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BOUGHT WITH THE INCOME OF THE
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MATHEMATICS
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, ,_ Cornell University Library
HG8781.F53 1922
An elementary treatise on frequency curv
3 1924 001 546 971
The original of this book is in
the Cornell University Library.
There are no known copyright restrictions in
the United States on the use of the text.
http://www.archive.org/details/cu31924001546971
An Elementary Treatise
on
FREQUENCY CURVES
and their Application in the Analysis
of
Death Curves and Life Tables
by
Arne Fisher.
Translated from the Danish
by
E. A. Vigfusson.
With an Introduction
by
Raymond Pearl,
Professor of Biometry and Vital Statistics
.Johns Hopkins University, Baltimore.
i*
American Edition.
New York.
THE MACMILLAN COMPANY.
1922.
N
EM
k^\>KQ>%f
Printed by Bianco Luno, Copenhagen.
INTRODUCTION
1 he fact that actuarial science is fundamentally
a branch of biology rather than of mathematics is
overlooked far more generally than ought to be the
case. Most people, even those of education and wide
culture, are inclined to look upon an actuary as a
particularly crabbed, narrow, and intellectually dusty
kind of mathematician. In reality his subject is one
of the liveliest in the whole domain of biology, and
none surpasses it in its practical interest and import-
ance to mankind. Because, what the actuary is, or
at least should be, trying always to formulate more
and more definitely are the laws which determine
the duration of human life. Why the actuary in fact
is too often intellectually but little more than a sort
of glorified computer, is really only the result of a
defect in the teaching of biology in our colleges and
universities. It has only lately come to be recognized
anywhere that a biologist needed a substantial founda-
tion in mathematics in order successfully to practice
a biological profession. It is' not too rash a prediction
to say that presently the time is coming when no
important actuarial post will be held by a mathe-
matician who knows little or no biology. The vigor
and originality of his biological outlook will be valued
as highly as the rigidity of his mathematical sub-
structure now is.
II Introduction.
The thing which chiefly makes this book by my
friend Arne Fisher notable, lies, in a broad sense,
in the fact that it is a highly original and absolutely
novel essay in general biology. The language is to a
considerable extent mathematical, to be sure, but the
subject matter, the mode of logical approach, and the
significant conclusion — all these are pure biology.
Unfortunately many biologists will not be able to
appreciate its significance, or even to read it intel-
ligently. But this is their loss, and at the same time
an exposure of the dire poverty of their intellectual
equipment for dealing with the problems of their
science.
There are two broad features of Fisher's work
which want emphasis. The first is the successful
construction of a life table from a knowledge of deaths
alone. That the construction is successful his results
set forth in this book abundantly demonstrate. To
have done this is a mathematical and actuarial
achievement of the first rank. It may fairly be
regarded as fundamentally the most significant ad-
vance in actuarial theory since Halley. It opens out
wonderful possibilities of research on the laws of
mortality, in directions which have hitherto been
wholly impossible of attack. The criterion by which
the significance of a new technique in any branch of
science is evaluated, is just this of the degree to which
it opens up new fields of research. By this criterion
Fisher's work stands in a high and secure position.
But of vastly more significance considered purely
as an intellectual achievement is his discovery of
the fundamental biological law relating the several
causes of death to each other, which made the tech-
nical accomplishment possible. More than one accepted
Introduction. JJT_
text book on vital- statistics has scornfully instructed
its readers that no good whatever could come from
any tabulation or study of death ratios; that they must
be avoided as the pestilence by any statistician who
would be orthodox. But orthodoxy and discovery are
as incompatible intellectually as oil and water are
physically, a cosmic law often overlooked by our
" safe and sane" scientific gentry. This book is an
outstanding demonstration that this law is still in
operation. Fisher has had the temerity to study the
ratios of deaths from- one cause or group of causes
to those from another group, or to all causes together,
and 1 has discovered that there abides a real and
hitherto unsuspected lawfulness in these ratios. Here
again his pioneer work opens out alluring vistas to
the thoughtful biometrican.
Altogether we of America are to be warmly
congratulated that this brilliant Danish mathematical
biologist has chosen to come and live with us.
Baltimore, November 1921.
Raymond Pearl.
AUTHOR'S PREFACE
1 he classical method of measuring mortality rests
essentially upon the fundamental principles first
enunciated by the British astronomer, Halley, in his
construction of the famous Breslau Life Table. Since
the time of Halley this method has been so thoroughly
investigated and has been perfected to such an extent
that new developments along this line cannot be
expected. Any improvements on the original principles
of Halley are after all nothing but refinements in
graduating methods; and even in this line it appears
that the limit of further perfection has been reached.
Halley's method, which is purely empirical in
scope and principle, rests primarily upon the know-
ledge of the number of persons exposed to risk at
various ages and the correlated number of deaths
among such exposures. In all cases where such
information is at hand the old and tried method meets
all requirements to our full satisfaction; and it would
appear superfluous to try to supplant it with fun-
damentally different principles.
In presenting the new method outlined in this
little book I wish to state most emphatically that it
has never been my intention to try to supersede the
conventional methods of constructon of mortality
tables wherever such methods are applicable. My
proposed method is only a supplement to the former
Author's Preface. V
tools of statisticians and actuaries, and aims to
utilize numerous statistical materials to which the
older system of Halley is not applicable. The idea,
whether it is new or not, meets in reality a very
frequent need in mortality investigations. It is a well
known fact that in the determination of certain
statistical ratios, it is easier to determine the nume-
rator than the denominator, as for instance in life
or sickness assurance, where the losses can be
ascertained with a very close degree of accuracy,
while the collection of persons exposed to risk at
various ages is often difficult to obtain. Similar
remarks hold true in the case of numerous statistical
summaries of mortuary records as published in most
government reports on vital statistics. The desire to
utilize this enormous statistical material was what
led me to try the proposed method.
In principle the plan is fundamentally different
from that of the empirical method of Halley, inasmuch
as I have attempted to substitute the inductive
principle for that of pure empiricism.
In the first place, I consider the d x curve, or the
number of deaths by attained ages among the
survivors of an original cohort of say 1,000,000
entrants at age 10, as being generated as a compound
curve of a limited number (say 8 or less) of subsidiary
component curves of either the Laplacean-Charlier or
Poissori-Charlier type.
The method of induction now consists in deter-
mining the constants or parameters of these sub-
sidiary curves. These parameters fall into two
separate categories: —
A. The statistical characteristics or semi-invari-
ants which determine the relative frequency distribu-
VI Author's Preface.
tion by attained age at death, as expressed by the
mean, the dispersion, the skewness and the excess
of each subsidiary or component curve.
B. The areas of each subsidiary or component
curve.
The working hypothesis which I have put forward
is that the relative frequency distribution of deaths by at-
tained ages, classified according to a limited number of
groups (generally 8 or less) of causes of death among the
survivors of the original cohort of entrants, tend to cluster
around certain ages in such a way that it is possible from
biological considerations to estimate in practice with a
sufficiently close degree of approximation the statistical
characteristics or semi-invariants of the relative frequency
distributions of the component curves, corresponding to a
previously chosen classification of causes of death (into 8
or less subsidiary groups).
This implies briefly that I suppose it is possible
from biological considerations to select a priori the
statistical characteristics of the category as mentioned
above under A.
Once this hypothesis is accepted as a true supposi-
tion, the areas of each of the component curves can
be determined by purely deductive methods (as for
instance the method of least squares) from the
observed values of the proportionate death ratios
R B (x) (x = 10, 11, 12, 100; B =1, II, III,
) corresponding to the groups of causes
of death.
Thus the parameters as determined in this
manner exhaust the given statistical material, i.e.
the observed proportionate death ratios R B (x). A
mere addition of the subsidiary or component curves
Author's Preface. VII
gives us then the compound d x curve from which it
is an easy task to find the functions, i x and q x .
The scheme as we have briefly outlined it above
is, therefore, not a cut-and-dried doctrine or a sort
of "mathematical alchemy" as some of my critics
have implied. Nor is it an authoritative or infallible
dogma. The keystone upon which its success depends
is merely a working hypothesis; i.e. a temporary or
preliminary supposition. I suppose something to be
true and try to ascertain whether, in the light of that
supposed truth, certain facts fit together better than
they do with any other supposition hitherto tried.
The validity of the working hypothesis must, in
my opinion, be proved or disproved either by-
independent methods and principles of construction
of mortality tables, such as for instance the empirical
principle of Halley, hitherto exclusively used by the
actuaries, or through additional biological studies. l
1 The biological basis of Mr. Fisher's working hypothesis, which is
of far greater importance than the purely ancillary mathematical deduc-
tion, has apparently been overlooked by many of his American critics,
such as Little, Thompson and Carver. Dr. Carver in the Proceedings
of the Casualty Actuarial Society of America (Vol. VI, page 357)
remarks that "if we can construct a table from death alone as in Proc.
Vol. IV, and by dividing these deaths by q x , determine the unenumer-
ated population — why not the converse?"
The answer to this remark is obvious. In the case of mortuary
records, Fisher considered two different and distinct attributes, namely
1) the purely quantitative attribute of attained age at death, and 2) the
purely biological attribute of cause of death, which in conjunction with
the working hypothesis to a certain extent aims to replace the unknown
exposures. If we were to follow Dr. Carver's facetious suggestion and, to
use his phrase, "go the proposed plan one better by using enumerated
populations only", we should, however, encounter a statistical series with
the single attribute of attained age only, but no second attribute corres-
ponding to that of the biological factor of the cause of death. Criticisms
VIII Author's Preface.
In the meantime I feel justified in presenting to
my readers the practical results obtained by this
method, which although perhaps not unimpeachable
in respect to mathematical rigour, neverthelees in my
opinion offers a means to attack a vast bulk of
collected statistical data against which our former
actuarial tools proved useless. The celebrated Russian
mathematician Tchebycheff, once made a remark to
the effect that in the antique past the Gods proposed
certain problems to be solved by man, later on the
problems were presented by halfgods and great men,
while now dire necessity fo.rces us to seek some
solution to numerous practical problems connected
with our daily conduct. The problem towards which
I have made an attempt to offer a sort of solution in
the present little essay is one of these numerous
problems of dire necessity mentioned by Tchebycheff,
and I hope that my work along this line, imperfect
as it is, may nevertheless prove a beginning towards
more improved methods in the same direction.
In conclusion I wish to extend my thanks to a
number of friends and colleagues both in America
and Europe and Japan who have kept on encouraging
me in my work along these lines in spite of much
adverse criticism from certain statistical and actuarial
circles. I wish in this connection to thank Mr. F. L.
Hoffman, Statistician of the Prudential Insurance
Company, for permitting me to apply the method to
various collections of mortuary records while working
as a computer in his department. My thanks are also
of the sort of Dr. Carver's brings to light the fundamentally different
principles applied by Mr. Fisher in sharp contradistinction to the purely
empirical methods of the orthodox actuary and statistician.
Translator.
Author's Preface. IX
due to Mr. E. A. Vigfusson for. making the trans-
lation from my rough Danish notes. If the resulting
English is perhaps open to criticsm, I beg to remind
the reader that my original manuscript was written
in Danish and translated into English by an Icelander,
while the composition and proof reading was done
by a Copenhagen firm.
To Professor Glover of the University of Michigan
I also wish to extend my thanks for inviting me to
deliver a series of lectures on the construction of
mortality tables before his classes in actuarial
methods during the month of March 1919. This
invitation afforded me the first opportunity to bring
the proposed method before a professional body of
statistical readers.
Last but not least I desire to acknowledge my
obligations to Professor Pearl whose introductory
note I consider the strongest part of the book. In
these departments of knowledge the appreciation of
one's peers is after all the only real reward one can
possibly expect. The fact that this eminent biologist
has recognized that the nucleus of the whole problem
is of a purely biological nature, and that the
mathematical analysis is merely ancillary, is
particularly pleasing to me, because it represents my
own view in this particular matter.
p. t. Newark, U. S. A., November 1921.
