution, showing greater deviations from the distribu- 
tion given by theory than those which are actually observed. 

Now Professor Karl Pearson has shown how this proportion can 
be calculated [Phil. Mag., vol. 1., pp. 157-175 (1900)] ; he finds the 
probabiHty that a random sample should give a frequency distribu- 
tion differing from that which theory proposes by as much as or by 
more than the distribution actually observed. This probability, P, 
is a function of ^, where 

y and y' representing the theoretical and observed frequencies for 
any particular group and the summation is to include all groups. 
It will be noted that this expression gives each difference {y—y') 
its appropriate importance by relating it to the frequency y of its 
own group. 

A table in Biometrika (vol. i., pp. 155 et seq.) gives the values of P 
corresponding to different values of ^^ (including all integral values 
from 1 to 30) and to values of n' , the total number of frequency 
groups, from 3 to 30 (see also p. 285). The mathematics in- 
volved in finding P is difficult, and the reader who wishes to enter 
into it must consult the original memoir, but the utiUty of the 
function has been proved by experience and it is readily applied 
in a particular case. 

In the above example ^- is found from col. (7) : it equals 5-04, 
and from the table of values of P, when ti' =13, we have 

P=0-957979 when x^=^^ 
and P =0-916082 when x^=^' 

Therefore, by proportional, interpolation, when '^^=5-04:, 
p =0-956303. Thus, supposing our data to follow the normal curve, 
in 956 random samples out of 1000 we should expect to get a 
worse-fitting distribution than that given by the sample actually 
observed. We may therefore conclude without hesitation that 
the normal curve provides an excellent fit in this particular instance. 



APPLICATIONS OF CURVE FITTING 



213 



We pass on now to fresh distributions to illustrate some of the 
other types of frequency curve. 

Example (2) deals with the percentage of trade union members 
unemployed at the end of each month for the years 1898 to 1912 
[data from the Sixteenth Abstract of Labour Statistics of the United 
Kingdom, Cd. 7131]. Table (43) shows the distribution of the 
180 records according to the percentage unemployed. 

The deviations are measured from the centre of the group (3-9— 5-2) 
as origin, and the class interval (1-3 per cent.) is taken as unit of 
deviation as usual. 

The first four moments are : — 



I.e. 



-29/180(=:c), 425/180, 397/180, 3053/180 ; 
-01611111, 2-3611111, 2-2055556, 16-9611111. 



Table (43). Distribution of Unemployed Percentages 
OF Trade Union Members 



Percentage 


Devia- 


Fre- 


First 


Second 


Third 


Fourth 


Unemployed. 


tion. 


quency. 


Moment. 


Moment. 


Moment. 


]Moment. 


0— 


-3 

















1-3— 


-2 


33 


-66 


132 


-264 


528 


2-6— 


-1 


57 


-57 


57 


- 57 


57 


3-9— 


. , 


41 




. . 




, . 


5-2— 


+ 1 


24 


4-24 


24 


+ 24 


24 


6-5— 


+2 


10 


+ 20 


40 


+ 80 


160 


7-8— 


+3 


11 


+ 33 


99 


+ 297 


891 


91— 


+4 


3 


+ 12 


48 


+ 192 


768 


10-4— 


+ 5 


1 


+ 5 


25 


+ 125 


625 


•• 


•• 


180 


-29 


425 


+ 397 


3053 



Referred to the mean, 

4-55+ l-3:r =4-3405556, 

the second, third, and fourth moments are (see Appendix, Note 5), 
i.2=^2-3611111-a|2=2-3351543, 
i/3=2-2055556-3:fc'i/2-x3=3-338395, 
v^=lQ'96lUn-^xPs-ex^V2-x^=lS-1^8ll. 

Owing to the very doubtful contact at the beginning of the curve 
Sheppard's adjustments were not made in this case, but the rough 
moments as calculated above were used. 



214 STATISTICS 

Thus ^1 = vyv""^ =0-875242 

j32=i;4/z/22=3-43817 
and Ac=ft(ft+3)V4(4i82-3i3i)(2ft-3j3i-6)=-0-466. 

Since k is negative the fitting curve should be of Type /.,the equation 
of which is 

where mja^=m2la^, and (a^-\-a<^—h, say. 

It is therefore necessary before going further to determine ?/o, a^, 
ag, h, m^ and m^ in terms of v^, v^, v^, or jS^ and jSgj the constants of 
the distribution. 

The value of 2/0 is found to be most conveniently expressed as a 
Gamma function which is defined, with the usual notation, thus : — 

whence it follows that T{lc-\-\)=kT{k). [See Appendix, Note 9, 
also p. 285.] 
Also, if 



B(m, n)=j^ x^-^ {\-xY-Hx, 



it may be easily shown that 

B(m, n)=T(m)T(n)IT(m-[-n). [See Appendix, Note 9.] 

The general method of procedure in determining the constants 
for all the different types is : — 

1. Express the fact that the area of the curve is a measure of 

the total frequency of the distribution — this enables us to 
find 2/0. 

2. Find the 71th moment of the curve with regard to some fixed 

origin — giving n particular values, 1, 2, 3, 4, this leads to 
the determination of /Xg, ix^, fi^, pi, ^2 i^ terms of the con- 
stants of the curve, and thence to formulae for calculating 
the constants. 

Once found, the same formulae may be used, of course, in all 
cases of the same type : we have only to replace letters by the 
numbers for which they stand. 

Applying this method to the Type I. curve, we have 

•+aa 

= / ydx 

2/0 '^'' 



-/.: 



«l""«; 






APPLICATIONS OF CURVE FITTING 215 

Put (ai+a;)=(%+«2K so that (a2— ^)=(^i+«2)(l~25) and 

dx 

— ={ai-\-a2)=b ; therefore 

dz 



^^.o6K+.2r-"YV.(i_,)^,, ... (2) 
a/^'a^"^ Jo 



B(mi+1, m2+l). 



Hence yA=- . ; 



■ + «2 



/ -r«2 

Again, N/x'„=/ ?/(ai+a:)«(Zir 

is the nth moment of the distribution referred to (— a^, 0), the 
point where the curve starts from the axis on the left-hand side, 
as origin. 

Therefore, as above, 

a^^a^"^ Jo 

=6«nJ 2™i+^(1-2)"WJ z'''^{l-z)'^dz, by (2). 

Hence, 

^'«=6«r(Wi+n+l)rK+m2+2)/r(mi+l)r(mi+W2+w+2) 

=b^(mj^-\-n)(mj^-\-n—l). ... (Wi+l)/(mi+m2+w4-l)(mi+m2+n) 
. . ..(mi+m2+2), 
by repeated appUcation of the relation r(k-\-l)=kr(k). 

Putting n=l, 2, 3, 4 in succession, we have 
/x'i=6K+l)/K+m2+2), 
^'2=62(mi+2)K+l)/(mi+m2+3)(mi+m2+2), 
/Lt'3=63(^^_^3)(^^_^2)(mi+l)/K+m2+4)K+m2+3)(mi+m2+2), 
^'^=64(^j+4)(mi+3)(mi+2)(mi+l)/K+m2+5)K+m2+4) 
(mi+m2+3)(mi+m2+2). 
These relations are rendered more concise if we write 
mi-\'l==m\, m2-\-l==m'2, m^-^m2-\-2=r ; 
thus fjL\=bm'Jr 

^\=b^m\(m\+l)/r{r+l) > 

/x'3=63m'i(m\+l)(m\+2)/r(r+l)(r+2) 

/x'4=6*m'i(m\+l)(m\+2)(m\+3)/r(r+l)(r+2)(r+3). 



216 STATISTICS 

To get the corresponding moments referred to the mean as 
origin we have the relations : — 

/ii=0, /X3==/x'3— 3/XV2— /^'\» 

H'2=H''2—H''\> /^4=/^'4— Vl/^3— W2— /^'^> 

which, after some straightforward reduction, give 
/i3=263m\m'2(m'2-m'i)/r3(r+l)(r+2) 

Thus B =u^ /a3 _ '^b^rn'\m'^(m\-m\) ^ i ¥m\m'\ 
HI /*3/A*2 r6(r+l)2(r+2)2 / r^r+l)^ 

=4(m'2-m\)2(r+l)/m'im'2(r+2)2 

Therefore, ^^ ^ft(^+2)^ ... (3) 

m\m\ 4(r+l) ^ ^ 

Again 8 =a /a2 _ 36^^\^^2Ki^^2(^-6)+2r-i ibhn^^^m^ 
' ^' '^^^^ r*(r+l)(r+2)(r+3) / r*(r+l)2 

3[m'im'2(r-6)+2r2] (r-fl) 



m'im'2 (r+2)(r+3) 

Therefore, -JTL =-.r+6+^J^±')^-^ . . . (4) 
m\m\ 3(r+l) ^ ^ 

Combining (3) and (4), 2ft(r+2)^ _ (r+2)(.+3) 

whence r=6(i8,-^,-l)/(3ft-2ft+6) . . (5) 

Again, since iJL2=b^m\m' 2/r^{r-\-l), 
therefore 62=^2(^+1) • lj8i(r+2)24-16(r+l)]/4(r+l), by (3), 

i.e. b=-y;W[ft(r+2)^+16(r+l)] . . (6) 

And m\m'2=4r2(r+l)/08i(r+2)2+16(r+l)], 

while m' i-\-m' 2=r ; hence m'j and m'2 are roots of 

^2_^^ I V — \ /_ _Q 

^^i(r+2)2+16(r+l) 

the solution of which quadratic is --hi / r-— ^ ^^"^ ^ ; 

2 WL i8i(r4-2)24-16(r+l)J' 



APPLICATIONS OF CURVE FITTING 



217 



therefore, m^ and mj* are respectively equal to 



C' 






and a^ and ctg follow from 



(7) 



(8) 



nil nig mi+nia 

Applying these formulae to the ' unemployed ' example, we find 

r=5-36048. mj =0-169185. m2=3-191295. 

6=9-33236. ai=0-469842. a2=8-86252. 

Also 2/0=58-1282, and the equation of the curve is therefore 

0169 / «. \ 3-19 



y=58-l(l+-^) (1- 
0470/ 



8-86 



The position of the origin, which is at the mode, is given by 




-<--- 



(mean-mode) =/x'i—% 

_bm\ bnii 
r mi-\-m2 
■m\ m\—V 



\ r r—2 I 



m 



m 



r{r-2) 

V.r-2' 

mode =4-3405556- i . -^ . !^, 
Vi r—2 

in this particular case, 

=2-3052009. 



(9) 



thus, 



^ [* When fx^ is positive Wg goes with the positive root of the quadratic, and 
vice versa.} 



218 STATISTICS 

This enables us to write down any x, and thence y by substituting 
for X in the equation of the curve, which, by taking logs, may be 
written 

log y=\o% yo-^m^ log ( l+^j+mg log ( 1— - 

e.g. for the x of the group (2-6— 3-9), bearing in mind that 1-3 is the 
unit of measurement for x, we have 

^3-25=(3-25-2-3052009)/l-3=0-9447991/l-3. 
Hence ("1+^^^^ =2-546835 ; (^1 -^^^=0-9179953 ; 

mj log ^+?i:^'^j =0-0686892 ; m^log ("l-?^^ = -0-118587 ; 

so that log 2/=l-714489, 
and y ^.2^=51-82. 

Similarly the ordinates at the centre points of the other groups 
may be calculated, but it must be remembered that the resulting 
values are only a first approximation to the observed frequencies, 
and a better series is obtained if, by using some good quadrature 
formula, we calculate the areas for the successive groups between 
the curve, the bounding ordinates, and the axis of x. Indeed in 
the case of the group (1-3—2-6) it is essential to do this, because 
(1) the rise of the curve is so very abrupt as to render the deter- 
mination of the single ordinate at the centre quite inadequate for 
an accurate measure of the frequency in that group, and (2) a 
portion of the group falls outside the range of the curve which only 
starts at 1-6944063 {i.e. mode— l-3ai), and this has to be allowed 
for in finding the frequency as represented by the area between the 
curve and axis. 

The base of the required area, range (1-6944063 to 2-6), was 
therefore divided into eight equal parts and the ordinates at the 
points of division were determined. The area was then found by 
using Simpson's weU-known formula : — 

Area=P[(i/o+2/2p)+2(2/2+2/4+ • • • +2/2p-2)+4(2/i+2/3+ . . • +y2p-i)l 

where h denotes the length of one of the equal parts into which 
the base is divided and 2p is their number ; in our case p=4: and 
h=^, the class interval being the unit, and the result is to be 
reduced in the ratio 

0-9055937 : 1-3 



APPLICATIONS OF CURVE FITTING 



219 



in order to allow for the smaller range of this group ; we thus get 
as the area for the group 

A.QQKKQO'T' 1 

—5^3— X ^^[(2/0+2/8) +2(2/24-2/4+2/6)+4(2/i+2/3+2/5+2/7)] =37.39. 

The observed and calculated frequencies for the whole series are 
compared in Table (44), the remaining areas in col. (4) being calcu- 
lated by the simpler but somewhat less accurate form of Simpson's 
formula, when only three ordinates are used, namely, 



/. 



+1 



2/<^^=i(2/-i+42/o+2/i). 



Table (44). Comparison of Observed and Theoretical 
Frequencies of Unemployed Percentages 

(1) (2) (3) (4) (5) (6) (7) 



Percentage 
Unemployed. 


Observed 
Frequency. 


Theoretical Frequency. 


Deviation. 


Square of 
Deviation. 


Ratio of No. 
in Col. (6) to 
No. in Col. (4). 


Ordinates. 


Areas. 


1-3— 
2.a- 
3-9- 
5-2— 
6-5— 
7-8— 
91— 
10-4— 


33 
57 
41 
24 
10 
11 
3 
1 


55-3* 

51-8 

37-8 

24-9 

14-8 

7-7 

3-3 

10 


37-4 

51-6 

37-8 

250 

14-9 

7-8 

3-4 

1-2 


+4-4 

-6-4 
-3-2 

+ 10 

+4-9 

-3-2 

+0-4 

+0-2 


19-36 

2916 

10-24 

100 

24-01 

10-24 

0-16 

004 


0-52 
0-57 
0-27 
0-04 
1-61 
1-31 
0-05 
0-03 


•• 


180 


•• 


1791 


•• 


•• 


X*=4-40 



To test the goodness of fit we have n^=S, ^2—4.49^ whence, by 
means of the P table, P =0-731852. Thus, roughly, we may say that 
three out of every four random samples of 180 records would give a 
worse fit with the proposed curve than is given by the actual distribu- 
tion observed, so that the fit may be regarded as quite a reasonably 
good one. This conclusion is also supported by an examination of 
the curve which has been drawn, fig. (36), with the histogram of 
the given statistics. 

Example (3). — The data for this example concerning infectioils 
diseases will be found in Table (16), p. 62 (or, see p. 224) ; the 
reader should work out the moments for himself and verify the 
following results : — 

[* The ordinate in this case cannot be accepted as an approximation to the 
frequency given by the curve.] 



220 



STATISTICS 



The first four moments referred to 7 as origin are 

0-282158, 4-86307, 17-4855, 129-394. 
Referred to the mean, 7-564316, the three latter become 
7^2=4-78346, 1/3=13-4140, 7/4=111-964. 

If we do not assume high contact at the terminals, and certainly at 
the lower end it is doubtful, we deduce from the above values of 
the moments that 

jSi=l-64396, j32=4-89321, a^=-1-53. 

Thus the fitting curve is of Type I. and its constants, when calcu^ 
lated, are 

r=ll-7819. mi=:0-31171. m2=9-47020. 



ai=0-79216. 



a2=24-0671. 2/o=60-363. 



Dw ---r 




t^'^z : 








: i + s^ __ . 




en --I t-t S^- 




50 __- _jL_^ _ ^ __ 




T ' -S _. . 