Arne Fisher.
TRANSLATOR'S PREFACE
During the spring of 1919 the attention of the
present writer was called to a brief paper entitled
Note on the Construction of Mortality Tables by means of
Compound Frequency Curves by the Danish statisticican,
Mr. Arne Fisher. The novelty and originality of this
paper impressed me to such an extent that I became
desirous' of obtaining more detailed information about
the process than that which necessarily was contained
in the above summary note, originally printed in the
Proceedings of the Casualty and Acturial Society of
America.
I wrote therefore to Mr. Fisher and inquired
whether he intended to publish any further studies
on this 1 subject. From his reply I learned that he had
delivered a series of lectures on this very topic before
Professor Glover's insurance classes at the University
of Michigan during the month of March 1919, but that
the proposed method had been met with such captious
opposition in certain actuarial circles that he had
decided to abandon the plan of publishing anything
further on the subject and had even destroyed the
English notes prepared for the Michigan lectures.
In the meantime the proposed scheme had
received considerable attention in actuarial circles in
Europe and Japan and several highly commendatory
Translator's Preface. XI
reviews had appeared in the English and Continental
insurance periodicals and various scientific journals,
notably the Journal of the Royal Statistical Society and
the Bulletin de V Association ales Actuaires Suisses. The
proposed method seemed indeed so novel and unique
that I could not help feeling that it deserved a
better fate than that of being forgotten. I sug-
gested therefore to Mr. Fisher that he prepare a
new manuscript. But unfortunately his time did not
allow this. He consented, however, to turn over to
rne his original Danish notes on the subject from
which he had prepared his Michigan lectures and
permitted me to make an English translation for the
Scandinavian Insurance Magazine. I gladly availed
myself of this opportunity to bring this fundamental
work before an international body of readers and
started on the translation in the summer of 1919.
At the same time Mr. Fisher decided to put the
proposed method and working hypothesis to a very
severe test, which would meet even the most stringent
requirements of some of his critics and their conten-
tion that the method would fail in the case of a
rapidly changing population group. For this purpose
he selected a- series of statistical data contained in the
annual reports and statements of a number of the
leading Japanese Life Assurance Offices, relating to
their mortuary records for the four year period from
1914—1917. More than 35,000 records of male lives,
arranged according to the Japanese list of causes of
death and grouped in quinquennial age intervals
formed the basis for the construction of the final
life table which was completed in November 1919.
This table, which like Mr. Fisher's other tables was
derived without anv information of the number of
XII Translator's Preface.
lives exposed to risk at various ages, is shown in the
addenda of this treatise.
Immediately after its construction Mr. Fisher isent
this table to the well known Japanese actuary, Mr.
T. Yano, and asked him for an opinion regarding the
trustworthiness of the final death rates of q x as
derived by his new method. The Japanese actuary's
answer arrived in April 1920. Mr. Yano had after
the receipt of Mr. Fisher's letter ascertained the
exposures and deaths among male lives at each
seperate age for about 40 Japanese life offices during
the period 1914 — 1917 and constructed by means of
the conventional methods a complete series of q x by
integral ages from age 10 to 90. These ungraduated
data are shown as a broken line polygon in the
appended diagram (Figure 1). In spite of the fact that
Mr. Fisher had no information whatever about the
exposed to risk the agreement of the continuous curve
of q x as determined by the frequency curve method
with Mr. Yano's ungraduated data is so close that
I think further comments superfluous. The slight
differences in younger ages might indeed rise from
the fact that Mr. Yano had access to all the experience
(containing more than 45,000 deaths) of all the Ja-
penese companies, whereas Fisher only used the
mortuary records as published by some of the leading
Japanese companies.
Like all scientific methods of induction Mr. Fi-
sher's proposed plan rests upon a working hypothesis,
namely that it is possible from biological considera-
tions to group the deaths among the survivors at
various ages in any mortality table according to
causes in such a maimer that their percentage or
relative frequency distribution according to attained
Translator's Preface.
XIII
age at death will conform to a previously selected
system or family of Laplacean-Charlier or Poisson-
A-L
-as.
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Fig. 1.
Charlier frequency curves. Mr. Fisher himself is very
frank in ■ stating that this is a working hypothesis
XIV Translator's Preface.
upon which hinges the success of the whole method.
One of the main objections of his critics is that it
seems impossible to prove the truth of this working
hypothesis. Naturally its truth cannot be proved by
mathematics or logic any more that we can prove
or disprove the existence of Euclidean space, which
in itself constitutes a working hypothesis for most
of our applied mathematics. Mr. Fisher's critics might
as well be asked to prove or disprove Newton's
hypothetical laws of motion and attraction as
extended by Maxwell and Hertz, or the newer
hypothesis recently put forwards by the relativists,
or the Lorentz hypothesis of contraction. It would
indeed be a terriffic blow to science and the extension
af knowledge if it was required that no working
hypothesis would be alloved in scientific work unless
such hypothesis could be proved to be true. What
position would biology occupy to-day if biologists had
insisted that Darwin's great hypothesis be proved
before it could be allowed 1 as a foundation in the study
of evolution?
The most convincing answer to Mr. Fisher's
captious critics among the old school of actuaries
and statisticians is, however, the undisputed fact that
his working hypothesis as such really does work.
As pointed out by Dr. Pearl in the introductory note
of this book the results set forth in the present
treatise abundantly demonstrate this fact. The 6
widely different mortality tables as shown in the
addenda stand as mute and yet as the most eloquent
evidence to the fact that the method works. It might
indeed' not appear impertinent to suggest that Mr.
Fisher's actuarial critics would render a greater
service to their profession by proving that these six
Translator's Preface. XV
mortality tables cannot be considered as reasonable
approximator to tables derived by orthodox means
from the same population groups than by starting
to poohpooh and ridicule his proposed method.
Winnipeg, Canada, November 1921.
E. A. Vigfusson.
"Nothing is less warranted in science than an uninqui-
ring and unhoping spirit. In matters of this kind, those
who despair are almost invariably those who have never
tried to succeed."
W. Stanley Jevons.
CHAPTER I
(TRANSLATED BY MISS DICKSON)
AN INTEODUCTION TO THE THEOEY OF
FEEQUENCY CUEVES
1. introduction The following method of con-
structing mortality tables from
mortuary records by sex, age
and cause of death rests essentially upon the
theory of frequency curves originally introduced
by the great Laplace and of recent years further
developed and extended through the elegant and
far reaching researches of the Scandinavian school
of statisticians under the leadership of Gram,
Charlier and Thiele and their disciples. This
method is, however, comparatively little known
and unfortunately not always fully appreciated
by the majority of English statisticians and ac-
tuaries, who prefer to apply the well known
methods of the eminent English biometrician,
Karl Pearson. For this reason it may be advisable
to give a preliminary sketch of Charlier 's methods
so as to obtain a better understanding of the
1
2 Frequency Curves.
following chapters dealing with the more specific
problem of mortality tables. The treatment must
necessarily be brief and represents essentially an
outline of the more detailed theory which I hope
to present in my forthcoming second volume of
the Mathematical Theory of Probabilities.
By the method of Charlier any frequency
function is expressed as an infinite series rather
than as a closed and compact algebraic or tran-
scendental expression by the Pearsonian methods.
By power series the thoughts of the majority of
students are associated with the famous series
which bear the names of Taylor and Maclaurin.
In these series the function is derived as an in-
finite series of ascending powers of the inde-
pendent variable whose coefficients are expressed
by means of the correlated successive derivatives
of the function for specific values of f(x). Thus
for instance we know that the Maclaurin series
may be written as follows :
m = /<o) + g-f (0) + ^/-(O) + . . .~no) + ...
where /"(0) is the symbol for the value of the n th
derivative when x = and n = 1, 2, 3, 4 . . . . n.
There are, however, contrary to the belief of
many immature students, only comparatively few
functions which allow a rigorous expansion by this
Introductory Remarks. 3
method, in which the derived functions and the
differential calculus play the leading roles.
But on the other hand there are other methods
of expansions in infinite series which are more
general and by which the coefficients of the in-
dependent variable are expressed by operations
other than those of differentiation. One of these
methods is to express the coefficients as definite
integrals either of the unknown function itself or
some auxiliary function.
The range of practical problems which lay
themselves open to a successful attack along those
lines is much wider than the corresponding range
of practical problems to which we may apply the
Taylor series.
Speaking generally as a layman (who continu-
ously has to face practical rather than abstract
problems) and specifically as a mathematical
novice (who considers mathematics as a means
rather than as an end) this fact appears to me
quite obvious from a purely philosophical point of
view. In nature and in all practical observations
we encounter finite and not infinitesimal quantit-
ies. In other words, what we actually observe are
finite sums or definite integrals, i. e. the limit of
a sum of infinitely small component parts.
The definite integral rather than the derivative
and the differential seems, therefore, to be the
4 Frequency Curves.
more elementary and primitive operation and the
one which suggests itself first hand. History of
Mathematics indeed proves this contention. Ar-
chimedes had (as shown by the researches of the
Danish scholar, Heiberg) laid the essential foun-
dation for an integral calculus about 500 B. C.
And nearly 25 centuries later, almost simultane-
ously with the historical discovery of Heiberg an-
other Scandinavian, the Swedish mathematician
and actuary, Fredholm, gave to the world his
epochmaking work on integral equations. Fred-
bolm's monumental memoir "Sur une nouvelle
methode pour la resolution du problems de Dirich-
let" was first published in the "Ofversigt af aka-
demiens forhandlinglar" (Stockholm 1900). Mea-
sured by time the subject of integral equations is
thus a mere infant in the history of mathematical
discoveries. Measured by its importance it has
already become a classic. Its application to a
steadily increasing number of essentially practical
problems in almost every branch of science has
placed it in a central position of modern mathe-
matical research and it bids fair to become the
most important branch of mathematics.
Fredholm in introducing his now famous in-
finite determinants, known as the Fredholmean
determinants, had a forerunner in the Danish
actuary, Gram, whose Doctor's dissertation "Om
Introductory Remarks. o
Rsekkeudviklinger ved de mindste Kvadraters Me-
tode" (Copenhagen 1879) gave prominence to a
certain class of functions which later on have
become known as orthogonal functions, and by
which Gram actually gave the first expansion of
a frequency distribution or frequency curve in
an infinite series. Scandinavians in general and
Scandinavian actuaries in particular may, there-
fore, feel proud of their share of imparting know-
ledge on this important subject, which makes a
strong bid to place mathematics on a higher plane
than ever before, not alone as an abstract but
equally well as an applied science. The genius
of the Italian renaissance Leonardo da Vinci, as
early as 1479 proclaimed "that no part of human
knowledge could lay claim to the title of science
before it had passed through the stage of mathe-
matical demonstration". Comparatively few bran-
ches of learning measure up to the standard of
Leonardo da Vinci, and our learned friends among
the economists and sociologists have a long road
to travel before they succeed in placing their
methods in the coveted niche of science. But the
new vistas of possibilities opened up to them by
means of M. Fredholm's discovery ought to
furnish them a powerful tool towards the attain-
ment of the high standard set by the great Italian.
The principal theorems of integral equations
6 Frequency Curves.
are bound to be especially fruitful in their ap-
plication to mathematical statistics and the pro-
blems of frequency curves and frequency surfaces
together with the associated problems of mathe-
matical correlation.
2. frequency If N successive observations
DISTRIBUTIONS originating from the game eg _
functions sen tial circumstances or the
same source of causes are made in respect to a
certain statistical variate, x, and if the individual
observations o. (i = l, 2, 3, . . . . N) are permuted
in an ascending order then this particular per-
mutation is said to form a frequency distribution
of x and is denoted by the symbol F(x).
The relative frequencies of this specific per-
mutation, that is the ratio which each absolute
frequency or group of frequencies bear to the
total number of observations, is called a relative
frequency function or probability function and is
denoted by the symbol cp(aO.
If the statistical variate is continuous or a
graduated variate, such as heights of soldiers,
ages at death of assured lives, physical and astro-
nomical precision measurements, etc., then
dzcp(z)
is the probability that the variate x satisfies the
following relation
Frequency Functions. 7
z — -^-dz<x<z + -^dz
or that x falls between the above limits.
If the statistical variate assumes integral (dis-
crete) values only such as the number of alpha
particles radiated from certain metals and radio-
active gases as polonium and helium, number of
fin rays in fishes, or number of petal flowers in
plants, then cp(z) is the probability that x assumes
the value z. From the above definitions it follows
a fortiori that
(a) F(z) = Nq(z) (Integral variates)
(b) dz F(z) =N(p(z)dz (Integrated variates)
Interpreting the above results graphically we
find that (a) will be represented by a series of
disconnected or discrete points while (b) will be
represented by a continuous curve.
As to the function <p (z) we make for the
present no other assumptions than those follow-
ing immediately from the customary definition of
a mathematical probability. That is to say the
function 9 (z) must be real and positive.
Moreover it must, also satisfy the relation
+ »
\ cp (z) dz = 1 ,
— 00
or in the case of discrete variates :
8 Frequency Curves.
'!>(*) = i
which is but the mathematical way of expressing
the simple hypothetical disjunctive judgment that
the variate is sure to assume some one or several
values in the interval from — go to + oo. The
zero point is arbitrarily chosen and need not coin-
cide with the natural zero of the number scale.
Thus for instance if we in the case of height of
recruits choose the zero point of the frequency
curve at 170 centimeters an observation of 180
centimeters would be recorded as +10 and an
observation of 160 centimeters as — 10.
3. property of In regard to a frequency func-
CONSTANTS OR , • • •
parameters tion we may assume a prion
that it will depend only upon
the variate x and certain mathematical relations
into which this variate enters with a number of
constants \, A 2 , A 3 , A 4 , symbolically ex-
pressed by the notation
F(x, \, A,, A 3 , A 4 . . . .)
where the A's are the constants and x the variate.
All these constants or parameters are naturally
independent of x and represent some peculiar pro-
perties or characteristic essentials of the frequency
Property of Parameters. 9
function as expressed in the original observations
o i (i=l, 2, 3, N). We may, therefore,
say that each constant or statistical parameter
entering into the final mathematical form for the
frequency function is a function of the observa-
tions o v This fact may be expressed in the follow-
ing symbolic form : —
\ = S 1 (o 1? o 2 , 0.,, ... 0^)
X N = S n(°1> °2,0 a , . . . N ).
But from purely a priori considerations we
are able to tell something else about the function
S . (i=l, 2, 3 .... N). It is only when per-
muting the various o's in an ascending magnitude
according to the natural number scale that we
obtain a frequency function. This arrangement
itself has, however, no influence upon any one
of the o's which were generated before this purely
arbitrary permutation took place. The ultimate
and previously measured effects of the causes as
reflected in each individual numerical observa-
tions, 0., depend only upon the origin of causes
which form the fundamental basis for the stati-
stical object under investigation and do not depend
10 Frequency Curves.
upon the order in which the individual o'e occur
in the series of observations.
Suppose for instance that the observations
occurred in the following order
o lt o 2 , o 3 , o
X'
By permuting these elements in their natural or-
der we obtain the frequency distribution F(x).
But the very same distribution could have been
obtained if the observations had occurred in any
other order as for instance
o 7 , o 9 , o N , . . . o 3 . . . . o x .
so long as all of the individual o's were retained
in the original records. Or to take a concrete ex-
ample as the study of the number of policyholders
according to attained ages in a life assurance
office. We write the age of each individual policy-
holder on a small card. When all the ages have
been written on individual cards they may be per-
muted according to attained age and the resulting
series is a frequency function of the age x. We
may now mix these cards just as we mix ordinary
playing cards in a game of whist, and we get an-
other permutation — in general different from the
order in which we originally recorded the ages on
the cards. But this new permutation can equally
Symmetric Functions. 11
well be used to produce the frequency function if
we are only sure to retain all the cards and do
not add any new cards.
4. parameters- The various functions S (o lt
symmetric o 2 , °3 °jy) are there-
fore, symmetric functions, that
is functions which are left unaltered by arbitrarily
permuting the N elements o, and no interchange
whatever of the values of the various o's in those
symmetric functions can have any influence upon
the final form of the frequency function or fre-
quency curve, F(x).
We now introduce under the name of power
sums a certain well known form of fundamental
symmetrical functions denned by the following
relations
5
= 0°
+ 0%
+ o° 3 + -
■■o° N
= N
s l
= 0]
+ o\
+ o\+..
■ °\
=z°\
S 2
= 0\
+ o\
+ <%+■■
o 2
1 • u s
= Z°i
S X
= f
+ 0»
+ of+ ■
N
= Z°!
Moreover, a well known theorem in elementary
algebra tells us that every symmetric function
may be expressed as a function of s lt s 2 , s 3 . .
. . . s N .
12 Frequency Curves.
From this theorem it follows a fortiori that
we are able to express the constants A in the fre-
quency curve as functions of the power sums of
the observations. While such a procedure is pos-
sible, theoretically at least, we should, however,
in most cases find it a very tedious and laborious
task in actual practice. It, therefore, remains to
be seen whether it is possible to transform these
symmetrical functions of the power sums of the
observations into some other symmetric functions,
which are more flexible and workable in practical
computations and which can be expressed in terms
of the various values of s.
5. THiELE-s It is the great achievement of
invariants Thiele to have been the first
mathematician to realize this
possibility and make this transformation by intro-
ducing into the theory of frequency curves a pe-
culiar system of symmetrical functions which he
called semi invariants and denoted by the symbols
^i, \, \ • ■ ■
Starting with power sums, s ; . Thiele defines
these by the following identity
XjOT X 2 oo 2 X 3 ro 3
e TL + Hr + ~pr
which is identical in respect to co
■^ ^^ =*o+f H-f + S -F + - (1)
Semi-Invariants.
13
Since s { =^o i the right hand side of the equa-
tion may also be written as e i ra + e°* co + eP3 m +...=
ST 0,-co
= Z«' ■
Differentiating (1) with respect to co we have
A, a> X,co 2 X,co 3
* n e
\1_ |2_
■ +...
A 2 co XgCO 2
Xi+ TT _ + T
+
s o + jY co +jy co2 +iy M3 +-
, AnCO Ao „
Multiplying out and equating the various
coefficients of equal powers of co we finally have
s x = \s
So = \s x + \ 2 s
s s = \s 2 + 2 \ 2 s x + X s s
s i = \ x s 3 + 3A 2 s 2 + 3a 3 s x + X 4 s
where the coefficients follow the law of the
binomial theorem.
Solving for A we have
\ = s t : s
X 2 = (s 2 s — sl):sl
a 3 = ( s 3*o — 3s 2 s 1 s + 2sl):sl
14 Frequency Curves.
x 4 = Si si-4s sSl si — 3*;*; + i2s 2 *;*o — 6s t)=^
The semi-invariants X in respect to an ar-
bitrary origin and unit are as we noted denned
by the relation
A,co \,co 2 Xoco 11
_1 |_ _? L _? L . . .
11 1 2 1 3 o,a> o,co o,ct>
s e>- — = e 1 +e 2 +e 3 +...
where o 1 , o 2 , o 3 . . . are the individual observa-
tions.
Let us now change to another coordinate
system with another unit and origin defined by
the following linear transformations : —
o'i = aoi + c (i = 1, 2,3,.. .).
The semi-invariants in this new system are
given by the relation
A' to X' oo 2 X'„a>3
-A | ? 1 § 1- ...
1 1 1 2 1 3 • o', ro o'„co o'„a>
s e — — = e 1 +e J +e 3 + ... =
(aoj + tOco (ao 2 +c) co
= e +e + ...
Since the various values of X' do not depend upon
the quantity co we may without changing the
value of the semi-invariants replace co by co : a
in the above equations, which gives
Semi-Invariants. 15
\\ m X'„co 2 X'-co 8
s e =
(aoj + c) — (oo 2 + c) — (ao 3 + c) —
a a a
e + e + e + . . . =
a T o,co o„co o,co
ceo XjCO X 2 co 2 X 3 co
"a" ~[l + l2~ ¥ ~\*
= e 5 e
.] =
Taking the logarithms on both sides of the equa-
tion we have
a^ o«[2_ o 8 [3_
CCO XtCO X,C0 2 XotO 3
~a + |l L + [2_|3_ +
Differentiating successively with respect to co we
have
X' X' to X'.to 3 c , , , , , X3C0 2
a\l_ a 2 2a 3 a d>
* + *= + *S? + ...-». + *. + f + ...
5 + ^ + ...-x. + w..
16 Frequency Curves.
Letting co = we therefore have
A, or \\ = aXj + c
a
a
J. X
= X 2 or X',
= a 2 X 2
K
a 3
= X 3 or X' 3
= o»X 8
from which we deduce the following relations
Xj (ax + c) = aX x (x) + c
X r (a#+ c) = a r X r (x) for r > 1,
which shows how the semi-invariants change by
introducing a new origin and a new unit.
We shall for the present leave the semi in-
variants and only ask the reader to bear in mind
the above relations between X and s, of which we
shall later on make use in determining the con-
stants in the frequency curve cp (x) .
6. the fourier Before discussing the genera-
INTEGRALS ,■ £ ,-. , , , »
tion of the total frequency
curve it will, however, be nec-
essary to demonstrate some auxiliary mathema-
tical formulae from the theory of definite integrals
and integral equations which will be of use in the
Fourier's Integrals. 17
following discussion as mathematical tools with
which to attack the collected statistical data or
the numerical observations.
One of these tools is found in the celebrated
integral theorem by Fourier, which was the first
integral equation to be successfully treated. We
shall in the following demonstration adhere to
the elegant and simple solution by M. Charlier.
Charlier in his proof supposes that a function,
F(co) , is defined through the following convergent
series.
F(v) = a[/(o) + /(a)e + /(2a)e +...
+ /(a)e +/(— 2a)e 4-...
or
in = <w
^(oo) = a ^/(cwi)e amtoi (2)
where / = \ — 1.
We then see by the well known theorem of
Cauchy that the integral
+ x
/(o9) = < ^f(x)e x ' oi dx (3)
is finite and convergent. If we now let ma = x
and let a = as a limiting value, a, becomes
equal to dx and /(am) = fix). Consequently we
may write
18 Frequency Curves.
lim F(o) = jT(co).
a =
Multiplying (2) by e~ rami da and integrating
between the limits — n/a and + n/a we get on
the left an expression of the form
+ */<*
{F(a)e- ra<oi dco
— ?t/a
and on the right a sum of definite integrals of
which, however, all but the term containing
f(ra) as a factor will vanish. This particular term
reduces to
a\f(ra)d(o or 2nf(ra).
— -x/a
Hence we have
+ 3t/a
%*)
-rami
f(ra) = ^F(a,)e "*""*». (4a)
By letting a converge toward zero and by the
substitution ret = x this equation reduces to
8»J
— X03i
/(*) = izVW* **■ (4b)
Fourier's Integrals. 19
Charlier has suggested the name conjugated
Fourier function of f{x) for the expression F (co).
We then have, if we introduce a new function
ib (to) defined by the simple relation :
j/2jr\|>(co) = limF(co)
a =
ib (to) = 77 =C/(a:)c* Di dx. (5 a)
\/2:
+ 00
J/2J
/(*) = i -^=\i|)(a))e- xa,i doo. (5b)
The equations (5a) and (5b) are known as
integral equations of the first kind. The eXpreS-
sion e (or e ) is known as the nucleus of
the equation. If in (5b) we know the value of
i]' (co) we are able to determine fix). Inversely,
if we know f(x) we may find i|> (co) from (5a).