+ K 




40 T -^ = -" 












Qn -_: -- . --tiz a__:!:_ _ 





30 — IT — J ::?: 




± : - 4 ^v 










s 


nn I "• 










s^ - - . ... . .. 








::: : i^s": : :::::::::::::: :: 


IQ --I -, 


"■^^ 








"= .^ 


::::::::: :::ffi:::::::j:::::: 




1 2 1 3 4^5 


6 7 8 9 10 11 12 



Percentage Unemployed 
Fig. (36). 

The equation of the curve is therefore, retaining three significant 
figures throughout, 

y=60.4(l+JLy7i_JL)'-. 
\ 0-792/ \ 24-1/ 

The curve starts at 2-02904 (so that the first group of observations 
lies whoUy outside its range) and ends at 51-7475. It is drawn, 
together with the corresponding histogram, in fig. (37). 

Supposing, just for the sake of comparison, we assume high 
contact at the terminals and attempt to fit the given distribution 
with a Type III. curve, to which Type I. is closely related. 

We then have, after making Sheppard's adjustments, 
;x2=:4-70013, /X3=13-4140, ix^=l0d-60l, 
whence j3i=l-73295, ^2=4-96129, a:=-1-47. 

It will be noted that the theoretically correct type to take here 
again is Type I., but this was discarded because, when attempted, 



APPLICATIONS OF CURVE FITTING 



221 



it led to a curve starting at a point corresponding to a disease rate 
of 3-385, so that the central ordinates of each of the first two 
observed groups lay outside the curve altogether. 

Type III. curve is of the form 



y^y^e-v^ 1+ 



70 



■eso 



S40 



30 



CO 20 



% 



% 



I 



;Pype 



I 



Xypel-II 



^i 



't^ 



ro 



10 



15 



20 



25 



30 



Disease Rate per 1000 persons liuing 
Fm. (37). 



To express the constants in terms of the moments, noting that the 
curve starts from a; = — a on one side and goes off to infinity on the 
other, we have 



N-=| ydx 

J -a 

=2/oj e-^-(l+-l dx 



^Vo f e-y%a-\-xfdx (where ya=p) 

= f-,re-y-(ya+yxrdx 

=^6^1 e'^y''+y''\ya+yxydx 

V eP /°^ 
=^o_ t-'z^dz (where ya^yx=z) 

yp^ k 



Therefore, y,=Np''+'/ae''r(p+l) . 



(10) 



222 STATISTICS 

Again, the nth. moment of the distribution referred to (—a, 0) 
as origin is 

Nja'„=l y{a'\-x)*^dx 

J -a 



J -a 

vo 



.2/0 e^ 



Therefore, by (10), 

Hence, 

i^'i=rtp+2)/yr(i>+l) = (^+l)/y 
i^'2=r(^+3)/y2r(^+l) = (^+2)(^+l)/y2 

i^'3=r(i)+4)/y3r(p+l)=(i>4-3)(i9+2)(p+l)/y^ 

Transferring to the mean as origin we have for the moments, since 

fJi3=H'' 3—Sxfji^—x^=2(p-{-l)/y^. 
Hence, combining these last two equations, 

y=V2//^3. v=(^f^Mfi\)-i . . . (11) 

In our particular case these equations give 

y=0-700780, :p=l-30820, a=l-86678, 

and, therefore, by (10), 

2/0=55-3323. 

Hence the curve is 



y=55-3e-°"'ni+^^ 
\ 1-87 

The equation of the curve, on taking logs, gives 
log y=log yQ—y log io« . x+p log 1 1+: 

=l-742979-0-304345a;+ 1-30820 log (l+a:/l-86678). 



APPLICATIONS OF CURVE FITTING 223 

Before we can go on to calculate the ordinates of the curve we 
need to know where the origin lies, and since it coincides with the 
mode it may be found from 

mean-mode =yJ ^—a 

=(p+l)y-p/y 

='-^ (12) 

Thus, mode=7-564316-2-853960=4-7I036. 



Mode Mean 



Suppose now we wish to calculate the ordinate corresponding to 
the X of the centre point of group (6—8), we have 

a;7=J(7-4-71036) 
=114482, 

bearing in mind that the unit is a rate of 2 per 1000. 
Hence, substituting this value in the equation for log y, 

log 2/7=1-666278 
2/7-46-374, 

and similarly any other y may be found. 
The curve starts at 

mode-a=4-71036-2(l-86678)=0-97680, 

so that the range of the first group as determined from the curve is 
(0-9768—2), and not (0—2) as in the observations. 

The ordinates and afterwards the areas, calculated by a method 
somewhat similar to that indicated in Example (2), were determined 
for each separate group of observations, and the results for both 
Type I. and Type III. curves are compared in Table (45). 

Type III. curve is drawn on the same diagram, fig. (37), as Type I. 
curve and the observation histogram, and the result lends emphasis 
to an important point, namely, the necessity for replacing ordinates 
by areas to obtain the frequency proper to any group. 

In order to get a measure of the goodness of fit in each case, 
the function P was calculated, but in the Type I. comparison the 
first group had to be omitted to avoid the infinite term which would 
have resulted in ^^^^ owing to this group falling right outside the 
curve, that is to say, the test had to be confined to towns in which 



224 



STATISTICS 



the observed case rate was not less than 2. The values found for 
P were : — 

Type I.— P=0-34307, 
Type III.— P=0-46298, 

so that in every 100 samples containing 241 observations each, we 
should get, roughly, 34 deviating from the Type* I. curve and 46 
deviating from the Type III. curve, at least as widely as the given 
distribution. In neither case can the fit be regarded as a very 
good one, but the failure is only marked in one or two groups, such 
as that of maximum frequency, where there may be other than 
random causes to account for it ; e.g. where isolation is inefficient 
the disease is likely to spread, one case infects another : in other 
words, the events are not independent. 



Table (45). Comparison of Observed Distribution of In- 
fectious Disease Kates, notified in 241 large Towns of 
England and Wales, with Theoretical Distribution. 

(1) (2) (3) (4) (5) (6) 





Observed 
Frequency. 


Theoretical Frequency. 






Case Rate. 




(/i-/)V/i. 


{fz-mu 










Type I. 


Type in. 








(/) 


(/i) 


(/a) 






0— 


5 




6-6 


, , 


0-39 


2 


39 


52-6 


43-7 


3-52 


0-51 


4 


69 


55-4 


54-3 


3-34 


3-98 


6— 


41 


43-2 


46-2 


Oil 


0-59 


8— 


29' 


31-2 


33-6 


015 


0-63 


10— 


22 


21-5 


22-4 


0-01 


001 


12— 


16 


14-2 


141 


0-23 


0-26 


14— 


7 


91 


8-6 


0-48 


0-30 


16— 


5 


5-6 


51 


0-06 


0-00 


18— 


3 


3-3 


2-9 


0-03 


000 


20— 


4 


1-9 


1-7 


2-32 


3-11 


22— 





10 


0-9 


100 


0-90 


24 





0-5 


0-5 


0-50 


0-50 


26— 


1 


0-3 


0-3 


1-63 


1-63 


•• 


241 


239-8 


240-9 


X\ = 13-38 


X^3= 12-81 



Example (4) refers to the wages of certain women tailors previ- 
ously recorded in Table (II), p. 41. The data as given in the 
original suffered a disadvantage common to such statistics : at 



APPLICATIONS OF CURVE FITTING 225 

either end the grouping differed from that in the centre, two or three 
classes being lumped together owing to the smallness of frequency 
in each. The figures ran thus : — Under 5s., 19 ; 5s. and under 6s., 
180 ; 6s. and under 7s., 384 ; ... ; 23s. and under 24s., 64 ; 
24s. and under 25s., 54 ; 25s. and under 30s., 122"; 30s. and over, 
36. They were recast in the form shown in Table (46), suggested 
by an examination of the histogram, in order to make the fitting 
simpler. 

The first four moments calculated from this adapted table and 
referred to 12s. as origin are : — 

z/'i=0-556718, i;'2=5-056373, i;'3=16-70163, i;'4=123-7691. 
When referred to the mean, 13-113436, the last three become 

1/2=4-746438, 1^3=8-60179, i/4=95-6914 ; 
or, after making Sheppard's adjustments, 

/X2=4-663105, />t3=8-60179, /x^ =93-3474 ; 
therefore, ft =0-7297 13, ^=4-29291, a:=1-63. 
The curve is thus of Type VI., 

y=yo(x-a)'^/x'^i. 

To calculate the constants, the nth moment about the origin is 
given by 

NjLt'„=l yx'^dx 

Ja 

=yJ'^(x—af^x''-^^dx 

Ja , 

-Vol <^"^-^ • ,-;^«(-~2J^^(^here ^=-j 

fVi-'^2-n-2n_2y2e^2 

n-ljo 



2/0 






-Biqi-q^-n-l, g^+l). 



Thus, putting n=0, 

qQI-'12-1 

and fJi'n=a^r{q^-q2-'^-n)r{q,)ir{q^-n)r{qi-q>,-l) ; 

therefore, ix\=ar{qi-q2-^)T{qi)ir{qi-l)r(qi-qi-l) 

=«(?!- l)/(9'i-9'2-2)- 
Also iJL\l^\.i=ar(q,-q2-'^--n)r{q,-n+l)ir{qi-n)r{qi-q2-n) 

=a{qi-n)l{q^-q2-n-l). 



226 STATISTICS 

Hence fJ^' 2=a^qj,-l){q^-2)l{qi-q^-2)(qj^-q.^-3) 

/^'3=«'to-l)fe-2)fe-3)/tei-^2-2)fe-g2-3)(gi-?2-4) 

But these relations are precisely the same as those of Type I. with a 
in place of b, —q^ in place of m^, and q^ in place of mg, so that 
(l-fQa), (1— Qi)* are the roots of 

q2_rq_|_4r2(r^l)yf-^^(r_|_2)2_^16(r+l)]=0 . -. (14) 

where r=603,-j8,-l)/(6+3ft-2ft) .... (15) 

Also yo=Na'^i-'^2-T(qi)/r(ai-a2-l)r(q2+l), by (13) . (16) 

and a is given by 

/.,=a=^(l-aJ(H-a2)/r2(r+l), .... (17) 

/X2 being the second moment of the given distribution referred to 
its mean as origin. 

The distance of the mean from the origin is 

/^'i=a(ai-i)/(qi-a2-2), 

and this fixes the origin, for the mean is known directly from the 
statistics. 

To get the mode, use the equation of the curve, putting — =0, 

dx 

and we have 

origin =mode — ag'i/(g'i — g'g) • 

Combining this with 

origin =mean— a(gi— 1 )/(g'i— ^2— 2) 
we have 

mean.mode=a(ai+a2)/(ai-a2)(qi-q2-2) • • (18) 
Applying these formulae to the case of the women tailors, 
r=-38-7698, ^1=51-5269, ^2-10-7571, a=2M1018, 

and the equation of the curve is 

y=yo(x-21-l)^»Vx"^ 

where log 2/0 =68 -8254. 

Also the origin is at —41-9104, the mode at 11-4498, and the maxi- 
mum theoretical frequency is 2299. 

[* When Ms is positive (1 +5^2) goes with the positive root of the quadratic, and 
vice versa. ] 



APPLICATIONS OF CURVE FITTING 



227 



Table (46). Distribution of Wages of cebtain 
Women Tailors, Actual and Theoretical. 



Wages. 


Frequency. 


Wages. 


Frequency. 


Actual. 


Theoretical. 


Actual. Theoretical. 


Is.— 

3s.— 

6s.— 

7s.— 

9s.— 

lis.— 

13s.— 

16s.— 

17s.— 


5 

14 

564 

1243 

2045 

2339 

1815 

1432 

854 


1 

52 

462 1 

1332 j 

2096 

2255 

1898 

1353 

859 


19s.— 
21s.— 
23s.— 
25s.— 

■ 27s.— 
29s.— 
31s.— 
33s.— 


523 

262 

118 

64 

43 

27 

15 

9 


503 

278 

147 

75 

38 

19 

9 

5 


•• 


•• 


.. 


•• 


11,372 i 11,372 

i 



The theoretical and actual frequencies are compared in Table (46) 
and the curve is drawn with the histogram in fig. (38). 



2500 



^^ 



2000 



J 1500 



1000 



500 



ft 



m 



s: 



0/. 



5/- 



10/^1 S 

' O a) 



15/- 20/- 

2 Rate of Wages 

Fiu. (38). 



25/- 



30/- 



35/- 



Example (5) discusses the distribution of frequencies of specimens 
of Anemone nemorosa with different numbers of sepals, recorded by 
G. U. Yule {Bicmetrika, vol. i., p. 307). 



Wn=j; 



228 STATISTICS 

The first four moments referred to 6 as origin are 

^^=0-508, i.'2=l-012, i/'3=2-476, i,'4=9-124. 
Referred to the mean, 6-508, the last three become 

^2=0-7539360, z;3=M95905, 1/4=5-459941. 

The contact, at one extremity certainly, being doubtful, Sheppard's 
adjustments were not made in this case. Hence, 

j8i=3-337259, j32=9-605476, /c=l-46. 

Since k does not differ greatly from unity an attempt was made to 
fit the observations with a Type. V. curve, namely, 
y=yoX-Pe-^^ 
The wth moment about the origin is given by 

yx^dx 

(since, p and y being positive, y vanishes at x—0 and at a;=oo) 

=yQy''-P+^rzP-''-h-'dz (where z=y/x) 

=y^y--P+^r{p-n-l). 
Thus N=2/oy-^+'r(2)-l). 

And ^' Jii\-x=yl{p-n-\). 
Hence />t'\=y/(^— 2) 

/^'2=y7(2>-2)Cp-3) 
/x'3=yV(p-2)(2>-3)(2?-4). 

Referred to the mean as origin, the last two moments become 

/^2=yV(f'-2)^(p-3), 

/^3=V/(i'-2)'{l>-3)(i'-4), 
whence 

this gives a quadratic for (i?— 4), one solution of which is 

p_4=[8+4V(4+i3.)]/i3., . . . (19) 
the positive root being taken in order to get a real y. 

Thus y*-(P-2)V[(P-3)/^J . . . (20) 

and y,=Ny^-Vr(p-l) .... (21) 

Since /^'i=y/(2'-~2), the position of the origin is given by 

Origin=Mean— y/(p— 2) . . . (22) 

Also the distance of the mode from the origin is y/p, so that all 
the constants of the curve are readily determined. 

[* The sign of 7 is taken to be the same as that of /Xg.] 



APPLICATIONS OF CURVE FITTING 



229 



In our particular case, we get 

^=9-643840, y=1710768, 
and the curve is 



y=yoX-'-e--^/^ 




__ 




600 --- f--J- - - - --__ 


X 


- Jt 


It 


H -A 


± i :::::: 


1 T _ 


... TIl 




500 f tt 


t iJ 


i il 


r 


-S2 X : it 


« - ^i T 


§• - 4 - 


<^ 1- i 


si t IT 






O - - . - 


~ : : ::: 


S it _ _ T IT 


^ _ - 4^ 


o> 4t - it 


1^ t - 41 


^ 1.1 


'5 1 a 


^^nn - t - S 


03300 I jr 


§ " t 


1 . .. i 


•p . . V 


§ . . \ 


^ -t 


^ : : " jl 


H^ - - ^ 




|2oo -:---- -----■■---£---------------- 


S ja 


a jn 


1" ___ L^::_: 


^ ::i:±:::::::::-iin::i:i:i:::::::ii:i:i:::i:i 


^i 


- "■ \i- 


tx 


ion - - 4- -- ■ -A- _ -_ _ _ _ _ 


+ tr- 


_t ^S 


t ^N 


t - ^S 


^^. 


If _ ^^ a ^S 


- J "3 jrt ^S«, 


::;±±::;i:i::::-:::::2:!±:Es«;;i==;;;:: 



8 

Number 0^ Sepals 

Fia. (39). 



10 



12 



where log 2/0 —9-38179. The origin is at 4-27 and the mode at 6-04. 
The greatest frequency is 620 approximately, and the frequency dis- 
tribution, calculating areas for the several groups as if they ranged 
between (4-5— 5-5), (5-5— 6-5), etc., is shown alongside the observed 



236 



STATISTICS 



distribution in Table (47). The curve is plotted in fig. (39) from the 
ordinates which were calculated at the centre and extremities of 
each group so as to enable Simpson's simple quadrature formula 
to be used to get the areas. 



Table (47). Distribution of Sepals of Anemone 
Nemorosa, observed and calculated. 



No. of 
Sepals. 


Frequency. 


No. of 
Sepals. 


Frequency. 


Observed. 


Calculated. 


Observed. | Calculated. 


5 
6 

7 
8 


34 
576 
276 

92 


51 
544 
296 

81 


9 ' 
10 
11 
12 

1 


1 

14 22 

4 G 

2 

4 1 


•• 


.. 


.. 


1000 , 1003 



[Examples have been given above of five out of the seven different types 
of frequency curve that have been enumerated. For further examples of 
all the types and a complete account of the method reference should be 
made to Professor Pearson's memoirs, especially the following : — 

Roy. Soc. Phil Trans., vol. 186a, pp. 343-414 (1895), On Skew Variation 
in Homogeneous Material ; and a Supplementary Memoir in vol. 197a, pp. 443- 
459 (1901). 

Biometrika, vol. i., pp. 265 et seq., On the Systematic Fitting of Curves to 
Observations and Measurements, continued in vol. ii., pp. 1-23. Also vol. iv., 
pp. 169-212, which discusses various historical hypotheses made to generaUze 
the Gaussian Law, the basis of the symmetrical normal curve. 

A large number of highly interesting practical illustrations of Pearsonian 
curve fitting occur throughout the pages of Biometrika, while W. P. Elderton's 
Frequency Curves and Correlation contains an admirably concise treatment of 
the theory, with applications to meet more particularly the actuarial point 
of view. 

It should be stated that rival curves and methods have been proposed as 
suitable for fitting certain types of frequency distribution, some of which have 
scarcely received the attention and the trial they deserve. Among the most 
interesting are those developed by Professor Edgeworth ; for some account of 
his voluminous work upon the subject the reader may refer to several memoirs 
in the Journal of the Royal Statistical Society, beginning December 1898 
(the Method of Translation), among which the following are important as 
giving more recent results of his researches : — 

Vol. Ixix. (1906), The Generalized Law of Error or Law of Great Numbers. 

Vol. Ixxvii. (1914), On the Use of Analytical Geometry to Represent Certain 
Kinds of Statistics. 

Vol. Ixxix. (1916), On the Mathematical Representations of Statistical Data; 
continued in vol. Ixxx. (1917). 

Two memoirs may be cited as of particular interest — those of May 1917 
and March 1918 — because they reply to criticism and draw a comparison from 
their author's point of view between his curves and those of Professor Pearson.] 



CHAPTER XVIII 

THE NORMAL CURVE OF ERROR 

Let us return for a moment to the general statement on p. 143, 
that ' whenever we have n similar but independent events happen- 
ing in which the probability of success for each is jp, the different 
resulting possibilities as to success are given by the successive 
terms in (s-f/)", namely, 

and their correspondent probabilities by the successive terms in 
0)+^)", namely, 

When we come to try and apply this theory directly to cases 
other than those of random sampling in artificial experiments with 
coins, dice, etc., we are faced at once with difficulties because of 
the limiting character of the assumption on which the theory rests, 
namely, that all the events are to he similar and independent. The 
similarity demanded is of the same radical type as that existing 
when we throw the same die or spin the same coin twice running, 
and the test for it is that p, the chance of success, is to be the same 
for every individual event. The independence is to be such that 
no single event and no combination of events is to have any influence 
upon any of the rest. 

Now for most classes of events it is impossible to assign any 
a priori value to p at all, still less can we be sure that p does not 
change from one event to the next. For example, the chance of 
death for soldiers in war-time varies from regiment to regiment 
according to where they happen to be located ; for the same regi- 
ment it varies from battaUon to battaUon according to whether 
they are in the trenches or behind the lines ; and from individual 
to individual according to innumerable little accidents of time, place, 

231 



232 



STATISTICS 



and condition. Also, where the shells burst thickest, p increases 
for any soldier there, but it increases also for his neighbour. Thus 
the events in such a case are not similar, neither are they inde- 
pendent. 

Moreover, as it stands, the theory cannot be appUed to any 
distribution in which the character observed is capable of continu- 
oiLS variation. This difficulty, however, has been overcome, as we 
have seen, by replacing the histogram representative of the binomial 
by a continuous curve which at the same time serves to describe 
the discontinuous series to a high degree of accuracy. 

To illustrate how close this 
description can be, even when n 
is comparatively small, we will 
fit with its appropriate normal 
curve the symmetrical binomial 
polygon formed by joining up 
the summits of the ordinates 
representing successive terms of 
the series 
— ^ 2io(HJ)i«, 



A 



A 



A 



K 



N 



\ 



erected at unit distance apart. 
The total area bounded by the polygon, the extreme ordinates, 
and the axis of x is practically 

= (2/0+2/1+^2+ • • • +2/'i+2/'2+ • • Oxil) 
=sum of toe given ordinates 

=1024. 

The equation of the normal curve is 



where 
and 



Yo=N/V2^ • (7=1024/V(5-57r). 



1 % 



Hence, taking logs, we have 

logi/=log Yo-— -logioe 

=2-39I5437-x2(0-0789626). 

It is easy from this equation to calculate the normal curve ordinates 
corresponding to x=0, 1, 2, 3, 4, 5, and the results, compared with 
the polygon ordinates, are as follows : — 



THE NORMAL CURVE OP ERROR 



233 



X 


Ordinate of Polygon. 


Normal Curve Ordinate. 






(J , r^ -*,.., „ 





262 


246-3 <^*«^ 


±1 


210 


205-4' -^ ' ' ^' 


±2 


120 


119-0 / m; ' 


±3 


46 


480 *"" ^ 


±4 


10 


13-4 ^^^' 


±5 


1 


2-6 / 7 



Now although the circumstances in which the series 

may be taken to represent the frequency distribution resulting 
from a particular kind of experiment were so stringently defined, 
there is no reason why the normal curve itself to which the theory 
led should be subjected to precisely the same limitations. After 
all, the real and only justification for choosing one curve rather 
than another to fit any given observations is that it does succeed 
in fitting them better. But when the further question is asked 
why the normal curve should succeed in describing some results 
so well, we must not be tempted by analogy to rush to the con- 
clusion that the causes at work are necessarily independent, and 
equal, and so on. In short, the theoretical justification and the 
empirical use of the normal curve are two quite different matters. 

Experience shows that the normal curve suffices to fit certain 
types of distribijtion, besides those which arise in tossing coins and 
in similar experiments, with remarkable accuracy ; among these 
may be noted : — 

1. Certain biological statistics ; for instance, the proportions of 
male to female births taken over a series of years for a large com- 
munity such as the population of a country ; also the propor- 
tions of different types of plants and animals resulting from cross- 
fertilization. 

2. Certain anthropometrical, particularly craniometrical and allied 
statistics^ such as the height, weight, lengths of various bones, skull 
measurements, etc., of a large group of persons, and the agreement 
is the closer if the group be reasonably homogeneous, i.e. composed 
of individuals of the same nationality and sex between the same 
narrow age limits, etc. ; also measurements of a similar character 
in animals and plants. 

3. Errors of observation in experimental work ; for example, 



234 STATISTICS 

several measurements of the same quantity — length, weight, speed, 
temperature, or whatever it be — will contain errors of this kind 
which are equally liable to be above or below the true value. 

4. The marks of shots upon a given target, assuming that the 
shots are equally liable to err in any given direction. This is an 
interesting case of the normal law in two dimensions, for the north 
and south line and the east and west line through the centre of 
the target may both be regarded as axes of normal curves of error.* 

5. Certain sociological statistics of a comparatively stationary char- 
acter ; for example, rates of birth, marriage, or death at neighbour- 
ing times or like places ; also the wages (and possibly the output 
if it could be satisfactorily measured) of large numbers of workers 
engaged in the same occupation under the same general conditions. 

6. Any statistics or quantities that are individually compounded of 
a large number of elements, mostly independent of one another, which 
themselves vary between limits not very widely divergent, and none 
of which exert a preponderating influence upon their resultant 
statistic. The latter may be simply the sum of its elements, or, 
more generally, it may be any function of the elements which, to 
the first degree of approximation, can be expressed in linear form. 

Now it would be a difficult matter in most of these cases to satisfy 
ourselves as to the fulfilment or non-fulfilment of conditions like 
those on which the binomial distribution rests. It is not easy 
indeed to visuaHze them perfectly, except in artificial experiments 
where they are largely under control. If anything, the chances 
seem almost hopelessly against their fulfilment in ordinary life, 
so closely must we hedge round our sample to keep out unequal 
influences. For example, to use a frequently quoted illustration, 
if p measures the chance of death for an individual, the death rate 
varies, as we know, considerably from place to place according to 
the age and sex constitution of the population ; it is influenced by 
differences in class, and occupation, and manner of life ; it is 
altered from time to time, violently by the ravages of war or disease, 
more gradually by improvement in general sanitation, housing 
conditions, etc. We should only expect to get the binomial distri- 
bution (and consequently the normal law if it depended upon the 

[* Sir John Herschel published in the Edinburgh Review (1850) an a priori 
proof of the normal law from a consideration of this problem. Taking <t>{.x'^) as 
the expression of the law for one dimension and <t>{x^ + y^) for two dimensions, 
the independence of errors in perpendicular directions leads to the functional 
equation <p{x^-\-y^)='(p{x'^)>^(p{y^), the solution of which is of the form 

0(x-) = — p c " ^^^. It should be added that the assumptions underlying the proof 
are not entirely above criticism. ] 



THE NORMAL CtJRVE OF ERROR 235 

same postulates) exactly verified if we were dealing with the same 
stationary population existing under the same stable conditions 
over a long period of time ; moreover, since jp is to be identical for 
each individual event in the ideal case, it would be further necessary 
that every family and every individual in our population should 
also remain in the same stationary and stable state. This is mani- 
festly impossible, especially after the industrial revolution which 
the advent of machine power created. 

These considerations suggest the interesting question whether the 
various types of statistics we have enumerated, as being approxi- 
mately subject to the normal law, could not, if we knew more 
about them, really all be included under heading number (6), repre- 
senting a further development from the binomial theory and an 
enlargement of the field in which it holds good. 

In an earlier chapter, when we were discussing the connection 
between marriage rate and prices, we showed how it was possible 
by a method of averaging to differentiate between long-time and 
short- time effects. The more transient fluctuations, only super- 
ficial in character, were removed and the real nature of any per- 
manent change in the figures was revealed. In much the same 
way, when we have a group of statistics which do not perhaps fit 
a normal curve of error at all closely, it may be possible by random 
averaging to get rid of some of the fluctuations which cause the 
badness of fit and to obtain a new group of statistics which more 
nearly obey the normal law. Averaging, that is to say, tends 
to smooth away the rough outstanding abnormalities ; and we shall 
presently show that if two variables, X^, Xg, which are independent, 
obey the normal law, any linear function of the variables 
{w-^-y^-\-w^,^, obeys the same law. This may throw some light 
on Class (6) where each statistic represents a compound, that is, 
in a broad sense, a kind of an average of a large number of elements 
which partially neutralize one another's infiuence, or rub the corners 
off one another, so to speak, since no single element is, by hypothesis, 
to exert an overwhelming infiuence upon the compound itself. 

But although the normal curve does serve to describe a consider- 
able number of frequency distributions within reasonable limits, 
there are many more cases in which it fails : for example, the 
greater part of those bearing on economic matters ; also statistics 
relating to the incidence of disease and degree of fertility are, as 
a rule, very markedly skew. Hence arose the necessity for an 
extension from the symmetrical normal to some kind of skew 
variation curves to fit such distributions. 



236 



STATISTICS 



The normal curve, however, has an importance of its own to 
which we must now draw special attention. It is the foundation 
of the theory of errors and provides us with an invaluable method 
of estimating the importance of one error in comparison with 
another, or of determining the probability that an error shall lie 
between stated limits. Upon it we depend for several most 
important approximations which are in constant use. 

The term ' error ' is used here in the sense that if we take the 
mean of a number of observations, the deviation of any one of 
them from the mean may be termed its error. When such devia- 
tions can be satisfactorily fitted, that is, within the limits of random 
sampling, by means of a normal curve, they are said to be subject 

to the normal law of error. 
This law is expressed, as we 
have seen, by the equation 

a 

where y . Sx measures the fre- 
quency with which an observed 
organ or character deviates from 
the mean by an amount lying between x and (ic-f-Sx) in a large 
population, i.e. y . hx registers the frequency of an error of size x 
to (x-\-hx), and N and a are constants dependent upon the particular 
application of the law. 

The 'probability curve or normal curve of error. As a guide to the 
drawing of the above curve it may be worth while plotting 

y=e-^. 
This is readily done by writing the equation in the form 

—x^=\og^y. 
Giving now to y the values 0, 0-1, 0-2, etc., we can find values of 
loge y a-s shown in Table (48), and, by means of a square root table, 
X is then determined. 

Table (48). Corresponding Values of x and y to plot y=e-^\ 




N 

2/ = — =- 

V2^ 



y 


logey 


X 

±00 


V 


logey 

-0-5108 


X 

±0-71 





— 00 


0-6 


01 


-2-3026 


±1-52 


0-7 


-0-3567 


±0-60 


0-2 


-1-6096 


±1-27 1 


0-8 


-0-2232 


±0-47 


0-3 


-1-2040 


±M0 : 


0-9 


-0-1054 


±0-32 


0-4 


-0-9163 


±0-96 


10 








0-5 


-0-6932 


±0-83 









THE NORMAL CURVE OF ERROR 



237 



This enables us to plot the graph as shown in fig. (40). Since 
logg 1=0, and the logarithm of any number greater than 1 is 
positive and thus cannot be equal to —x^, it follows that y cannot be 
greater than 1 . Moreover y cannot be less than 0, for the logarithm 
of a negative quantity is meaningless, but, as y approaches 0, 
X approaches cxD. 

Also the curve is symmetrical about OY because for any possible 
value of y there are two values of x, equal and opposite. 

Returning now to the curve 

y — 7= — ^ > 

V27T . (7 

it must be of the same general shape as y^er^^ because the two 
only differ in their constants. It is clearly symmetrical, for 




-200 -1-75 -1-50 -1-25 -100 -0-75-0-50 -0-25 0-25 0-50 0-75 100 1-25 1-50 1-75 200 
Fio. (40). The graph of ?/=c-*'. 



instance, about the axis of y, because, in this case also, to any value 
of y there are two values of x equal and opposite. Moreover it 
tails off to the right and left from OY, the axis of x being an 
asymptote,. for as x tends to ioo? V tends to zero as before. 



When 



a;=0, 2/=N/\/27r . cr, 



giving the point B, fig. (41), where the curve cuts the axis of y. 
This is evidently the highest point on the curve, for 



dy 



Na: 



■a;2/2<r2 



dx V27r . a^ 
and this vanishes when a;=0. 

d^y N 






Again, 

d^^ V27r(T» 

which vanishes when iC=ior, and at these two points, H, H', we 



238 



STATISTICS 



therefore have ' points of inflexion ' where the bend of the curve 
changes its direction. 

The axis of y about which there is symmetry evidently locates 
the mean error, in this case zero ; in fact the mean and mode 
coincide, so that the mean or zero error is also the one which most 
frequently occurs, and any two other errors which are equal in 
magnitude but above and below the mean respectively occur with 
equal frequency : i.e. the frequency of positive errors is balanced 
by the equal frequency of negative errors on the other side of the 
mean, making the median error likewise zero. 