7 cv^e'asVhe ^ e are now * n a P° s iti° n *°
a^Yntegral ma ke use OI * ne semi-invariants
equation f Thiele, which hitherto in
our discussion have appeared as a rather discon-
nected and alien member. On page 13 we saw
that the semi-invariants could be expressed by
the relation
20 Frequency Curves.
■ CO + ttt CO 2 + -
Q. | 2 I :
^3 <i i
e— = 2^e ■
where 0; (i = 1, 2, 3 ) denotes the in-
dividual observations.
The definition of the semi-invariants does not
necessitate that all the o's must be different. If
some of the o's are exactly alike it is self-evident
that the term e i must be repeated as often as
o occurs among all of the observations. If there-
fore Ny(oi) denotes the absolute frequency of o,
where cp (o;) is the relative frequency function,
then the definition of the semi-invariants may be
written as : —
V / n Ll Li LL v i \ "i
For continuous variates, x, the above sums
are transformed into definite integrals of the form
■co 2 + -ro 3 +.
e \ cp(x)aa = \ <p(a:)e
rfx.
Let us now substitute the quantity co \ — ] , or
ica, for co in the above identity. We then have : —
X l • , X 2 -2 2 - A 3 .3 3 . + <* +""°
|1_ ' [2_ |_3_
\ cp(*)rfx = \ (p(aj)e 1M °da;
Approximate Solution. 21
under the supposition that this transformation
holds in the complex region in which the func-
tion is denned.
In this equation the definite integrals are of
special importance. The factor \ y(x)dx is, of
course, equal to unity according to the simple
considerations set forth on page seven. The in-
tegral on the right hand side of the equation is,
however, apart from the constant factor j/2ji
nothing more than the i|) function in the conjugate
Fourier function if we let cp(#) = f(x), and
e {± ^ ^ = l/2^(co).
According to (5b) we may, therefore write f(x)
or cp(a;) as
„ i + { £«>+&**+&**+- -«,.
cp(*) = ^ Je e An
as the most general form of the frequency func-
tion cp (x) expressed by means of semi-invariants.
8. first approx- The exactness with which
solutwn 9 0*0 is reproduced depends,
of course, upon the number of
A's we decide to consider in the above formula.
As a first approximation we may omit all X's
22 Frequency Curves.
above the order 2 or all terms in the exponent
with indices higher than 2. Bearing in mind
that i 2 = — 1 we therefore have as a first ap-
proximation
^ /, tro^! — *)-j2-co 2
«Po(*)=2^Jc - *»■
— CO
The above definite integral was first evaluated
by Laplace by means of the following elegant
analysis. Using the well known Eulerean relation
for complex quantities the above integral may be
written as
+ °° \ 2 a> 2
\ e cos [(X 1 — x'jcoj cko +
+ co \2
. C ~^ (
+ I
sin [(X 1 — :r)co] dco.
The imaginary member vanishes because the
factor e is an even function and sin|(X 1 — a;)coj
an uneven function, the area from — oo to will
therefore equal the area from to + oo , but be
opposite in sign, which reduces the total area
from — oo to + oo or the integral in question to
zero.
Approximate Solution. 23
In regard to the first term, similar conditions
hold except that cos [(A 1 — a;) col is an even func-
tion and the integral may hence be written as
V-i An
f -IT CD 2
I = 2 \ e cos (rco) dm where r = X x — ■?.
Regarding the parameter r as a variable and dif-
ferentiating 7 in respect to this variable we have
dI 2 f ( ^ ~
) sin (rco)dco.
From this we have by partial integration : —
dl_ 2
dr X 2
r - v raJ T " - - ro '
e sin (rco) dco — — \ e cos ( rco ) ^
(1 "
= — -r— or
A
Id/ r
/ rfr ~" X '
From which we find
log / = -j^- + log A
where log A is a constant. Hence we have : —
/ = Ae 2 ^
24 Frequency Curves.
In erder to determine A we let r =
= and we
have
/„ = A = 2 \ e dco = 2 /^-
= !/?■
This finally gives the expression for cp (cc) in the
following form :
as a preliminary approximation for the frequency
curve 9(33).
The first mathematical deduction of this ap-
proximate expression for a frequency curve is
found in the monumental work by Laplace on
Probabilities, and the function cp (a;) entering in
the expression cp (a;) dx, which gives the probab-
ility that the variate will fall between x — \dx
and x +\ dx, is therefore known as the Lapla-
cean probability function or sometimes as the
Normal Frequency Curve of Laplace. The same
curve was, as we have mentioned also previously
deduced independently by Gauss in connection
with his studies on the distribution of accidental
errors in precision measurements.
Laplace's probability function, cp (x) posses-
ses some remarkable properties which it might
Approximate Solution. 25
be well worth while to consider. Introducing a
slightly different system of notation by writing
\ = M and \/\ 2 = a, q> (x) reduces to the fol-
lowing form.
o|/2tt
which is the form introduced by Pearson.
The frequency curve, cp (a;), is here expressed
in reference to a Cartesian coordinate system with
origin at the zero point of the natural number
system and whose unit of measurement is also
equivalent to the natural number unit. It is,
however, not necessary to use this system in pre-
ference to any other system. In fact, we may
choose arbitrarily any other origin and any other
unit standard without altering the properties of
the curve. Suppose, therefore, that we take M
as the origin and c as the unit of the system. The
frequency function then reduces to
1 - x' : 2
Since the integral of cp (x) from — oo to + oo
equals unity the following equation must neces-
sarily hold.
+*
26 Frequency Curves.
9. development The Laplacean Probability
BY POLYNOMIALS ^^ ^^^ howeyev ,
some other remarkable proper-
ties which are of great use in expanding a func-
tion in a series. Starting with cp (x) we may by
repeated differentiation obtain its various der-
ivaties. Denoting such derivatives by cp x (x) ,
<p 2 (x), cp 3 (x) . . . respectively we have the fol-
lowing relations. 1 )
— x': 2
cp (a;) = e
<Pi(z) = —xy (x)
(p 2 (z) = (z 3 — l)cp (a;)
Vsfa) = — (« 3 — 3x)cp (a;)
(p 4 (a;) = (j? — Gx* + 3)y Q (x)
and in general for the nth derivative :
cp B (a;) = (-ir
n(n — l)(w — 2)(w — S)x
_ n(n~l) n ~ 2
X> 1 y-^ X +
2-4
ra (n-1) (/i- 2) (ra-3) (re-4) (rc-5) aT~ 6
2-4,-6 + "
cp (aj).
1 In the following computations we have omitted
temporarily the constant factor l;j/"2ir of <p (a:) and its
derivatives.
Hermite's Polynomials. 27
It can be readily seen that the derivatives of
<p (x) are represented throughout as products of
polynomials of x and the function cp (x) itself.
The various polynomials
H (x) = 1
H^x) = — x
H a (x) = x 2 — l
H a (x) = -(x*-3x)
H^r) = (.,<* — 6 a* + 3)
and so forth are generally known as Hermite's
polynomials from the name of the French mathe-
matician, Her mite, who first introduced these
polynomials in mathematical analysis.
The following relations can be shown to exist
between the three polynomials
H n+ i(x) — xH n (x) + nH n --i.{x) =
and
d 2 H n (x ) xdH n (x) _
A numerical 10 decimal place tabulation of the
first six Hermite polynomials for values of x up
to 4 and progressing by intervals of 0.01 is given
by J0rgensen in his Danish work "Frekvens-
flader og Korrelation" .
There exist now some very important relations
between the Hermite polynomials and the deriva-
tives of <p (x) , or between H n (x) and y n (x).
28 Frequency Curves.
Consider for the moment the two following
series of functions
ToO"). M^Di %(*)> <?s( x h <?i( x ), ■ • •
H (x), H x {x), H 2 (x), H,{x\ H t (x),. . .
where cp„(a;) = i/„ (a;) cp (a;) and where lim y„(x) =
for ./' = ± oo.
We shall now prove that the two series cp„ (x)
and H n (x) form a biorthogonal system in the
interval — oo to + oo , that is to say that they are
(1) real and continuous in the whole plane
(2) no one of them is identically zero in the
plane
(3) every pair of them cp n (x) and H m {x) ,
satisfy the relation.
+.<* >
\ <p n (x)H m (x)dx = (n < m).
We have the self evident relation (letting x = z)
-f-eo -j-OO
5 H m (z)y n (z)dz = $ # m (z).ff„(z)(p (z)dz =
CC CO
+«
= jj #„(z)cp m (z)dz.
Since this relation holds for all values of m and n
it is only necessary to prove the proposition for
n>m. For if it holds for n>m it will according
to the above relation also hold for n<m.
Hermite's Polynomials. 29
By partial integration we have : —
jj H m (z)(p n (z)dz =
00
+ 00 + 00
= H m {z)y n -i{z) ] — $ H' m (z)y n -i(z)dz
— Go — or.
when H' m {z) is the first derivative of H m {z).
The first member on the right reduces to
since <p„_i(z) = for z = ± qo. We have therefore : —
+ 00 , - tt
$ # m (z)cp„(z)dz = — jj H' m (z)y n -i(z)dz
— Co — 00
-j-co -pOo
jj H' m (z)<? n _i(z)dz = — jj H&(z)y n -2(z)dz
— co — Co
-}-00 -f-ao
J ^m(z)(p„_2(z)rfz = — $ /7£(z)q> n _ 3 (z)dz.
— 00 OC
Continuing this process we obtain finally an ex-
pression of the form
+ (lf m (z)<? n (z)dz = (-ir +1+ ^V +1, 9 n _ w _,( Z )&,
— Co — -oc
when #°" +1) (z) is the m + 1 derivative of # (z)
and n — to — 1>0. Since H m (z) is a polynomial in
the TO.th degree its w + 1 derivative is zero and
we have finally that
30 Frequency Curves.
+ »
jj H m (z)y n (z)dz =
for all values of m and n where ^^ /h .
For m = n we proceed in exactly the same
manner, but stop at the mth integration. We
have, therefore, by replacing m by n in the above
partial integrations
+ (HA*)Vn{z)dZ = (-l)"'f< ) ( Z )«p,_ n («)& =
— 00 — CO
The nth derivative of H n (z) is, however, nothing
but a constant and equal to ( — l)"|_ra_. Hence we
have finally
'fjy n (8)q. n (8)cfe = {-lf{-lf\±\ e -^dz =
00 CO
= |_ra |/2jt.
The above analysis thus proves that the func-
tions H m (z) and <p B (s) are biorthogonal to each
other for all values of n different from m through-
out the whole plane.
We can now make use of these relations be-
tween the infinite set of biorthogonal functions
H m (z) and <p„(z) in solving the problem of ex-
Hermite's Polynomials. 31
panding an arbitrary function cp (z) in a series
of the form
9(0) = C <p (2) + Cl Cp, (z) + C,cp 2 (3) + . . .
the series to hold in the interval from — 00 to
+ 00.
If we know that 9(2) can be developed into
a series of this form, which after multiplication
by any continuous function can be integrated
term for term, then we are are able to give a
formal determination of the coefficients c.
This formal determination of any one of the
c's, say C{ consists in multiplying the above
series by Ht(z) and integrating each term from
— 00 to co. All the terms except the one con-
taining the product Hi (z) <pi vanish and we have
for Ci. + oo +00
CO — CO
°i = +S ~ •
\yi(z)Hi(z)dz |J_|/2^
CO
If we define the Hermite functions as
H (z) =1
HAz) = z 2 — 1
if,(«) = z s — 3z
HAz) - 2 4 — 6« 2 + 3
32 Frequency Curves,
the above formula takes on the form
+ 00 + CO
jj cp (z) Hi (z) dz § <p (z) #i (z) rfz
00 00
|j cpi (z) #* (z) <fe (— l) r [i_ \/ 2 JT
— CO
which we shall prefer to use in the following
discussion.
It will be noted that this purely formal cal-
culation of the coefficients c is very similar to the
determination of the constants in a Fourier Series,
where as a matter of fact the system of functions
cosz, cos 2«, cos 32,
sin;-;, sin 22, sin Zz,
is biorthogonal in the interval 0<z<l-
But the reader must not forget that the above
representation is only a formal one, and we do
not know if it is valid. To prove its validity
we must first show that the series is convergent
and secondly that it actually represents 9(2) for
all values of 2.
This is by no means a simple task and it can-
not be done by elementary methods. A Russian
mathematician, Vera Myller-Lebedeff, has, how-
ever, given an elegant solution by means of some
well known theorems from the Fredholm integral
Gram's Series. 33
equations. She has among other things proved
the following criterion : —
"Every function cp (z) which together with its
first two derivatives is finite and continuous in the
interval from — co to + oo and which vanishes
together with its derivatives f or z = ± oo can be
developed into an infinite series of the form : —
cp( 2 )=^>- z ' :2 #.( 0)
where Hi(z) is the Hermite polynomial of
order i" .
10. gram's series It is, however, not our inten-
tion to follow up this treatment
which is outside the scope of an
elementary treatise like this and shall in its place
give an approximate representation of the fre-
quency function, cp(z), by a method, which in
many respects is similar to that introduced by
the Danish actuary Gram in his epochmaking
work "Udviklingsrsekker" , which contains the
first known systematic development of a skew
frequency function. Gram's problem in a some-
what modified form may briefly be stated as
follows : — Being given an arbitrary relative fre-
quency function, cp (z), continuous and finite in
the interval — oo to + oo (and which vanishes
3
34 Frequency Curves.
for z = ± oo J to determine the constant coeffi-
cients c , c 1 , c 2 , c 3 in such a way that
the series
c 9o(g) + Ci9i(g) + c 2 cp 2 (z) + + c n yn($ =
|/<Po(z) l/<PoO) l/?o( 3 ) |/9o( 2 )
gifles ifte besi approximation to the quantity
cp fa;,) : )/cp (zj in ifoe sense of the method of least
squares. That is to say we wish to determine the
constants c in such a manner that the sum of
the squares of the differences between the func-
tion and the approximate series becomes a mini-
mum. This means that the expression
^ C 9(2) X'^c^iiz)
y\
|/?o( 2 ) ^—> 1/<PoO)-
dz
must be a minimum.
On the basis of this condition we have
j^f m<) Zci(?i{2) = ^^^ = U(s}
where the unknown coefficients c must be so de-
termined that
Gram's Series. 35
/ =
nVvM
dz equals a minimum.
+»
Taking the partial derivatives in respect to Ci we
have
— CD —00
Now since
-i- CO
\ [U{z)] 2 dz =
05
{{cl [H,{z)}*+< [#:(*)]'+ • ..cl[H n {z)Y}^{z)dz,
we get
4-co -f 00
¥-= -2 [ ^=H i (z)]/^)dz+2c i \ [ff,(s)]'<p («)t
where the latter integral equals
$ <p t (e)Hi(e)dz = (— l)*[i|/2«.
Equating to zero and solving for c* we finally
obtain the following value for d —
.+00
d = ,^=U y{z)Hi{z)dz (1=1,2,3,...)-
|i J/2n J
36 Frequency Curves.
This solution is gotten by the introduction of
|/cp (z) which serves to make all terms of the
form Cicpi(0):|/<p o (0) = |Ap (» C;#»(z) (i = 1, 2,
3 . . . n) orthogonal to each other in the interval
— oo to + oo.
In all the above expansions of a frequency
series we have used the expression % (z) = e~ za/a
as the generating function (see footnote on page
26) , while as a matter of fact the true value of
<p (z) is given by the equation <p (z) = e~ z " /2 : |/2ji.
The definite integral on page 32
+ 00 -t~°°
(- 1)* \ H t (e) Vi {z)de = \i_ $ e-^dz = \£fte
will therefore have to be divided by |/2jt, and
the value of the gen
forth be reduced to
the value of the general coefficient c$ will hence-
$ ^{z)H i {z)dz
Ci== ""(_l) i li
where Sj (z) is the Hermite polynomial of order
i defined by the relation
%K) 2 2-4
i (t — 1) (t — 2) (i — 3) (t — 4) (t — 5) z f ~ 6
2-4-6 + '"'
Gram's Series. 37
On this basis we obtain the following values
for the first four coefficients : —
+»
c = jj y(s)dz = 1
— cc
+»
c x = (— l) 1 $ cp(z)zefe : |l_
— CO
+«>
c, = (— l) 2 jj (z 2 — 1) cp(z) & : |_2_
— CO
+°°
c 3 = (_ 1)3 J 2 s_ 3z)cp(z)<fe:|3_
— Co
c 4 = (— 1)*^ (z 4 — 6z 2 + 3 e) 9 (2) cfe : |5_
— CO
While the above development of an arbitrary
frequency distribution has reference to 9 (z) , or
the relative frequency function, it is, however,
equally well adapted to the representation of ab-
solute frequencies as expressed by the function,
F(z). If N is the total number of individual
observations, or in other words the area of the
frequency curve, we evidently have
-{-Go "j~°o
F(z) = iVcp(z) or $ F{z)dz = N J y(z)dz = N.
00 — CO
Since N is a constant quantity we may, there-
fore, write the expansion of F(z) as follows:
38 Frequency Curves.
F(z) = iV[c <p (z) + c 1 cp 1 (z) + c 2 cp 2 (2)+ . . .] =
= NZctHMe-**
where the coefficients ci have the value
+ CO
d = t~Z J F(z)H i (z)dz for i = 1, 2, 3, . . .
• CO
and where
N = \ F{z)dz.
CO
Since all the Hermite functions are polynom-
ials in z, it can be readily seen that the coeffi-
cients c may be expressed as functions of the
power sums or of the previously mentioned sym-
metrical functions s, where
s r = jj z r F{z)dz.
— Co
These particular integrals originally introduced
by Thiele in the development of the semi-in-
variants have been called by Pearson the
"moments" of the frequency function, F(z), and
s r is called the r* A moment of the variate z with
respect to an arbitrary origin.
It can be readily seen that the moment of
order zero, or s is
Gram's Series. 39
-f-CD -|-Q0
s = \ z°F(z)dz = N = N \ y{z)dz.
— Co — co
Hence we have for the first coefficient c .
+ 00 -j-00
c = $ F(z)dz: $ F(z)dz = 1.
CC -)~Q0
We are, however, in a position to further
simplify the expression for F(z).
As already mentioned we are at liberty to
choose arbitrarily both the origin and the unit
of the Cartesian coordinate system for the fre-
quency curve without changing the properties of
this curve. Now by making a proper choice of
the Cartesian system of reference we can make
the coefficients c 1 and c 2 vanish. In order to ob-
tain this object the origin of the system must be
so chosen that
^ \ zF(z)dz : \ F(z)dz = 0.
c, =
This means that the semi invariant s r : s = A x
must vanish. It can be readily seen that the above
expression for X u is nothing more than the usual
form for the mean value of a series of variates.
Moreover, we know that the algebraic sum (or
in the case of continuous variates, the integral)
of the variates around the mean value is always
40 Frequency Curves.
equal to zero. Hence by writing for z the expres-
sion (z — M) when M equals the mean value or
\j we can always make c x vanish.
To attain our second object of making c 2
vanish we must choose the unit of the coordinate
system in such a way that the expression
+ 00 -{-00
c 2 = t~^ jj F(z)R 2 (z)dz : ^ F{z)dz =
which implies that
-(-03 -}-O0
+ «
\ F(z)z 2 dz — $ F(e)de : $ F(z)dz =
or that s 2 : s — 1 = 0, or when expressed in terms
of the semi-invariants that
X 2 = (s 2 s — s\):sl = 1.
But by choosing the mean as the origin of the
system the term s x : s is equal to and we have
therefore X 2 = 2 = s 2 : s = 1. Hence, by selec-
ting as the unit of our coordinate system j/X 2 or
o, where o is technically known as the dispersion
or standard deviation of the series of variates, we
can make the second coefficient c 2 vanish.
In respect to the coefficients c 3 and c 4 we
have now
c* =
(-1)3
Semi-Invariants.
+ 00 + C0
41
J z*F(z)dz — 3 ^ 2i?(2)cfe : J i?(g)<fe
+ 00
which reduces to
(-1) 4
A^- while
r+ao
C, =
|4
^ 2 + F(z)cfe— 6 J 2 2 F(s)& +
+ 00 -I -|-00
+ 3 $ F{z)dz : J i^(2)rf2
which reduces to
A ± D Siy Oft
5 $ Q S Q
14 =
— 3
While the coefficients of higher order may be
determined with equal ease, it will in general be
found that the majority of moderately skew fre-
quency distributions can be expressed by means
of the first 4 parameters or coefficients.
n. coefficients We shall now show how the
semi-invariants same results for the values of
the coefficients may be ob-
tained from the definition of the semi-invariants.
Since we have proven that a frequency function,
F(z), may be expressed by the series
42 Frequency Curves.
F ( z ) =JEciyi(z)
we may from the definition of the semi-invariants
write down the following identity : —
X,co X./o z
— \-— 1-
\U + [2_ +■"
s e =
+»
= N $ e 0ra (c o cp o (2) + c 1 cp 1 (z) + c 2 cp 2 (s) + ...)d2
where N is the area of the frequency curve.
The general term on the right hand side of
the equation will be of the form
+»
c r $ e zw Q? r (z)ds
where the integral may be evaluated by partial
integration as follows : —
-(-00 ^\~ x H -00
$ e z< °y r (z)dz = e'Vi(«) ] — ra $ e? a y r - X {z)dz,
— oo — co — oo
and where the first term on the right vanishes
leaving
+ 00 -j-00
$ e 20 > r (2)cfe = (-co) 1 $ e"°<pr-i(e)de.
— 00 CO
Continuing in the same manner we obtain by
successive integrations
Semi-Invariants. 43
+» +00
(_„)! J e "°y r -.^z)dx = (-co) 2 J e 2m cp r _ 2 (z)dz
— oc —00
+ CO -[-GO
(-co) 2 5 e 2m cp r _ 2 (2)(fe = (— co)8 J e zro (p r _ 3 (0)d2
from which we finally obtain the relation
+ 0= +00
ij e zw <p r (z)dz = (-co)' J e za> <p (z)dz =
— 00 — 05
+» z *
1 "IT.'-'*.
^
?s
This latter integral may be written as
1/2* 3
Consequently the relation between the semi-in-
variants and the frequency function may be writ-
ten as follows : —
CO"
~2~
44 Frequency Curves.
X,co \,co z
— U_J 1-
LL + LL +
s e =
= N [c — c ± co + c, co 2 — c 3 co 3 + .
or
J^CO CO 2
lr + [I (x,-i) + ...
s e =
= iV [c — q CO + c 2 CO — c 3 CO 3 + . . .] .
By successive differentiation with respect to co
and by equating the coefficients of equal powers
of co we get in a manner similar to that shown
on page 13 the following results : —
. _ £o _ fo _ 1
C ° - N ~ s -
c x = — \
= ri[(*2-l) + ^
° 3 = ^f A 3 + 3(X 2 -l)A 1 + X^]
c* = rjk+4\3X 1 + 3(A 2 -l) 2 + 6(X 2 -l)^+Xt]-
If we now again choose the origin at A x , or
let Aj = 0, and choose j/A 2 = 1 as the unit of our
coordinate system we have : —
c o = 1, <h = °) C 2 = 0. c 3 = -ry- A 3 , c 4 = .-^-A 4 .
Linear Transformation. 45
12. linear trans- The theoretical development of
the above formulae explicitly
assumes that the variate, z, is
measured in terms of the dispersion or |A 2 (z) and
with X x (z) as the origin of the coordinate system.
In practice the observations or statistical data are,
however, invariably expressed with reference to
an arbitrarily chosen origin (in the majority of
cases the natural zero of the number scale) and
expressed in terms of standard units, such as
centimeters, grams, years, integral numbers, etc.