Again, the area / ydx measures the frequency of errors lying 

r + X 

between x^ and X2 above the mean ; I ydx registers the frequency 




Fio. (41). 



of errors between and x, or of deviations up to this magnitude, 
on either side of the mean ; and, in particular, for all errors 

the total frequency = I ydx 



V2. 



N r^ 
277 . aJ-^ 



^'I'-'dx 



N 



V27r.c7 



(V27r . a) (as on p. 206) 



This enables us, by means of the fundamental definition, at once 
to write down the probability of errors between any stated limits 
and explains the origin of the name, the probability curve, which 



THE NORMAL CURVE OF ERROR 



239 



is sometimes given to the equation. Thus we have the probability 
of an error between -\-Xi and +^2 

_frequency of errors between the given limits 
frequency of all errors 

= / ydx/i ydx • 






(1) 



Incidentally, the probability of an error between x and 

N 
8x 



(x+Sx) 



x2/2(7-2 



VS-; 



(2) 




Fig. (42). 

Greometrically, the area represented by the shaded portion of 

fig. (42) measures the frequency of errors between -{-x^ and +a;j, 

while the complete area between the curve and axis X'OX measures 

the total frequency, so that the probability of an error between 

-\-Xi and -\-X2 is measured by the proportion which the area of the 

shaded portion bears to the whole area. 

dx 
If in the above expression (1) we put x/a=$, so that — =ct, 

d^ 

it becomes 



V27r4 



(3) 



which is known as the probability integral, ^^ and ^2 being the 



240 



STATISTICS 



values of f which correspond to the values x^ and x^ of x. But 
this integral measures the area of the shaded portion of the curve 

1 



y= 



■u' 



V27T 



(4) 



shown in fig. (43), which is really the normal curve over again, but 
drawn on a different scale, namely, with the ordinates reduced in 
the ratio N : a and with the standard deviation a taken as the 
unitof measurement for a;, for f=: 1,2, 3 . . . whena:=cr, 2cr, 3a, . . . 
This has the effect of making the total area unity and the area 
given by 

1 r^2 . .„ 

. (3) bis 



27rJh 



V2- 

now directly measures the probability of an error between o-f ^ and a^^. 

Tables have been prepared 
(see pp. 284, 285) which enable 
us to write down the value of 
this integral for different values 
of fi and ^2 between certain 
limits (see Appendix, Note 10). 
Let us take an example to 
show how the curve may be 
used, and we choose one leading 
to a binomial distribution, so 
giving an expression for the 
probability by first principles, 




Fig. (43). 

in order to compare the two methods 



Example. — Suppose we toss simultaneously 100 coins, and sup- 
pose the chance of success, say ' heads,' is the same for each coin 
and equal to 1/2. In that case, according to the binomial theory, 

the probabiUty of 100 heads =(l/2)ioo, 

„ 99 heads and 1 tail =iooCi( 1/2)99 (1/2), 

„ 98 heads and 2 tails =iooC2(l/2)98(l/2)2,andso on. 

The most probable number of heads==7ip=(100)(l/2)=50. This 
does not mean, as explained before, that if we perform the 
experiment once we are sure on that one occasion to get exactly 
50 heads and 50 tails, but that if we go on repeating the experiment 
we shall in the long run get 50 heads and 50 tails turning up more 
often than any other combination. 

Let it be required to find the probability of getting at least 65 



THE NORMAL CURVE OF ERROR 241 

heads, that is, we want the probability of getting 55 heads or 
more, and this is given by 

a sum not very readily calculated if we have to go at it in a straight- 
forward manner. 

Now let us turn to the curve of error method. The standard 
deviation for the distribution is given by 

Since the mean number of heads to be expected if the experiment 
is repeated a considerable number of times =50, we want to find 
the probability of an error equal to or greater than 5, i.e. an error 
lying between a and +CX), because a =5. 

But the probability of an error between cr^i and erf g 

Hence the required probability 

=0-15866, by the probability integral tables. 

In other words, if we repeated the experiment 100 times, we might 
expect 55 or more heads about 16 times. 

We can now show that if X^, Xg are two uncorrelated variables 
obeying the normal law, then (w-^-^-\-w^2) ^^'^^ ^^^V ^^^ same law. 

Suppose x^, X2 are observed deviations from the mean values 
Xi, Xg in one particular record, a^, o-g being the respective S.D.'s. 

Let X=i<;iXi+^2-^2' ^^^ ^^^ ^ ^® ^^® deviation in X corre- 
sponding to deviations x^, x^ in the given variables. 

Thus X-\-x=w^{X,+Xi)-{-w^{X^+X2) 

=KXi+w;2X2)+ Kiri+w;2^2)- 
Therefore, x=W;iXi-]-W2X2' 

But the same error x may be obtained by giving x^, Xg many different 
values provided their weighted sum is unaltered. Let us first 
keep x^^ constant, so that the corresponding value of X2 required 



242 STATISTICS 

to produce an error lying between x and (x-\-Bx), where Bx is small, 
must be such that 

X<WjXj^-\-W2X2<X-\- SiC, 

i.e. x—WiXi<W2X2<x—WiXj^-]-Sx, 

i.e. X2 lies between (x—WiX-^fw^ and (x—w-^x^-\-hx)lw2, and the 
probability for this 

Wi ' V27r . (72 

Now this is in a form which only involves 8a;, x, and a^^, and we 
get the total probabiUty for an error lying between x and (x-^bx) 
by giving all possible values to the error Xj^. 

But the probability for x^ itself to lie between x^^ and (iCi+^^i) 

.Xi+Sxi 

V27rar 



= _ f e-'^"'^'^dx 



0^1 ^-a;2j/2o-2 



e -^1/2.-1^ by (2), 



V'27r(Ti 

and the probability for this to. concur with a suitable a^g to produce 
an error in the weighted sum lying between x and (x-^Sx), on the 
assumption that X^ and X2 are independent, is therefore 

'Bx 1 



L(JiV27T 



_^2 G2V27T 

^ _ x^i _ ix-u'ixiy^ 

6^ c 2<''i -'^'^t^'^ 8a;i. 



g-(x-wia;i)'-i/2(r22Uf-'2 



2^227ro-iCr2 

Hence the total probability for an error lying between x and {x-\-Bx) 
is obtained by integrating this result, that is, summing all possible 
probabilities, between a;i=— 00 and x^=^-]-co. This gives 

_^____/ e ^''^'^l 2(r-'2W-i2' 2<rV<^vi2 ^ '^"■^^"'^f^a; 
lyg . 273-Cri(T2y-oo 

^^ .+00 .a:% ^^^ +J!M_:ci-_^!_ 

_ "-^ / g ^2(r2ia22w22^2a22t(;'^ ^ ^"^'^^^"^dx 

where a^=w\cr2j^-(-2i;22or'i2 
2ttg^2^-'=° 



W9 . 27ro-i 






W2 - 2770-1(72 



g 2o-22iyV^ 



J-co J <7 



THE NORMAL CURVE OF ERROR 243 



Where t = —pi — — x 

V2\o'iO"2^2 0-2^20' 



8^ ,-' • ..^V^/- /2.a,a,t.^ 



It^g • 27r(TiO'2 



8X -^/2<r2 

-e 



V27r.cr 
which proves that the error x obeys the normal law with 

S.D.=V(wVi+wV.) .... (5) 

The above principle is readily extended, for if 

X=w;iXi+i/;2X2+ . . . +?^^„X„, 

Xi, X,, . . . X„ being independent variables obeying the normal 
law, then X also obeys the normal law and its 

S.D.=V(wVi+wV.+ • • . +wV\) . . (6) 

In discussing the results of random sampling we worked upon 
the principle that, given a number of sample observations of any 
statistical constant, a mean or a percentage or a coefficient of 
regression or anything else, an error or deviation as large as cr, 
the standard deviation, from the true value for the whole population 
might quite likely occur, but that an error exceeding 3cr would be 
unlikely, and we explained that, as a result of convention, the 
probable error, equal to fcr roughly, was largely used in place of a 
by many writers. We have now to examine the basis of this 
principle, and the first point to notice is that it only strictly applies 
to a normal distribution. 

To fiTid the probability of an error lying between —a and -\-a in a 
normal distribution. 

The required probability =-y= — I e'^^'^'^'^dx 

1 r+i 
=-^= e-^i^d^ (where x =(7^ ) 

V2W-1 

^-^(e-P'^d^ 
V27tJo ^ 

=0-6827, by means of the tables. 

This then is the probability that the error in a given sample shall 
not exceed the S.D., cr. The probability that the error shall exceed 



244 



STATISTICS 



cr is accordingly (1— 0-68)=0-32,' It therefore appears that the 
odds against an error exceeding this amount are 68 to 32, or about 
2 to I. 

The probability of an error between —2g and +2(7 

1 /•+2 

V27tJ-2 

=0-9545, 

and the probabiHty of an error outside these limits =0-0455. 

Hence the odds against an error exceeding 2cr are about 21 to 1. 

The probability of an error between — 3cr and +3(7 

1 /• + 3 

=-= e-^'^Hi 

=0-9973. 
Hence the odds against an error exceeding 3cr are about 370 to 1 . 







_ 


B 


rf'-r!!!""*,- _ _ _ 


- *^ _ __ _ _ __ 


/-- s^l It 


.. -.^ ^ 'S 


P Z S I !v , 


iizx " ^ jiE: 


7 : X" -\- ^ - "" "~~ "- :" " : 


/ ^ -IS 


/- - ^ 1 "V 


-^ t j ^ 


l__ ir j "^ s^ it 


-__ — __ _g J ^ — ^__ — _ _ _ _ 


CO J •'^ "-'^ F 


_» J ^tik 


:;;:::--:: ::-±j:gi2i-: :::::::: J S::i=i-S,i 



Fig. (44). 

That these results are reasonable can be seen by an examination 
of the curve of error 

N 



,-a;2/2a2 



the graph of which is drawn, fig. (44), in the particular case when 
(7=5, N=100. The maximum ordinate is thus=20/V27r=7'98, 
and the curve becomes 

2/=7-98e-*'/5o. 
When x^ a^ 5, 2/=(7-98)(0-606)=4-84, P^Ni in the figure. 
„ a:=2(7=I0, 2/=(7-98)(0'135)=l-08, PgN^ „ 
„ a-=3(7=15, y=(7-98)(0-011)=0-09, P3N3 „ 



THE NORMAL CURVE OF ERROR 245 

There is a point of inflexion where the curve changes its 
direction at P^, also at the companion point P'^ on the other side 
of OB. 

The areas ONiPjB, ONgPaB, ON3P3B, P3N3X represent respec- 
tively the frequencies of errors to cr, to 2ct, to Sa, Sa and over 
(considering only errors on the positive side, that is, deviations 
above the mean), and the figure shows how very improbable is a 
deviation from the mean exceeding Scr, for the area between the 
curve and axis beyond this limit is negligible. Put in another 
way, a range of 6cr should include practically all the observations 
in the sample. 

The probable error has in the past received various names, such 
as mean error, median error, quartile deviation, and although some 
of these may seem more applicable and less confusing than the 
name to which it has settled down, there is perhaps not sufficient 
excuse -for unsettling it again, even had we the power to do so, 
by attempting a return to one of these old names. 

If its magnitude be r it is defined to be such that the chance 
of an error falling within the limits —r and +r is exactly equal to 
the chance of an error falling outside these limits, in fact it is an 
even chance whether a particular error falls within these limits 
or not. 

Since area measures frequency it follows that the ordinates 
drawn through the probable errors divide both halves of the normal 
curve (above and below the mean) into two equal parts ; the one 
above the mean, QR, is shown in fig. (44), and consequently the 
area OBQR=the area QRX, in that figure. These ordinates there- 
fore coincide with the quartiles, and the probable error is precisely 
the same measure as the quartile deviation. 

The magnitude of the error is readily calculated from the proba- 
bility integral table, for, by definition, we have 

1 r+^ 
i=— =— e-^'/'^'^'dx 

V27T.aJ-r 
1 f+rl<r 

=-7= e~^''H^ (where x=a^). 

and the probability integral table at once gives 
r=0-6745o-=approximately |a, 



246 STATISTICS 

Thus we have the frequently quoted rule that the 

quartile deviation =?(standaxd deviation), . • (7) 

or probable error =0*6745 (S.D.) 

The probability of an error lying between — 3r and +3r 
1 r+^r 






2 /-SCO -6745) 

= —=1 e-^^'H^ (where x=a^ as before) 

=0-9570. 

Thus the odds against a deviation exceeding three times the probable 
error occurring in a single trial are about 22 to 1, or much the same 
as the odds against a deviation exceeding twice the S.D. 

There remains one other standard of measurement in connection 
with errors which is at least deserving of mention, namely, what 
we have previously called the mean deviation, which may be denoted 
by t;. It is simply the mean of all errors without regard to sign ; 
thus, since yhx measures the frequency of an error lying between 
X and {x-\-hx) 

rj=2 j xydx 2 1 ydx 
= l^xe-^l^-'dx/ Te-^'l-^'^'dx 
=ar$e-^'lMdfe-^''lH^ (where x=g^) 

rco I rco 

=^/2aj te-'^dt/j e-^^dt (where ^'^=2t*) 

=v2c7[-cn^ 



2 Jo'' 2 



=aV2/7r 
=0-7979c7, 
hence the rough rule that the 

mean deviation =g(standard deviation) . . . (8) 

It must be borne in mind that all the above rules relating to 
errors — using the term as synonymous with the deviations of single 
or sample observations from the mean of a considerable number of 
the same character — strictly apply, as we said before, to the normal 



THE NORMAL CURVE OF ERROR 



247 



curve of error and are only approximately true for other distribu- 
tions, the approximation being the closer the nearer they approach 
to the normal form and the larger the number of observations 
involved. They have been tested in some cases in earUer chapters 
(see, for example, Chapter VII.), and the results obtained, even 
with very skew distributions of comparatively small numbers of 
observations, are at all events close enough to suggest the utility 
of the rules in more favourable cases. 

The effect of variaf)ility on errors. The probability of an error 
lying between and t 



1 /•« 



V2, 



TT . CJ.'O 



Put x—x'jm, and this becomes 



1 p 



,-X''H2<T^)'l'^^ 



m 



a/27t . (ma) 



1 /-("lO 

. inujyo 




Fio. (45). 



Thus, if the variability be increased m-fold the range of error (of 
equal probability) is increased m-fold, so that if we have two sets 
of N observations, with the variability of one set double that of 
the other, the range of error also in the one set is double that which 
is equally likely to occur in the other. This is brought out fairly 
clearly in fig. (45), which is the result of plotting the curve 



N 
2/=— =-c 
V27r<T 



-arhl<r- 



248 STATISTICS 

in the two cases. The variability a of curve (1) is double that 
of curve (2) ; if then we measure along OX in the figure 

ONi=:20N2=2^, 

the area B^ONiPi will be equal to the area B2ON2P2, showing that 
the probability for an error between and 2^ in the one case is equal 
to the probability for an error between and t in the other case. 

[James Bernoulli (1654-1705), the eldest of three remarkable brothers, 
showed how the binomial theorem could be used to estimate the probability 
that the ratio of the number of successes to the number of failures under 
defined conditions should lie between set limits, where success means that a 
certain event happens and failure means that it f aUs to happen. 

It was Gauss who first actually published a proof (1809) of the equation of the 
normal curve, although Laplace had suggested as early as 1783 the utility 
of a probability integral table, ^e-^Ht. Gauss's proof depended upon certain 
axioms which cannot be established and are not necessarily true, one of which 
was that ' errors above and below the mean are equally probable.' Laplace 
and Poisson improved upon Gauss and succeeded without assuming this 
axiom, but with the aid of theorems due to Euler and Stirling, in developing 
the continuous probability integral from the discontinuous binomial series. 