Let us denote the general variate in such ar-
bitrarily selected systems of reference by x. Our
problem then consists in transforming the various
semi-invariants, \(x), X 2 (x), \(x), \(%)
to the z system of reference with \ (z)
as its origin and |A 2 (z) as its unit. Such a trans-
formation may always be brought about by means
of the linear substitution
z = ax+b 1
which in a purely geometrical sense implies both
a change of origin and unit. On page 16 we
proved the following general properties of the
semi-invariants
\ t (s) = X 1 (ax + b) = a\(x) + b
\ r ( 2 ) = X r (ax+b) = a r X r (x).
46 Frequency Curves.
Let us now write \ (x) = M and A 2 (x) = d 2 ,
we then have the following relations : —
X^z) = aM + b
XjjOO = a 2 c 2 .
Since the coordinate system of reference must
be chosen in such a manner that \ (z) = and
)A 2 (z) =1 we have : —
aM + b =
ad = 1
, • 1 , l — M
from which we obtain a = — and o = — - — ,
o <3
which brings z on the form : z= (x — M) : c while
cp (2) becomes
, •. 1 — (i — ilf 2 ):2 !
J/ZTTd
Moreover, we have \ r (z) = X r (a;) : C for all
values of r > 2. We are now able to epitomize
the computations of the semi-invariants under the
following simple rules.
(1) Compute \ (x) in respect to an arbitrary
origin. The numerical value of this parameter
with opposite sign is the origin of the fre-
quency curve.
(2) Compute A, (as) for all values of r > 2. The
numerical values of those parameters divided
with (J/X 2 (x) r , or cr, for r = 2, 3, 4, . . .
.... are the semi-invariants of the frequency
curve.
Charlier's Scheme. 47
13. chablier's The general formulae for the
SCHEME OF ■ • • ,
computation semi-invariants were given on
page 13. In practical work
it is, however, of importance to proceed along
systematic lines and to furnish an automatic check
for the correctness of the computations. Several
systems facilitating such work have been proposed
by various writers, but the most simple and
elegant is probably the one proposed by M. Char-
lier and which is shown in detail with the neces-
sary control checks on the following page. Char-
lier employs moments, while we in the following
demonstration shall prefer the use of the semi-
invariants.
If we define the power sums of the relative
frequencies 9(2;) by the relation
-j-00 "h °°
m r = \ x r F(x)dx : jj F(x)dx (r = 0, 1, 2, 3, . . .),
— 00 — CO
we find that the expressions for the semi-invariants
as given on page 13 may be written as fol-
lows : — ■
Aj = m 1
A, = m 2 — m l
A 3 = m 3 — 3m 2 w 1 + 2m^
A 4 = m 4 — 4m 3 m 1 — 3ml + 12m 2 m[ — 0>m l
48 Frequency Curves.
The advantage of the Charlier scheme for the
computation of the semi-invariants lies in the fact
that it furnishes an automatic check of the
final results. If we expand the expression
(x + l) 4 F(x) we have: —
x i F(x) + ix 3 F{x) + 6x 2 F(x) + 4:xF(x) + F(x)
or
^(x+l) 4 F(x) = s i + 4:S 3 + Qs 2 + 4:S 1 + s ,
which serves as an independent control check of
the computations. Moreover, another check is
furnished by the relation
m i = A 4 + 4m 1 A 3 + §m\ \ 2 + 3\ 2 2 +m\.
In order to illustrate the scheme we choose the
following age distribution of 1130 pensioned func-
tionaries in a large American Public Utility cor-
poration.
Ages
No. of Pensioners
Ages
No. of Pensioners
35—39
i
65—69
286
40—44
6
70—74
248
45—49
17
75—79
128
50—54
48
80—84
38
55—59
118
85—89
13
60—64
224
over 90
3
The complete calculations of the coefficients c
are shown in the appended scheme by Charlier.
Charlier's Scheme.
49
fc<mcot~cooooco
s» 02 eo r- co i-H oo
+ T-H y-h
8
ioj w in w oo ■*
t N n eo oo oo (M
w 1 T"H CO "^ OO T"^
«CDO00O"*«*O
"in m » o» ■* oi
g< O-l r- O OJ 05 OJ
% 1-1 I-"
«co©oiOioj-«<o
"co in n to n w
«( T-l OJ "* "* OJ
«CO©00»*CO-*0
ST CO JO ■* CO N
•a i-i c^ w
CtH tONOO 00 ■*«
•3- H ■■* rt N M
fc, rn (M W
to in ■* COM th o
C5 »* os ^ cs •* as
■■ ■* in in tD CD
_ in 6 in Q in
■<*( •* in in co co
CD to .,
C) CO N H in
woo moJ
00 00 00 CO O CO
•* ■* n oi inos
IMOO COIMOJ
Oi CO CO rH rH
OO ■* to w o to
■* nn coin «
NO O OOOitM
CO
OONOiOOO CD
CO
OJ CO
00
m
-^ r-t -* o in co
(N in CO N
CO
CO
Op t-
cS co
i— i
T-H
Oi CO
00 CO ■* CN O CD
"* in i—i in t-h
00 00 00 CO tN
rt N CO ■* in CO
+ + + + + +
Oi
00
28
CO
3
en
OS
CO
CO
in
T-H
Oi
CO
CO
OJ
CO
OS
Oi
in
m
OS
T-*
CO
CO
•* O. ^
in
50
Frequency Curves.
>• >• >-
fw 4
CO ^
3 § 3
S §
I I
© l-» 05
CO en 05
Cn *. h*
00 Oi v]
o k* go
© Cn t-*
o co m
O CO 05
© o o o
© O O M
o o o co
O © *. Cn
p£ a
CO
s
I-*
1
CO
CO |
1
2
ffi if U
§ s §
-• u- H t* 10 to
OS
1
II
CO
CO
o
1
CO
CO
to to
to
Cn
en
b
to
CO
o
o
GO
b
en
..
h^
CD
*>
o
h--
o
OS
05
GO
Cn
O
Cn
o
CO
*~
eo
CO
00
W
O
o
^J
o
05
to
05
CO
Cn
00
©
*• f-k CO
CO C5 05
go to eo
eo #• -j
w. o **
CO
05
CO
^J
CO
CO
CO
en
o
CO
o
©
o
o
o
o
er
© _
en *.
05 © IV
o
o
iP
he-
Co
CO
Co
CO
O
O
o
va
CO
rfi
a
II
II
1
II
II
I
II
1
^>
1
CO
O
i-»-
m
CO
C5
tr
H*
GO
(35
cn
tc
CO
CO
CD
00
>*»
o
o
CO
CO
CO
C5
w
Observed and Theoretical Data. 51
The above computations give the numerical
values of the frequency function which now may
may be written as follows :
F(x) = 1130 [(cpoCz) + .0258 cp 3 (z). 0158 <p 4 (re)]
where _ ^ / x + .oi95 \'
1 2 V 1.6240 )
"' betwUnob- The next ste P is now to work
SE Yhe D ob^¥ica1 ND out the numerical values of
values F(x) for various values of x
and compare such values with the ones originally
observed. This process is shown in detail in the
following scheme .
Column (1) gives the values of the variate x
reckoned from the provisional origin, or the centre
of the age interval 65-69. (2) is x less the first
semi-invariant, whereby the origin is shifted to
the mean or X. Column (3) represents the final
linear transformation : z =(x — A-,): d.
Columns (4), (5) and (6) are copied directly
from the standard tables of J0rgensen or Charlier.
Column (7) is (5) multiplied by 0.0258 or the
product — [c 3 <? 3 (z)]:{3_, while (8) is [c 4 cp 4 (z)]: [4.
Column (9) is the sum of (4), (7) and (8).
If we now distribute the area N = s or 1130 pro
52
Frequency Curves.
+ + +
en en *•
+ + +
COM w
M M CO lis. Ol C
tn a -J ' ' — •
•q Ol Ul *. W M K
+ I
© o
H»to*.oioi
*. W CO iO w
©_
o
-3
*- CO M CO
■^ © *. »
cnofr S
+
po6. M ^MCoco
2 b io bo *-
O ^ co en
mm © en i- 1
i
CT> CO
OO ©
JO ©
>- 05
8
o
SOO^-COCOCOi-'OOOOO-e
t-A~J0OM)CC>COCD-*i^©©©©
coooou-qooto<»*.toco5oo
^OJ^UlCOCDCl^^OOCfJOl^O
+ 11 +
O UI O CO
h-i o cc en
f£- Oi to CO .
" © £0 C7t $K 00
■ ?^ ot cTt ^ ^1 h3 p -^ ,
i I
OOi-^O^-^IfOi-J-CO-^ifc-.©
?o 5; y -rr 3
OS -si -4 nS,
§88888282888888
OP^WHOOCOOW^^WMOO
88888282828888
Oo»^ffiw*-oocJ^aoMo
W"40HU1COKCDO^CC!OOM
fOOOOi-^CO^-COi-^OOO
OWhi(OOU , CO)&.CD»*-vl-
CO^O5tO--iaoN>rO©OlC3Cn>ft.0O
CO
^ K) to ro 1- 1
C001*-01HiMfl)0000^
" Ml ?\3 ?0 i- 1
N!C8l»ai00CSP<CSC<)
Method of Least Squares. 53
rata according to (9) , we finally reach the theore-
tical frequency distribution expressed in 5-year
age intervals and shown in column (10) alongside
which we have inserted the originally observed
values. Evidently the fit is satisfactory. It will
be noted that the final frequency series is expres-
sed in units of 5-year age intervals. This, how-
ever, is only a formal representation. By sub-
dividing the unit intervals of column (1) in 5
equal parts, and by computing all the other
columns accordingly, we get the theoretical fre-
quency series expressed in single year age inter-
vals.
is. the principle The following paragraph pur-
OF METHOD OF , , ■ , • n •, ■
least squares ports to give a brief exposition
of the determination of the co-
efficients in the Gram or Laplacean — Charlier
series in the sense of the method of least squares
as a strict problem of maxima and minima, wholly
independent of the connection between the method
of least squares and the error laws of precision
measurements. l
The simple problem in maxima and minima
which forms the fundamental basis of the method
1 In the following demonstration I am adhering to
the brief and lucid exposition of the Argentinean actuary,
U. Broggi, in his exellent Traite d' Assurances sur la Vie.
54 Frequency Curves.
of least squares is the following : Let m unknown
quantities be determined by observations in such
a manner that they are not observed directly but
enter into certain known functional relations,
fi(x 1 , x 2 , x 3 , . . . . x m ) , containing the unknown
independent variables, x lt x 2 , x 3 , . x m . Let
furthermore the number of observations on such
functional relations be n (where n is greater than
m). The problem is then to determine the most
plausible system of the values of the unknowns
from the observed system.
11 \%1 ) ^"11 ^3 l ■ • • %m) = #1
fn V^i j ^2 ? *^3 1 ' • • ^m) — On
when f lt f 2 , . . . f n are the known functional
relations and o x , o 2 , . . . o n their observed values.
Such equations are known as observation equa-
tions.
In order to further simplify our problem we
shall also assume that
1 All the equations of the system have the
same weight, and
2 All the equations are reduced to linear form.
By these assumptions the problem is reduced
to find m unknowns from n linear equations.
Method of Least Squares. 55
a 1 x 1 + b 1 x 2 + .
, . = o 1
a 2 x x + b 2 x 2 + .
. . = o 2
a s .x x + b 3 x 2 + .
■ • = ° 3
&n %\ i O n X% + . .
. = 0„
Since n is greater than m we find the problem
over-determined, and we therefore seek to deter-
mine -the unknown quantites, x lt x 2 , . . . x,„ in
such a way that the sum of the squares of the
differences between the functional relations and
the observed values, o becomes a minimum. This
implies that the expression
i = m
£(a i x 1 + b i x 2 + . . . — oif = ^(^n x ii ■ ■ - x m)
i = l
must be a minimum or the simultaneous existence
of the equations.