Further extensions of the normal curve applicable to skew distributions 
have been worked out by other writers, such as Galton and Mc Alister, Fechner, 
Lipps, Werner, Charlier, Kapteyn, and finally by Edgeworth, who has contri- 
buted materially to the development of the idea of ' the Law of Great 
Numbers.' Karl Pearson approaching the subject of skew variation from 
the same point but by an original route, has discovered a complete system of 
curves suitable for fitting almost all kinds of distributions in homogeneous 
material, especially such as are met with in the biological world. 

(See Todhunter, History of Probability. 

Edgeworth, Law of Error in the Encyclopaedia Britannica (10th edition). 
Pearson, Das Fehlergesetz und seine Verallgemeinerungen durch Fechner 
und Pearson : A Rejoinder ; Biometrika, vol. iv., pp. 169-212).] 



CHAPTER XIX 

FREQUENCY SURFACE FOR TWO CORRELATED VARIABLES 

It may serve at this stage to widen the outlook upon the subject 
of correlation for those who are able to follow it up on mathe- 
matical lines if we briefly consider the algebraical expression for 
the combined distribution of two variables. 

Let the variables be X^, Xg. They may be absolutely independent 
or they may be related in some way, but in either case we shall 
assume it possible to set up a one-to-one correspondence between 
them : thus, X^ might represent the marriage rate and Xg the 
index number for wholesale prices, and we might always pair 
together the X^ and the X2 which refer to the same year, as in the 
correlation example in a previous chapter ; moreover this pairing 
might still be effected even if there were really no other connection 
at all between X^ and Xg. 

If then a^i, x^ typify the deviations of X^, Xg from their respective 
means (the means in the above case being derived by averaging 
the figures for a number of years), it is possible to write down an 
expression of the form 

for determining the probability of deviations between x^ and 
(o^i-f Sajj), X2 and (^Cg+S^Tg), occurring simultaneously (in the same 
year, in the above case) ; or, to put the same thing in another way, 
ySx^Sx^ would represent the proportional frequency with which 
such deviations might be expected to occur together in a large 
number of observations. 

The frequency curve y=f{x), where ySx denotes the frequency 
with which a variable with deviation lying between x and {x-\-8x) 
from its mean value is observed in a given distribution, was repre- 
sented by plotting corresponding pairs of values of x and y as 
points in a plane. In the expression y=Y{Xi, Xg), however, we have 
three variables to consider, x^ and x^j and y which measures the 
frequency of the simultaneous appearance of x^ and x^. Such a 
trio may geometrically be represented by a point P {x^, x^, y) in 

249 



250 



STATISTICS 




Fig. (46). 



space of three dimensions, for (xj, a^g) can first be located as a point 
in a fixed plane and a height y may then be measured above this 
plane as in fig. (46). Clearly as x^ and a^g vary, y also varies, and 
consequently the point P moves about in space, but it moves always 
in obedience to the relation 

y=¥{x^, x^). 

This relation is called the equation of the surface along which 
P travels, showing that it holds good for 
the co-ordinates (x-^, x^, y) of any position 
wl\ich the point can take up on that surface. 
It is convenient, however, to use the notation 
z=F(x, y) 

in preference to y=F{x-i^, x^ for the 'fre- 
quency surface,' because OX, OY are nearly 
always taken to represent the axes of refer- 
ence in space of two dimensions (i.e. in a plane), and by a natural 
extension OX, OY, OZ are taken to represent the axes of reference 
in space of three dimensions, fig. (47). 

We proceed to discuss the frequency surface for two variables, 
and we shall start with the comparatively simple case when the 
variables are completely independent. 

Frequency surface showing distribution of 
two completely independent variables each 
subject to the normal law. 

Let X, Y be the variables, and let x, y de- 
note deviations from their means X, Y, the 
point (X, Y) being taken as origin of co-ordi- 
nates and the usual notation being adopted. 

Thus the probability of a deviation between^ and (a:+Sa:) occurring 

— g 0^*" 

V277 . G^ 

and the probability of a deviation between y and (y-\-^y) occurring 




Fig. (47). 



hy 



i^o 



V27r . Gy 
Therefore the probability of such deviations occurring together 
since the variables are supposed completely independent 



hx 



g-«2/2<r.^ 



hy 



\V27T.G^ JW27T.G^ 

27rO'j.CTy 



_g-2/2/2<r,5 



FREQUENCY SURFACE 251 

Hence the frequency with which such pairs of deviations are 
observed together if n be the total number of observations 

Denoting this by zSxSy, we get for the required frequency surface. 

z=n/27TC7^y . e ^''' ''"'^ . . . (1) 

If we give y some particular value, 2/1, we find from the above 
equation that the law of frequency for the corresponding x is 



2i7TayXjy 



\_27ra^y J 



g-xa/2crx2 



where n^ has been written in place of 

\V27r.(Ty 

But this is evidently a normal curve in the plane XjOZj, having 
the same mean, X, and the same S.D., erg., whatever be the value 
of y^. 

Hence all arrays of X are similar, having the same mean and the 
same standard deviation, and this, by symmetry, also applies to 
all arrays of y. 
. Now put z equal to some constant, k, in equation (1), so that 

k—— —6 '' '^'^ 

n 

Since the left-hand side of this equation is constant for different 
values of (x, y), it follows that the right-hand side is also constant, 
and hence 

^+i^,=c, ... (2) 

where c is a constant. 

We conclude that the values of x and y which can occur together 
with a given frequency, k, are such that the point {x, y) always lies 



252 



STATISTICS 



somewhere on the ellipse (2) in the plane z^k, fig. (48) ; e.g. values 
in the neighbourhood of x^^ and y^ occur with the same frequency as 
values in the neighbourhood of x^ and 0, because in the figure the 
points (x^, 2/i5 ^) ^-nd (x^, 0, h) both lie on the ellipse defined by 



z=k, 



,+- 



The different ellipses which can be obtained by varying the 
frequency, and consequently varying c, are clearly concentric, 
similar, and similarly situated if they are orthogonally projected 
on to the plane z=0, for the effect of such projection is that any 




Fig. (48). 

point (x, y, z) drops down on to the point (x, y, 0) which stands 
immediately below it in the plane XOY. 

The general shape of the surface can be gathered from fig. (48) 
where the ellipse z^=k, and the normal curves a;=0, 2/=0, and 2/=2/i 
have been drawn. 

It will also be noted that if the scales of x and y are altered by 

X u 

writing — —x' and —=y', so that unit change in each may be the 

same, the ellipse (2) becomes a circle 
x'^-\-y'^=c. 

This change of scales is equivalent geometrically to projecting 
orthogonally the ellipse into a circle ; of course the planes of pro- 
jection are not the same as in the previous orthogonal projection 
mentioned, 



FREQITENCY SURFACE 



553 



Frequency surface for two correlated variables. Let the variables 
be X and Y, and let us work as before with their deviations x and y, 
whichis equivalent to taking the mean point (X, Y)of all the observa- 
tions as origin. 

Now the line of regression giving the best y, or the y of greatest 
frequency, corresponding to any x is 



y=r- 



with the usual notation, r being the coefficient of correlation 
between X and Y. 

Hence the error made in estimating any y from this equation 
instead of taking the y given by observation is 

7]=y (observed) —y (estimated) 



=y-rJLx. [See fig. (49).] 



Thus, corresponding to every pair of observations (ic, y) there is 
an 77, and the same 77 will be repeated 
just as often as the same pair of 
observations (a;, y) is repeated. 

Therefore the frequency distribu- 
tion of (a;, 7;) must exactly correspond 
to that of {x, y). 

Further, the correlation of the 
variables x and rj is zero, for posi- 
tive and negative errors 77 are equally likely to occur for different 
values of x; in fact, this coefficient of correlation is E{xr^)ln<jy.(T^, and 



Y 


V 


4-' 







X 



Fio. (49). 



i:(xr^)=E\x[y 



r-^x 



■■E{xy)- 



S(x') 



P 



=np- 

^^np—rvp 
=0. 



na. 



Assuming then that the variables x and 7; are quite independent, 
the probability of them occurring together is readily \^Titten do^^^l, 
for it is simply the product of their separate probabilities. 



254 



STATISTICS 



But the probability of a deviation between x and {x-\-%x) occur- 
ring, if we consider this variable alone, is 



1^ .,-2S, 



V27r(T, 



and the probability of a deviation between 7; and (tz+S?;) occurring. \ 
if we consider this variable alone, is \ 



s^^-2:> 



a/27 
Hence the probability of a combined occurrence of such deviations 



a;2 



\V27r(7^ / \V27rc7^ 



277(7^(7, 
_ 83:87; 



27ror^CT„ 



U,2+ 0-2 j 



27ro-a^, 
But mj^^=E(y-r'^x 



2 



:2;(2/2)-2r . ^^ . 2:{xy)+r-^2(x^) 



jy 



2 



Similarly, no^^na^iX—^^ 

where f is the error made in estimating x from x=r—y 

... %=<=(l-r^). 

Thus ^^ =-L . °:v__L .^=Jl, 



FREQUENCY SURFACE 255 

/ 1 ." r^Gj\ 1 /, . . aJ\ 



and ^+!:^ =jL(i+,2.^\ 



1 



Hence the probability of the combined occurrence of deviations 
X to (x-\-hx), 7} to (77+ St;) 

= '- « ^o-„2 » o-tcTr,^ <rt2) • 



2TTG^.ayVl-f^ 

thus, if we denote by zhxSy the frequency of the combined occur- 
rence of deviations x to (x-{-Sx), y to {y-^hy), when ri is the total 
number of observations, we have * 



z=- 






27r\/l— r^ . CjcO-y 



When the variables X and Y are completely independent, so that 
r is zero, this reduces, as it should, to our previous result 

27r(JxCry 
In the surface z—fju.e ^<^^'^ '^f'^ <r««ry/i-r3 . , . (3) 

where /x = = , if we give y some particular value y^, 

27r\/l-r2 . <7^c7y 

we find that the law of frequency for the corresponding x is 



z=fx , e 2(1 



1 (yh^x'^ -gr'^J ) 



:/Lt.e 2(l-r2)|<r/ '^Va, a,/ 



=/x . e *-"'' e 



2<ry2 2(1 -r2)V<rr cry/ 



(4) 



[*" For an outline of Karl Pearson's method of reaching the Law of Frequency 
for two correlated variables, and certain deductions from it, see Appendix, 
Note 11.] 



256 



STATISTICS 



But just as 



y- 



-i 



(x-a)2 



V27rc7^ 

represents exactly the same normal curve as 

1 



y 



-A 




Fig. (50). 



A/27rC7a, I 

shifted through a distance a along 
the axis of x, fig. (50), so we con- 
clude that the curve (4) in x and 
z, in the plane y=yi, is exactly the 
same as the normal curve 

a:2 






yhlW- 



g-ior;.2(l_,.2) 



shifted through a distance ry^— along an axis parallel to OX. In fact , 

CTy 

(4) represents a normal distribution for x, the mean, corresponding 
to greatest frequency when z=-~-—-^, being determined by the] 

intersection with the surface (3) of the planes 

X y 
y=yv --=r-, ] 



and the standard deviation being a^Vl—r^, which we note is^ 
independent of y^, fig. (51). To put the same thing in another; 
way, the array of x's corresponding to a particular value 2/1 of y\ 

have a mean deviating from X by r— . y^, and a standard deviation? 



In particular, when y=0, z=fjbe ''<^'^^-^')^ a normal distribution^ 
for X, the mean, corresponding to greatest frequency with z=fjby\ 
being determined by the intersection with the surface (3) of the; 



y 



planes 2/=0, — =r— , and the standard deviation being Ur^^/\—r^\ 



as before. 

Similarly, when x=Xi, we get as in (4) a normal distribution for y. 



-fxe 






the mean, corresponding to greatest frequency when z- 



determined by the intersection with the surface (3) of the planes 



being] 



X — X-iy 



y 



FREQUENCY SURFACE 



257 



and the standard deviation being or^Vl— r^, which is independent 
of x^. In other words, the array of y'a corresponding to a particular 

value Xi of x have a mean deviating from Y by r— Xj, and a standard 



deviation GyVl—r^. 



In particular, when x=0, z=fjLe '^y'^^-^^)^ a normal distribution 
for y, the mean, corresponding to greatest frequency with 2;=yLt, 
being determined by the intersection with the surface (3) of the 



planes x=0, —=r—, and the standard deviation being CyVl—r^. 
By putting 2=some constant, k, and arguing just as we did in the 

2 




Fig. (51). 

case of two independent variables, we find that all values of x and y 
which occur together with the same frequency define points {x, y) 
which lie on the ellipse 

The different ellipses which can be obtained by varying the fre- 
quency, and consequently varying c, are concentric, similar, and 
similarly situated, if they are orthogonally projected on to the 
plane z=0. The planes giving the means of the x's, or the most 
frequent x's, corresponding to particular values of y, and the means 
of the 2/'s, or the most frequent 2/'s, corresponding to particular 
values of ic, meet 2=0 in the Unes of regression 



X y y 

7 , 

CflJ <Ty (Jy 



x 
-r — 



258 STATISTICS 

If we alter the scales of x and y by writing — ^=x' and — =?/'> 

so that unit change in each shall be of the same magnitude, the 
frequency surface takes the form 



z=^e 2(1-'^)' 



(x"^+y".i - 2r3fy') 



When y'=0, z=fie ^(i-^"^) ^ a normal distribution, the mean being 



on the plane x'=ry\ and the standard deviation being Vl—r^. 

Similarly for x'=0. When y'=y\, 2=jLte"*''%'2(i-r2)(^-''^'iV'^^ ^ 
normal distribution, the mean being on the plane x'=ry', and 
the standard deviation being Vl—r^ as before. Similarly for 

Again the ellipse which is the locus of the points {x'y') obtained 
by putting 2;=constant, k, corresponding to variables which occur 
with the same frequency, is (in the plane z=k) now 

x'^+y"^-2rx'y'=c, 
and, projecting on to the plane z=0, the lines of regression are 

x'=ry', y'=rx'. 

These lines are the intersections with 2=0 of the planes containing 
the means of the a;"s, or the most frequent x"s, corresponding to 
particular y"s, and vice versa. 

X 11 

Since, geometrically, the transformation —=x\ —=y', is equiva- 

CTa, (Jy 

lent to an orthogonal projection, we may learn something about 
the more general ellipse by considering properties of the simpler 
projected curve which are not changed by projection. 

Let us first, however, find the magnitude and direction of the 
axes of 

x'^-\-y"^—2rx'y'=c. 

By turning the axes through some angle 6 this equation is 
reducible to the form 

which is the ordinary form for an ellipse when its axes lie along 
the axes of co-ordinates. But the equation in x\ y' is clearly 
symmetrical about the lines y' ^=x' and y' ^—x\ because y' and x' 
or y' and —x' can be interchanged without the equation being 
affected. Hence these lines must give the directions of the major 
and minor axes. 



FREQUENCY SURFACE 



259 



To turn the axes of co-ordinates through an angle of 45°, fig. 
(52), we must write 

x' =x" cos 45°-/ sin 45°=^^~j^'' 

V2 



/' I « .// 



y'=x" sin 45°+^/" cos 45^ 
Y' 



x"+y 
' V2 




Fig. (52). 

The equation of the ellipse thus becomes 

(x"-y"f , {x"+y"f ^S^"-y"){x''+y'') 



2r^ 



I.e. 
i.e. 

i.e. 



2 ■ 2 V2V2 

a;"2(l-r)+2/"2(l+r)=c, 



-c. 



x"^ y^ 

c c 

l—r 1+r 



=1. 



Hence the semi-major axis is a= / , and the semi-minor axis 

SJ l—r 

is 6= ^ / We note that as r increases from to 1, a increases 

V 1+r 

from Vc to 00, while h decreases from Veto . / • Also, as r decreases 

from to —1, a decreases from ^/c to / — , while 6 increases from 
Vc to (X). 



260 



STATISTICS 



The ellipses, x"^-\'y"^—^rx'y'=c, corresponding to different values 
of r all pass through the points of intersection of | 

x"^^y'"—c and x'y'=0. i 

But x'^-\-y'^=c is what the equation of the ellipse becomes when r, ■ 
the coefficient of correlation, vanishes. The connection between ; 
these curves is shown in fig. (53), which represents their projection | 
on to the plane z— 0. A positive correlation between x and y i 
might be expected to increase the y corresponding to a particular i 
positive X, if the frequency be fixed beforehand, and that is the | 
effect which the figure also would suggest. j 




Fig. (53). 



Now, in x'^+y'^-2rxy=c, 

the lines of regression are 



y =rx , y =-x , 
r 



and the axes of the elHpse are 



y'=x', y' =—x' 



Hence the lines of regression are equally inclined to the axes of the 
ellipse as well as to the axes of co-ordinates, fig. (54). 
Further, the pair of lines 



y'=x\ y'=-x' 



form a harmonic pencil with the pair 
x'=0, y'=0, 
and also with the pair 

1 



y'=rx\ y'=-x 
r 



This is obvious from fig. (54). 



FREQUENCY SURFACE 



261 



Now project back to the ellipse 



^+— -2r-^=constaQt. 
The algebraical transformation for this is merely 




Fig. (54). 

Since the harmonic property is unaltered by projection we then 
have the pair of lines 

y _x y _ X 

Gy Gx Oy CTg. 

harmonic with the pair 

x=0, y=0, 
and also with the pair 

y _ X y _1 X 

Gy Gg, Gy T G y. 

Hence the two lines of regression corresponding to maximum 
correlation (r=+l and r=— 1) are harmonic with 

(1) the axes of co-ordinates ; 

(2) the lines of regression for any r. 

Again it may be easily seen that the lines 
y'=rx' and a;'=0 
are conjugate diameters of the ellipse 

x'^+y'^-2rx'y'=c, . . . (6) 
for they may be written as one equation thus : 

rx'^-x'y'=0, 



262 STATISTICS 

and this represents a pair of lines harmonic with the (imaginary) 
asymptotes of (5), namely, with 

x'^-\-y'^—2rx'y'=0. 
[The criterion for ax^-\-21ixy-\-hy'^=0 

to be harmonic with a' x'^ + 2h' xy-{-b'y^=0 
is ab' -i-ba' =2hh' .] 

But it is a well-known property of conies that any pair of lines 
harmonic with the asymptotes are conjugate dianieters of the 
conic. 

Similarly it may be shown that the lines 

y' =-x' and y' =0 
r 

are conjugate diameters of the ellipse (5). 

But, on projection, the conjugate property also is unaltered. 

1/ X 

Hence the lines — =r — , x=0, 

II \ oc 

and the lines — = , y=0 

^y 'f ^x 
are conjugate pairs of diameters of the ellipse 

But for conjugate diameters the midpoints of all chords parallel 
to either lie on the other. 

Thus we come back again by another route to the familiar line of 
regression theorems that, for a given r, all arrays parallel to a;=0 

have their means on— =r— , and all arrays parallel to y=0 have 

X 1J 

their means on _=r— • 



APPENDIX 

1. Compound Interest Law. If the capital increases continuously, 
instead of going up by jumps at the end of stated periods, the con- 
nection between the original principal S^, the rate per cent, per 
annum r, and the amount S^ at the end of t years is given by 

for the rate of increase is measured by 

dB_ rS 

which leads at once to the above equation on integrating. 
Other instances of the same law are : — 

(1) ^ particle moving against a resistance proportional to its 
velocity, v^=VQe~'^\ 

where v^ is the velocity at time t, v^ is the original velocity, and c is 
some constant. 

(2) The .variation of the pressure of the atmosphere with height, 

where pj^ is the pressure at height h above a surface level, p^ is the 
pressure at the surface, and c is some constant. 

{^) The rate of cooling, d^z=:0^e~'^\ 

where Of is the excess of temperature at time t of the hot body 
over that of surrounding bodies, 6^ is the excess when the measure- 
ment begins, and c is some constant. 

2. Weighted Mean. Let the observations be represented by the 
different values, Xj^, x^, . . . x^, of the variable concerned, and let 
the respective weights attached to these observations be /i,/2, - - • fn^ 
so that the average, by definition, 

_ a;j/i4-a:2/2+ » ♦ ♦ -\-Xnfn 

S68 



264 STATISTICS 

Now, suppose a different set of weights be chosen, namely, 
fv /'2» • • • /'n» giving a new average 

/1+/2+ • • • H"/w " ■ 

The difference between these two expressions 

_ ^l/l + a;2/2+ • • • _ a^l/'l + a^2f2+ • • • 
/1+/2+ . . . f l+/'2+ . . • 

(/1+/2+ . • •)(/'!+/ 2+ . . .) 

_ i(/lf2K-^2)-/2fl('^l-^2)j + j/lf3(^l-^3)-/3f 1(^1-^3)!+ • ' ♦ 
(/l+/2+ • • .)(f I+/2+ . . .) 

flf2(^l-^2)(^-^J+/lf3(^l-^3)(^^-^J+ . . . 
^ (/1+/2+ • • •)(/'l+/'2+ . . .) 

Hence this difference is very small and the averages are very 
nearly equal if the weights f-^, /g, fz • - • ^^^ replaced by others 
fi, /'a, fz ' ' • very nearly proportional to them, so that /i//'i, 
/2//2> /s/Z's • • • are not far from equality, and this is the more 
pronounced if the observations x^, x^, iCg . . . themselves are all 
of the same order of magnitude and the sums of their weights, 
27/ and 2*/', are large so that the expressions of ty^Q(x^—x^l(Ef){Sf') 
are small. 

3. Geometric and Harmonic Means. Given n numbers 
a, 6, c . . . 
their geometric mean, g, is defined by the formula 

g=^l/(ahc . . . ), 
and their harmonic mean, ^, is defined by 

1=-+.-+'+ • • • 

h a b c 

We note that when a=b—c= . . . =k, say, 
then g=l/(kkh . . .) = l/{k'')=k 

and _=_-}-_-|---[- . . . =_ 

ih fC i€ iC K 

so that h=k. 



7' 



APPENDIX 265 

It is worthy of remark that if the geometric mean be adopted as 
average in discussing the index numbers of prices it possesses an 
interestihg property which does not hold for any of the other means 
in common use. 

Suppose the prices of n standard commodities at three successive 
dates be represented by (a^, 6^, c^ . . . ), (a^, h^, Cg . . .)> (<^3j ^3. ^3 . . .)• 
Then the index numbers of the separate commodity prices at the 
third date, taking the prices at the first date as standard, are 

100-«, 100^, 100^ . . . 
a^ hi Ci 

Hence the geometric mean of these n index numbers together 

100?? X 100^ X 100?? X . . . 
«! bi Ci 

where g^, g^ denote the geometric means of the n prices at the two 
dates. 

It follows that the ratio 

index number of prices at 3rd date with prices at 1st date as standard 
index number of prices at 2nd date with prices at 1st date as standard 

lOOgJgi 

=9J92' 
It is therefore quite ifidependent of the particular date chosen as 
standard. 

4. The Mean of Combined Sets of Observations. (1) Suppose one 
variable x is expressed as the sum of a number of other variables, 

thus a;=a+6+c+ . . ., 

and suppose that we have n different values of the variables, giving 
equations of the type 



Xn=0'n+K-^Cn+ 



266 STATISTICS 

Hence, by addition, 

so that nx=^nd-\-nB-{-nc-\- ... 

x=d-\-h-\-c-Y . . ., 

where x, a, h . . . denote the means of the n values of the respec- 
tive variables. 

Thus the mean of a sum equals the sum of the means, and, if some 
of the positive signs in {a-\-b-\-c-\- . . .) are made negative, there 
will evidently be a corresponding change of sign in (a+6+ . . .). 

Example. — Suppose 100 family budgets are collected and the 
items in each are separated under five heads — rent, food, clothes, 
coals and light, sundries. The expenditure, x, in each budget would 
thus be expressed as the sum of five variables, a, b, c, d, e, and the 
mean of the 100 different re's would equal the sum of the means of 
the a's, the 6's, the c's, the d'a, and the c's. 

(2) Sets of observations are mxide which differ in locality or time or 
some other respect. To find the resultant mean. 

Let I observations of the variable x refer, say, to one date, 
„ m „ „ „ „ „ a second „ 

„ n „ „ „ „ „ a third „ 

and so on, and let the means of these successive groups of observa- 
tions be Xi, x^, :r^, . . . , so that we may write 

Xi=I!xi/l, x^=.UxJm, x^^ZxJn, . . . 

If then X be the resultant mean, we have 

Zxi+2x^-\- . . . _lxi+mx^+ . . . 



Z+m-j- . . . Z+mH- . . . 



Example. — If the school children in the different schools of a 
county are weighed, I children in one school, m in another, n in 
another, and so on, giving mean weights Xi, x^, x^ . - - » the 
resultant mean weight for the children in all the schools combined 
is then given by the above expression. 

5. Mean and Standard Deviation of a Distribution of Variables. 

Let Xi, X2, x^ . . . Xn denote the deviations of each value, or group 
mid- value, of the observed organ or character when measured from 
some fixed value, and let f^, /2, fz - - - fn denote the observed 
frequencies of these respective deviations. 



APPENDIX 



267 



The arithmetic mean of the variables is thus given by 

^ = (/l^l+/2^2+ . . . H-/„^„)/(/i+/2+ . . . +/„), 
referred to the fixed value as origin. 

We may conveniently represent the deviations x^, x^, x^ . . . hj 
lengths measured from an arbitrary origin along a straight Une, 
in which case the point defines the position of the fixed value 
from which the variables are measured. 

Let P mark the position corresponding to a typical variable and 

let G mark the position corre- _ ^ ^ ^ 

sponding to the mean, x. Thus g g ^ 

OV=x, OG=:r, and if we denote "^ '-^ ^ 

the distance of P from G by f , we have 

x=x-\-^. 
Hence 

^==(/l^l+/2^2+ . . . +/a)/(/i+/2+ . . . +/n) 
=[/l(^+^l)+/2(^+f2)+ • . • +/n(^+f J]/(/l+/2+ . . . +/n) 
= mfl+f2+ . • • +/J+(/lfl+/2^2+ . • . +/nf«)]//l+/2+ . . . +/n) 
=^+(/l^l+/2f2+ . . . +/nfn)/(/l+/2+ • • • +/„)• 

Therefore {Ai,+f,^,+ . . . -\-fnL)=0 . . . . (1) 

The expression {/liCi 4-/2^2+ • • • -\~fn^n) is called the first 
moment of the distribution referred to as origin. We conclude that 
when the distribution is referred to G as origin, i.e. when deviations 
are measured from the mean of the distribution, thefirst moment vanishes. 



Frequency Distribution Table. 

(1) (2) (3) (4) 



Deviations of Var- 
iables from some 
fixed value. 


Frequency of 
Deviations. 


Product of Nos. 

in Col. (1) and 

Col. (2). 


Product of Nos. 

in Col. (1) and 

Col. (3). 


Xq 


/i 

/3 
fn 


to 

f^2 
to 


f^\ 
f^\ . 


" 


N 


■N'l 


N'a 



In the notation of the above table, where the dashes are omitted 
in Nj, N2 when the mean is origin, we have 
;c=N'i/N and Ni=0. 



268 STATISTICS 

Again, the root-mean-square deviation, s, measured from the 
arbitrary origin 0, is given by 

■ «'=(A^\+AX\+ . . . +/nX„^)/(/i+/2+ . . . +/n) 

=N',/N, 

and N'2 is called the second moment of the distribution referred to 
as origin. 

Substituting as before we have 

_ xHf,^ . . . H-/J + 2:^(A^1+ • . . +/ngn)+(/lfl+ . . . +fnL') 

(/i+ . : . +/n) 

=^'+(/lf 1+ . . . +fnL')l{fl+ . . . +/n), 

since /i^i+ . . . +/„fn=0. 

Hence 8^=x'+g\ . . . (2) 

where a is the root-mean-square deviation measured from G as 
origin, or the standard deviation as it is called. 

From this result it is clear that o- is always less than s, or the root- 
mean-square deviation is least when measured from the arithmetic 
mean. 

Generally, if we write 

^'*'=(/AH • • • +/a')/(/i+ • • • +/„). 

V,c=(fA^+ ■ ■ ■ /nL')/(/l+ • • • +/n), 

where E{fx^) and Z{f§^) may be called the A;th moments referred to 
and to the mean as origins respectively, so that vi=0, v<i=a^, 
v\=s^, we have 

=vu^hv^^x^ ^^~ V fc-2 . ^24- . . . J^^, 

For example, when A; =2, since 1/0= 1 ^^^ 1^1=0, 

v^^v\-Ti'- . . . (2) bis 

Again, when A;=3, v^=^v\—^i'^—y^ . . • (3) 

and, when ib=4, v^=v\—^v^—^v^—y.^ . . (4) 

There are interesting statical analogues to the above results 
concerning the mean and standard deviation. 



APPENDIX 269 

Let us imagine a set of weights, /^ /g, /g . . . suspended at 
Pi, P2, P3 . . . from, a straight horizontal bar, and let the distance 
of any typical weight / from some arbitrary origin on the bar be x. 
Then the first moment, 

/l^l+/2^2+ • • • -^h^n 
(where some of the a;*s may be negative corresponding to weights 
suspended to the left of 0) measures the total turning effect of all 
the given weights about 0, and if we further imagine all these 

weights replaced by a single weight ^^v ^ ^ 

equal to their sum (/1+/2+ . . • ^ — ^— ^ Tp — 

4-/n), then, in order to produce X 

the same turning effect, it would / 

have to be placed at a point G, the distance of which from 
is given by 

^(/l+/2+ • • • +/n)=(/l^l+/2»^2+ • • • +/n^„). 

Thus x={S^x^-\-Ux^-^ . . . +/„:rJ/(A+/2+ . . . +/J, 

and, statically, this defines the position of the centre of gravity of 
the given weights, /i, /g, . . . /„, relative to 0. 

As before, x=Sf(x-{-^)ISf 

hence fiii+M^^ • • • +fnL=0, 

and, statically, this means that the turning effect of /j, /a . • • /« 
about G is zero, in other words, the bar would balance freely about G. 
Again, the second moment, 

/l^ l~l~/2^ 2+ • • • ~\~JnXn 5 

measures the moment of inertia of the weights /i, fz - - - fn about 0, 
and, if we imagine these different weights replaced by a single 
weight (/1+/2+ . • . +/«) as before, the moment of inertia will 
be unaltered if the latter be located at a distance 5 from 0, where 
(/1+/2+ . . . +fn)s'={fix\+f,x\-{- . . . +/„:r„2); 
therefore s^=(Ax\+ . . . +fnXn')l(fi+ • • • +/«) 

=i:f(x+irii:f 

=x^+g\ 

as before, and the interpretation of this is that the square of the< 
radius of gyration of the system of weights about equals the 
square of the radius of gyration about G, the centre of gravity of 
the system, together with the square of the distance of G from 0. 
Also, 5 is clearly least when it is measured from G. 



e 



o 


.r, 


>^i 


X 


f 








X, 


X 

1 



270 STATISTICS 

6. The Mean Deviation a Minimum when measured from the 
Median. Consider first the case when only two different values of 
the variable are observed, X^, Xg, and let their deviations from an 
arbitrary value, 0, chosen as origin, be respectively x-^, x^. 