£1 = 0,^ = 0,. ..^ = o. (/)
ox x ox 2 ox m
If we now introduce the following notation
OiX 1 + biX 2 + • • ■ — Oi = Xj for i = 1, 2, 3, . . . re,
the m equations in the above system (I) evidently
take on the following form
56 Frequency Curves.
X 1 a 1 + X z a z + . . . +X n a n =
\ x b x + \ 3 b 2 + . . . + X n K =
If we now again re-substitute the expressions
for A in terms of the linear relations
OiX 1 + biX 2 + . . . Oi = h, for i = 1, 2, 3, . . . n,
and collect the coefficients of x x , x 2 , . . . x„, these
equations may be expressed in the following sym-
bolical form :
[aa]^! + [af)]a; 2 + . . . . — [ao'] =
[ab^x 1 + \bb']x 2 + . . . . — \bo] =
[ak~]x 1 + [bk}x 2 + . . + \Jik~]x m — [feo]=0
where [aa] = a x 2 + a./ + . . . .
[ab~] = a x bj + a 2 b 2 + . . . .
is the Gaussian notation for the homogeneous sum
products.
The above equations are known as normal
equations, and it is readily seen that there is one
normal equation corresponding to each unknown.
Our problem is therefore reduced to the solution
of a system of simultaneous linear equations of m
Normal Equations. 57
unknowns. If m is a small number, or, what
amounts to the same thing, there are only two or
three unknowns the solution can be carried on
by simple algebraic methods or determinants. If
the number of unknowns is large these methods
become very laborious and impractical. It is one
of the achievements of the great German mathe-
matician, Gauss, to have given us a method of
solution which reduces this labor to a minimum
and which proceeds along well denned systematic
and practical lines. The method is known as the
Gaussian algorithmus of successive elimination.
is. gauss' solu- For the sake of simplicity we
TION OF NORMAL i nl ,- M. 1 i.
equations snail limit ourselves to a sy-
stem of four normal equations
of the form
[aa]^! + [ab]x 2 + [_ac]x s + [arf]^ — [ao~\ =
[ab]^! + \bb~]x 2 + [bc]x i + [bd]x i — [bo] =
[ac]^! + [bc]a: 2 + [cc]:r 3 + [cd']x i — [eo] =
[ad]x 1 + [bd]x 2 + [cd~\x 3 + [dd]x i — [cfo] =
The generalization to an arbitrary number of
unknowns offers no difficulties, however.
On account of their symmetrical form the
above equations may also be written in the more
convenient form, viz. :
58 Frequency Curves.
[aa~\ x 1 + [ab~\x 2 + [ac~\x 3 + [_ad~]x i — [ao] =
[bb~]x 2 + [bc]x 3 + [bd]a; 4 — [bo] =
[cc]a; 3 + [cd]x i — [co] =
[dd] Xi — [do] =
From the first equation we find
^ ~ [ao] [ao] 2 . [aa] 3 [aa] 4 '
Substituting this value in the following equa-
tions and by the introduction of the new symbol
[ik] — H[oft] = [ik.l]
[aa]
we now obtain a new system of equations of a
lower order and of the form
[bb.l]x 2 + [bc.l]x 3 + [bd.l]ir 4 — [bo.l] =
[cc. 1]» 3 + [cd. l]a; 4 — [co.l] =
[dd.l]x 4 — [do.l] =
Solving for x 2 we have
[bo.l] [bc.l] [bd.l]
X * == [bb.l] [bb.l] Xi [bb.l] Xi '
Substituting in the following equations and
writing
Normal Equations. 59
we have
[cc.2]x s + [cd.2]x 4 = fco.2]
[dd.2]x t = [do.2]
or
[co.2] [cd.2]
3 ~ [cc.2] [cc.2] Xi '
Moreover, by writing
[ik.2] = [ci.2]&±=[ik.S],
we have finally
[dd.S]x A = [do.3]
This gives us the final reduced normal equa-
tion of the lowest order. By successive substitu-
tion we therefore have :
[do.3]
4 _ [dd.S]
[co.2]
[cc.2] '
_ [bo.l] _ [bc.l] [bd.l]
x * ~ [bb.l] [bb.l] [bb.l]
_ [ao]_[ab] _\ac\ [ad]
Xl ~ [aa] [aa] 2 [aa] X * [aa] Xi
as the ultimate solution of the unknowns.
[co.2] [cd.2]
Xz ~ [cc.2] [cc.2] '
60 Frequency Curves.
17. arithmetical The example in paragraph 13
APP mbtho°d ° F gave an illustration of the ap-
plication of the method of mo-
ments. As previously stated this method works
quite well in cases of moderate skewness, but is
less successful in extremely skew curves and where
the excess is large. We shall now give an illustra-
tion of the calculation of the parameters by the
method of least squares. The example we choose
is the well-known statistical series by the disting-
uished Dutch botanist, de Vries, on the number
of petal flowers in Ranunculus Bulbosus. This
is also one of the classical examples of Karl Pearson
in his celebrated original memoirs on skew varia-
tion. Although the observations of de Vries lend
themselves more readily to the method of logarith-
mic transformation, which we shall discuss in a
following chapter, we have deliberately chosen to
use it here for two specific reasons. Firstly it is
a most striking illustration in refutation of the
immature criticism of the Gram-Charlier series
by a certain young and very incautious American
actuary, Mr. M. Davis, who has gone on record
with the positive statement, "that the Charlier
series fails completely in case of appreciable skew-
ness". Secondly (and this is the more important
reason) it offers an excellent drill for the student
in the practical applications of the method of least
Numerical Application. 61
squares because it gives in a very brief compass
all the essential arithmetical details. The observa-
tions of de Vries are as follows : i
No. of petals
X
F{x) = o.
5
133
6
1
55
7
2
23
8
3
7
9
4
2
10
5
2
where F(x) denotes the absolute frequencies. The
observed frequency distribution is well nigh as
skew as it can be and represents in fact a one-
sided curve, and should therefore — if the state-
ment by Mr. Davis is correct — show an absolute
defiance to a graduation by the Gram-Charlier
series.
The process we shall use in the attempted
mathematical representation of the above series is
a combination of the method of semi-invariants
and the method of least squares. Following
Thiele's advice we determine the first two semi-
invariants in the generating function directly from
the observations while the coefficients of this
function and its derivations are determined by
the least square method.
Choosing the provisional origin at 5, we obtain
the following values for the crude moments.
62 Frequency Curves.
s = 222, s 1 = 140, s 2 = 292, s 3 = 806, s 4 = 2,752,
s 5 = 10,790, s 6 = 46,072, s 7 = 207,226,
from which we find that
\ = 1, x x = 0.631, A 2 = 0.917, X 3 = 1.644,
A 4 = 3.377, A 5 = 5.972, X 6 = —2.911,
X 7 = 122.638.
All these semi-invariants with the exception
of the two first are, however, so greatly influenced
by random sampling in the small observation
series that it is hopeless to use them in the deter-
mination of the constants in the Gram-Charlier
series. In fact an actual calculation does not give
a very good result beyond that of a first rough
approximation. The generating function, on the
other hand, may be expressed by the aid of the
two first semi-invariants as follows :
]_ — 2 2 :2
9 ° w = m e '
where z is given by the linear transformation :
z = (3 — 0.631) : 0.9576. (\/)T 2 = 0.9576).
We now propose to express the observed func-
tion F(x) or 9(2) by a Gram-Charlier series of
the form :
Numerical Application. 63
F(x) = cp(z) = A; cp (z) + A: 3 cp3(z) + /c 4 cp 4 (z).
In this equation we know the values of the
generating function and its derivatives for various
values of the variate z as found in the tables of
J0rgensen and Charlier, while the quantities k are
unknowns. On the other hand we know 6 specific
values of F(x) as directly observed in de Vries's
observation series. We are thus dealing with a
system of typical linear observation equations of
the forms described in paragraphs 15 and 16
and which lend themselves so admirably to the
treatment by the method of least squares.
From the above linear relation between x and
z we can directly compute the following table for
the transformed variate z.
X
3
—0.688
1
+ 0.402
2
+ 1.493
3
+ 2.583
4
+ 3.674
5
+ 4.764.
The numerical values of <% (z) and its derivat-
ives as corresponding to the above values of z can
be taken directly from the standard tables of J0r-
gensen and Charlier. We may therefore write
down the following observation equations :
64
• Frequency Curves.
?0
<J>3
ft
.3148fc
— .5472fc 3
+ .1207fe 4
—133 =
.3679/c
+ .4198fe 3
+ .7566fe 4
— 55 =
.1308/c
+ .1506fc 3
— .7073fc 4
— 23 =
.0145fe„
— .1346fc 3
+ .1062fc 4
— 7 =
.0005fe
— .0180fc 3
+ .0486fc 4
— 2 =
.0001fc„
— .0005fc 3
+ .0020fe 4
— 2 =
for which we now propose to determine the un-
known values of 7c by the least square method.
While this method may of course be applied
directly to the above data, it will generally be
found of advantage to start with some approximate
values of the k's. It is found in practice that
this approximate step saves considerable labour
in the formation and ultimate solution of the
normal equations.
Although the first approximation in the case
of numerous unknowns must be in the nature of
a more or less shrewd guess, which facility can
only be attained by constant practice in routine
mathematical computing, we are, however, in this
specific instanoe able to tell something about the
nature -o fthe coefficients from purely a priori con-
siderations. We know for instance from the form
of the Gram-Charlier series that the coefficient k
of the generating function must be nearly equal
to the area of the curve, which in this particular
instance is 222. Moreover, a mere glance at the
observed series tells us that it has a decidedly
Numerical Application. 65
large skewness in negative direction from the
mean coupled with a tendency of being "top
heavy", indicating positive excess. We can there-
fore assume as a first approximation that the
coefficients of the derivatives of uneven order are
negative and the coefficients of derivatives of even
order are positive.
From such purely common sense a priori con-
siderations we therefore guess the following first
approximations, viz. :
k l = 222, k\ = — 25, k\ = 30.
The probable values of the various fc's may be
written as
h, = rik\ for i = 0, 3, 4,
and our problem is therefore to find the correction
factor r with which the approximate value k\
must be multiplied so as to give kt.
Applying the various values of k\ to the
original observation equations on page 64 we obtain
the following schedule for the numerical factors
of
a
b
c
s
69.9
+ 13.7
+ 3.6
—133.0
—45.8
81.7
—10.5
22.7
— 55.0
+ 38.9
29.1
— 3.8
—21.2
— 23.0
—18.9
3.3
+ 3.4
+ 3.2
— 7.0
+ 2.9
0.1
+ 0.5
+ 1.5
— 2.0
+ 0.1
0.0
+ 0.0
+ 0.0
— 2.0
— 2.0
184.1
+ 3.3
+ 9.8
—222.0
—24.8
66 Frequency Curves.
where the additional control column s serves as a
check.
The subsequent formation of the various sum-
products and normal equations is shown in the
following schedules together with the s columns
as a check.
aa
ab
ac
ao
as
+
4,886
+ 958
+ 252
— 9,297
—3201
+
6,675
—858
+ 1,855
— 4,494
+ 3178
+
847
—111
— 617
— 669
— 550
+
11
+ 11
+ 11
— 23
+ 10
+
+
+
—
+
+
+
+
—
+
+ 12,419 + +1,501 —14,483 — 563
bb be bo bs
+ 188 + 49 — 1,822 — 628
+110
—
238
— 578
— 408
+ 14
+
81
+ 87
+ 72
+ 12
+
11
— 24
+ 10
+
+
— 1
+
+
+
—
+
+m"
—
96~
— 1,182
— 954
cc
CO
cs
+
13
— 479
— 165
+
515
— 1,249
+ 883
+
449
+ 488
+ 401
_j_
10
— 22
+ 9
4-
2
— 3
+ 1
+
+
+
+
989
— 1,265
+ 1129
Numerical Application. 67
We may now write the normal equations in
schedule form as follows :
ORIGINAL NORMAL EQUATIONS
(a) +12,419 + + 1501 — 14483
(1) +0+0—0
(b) + 324 — 96 — 1182
(2) + 181 — 1750
(c) + 989 — 1265
(3) +.00000 +.12086 —1.16617
The sum-products from the observation equa-
tions are shown in the rows marked (a) , (b) , (c) .