If /i, /g be the observed frequencies of these values, the sum of 
their deviations from is 

which is clearly less when the 
value lies between X^, Xg 
than when it is smaller or 
greater than both of them. 
7 7 Choosing 0, therefore, be- 

^ X ^ tween X^, Xg, if /i be the 

greater frequency we write the deviation sum 

=f2^+{fl-f2K, 

where x is the deviation of either of the values X^, Xg from the 
other, and (/i— A) is positive since /i>/2. 

Now this is evidently least when (fi—f<^x-^ vanishes, i.e. when 
(1) x^=^, in which case coincides with X^, the more frequent of 
the two variables, or, when (2) /i=/2, and in this case, when the 
two observed values occur equally often, the deviation sum is 
constant for any origin between X^ and Xg. 

When several different values of the variable are observed, they 
may be arranged in order of magnitude, X^, Xg, Xg . . . X„, from 
the least to the greatest, with frequencies f^, /g, /s • • • fu- 
ll fi>fn we pair off f^ of the X„'s with /^ of the X^'s ; the devia- 
tion sum for this pair is least and remains constant when measured 
from any origin between X^ and ^x X X«-iX« 

X„. We next pair off some or all 4 -^ i ^' ^ 

of the Xi's which remain against ' ^ ^ 

an equal number of X„_i's and the deviation sum for this pair is 
least and remains constant when measured from any origin between 
Xj and X„_i. If some X^'s still remain, we pair them off so far 
as we can against an equal number of X^.g's but, if it be X„_i's 
that remain, we pair them off against an equal number of Xg's. 

This process can evidently be continued until ultimately we 
reach the origin from which the mean deviation of the whole 
distribution is a minimum, for if any X be left unpaired the origin 
will coincide with that X. Otherwise, the deviation is least when 



APPENDIX 271 

measured from any value between the last two X's paired off ' 
together, and within that range it is constant. 

Since, by definition, the median is the value of the variable half- 
way along the series of given observations, ranged in order of their > 
magnitude and assigning each its due weight or frequency, it is \ 
clearly such that a balance can be effected by pairing off the values I 
on either side of it against one another in the manner explained 
above ; it therefore follows that the mean deviation of a frequency 
distribution is a minimum when the deviations are measured from ; 
the median. ^x^ 

The statical analogy to the median also is worth noting. With j 
the same notation as before, the moment or turning effect of two 

forces, /i, /2, about is ^ .v ^ i 

But in this case, if be taken / f 

at some point in between X^ y -^ ! 
and Xg, since the mean devia- | <^.^ ^^ y 

tion sums the separate devia- x^ O I ; 
tions without regard to sign, v 

we must imagine /^ reversed -4 \ 

so as to produce a turning effect in the same direction as before, i 

The moment will then be still {fiX-^+f^^^^ ^^^ i* is ^^^^ when j 

occupies such a position than when it is on X^Xg produced in | 

either direction. I 

Taking 0, therefore, somewhere in between X^ and Xg, the moment | 

may be written \ 

=/2K+^2)+^i(/i-/2) ; ; 

and, iffi>f2, this is least when x^ vanishes, that is, when coincides I 

with Xj, but if /i=/2, the two forces constitute a couple, and the \ 

moment is the same whatever position occupies between Xj : 

and Xg. i 

7. The Method of Least Squares. To the student who is un- 
acquainted with the differential calculus, the following descriptive < 
argument, the basis of the principle of least squares, for determining | 
the values of m and c which make ■ 

(ma;i+c-2/i)2+(ma;2+c-2/2y'^+ • . . +(wa;„+c-2/„)2 ... (1) | 

a minimum, may prove instructive. 

Let us call the above expression E and let us suppose that different j 

values are given to m while c remains unchanged ; in that case E j 



272 



STATISTICS 



will vary with m, and we might imagine the different values obtained 
for E plotted against the corresponding values of m giving a curve 
of some type. Such a curve may rise and fall in wave-like fashion 
as in the figure, resulting in maximum points like A and C, and 
minimum points like B, where we define a maximum point to be 
such that, as we move away from it along the curve, whether to 
left or right, the size of the ordinate (and therefore the value of E) 
decreases ; likewise, a minimum point is such that, as we move 
away from it, the ordinate (and therefore also E) increases. In 
the neighbourhood of such points it is clear that the size of the 

ordinate, such as Aa or B6, 
changes so slowly as to be 
practically stationary. 

Suppose then that m and 
(m-f/x), fj, being very small, 
are two values of m respec- 
tively at and near a minimum 
position on the curve, i.e. a 
position like B corresponding 
to a minimum value for E. 
Since E near such a point 
does not differ appreciably from E at such a point, we may prac- 
tically equate the two expressions obtained for E by substituting 
(m-\-fjb) and m respectively for m in (1), thus 




(m+/ta;i+c-2/ir+(m+/xa;2+c-2/2)2+ • • . 

=(ma;i+c-2/i)2+(ma;2+c-2/2)2+ . . . 

=(ma;i+c-2/i)2+(ma:2+c-?/2)2+ . . . 

[{mx^-\-c—yif+2fiXi{mx^-{-c—yi)-\-fjL^x\]-{- . . . 
=:(ma;i+c— 2/i)2+ . . . 

Thus [2xi(mXi-\-c—yj)+ixx^j]-\- . . . =0. 

Now, the smaller we take /x, the nearer to the truth does this 
result become. Hence, by making fi tend to zero, we are led to 
the strictly true relation 

a;i(ma;i+c— 2/i)+ ... =0. 

This is one of the equations in the text. To obtain the second, 
we keep m constant and vary c. 

Suppose c and (c-\-y) are two values of c at and near a minimum 



APPENDIX 273 

position on the curve ; then, equating the two corresponding 
values of E, we have as before 

(maJi + C-fry— 2/i)2+ . . . =(^^^_^c-2/i)2+ . . . 

(ma:i+c-2/i+7)2+ . . . ={mXi-\-c-y^)^-{- . . . 
[{mXj^-^c-yJ^+2y(mXi-\-c-yj)+y^]+ . . . =(mxi+c-yj)^+ . . . 
Thus [2(m:^,+c-2/i)+7]+ ... =0, 

and, proceeding to the limit when y tends to zero, we reach the 
other equation in the text, namely, 

(ma^i+c— 2/i)+ ... =0. 

[The Method of Least Squares came first into prominence in 
Astronomy in connection with the determination of the best value 
to take when a number of observations, apparently equally reliable, 
give results not quite in agreement. If, for instance, x be the true 
value of some variable, and if x^, ajg, x^ . . . x^ he the results of 
n observations, the method of least squares assumes x to be given 
by making 

y^ix—xj^-i-ix—x^y^^ . . . -^{x—x^f 
a minimum. 

Now — =2(a;— a;i)+2(a;— iCg)^- • • • +2(ic—a;J, and this vanishes 
dx 

when {x—x-i)^(x—X2)-\- . . . -\-{x—Xn)=0, 

i.e. x=(Xj^-{-X2+ . . . +Xn)/n, 

so that in this case we are led to the ordinary arithmetic mean of 
the n observations as the best value. 

The method was used by Gauss as early as 1795.] 



8. To prove 


r+co 
J-co 


■^'6X=V7T. 


Let 




r+co 

1= e-^dx; 

J-co 


thus, also. 




r+co 

1= e-'Hy; 

J-CO 


therefore, 




r+co f+co 

P=/ e-^dx e-^'dy 

J-co J-co 



r+co r+co 

-. I e-(^+y-'^dxdy 

J -co J-co 
rco r2n 

e-'\drd6 

Jr=oJ0 = O 



274 STATISTICS 

(by changing to polar co-ordinates) 

= e-'^rdr\ dd 

Jo Jo 



=[-?]:m: 



„ i: 

=(i)(27r). 

.+00 



Hence 1=1 er^dx=^/'n. 

J-ca 

9. To prove : — 

(1) r(n+l)=nr(n). (2) B(m, n)=^|?^l \ 

r(m+n) 

rco j 

(1) r(w+l)= x^'erHx 

Jo \ 

rco I 

=— a;«c^(e-^) i 

Jx^O 

=^r(7i), j 

because the expression in square brackets vanishes at both Hmits. ^ 

/•CO rco ' 

(2) r(m)r(7i)= e-^^'^-Hil e-^Tj^'-^d'n \ 

Jo Jo 

= ( e-^x^^-'^2xdx\ e-y''y'^''-'^2ydy, 

Jo Jo I 

where x^=i, y^=7]. \ 

Hence r(m)r{n)=4:( f e-^^+y'^x^'^-^y'^''-Hxdy \ 

Jo Jo • 

= [ f " e-Vm+2n-2 cos^"*"!^ sin^^-i^ rdrdO \ 

Jr=oJe=o 

(by changing to polar co-ordinates) . • 

Thus T{m)T(n) = T e-''\^^+^^-^dr j ^cos^'^'-W sin^'^-Wdd i 

where p=T^ a,nd k=sin^B; \ 

therefore, r{m)r(n)=r{m-\-n)'B{n,m) \ 

=r(m+7i)B(m, 7i) 1 

by symmetry. i 



APPENDIX 



275 



10. Elementary Method of Testing the Probability Integral Table. 

The reader may find more satisfaction in using the probability 
integral table if he tests for himself one or two of its results by 
means of squared paper or in some other way. 

We have seen that the probability of an error between and g^ 
is given by the expression 






-^^di. 



V2ttj 
Put ^=V2Xf and this becomes 

If" f I f' 

-^ e-^dx= I e-^dx/ I e-'^dx, by Note (8) 

i/^Jo Jo / J-oo 

=area OBPN/area A'BA, in the figure. 



■+00 



V. 



Now the graph of y=e~^ is drawn in fig. (40) of the text, and it 
is possible therefore to get an 
approximation to the above 
result for any value of x by 
counting the number of small 
squares in that figure enclosed 
by the areas corresponding to 
OBPN and A'BA respectively. 
Each complete small square 
may be reckoned as 1, and each ^ .^,2 

portion of a square may be 

reckoned as 1 if it exceeds half a square and as zero if it is less 
than half a square. 

This gives, for example, 




1 /•0-25 

VttJO 



«^^a;=98/707 =0-139, 



whereas the tables give 0-138. 

For a value like a; =0-71, count the squares in the usual way 
between curve, axes, and ordinate a; =0-70 ; then add to the result 
one-fifth of the number of squares in the small slice of area between 
curve, axis, and ordinates a; =0-70 and .r=0-75. We get 



1 roTi 



e-^dx=2U)ll01 =0-339 



as compared with 0-342 from the tables. 

These results are not unsatisfactory considering the rough nature 
of the method followed to obtain them. 



276 STATISTICS 

11. Bravais' Law of Frequency in the case of two Correlated 
Variables with certain Deductions therefrom— [based on Professor 
Karl Pearson's memoir, Regression, Heredity and Panmixia {Phil. 
Trans., vol. 187a, pp. 253-318)]. 

Consider two variables whose deviations, x and y, from their 
respective means are due to a number of independent causes, the 
deviations in which from their means can be quantitatively denoted 
by 61, €2, . . . 6^. 

We assume that each e deviation is so small compared to the 
mean value from which it is measured that x and y can be sensibly 
expressed as linear functions, thus 

x=a^e^+a^e^-\- . . . -\-a^,,e^ . . . (I) 

2/ =6^6,+ 6262+ ... +6^6^ . . . (2) 

(Some of the a's and 6's may be zero, and if x only involved, say, 
^1) ^2 • . • €fc, and y only involved e^+i . . . e^, then it would be 
natural to expect no correlation between x and y.) 

We further assume that each e varies according to the normal 
law with S.D. a with appropriate suffix. 

Equations (1) and (2) show that the same x and y may arise in a 
multitude of different ways obtained by varying the e's so that 
their weighted sums (the a's and 6's being the weights) remain 
unaltered. The probability that the particular deviations Ijdng 
between 

^1^(^1+^61), 62.^(62+862), . . . e^S^^-\-he^) 

shall concur, since they are all independent, is 



z= 



^^I__ g-e3i/2<r2i ) , . / _^f«^g-em2/2crm2 
,C7iV27r / ' ' ' WV27T 



But, writing 

a3e3+ • • • +<^mem=a» ^3^3+ • • • +^m€m=ft 

equations (1) and (2) become 

a^ei-^a^e^-{-(a—x)=0 
6161+6262+ (iS-i/)=0. 



Therefore — . 



61 _ 62 _ 1 



0'2(?-y)-h(<^-^) 6i(a-a;)-ai(^-2/) a^h^-aj)^ 
And, for any function z/, 

J J J J \0€i O62 O62 061/ 

= (aib2—a2bi)jjvdeid€2' 



APPENDIX 277 

Hence 

_ BxSy e U^a^ • • • +2<r„2; 



g~ 2cr2i(oi&2-a2bi)2 " 2ayiaib2 - cu^hyi g^ . . . 8e . 

The total probability for deviations between x^{x-\-Sx) and 
y^{y+Sy) is obtained by integrating z between limits — 00 and -f-oo 
for all the e's from 63 to €„j, and it is not very difficult to see that 
this will ultimately lead to an expression of the form 
C . 8x8y . e-("^^+^^^y+^y''). 

This is Bravais' Law of Frequency. 

To find the meanings of the constants a, b,h. The total probability 
for a deviation between x^{x-\-8x) associated with any deviation y is 



=:C8xj 



00 






But if a; be subject to the normal law, the probability for a devia- 
tion between x^(x-\-8x) is 

V27r . (7a, ' 

where o-a. is the S.D. of x independent of y. 
Comparing these two results, we have 

if r=—h/Vab. 

Similarly, l/2<7/=(ab-h^)/a=b(l-0, 

so that h=— rVab=— r/2cT,(Ty(l— r*). 

Again, we may integrate z for all values of x and y, and so get 
the total frequency, N, of the (a;, y) pair. 

/+00 r+co 
^Ao^+2kxy^rmdxdy 
-00 J-co 

=Gy/nTb\^"e-^-'^-^'^"dx 

J-co 

^CV7T]bVWlbl{ab-h% 



278 


STATISTICS 


Hence 


7T 




=-V[«6(l-r2)] 




_ N 




2770r,(7,V(l-^') 


Thus 


1 rofi 2ra^ 3/2-1 



where C has the above value. 

It still remains to interpret r and to see that it is really the 
coefficient of correlation as defined in Chapter x. For this purpose 
let us suppose we have observed n pairs of associated x's and 2/'s, 
namely 

(^l2/l). (^2^2) • • • {^nVn)' 

The probability for such a concurrence, taken along with a given 
value for r and assuming the observations independent, is pro- 
portional to 

1 1 p% 2ra;iyi yi^-i •. 1 T^ 2r!CwVn yn^l 
e~2(l-r2)La:r2 <r;c<rs, +crj,2j X y g 2(l-r2)Lo-^ o-;ro-y +o-y2J 

V(i-r^) V(i-»-') 

1 1 r2a:2_2rSa:j/ 2l/2"l 

:= g ~ 2(1 - r2)L<rx^ tr;t<ry "^ <ry2 J 

(l_^2)«/2 

_(l_,2)-n/2g-.7rbjf^--2--^» 

where /c=.Exy/nagjcjy 



=e 



_|log(l-r2)--^^(l -Kr) 



Now the probability of this particular distribution is greatest 
when 

J log (l-r^)+p'^ 

is least, and, differentiating with respect to r, this leads to 
2r {l-r^){-K)+2r{l-Kr) _^ 

^1-7-2 (1-7-2)2 

i.e. -r(l-r^)-Kil-r^)+2r(l-fcr)=0, 

i.e. —r-\-r^—K-{-Kr^-{-2r—2Kr^=0, 

i.e. (r-/c)(l+r2)_0. 

It is not difficult to show that r=K gives a minimum ; hence the 
required probability is a maximum and we get the best value for 
the coefficient by taking 



APPENDIX 279 



CERTAIN CURRENT SOURCES OF SOCIAL STATISTICS 

Any one who is anxious to get reliable figures bearing upon some 
social matter is somewhat at a pause unless he is thoroughly con- 
versant with all the statistical ramifications of Government autho- 
rities, local and national, of trade unions, friendly societies, and 
hosts of other bodies of a public or semi-public character. 

While recognizing the lavish outpouring of statistics of all kinds 
upon a multitude of diverse topics every year, and appreciating the 
immense care and patience shown by those who are responsible for 
their collection and preparation, one cannot but deplore the lack 
of any co-ordinating principle in general between one body and 
another either in deciding what statistics shall be collected, by 
whom and when they shaU be collected, or how afterwards they 
shall be tabulated and presented to the public. Too often a narrow- 
minded jealousy prevents one authority from consulting with 
another, and such co-operation as does exist is due largely to the 
efforts of able and enlightened individuals. The result is that a 
vast amount of labour and expense goes waste and the loss to the 
public is incalculable, but the public do not care, and they do not 
care because they do not know. 

At present, to quote from an influential petition on the subject 
recently presented to His Majesty's Government, * It is almost 
universally the case that any serious investigation is reduced to 
roughly approximate estimates in relation to some factor which is 
essential for its result. ... It is not too much to say that there is 
hardly any reform, financial, social, or commercial, for which adequate 
information can be provided with our present machinery.' But 
this state of things would be partly remedied by adequate control 
such as might be secured by the establishment of a central statis- 
tical office with a minister in charge who should be responsible for 
unification so far as possible in the collection, tabulation, and issue 
of all public statistics. 

It is scarcely possible for a single private individual to make 
a quantitative investigation of any social question on a large enough 
scale to produce results of real value ; conspicuous instances like 
Booth and Rowntree might seem to be exceptions to this rule, but 
even they had a number of workers acting under their direction, 
without whose aid their task would have seemed almost hopeless. 



280 STATISTICS 

For such statistics as we have we are therefore dependent upon 
Government departments, local authorities, public officials, trade 
associations representing employers or labour, public companies, 
and so on. The reader who wishes to get some idea of the extent 
and the limitations of official British statistics is referred to the 
admirable introductory chapters of Bowley's Elements of Statistics. 
Here we cannot do more than mention a very few of the most 
important sources whence such statistics are derived. 

The most voluminous of all our records is probably the Census 
of the Population which is taken every ten years. Its scope is but 
faintly realized by enumerating the chief subjects on which the 
Registrar- General asked information from each householder in 1911, 
namely : 

(1) Numbers and Geographical Distribution of the Population. 

(2) Nationality and Birth-place. 

(3) Numbers at Different Ages, Male and Female. 

(4) Numbers Single, Married, and Widowed. 

(5) Sizes of FamiHes, including Children Dead. 

(6) Numbers engaged in different Professions and Occupations. 

(7) Numbers Blind, Deaf, Dumb, not in their Right Mind. 

(8) Numbers occupying Dwellings of Different Sizes as measured 
by the Number of Rooms. 

This may seem an ambitious scheme when it is stated that the 
mere enumeration of the people was successfully opposed less than 
two hundred years ago as ' subversive of the last remains of EngUsli 
liberty and likely to result in some public misfortune or an epidemi- 
cal disorder,' and the first census was only taken in 1801. [See 
Article in the Encyclopaedia Britannica on the subject.] 

The results of each census are published in bulky volumes as 
soon as they can be reduced and tabulated, a process which, of 
course, takes a considerable time even for an army of workers 
with calculating machines and every modern device to faciUtate 
their progress. It is to be regretted that more is not done to 
advertise so valuable a record of work by publication in a cheap 
and attractive form of a summary of matters which vitally affect 
the good of the commonwealth. As it is, the census volumes tend 
to be purchased only by pubUc authorities and officials who require 
to use them occasionally as books of reference. 

Neglect of the blandishments of advertisement — to be commended 
in general because such neglect is somehow associated with the 
presentation of all truth — may be perhaps carried too far in the 
issue of statistics. 



APPENDIX 281 

It will be noted that in the periodical census no mention is made 
of wages though the people are classified as regards occupation, 
and for information upon this point we must turn to another source. 
The last general census of wages was taken in 1906, following 
and improving upon an earlier inquiry twenty years before, but, 
in connection with an inquiry by the Board of Trade into the cost 
of living of the working classes, information was collected as to 
rates of wages in 1912 of workpeople in certain occupations in the 
building, engineering, and printing trades, these being selected as 
industries common to most towns, and because the time rates of 
wages paid in them are largely standardized. 

The 1906 inquiry into earnings and hours of labour, unlike the 
decennial census, was conducted on a voluntary basis and was 
never wholly completed. In brief it set out to discover from 
employers : — 

(1) The Numbers of Working-people Employed in Various 
Occupations, distinguishing Men, Women, Lads, and Girls. 

(2) The Nature of the Work done and the Rates of Wages Paid, 
distinguishing Time Rates from Piece Rates. 

(3) The Hours Worked, distinguishing Under- or Over-time from 
Normal Time. 

The ground actually covered by the inquiry embraces the fol- 
lowing trades : Textiles, Clothing, Building and Woodworking, Public 
Utility Services, Metal, Engineering, and Shipbuilding — in 1906 ; 
also Agriculture, and Railway Service — ^in 1907 ; the reports upon 
these trades were published separately at different dates between 
1909 and 1912, and the following trades were bulked together in 
one volume, pubHshed in 1913 — Paper and Printing ; Pottery, 
Brick, Glass, and Chemicals ; Food, Drink, and Tobacco ; and 
Miscellaneous Trades. 

The Cost of Living Inquiry of 1912 was in continuation of a 
similar inquiry in 1905, which in addition compared conditions in 
the United Kingdom and certain foreign countries. It dealt not 
only with wages but also with rents and retail prices. 

The report states that ' particulars as to the rent and accommo- 
dation of tjrpical working-class dwellings were obtained from 
officials of local authorities, surveyors of taxes, house owners and 
agents, and by house-to-house inquiry.' Also * returns of the 
prices most generally paid by working-class customers for a number 
of specified commodities were obtained in each town by personal 
inquiry from a number of retailers engaged in working-class trade.' 

Since then Lord Sumner's Committee and a Committee of the 



282 STATISTICS 

Agricultural Wages Board have examined the change in the cost of 
living between 1914 and 1919, as evidenced by a number of house- 
hold budgets collected from among urban working- classes and 
workers in rural districts respectively. 

One other highly important inquiry carried out by the Board of 
Trade deserves notice, namely, the First Census of Production of the 
United Kingdom (1907). 

The published report shows : — 

(1) The total Net Output in Money Value for each Trade Group 
in each Industry. 

(2) The Number of Persons Employed in each Trade Group 
(salaried persons and wage-earners exclusive of outworkers). 

(3) The Net Output per Person Employed in each Trade Group 
as deduced from (1) and (2). 

(4) The Horse-power of Engines in Mines, Quarries, or Factories 
Employed in each Trade Group. 

It is explained that the term ' net output ' here represents the 
value of the aggregate output of the factories, etc., from which 
returns were received in each trade group, after deducting the cost 
of materials purchased from factories, etc., not included in the 
group, or supplied by merchants or others not making returns to 
the Census of Production Office. 

Valuable as the results of these inquiries undoubtedly are, they 
would be of still more value were it only possible satisfactorily to 
collate the various returns of population, wages, and production. 
No record of wages was included, for example, in the Census of 
Production statistics, and it is quite impossible to deduce the number 
of wage-earners and those dependent upon them in any trade at 
any given time. 

Apart, however, from such special inquiries as we have instanced, 
and the ten-yearly census of the people, there are other periodical 
records issued which provide us with valuable information. The 
Ministry of Labour, until recently a special branch of the Board 
of Trade, charged with the duty of keeping in touch with labour 
conditions, issues each month a Labour Gazette giving particulars 
relating to the state of employment in the principal trades in the 
United Kingdom based on returns from employers, trade unions, 
and employment exchanges, besides information concerning trade 
disputes, changes in wages and hours, the course of prices, railway 
traffic receipts, foreign trade, etc. The Board of Trade also pub- 
lishes weekly a Journal and Commercial Gazette dealing with matters 
of interest to all who are engaged in commerce or finance ; while a 



APPENDIX 283 

Monthly Bulletin of Statistics of production, trade, finance, employ- 
ment, etc., at present issued under the name of the Supreme 
Economic Council, is an important recent addition to our knowledge 
of international statistics. 

Again the Registrar- General makes a quarterly return and annual 
summary of births, marriages, and deaths in the different counties 
of England and Wales, and of births, deaths, and infectious diseases 
in certain large towns. In each public health area the medical officer 
reports periodically upon the hygienic condition of the district and 
the health of the people under his care. The Board of Education 
is answerable for conditions in the schools, and the Home Office 
in factories and prisons ; they report from time to time. The 
Ministry of Health similarly issues returns relating to pauperism 
and to housing, while the Board of Agriculture and Fisheries registers 
the acreage under crops and the number of Uve stock in the United 
Kingdom, and the Commissioners of Customs record the expansion 
or contraction of foreign trade. 

In addition we have the endless accounts and statistics suppUed, 
some voluntarily and some compulsorily, by municipal bodies, 
public companies, banks, trade associations, co-operative societies, 
insurance companies, trade unions, etc. 

And yet, in spite of all this wealth of statistics, some surprising 
gaps occur, as we have already seen, in important particulars 
which cannot be traced. We shall quote only one more instance 
of such a hiatus — the income-tax returns provide a basis for measur- 
ing that part of the national income which is subject to taxation, 
some idea also can be formed of what the wage-earners receive, 
but as to the earnings of the portion of the community falling in 
between these two classes we are entirely ignorant. It is possible 
that war conditions during the years 1914-19 may have vastly 
increased the knowledge of the Government as to some matters 
such as internal resources and inland trade, of which little was 
known before, but, if so, the public, whom it concerns so closely, 
have not yet been permitted fully to share in this advantage. 

For an excellent summary of labour statistics compiled or col- 
lected by the Government the reader is recommended to consult 
the Annual Abstract of Labour Statistics of the United Kingdom, 
published in the past by the Labour Department of the Board of 
Trade. 



*\ix 



284 



STATISTICS 



A NOTE ON TABLES TO AID CALCULATION ' 

The short tables which follow are only inserted as specimens, as 
it is expected that the reader who wishes to make extensive use 
of such tables will have access to the fuller ones to which reference 
is made below. 




-1-00 



Fio. (55), 



Probability Integral Table, giving area of curve z- 
terms of corresponding abscissa, see fig. (55) : — 



V27T 



•00 
•10 
•20 
•30 
•40 

•45 
•50 
•55 
•60 
•65 
•70 

•71 

•72 
•73 
•74 
•75 



Ul + a) 



•50000 
•53983 
•57926 
•61791 
•65542 

•67364 
•69146 
•70884 
•72575 
•74215 
•75804 

•76115 
•76424 
•76730 
•77035 
•77337 



•00000 
•07966 
•15852 
•23582 
•31084 

•34728 
•38292 
•41768 
•45150 
•48430 
•51608 

•52230 

•52848 
•53460 
•54070 
•54674 



•76 

•77 
•78 
•79 
•80 

•85 

•90 

•95 

100 

105 

110 

150 
200 
2-50 
3^00 
350 



Ul + a) 



•77637 
•77935 
•78230 

•78524 
•78814 

•80234 
•81594 
•82894 
•84134 
•85314 
•86433 

•93319 
•97725 
•99379 
•99865 
•99977 



•55274 
•55870 
•56460 
•57048 
•57628 

•60468 
•63188 
•65788 
•68268 
•70628 
•72866 

•86638 
•95450 
•98758 
•99730 
•99954 



Fig. (56), the result of plotting a against |, enables us to estimate' 
the probability of an error Ijdng between any two limits. 



APPENDIX 



285 



Table giving P, to test * 

values of n' and -^ : — 



goodness of fit,' corresponding to certain 



n' 

7 


x2->4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


•67668 


•54381 


•42319 


•32085 


•23810 


•17358 


•12465 


•08838 


•06197 


•04304 


•02964 


•02026 


8 


•77978 


•65996 


•53975 


•42888 


•33259 


•25266 


•18857 


•13862 


•10056 ' ^07211 


•05118 


•03600 


9 


•85712 


•75758 


•64723 


•53663 


•43347 


•34230 


•26503 


•20170 


•15120 -11185 


•08176 , 05914 


10 


•91141 


•83431 


•73992 


•63712 


•53415 


•43727 


•35048 


•27571 


-21331 16261 


•12232 09094 


11 


•94735 


•89118 


•81526 


•72544 


•62884 


•53210 


•44049 


•35752 i •2850() -22367 


•17299 : 13206 


12 


•96992 


•93117 


•87336 


•79907 


•71330 


•62189 


•53039 


•4432(3 -36264 •29333 


•23299 -18250 


18 


•98344 


•95798 


•91608 


•85761 


•78513 


•70293 


•61596 


•52892 -44568 -36904 


-30071 i 24144 


14 


•99119 


•97519 


•94615 


•90215 i ^84360 


•77294 ^69393 


•61082 -52764 -44781 


•37384 


•30735 


15 


•99547 


•98581 


•96649 


•93471 ^88933 


•83105 ^76218 


•08604 j -60630 i -52652 


-44971 


•37815 



One of the earliest tables of the probability integral appeared in 
Kramp's Analyse des Refractions (Strasbourg, 1798), where the 
calculation of j^e-'^Hx was given to eight places from x=0 to a; =3 
at intervals of 0-01. Tables more recent and extensive are those 
due to J. Burgess {Trans. Roy. Soc. Edin. 1900) and to W. F. 
Sheppard (Biometrika, vol. ii., pp. 174-190). Of these the latter 



I -00 



•50 



::iS 



■apftW- 



it 



fnamm&l _ .a. 3^ 



5r 



m 



2'Sd 



50 



. i_-oq ^ 



1-60 

Fig. (56). 



2-00 



250 f 



is reproduced in the admirable Tables for Statisticians and Bio- 
metricians, edited by Karl Pearson (Camb. Univ. Press, 1914), and 
the same volume also contains Palin Elderton's P Tables for testing 
' goodness of fit ' which first appeared in Biometrika, vol. i., and 
Duffell's Tables of the Logarithms of the T Function from Biometrika, 
vol. vii., besides a large number of other valuable tables. 

It should be remarked in connection with the last-named table that 
the formula T(x-\-\)=x T{x) enables us to reduce the calculation 
of any T function to one in which x lies between 1 and 2, by repeated 
applications of the logarithmic relation, thus 
logr(a;+l)=log a:+log T(x) 

=log a;+log (a:-l)+log T(x-\), 



286 



STATISTICS 



and so on. When x is large, however, say greater than 10, the 
well-known approximate formula 

(see, for instance, Whittaker's Analysis, § 110) will be found useful, 
and it may also be written 

log ^:(^±i)=0.3990899+«:2??l!^+^ log x, 

x^e-^ X 

a form often convenient. 

It may be of service to record here the values of a few constants 
which frequently recur for speedy reference : 



6=2-718 2818 


7r=3-141 5926 


logio 2=0-301 0300 


i = 0-367 8794 

e 


logio7r= 0-497 1499 


logio 3=0-477 1213 


logio 6=0-434 2945 
logio(logioe) = 1-637 7843 


logio^^ 1-600 9101 

V27r 





The statistician who has Pearson's Tables, Barlow's Tables of 
Squares, ate, together with a good set of Tables of Logarithms 
(unless he is so fortunate as to have a mechanical calculator, for 
instance a Brunsviga, at his disposal) and of Trigonometrical 
Functions such as Chambers's Seven-Figure Tables, may consider 
himself amply provided for serious research and decidedly better 
off than his predecessors who prepared the way for him by doing 
great work with much poorer tools. 



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