The row marked (3) and printed in italics is
formed by dividing each of the figures in row (a)
with 12,419. The row marked (1) contains the
products of the figures in row (a) multiplied with
the factor .00000. All these products happen in
this case to be equal to zero. Eow (2) is the
products of the factor 0.12086 and the figures in
row (a) .
We next subtract row (1) from row (b) , row
(2) from row (c) , which results in the following
schedule, which is known as the first reduction
equation.
FIRST REDUCTION EQUATIONS
(0) +324 — 96 — 1182
(1) + 28 + 350
(b) + 808 + 485
]2)~~ —.29626 ~ —3764814
68 Frequency Curves.
The above equations are treated in a similar
manner as the original normal equations, and we
have therefore the 2nd reduction equation of the
form :
SECOND REDUCTION EQUATION
+ 780 +135
The solution for the unknown r's may now
be shown as follows :
r 4 = — 135 : 780 = —.17308
r 3 = 3.64814— (—.29626) (—.17308) = 3.59637
r = 1.16617— (0.0) 3.59637) — (.12086)
(—.17308) = 1.18709.
From which we find : —
k B = 263.5, K=— 89.9, fe 4 = — 5.1
Applying these factors to the values of 9 («),
y 3 (z) and <p 4 (2) we obtain the following re-
sult :— T
*0?0
hva
h9*
2 ^9i
Obs
82.9
+ 49.2
—0.6
131.5
133
96.9
—37.7
—3.9
55.3
55
34.5
—13.5
+3.6
24.6
23
3.8
+ 12.1
—0.5
15.4
7
0.1
+ 1.0
—0.2
0.9
2
0.0
+ 0.0
-0.0
0.0
2
1 For a closer approximation see my Mathematical
Theory of Probabilities (Second Edition, New York, 1921).
Transformation of Variates. 69
is. transforma- While it is always possible to
TION OF THE n £ i
variate express all frequency curves by
an expansion in Hermite poly-
nomials, the numerical labor when carried on by
the method of least squares often involves a large
amount of arithmetical work if we wish to retain
more than four or five terms of the series. Other
methods lessening the arithmetical work and ma-
king the actual calculations comparatively simple
have been offered by several authors and notably
by Thiele, who in his works discusses several
such methods. Among those we may mention the
method of the so-called free functions and ortho-
gonal substitution, the method of correlates and
the adjustment by elements. The chapters on
these methods in Thiele 's work are among some
of the most important, but also some of the
most difficult in the whole theory of observations
and have not always been understood and appre-
ciated by the mathematicians, chiefly on account
of Thiele 's peculiar style of writing. A close study
of the Danish scholar's investigations is, how-
ever, well worth while, and Thiele 's work along
these lines may still in the future become as
epochmaking in the theory of probability as some
of the researches of the great Laplace. The
theory of infinite determinants as used by M.
Fredholm in the solution of integral equations is
70 Frequency Curves.
another powerful tool which offers great advant-
ages in the way of rapid calculation. All these
methods require, however, that the student must
be thoroughly familiar with the difficult theory
upon which such methods rest, and they have
for this reason been omitted in an elementary
work such as the present treatise.
We wish, however, to mention another method
which in the majority of cases will make it pos-
sible to employ the Gram or Laplacean — Charlier
curves in cases with extreme skewness or excess.
We have here reference to the method of logarith-
mic transformation of the variate, x.
is. the general One of the simplest trans-
tr^s¥ormation formations is the previously
mentioned linear transforma-
tion of the form z = fix) = ax + b, by which
we can make two constants, c 1 and c 2 vanish.
Other transformations suggest themselves, how-
ever, such as fix) = ax 2 + bx + c, fix) = [/«,
fix) = logx and so forth. For this reason I pro-
pose to give a brief development of the general
method of transformations of the statistical
variates, mainly following the methods of Charlier
and J0rgensen.
Stated in its most general form our problem
Theory of Transformation. 71
is : If a frequency curve of a certain variate is
given by F(x) what will be the frequency curve
of a certain function of x, say /(a?) ?
The equation of the frequency curve is y =
F(x) , which means that F(x)dx is the probability
that x falls in the interval between x- — \dx and
* + %dx. The probability that a new variate z
after the transformation z = f(x) , or x (*0 = #i
falls in the interval z — \dz and z + ^dz is there-
fore simply
F[x(z)]y}(z)dz = F(x)dx,
which gives in symbolic form the equation of the
transformed frequency curve.
The frequency for z = i{x) is of course the
same as for x. The ordinates of the frequency
curve, or rather the areas between corresponding
ordinates, are therefore not changed, but the ab-
cissa axis is replaced by f(x). Equidistant inter-
vals of x will therefore not as a rule — except in
the linear transformation — correspond to equid-
istant intervals of fix).
If, for instance, the frequency curve F(x) is
the Laplacean normal curve
1 — x?:2o*
F(x) = —==, e
<3\/2n
72 Frequency Curves.
and if we let z = f(x) = x 2 or x = ]/z, we have
1 e
evidently h __ 2;2(j2
W =
oj/2n 2|/z
8« logarithmic Of the various transformations
™ ANSFOBMArJOiV the logarithmic is of special
importance. It happens that
even if the variate x forms an extremely skew
frequency distribution its logarithms will be
nearly normally distributed.
This fact was already noted by the eminent
German psychologist, Fechner, and also men-
tioned by Bruhns in his Kollektivmasslehre. But
neither Fechner nor Bruhns have given a satis-
factory theoretical explanation of the transforma-
tion and have limited themselves to use it as a
practical rule of thumb.
Thiele discusses the method under his adjust-
ment by elements, but in a rather brief manner.
The first satisfactory theory of logarithmic trans-
formation seems to have been given first by J0r-
gensen and later on by Wicksell. 1 ) Jgrgensen
1 The law of errors, leading to the geometric mean
as the most probable value of the variate as discovered
by Prof. Dr. Th. N. Thiele in 1867 may, however, be con-
sidered as a forerunner of Jgrgensen's work.
Logarithmic Transformation. 73
first begins with the transformation of the normal
Laplacean frequency curve. Letting z = logx and
bearing in mind that the frequency of x equals
that of logx we have
z — f(x) = log x, or x = x(z) = e z and dx = e?dz.
The continuous power sums or moments of
the rth order around the lower limit take on
the form
=J 1 /log x — «i\ !
{n]/'2n)- x N jj afe* l " ' dx =
u
+ f _w!=*y
= (n^2^) _1 iV \ e«e 2 ^ m Vdz.
on the assumption the logx is normally distrib-
uted.
The change in the lower limit in the second
integral from — 00 to zero arises simply from the
fact that the logarithm of zero equals minus in-
finity and the point — 00 is thus by the trans-
formation moved up to zero.
By a straightforward transformation we may
write the above integral as
74 Frequency Curves.
+»
iV mir + D + ikriHr+iy p — l Wdt
M r = -=e
„ T mCr + lJ + '/sM^r+l) 2
= Ne
Changing from moments to semi-variants by
means of the well-known relations
X = M
A 1 = M ± :M
X 2 = (M 2 M -Ml):Ml
X 3 = (M 3 Ml — 3M 2 M 1 M + 2M\):Ml
A 4 = (M K M\ — m z M r M\ — 3MIMI +
+ 12M a M\M — 6M\):M 4
we have
tn+'hn'
A = Ne
A l — e
A 2 = e 2m+3n '(e n *-l)
^ = e «- + e-' (6 *.-_ 4c »»'_ 3e ^ +12< ,.'_ 6)-
Mathematical Zero. 75
These equations give the semi-invariants ex-
pressed in terms of m and n. On the other hand
if we know the semi-invariants from statistical
data or are able to determine these semi-invariants
by a priori reasoning we may find the parameters
ra and n.
21. the mathema- A point which we must bear
in mind is that the above semi-
invariants on account of the
transformation are calculated around a zero point
which corresponds to a fixed lower limit of the
observations.
Very often the observations themselves in-
dicate such a lower limit beyond which the fre-
quencies of the variate vanish. In the case of
persons engaged in factory work there is in most
countries a well-defined legal age limit below
which it is illegal to employ persons for work.
Another example is offered in the number of
alpha particles radiated from certain radioactive
metals. Since the number of particles radiated
in a certain interval of time must either be zero
or a whole positive number it is evident that — 1
must be the lower limit because we can have no
negative radiations. Analogous limits exist in the
age limit for divorces and in the amount of
moneys assessed in the way of income tax.
76
Frequency Curves.
The lower limit allows, however, of a more
exact mathematical determination by means of
the following simple considerations. It is evident
that this lower limit must fall below the mean
value of the frequency curve. X/et us suppose that
it is located at a point, a, located say r\ units in
negative direction from the mean, M = \ , and
let us to begin with select \ as the origin of the
coordinate system in which case the first semi-
invariant, X 1; is equal to zero. Transferring the
origin to a the first semi-invariant equals n , while
the semi-invariants of higher order remain the
same as before the transformation and we have :
-. MJ+1.5B 8
Aj — - a = r\ = e
A 2 = n 2 (e K ' — 1) or e" ! = l+.\ 2 :n 2
\l 3X|
— H
n 6 n 4 .
which reduces to X 3 r\ 3 — SAjJn 2 — Xij = 0.
The solution of this cubic equation which has
one real and two imaginary roots gives us the
value of n or \ — a and thus determines the
mathematical zero or lower limit. We have in
fact :
m
log(l + X 2 :n 2 ) and
log t) — l.bn 2 , while
N = \ n :e
m-^jzn 2
Logarithmic Transformation. 77
22. logarithmic- We have already shown that
ALLY TRANS- . J
formed fre- the generalized frequency curve
QUENCY SERIES & 1 J
could be written as
+ ..
77/ \ / \ ^Wifa) <¥p 2 (x) c a y a (x)
F(x) = c cp (z) — J^-L + sn^J. — J^J
where the Laplacean probability function
— (»— My
<Po(«) = -77^= e
is the generating function with M and o as its
parameters.
The suggestion now immediately arises to use
an analogous series in the case of the logarithmic
transformation. In this case the frequency curve,
F(x), with a lower limit would be expressed as
follows :
F(x) = k % (x) ~jf-+ 2 , - --3'— + • ■ •
while the generating function now is
where m and n are the parameters.
1 n\ = \n.
78 Frequency Curves.
Using the usual definition of semi-invariants
we then have
XjCO \ro 2 X 3 a>3
p Tr + -2T + -3r+---_ c , £i» , ^ , S3C0 3
5 e — s -t- i! "^ 2! 3! '"
.3!
The general term on the right hand side in-
tegral is of the form
(— l) s k s :s\l e xco ® s (x)dx
h
where the integral may be evaluted by partial
integration as follows :
] e x(a <5> a (x)dx = e^O^Or)] — co "$ e x< °<$> s - X (x)dx.
00
Since both <& (x) and all its derivatives are
supposed to vanish for x = and x = 00 the first
term to the right becomes zero and
] e m ®.(x) dx= — co J e* 03 ^-! (as) dr.
By successive integrations we then obtain thp
following recursion formula
Transformed Frequency Series. 79
(— co) 1 1 e xca <P s - 1 (x)dx = (— to) 2 jj e x( °®^(x)dx
O
(_ 03(2 J e x( °<5> s ^ 2 (x) dx = (-co) 3 ] e xa <$> s - S (x)dx
(— to) 8-1 1 e xw ^(x)dx = (— co) s \ e xw %(x)dx.
Or finally
] e XC0 <P s (x)dx = (— to) ! ] e xm %(x)dx.
Expanding e x<a in a power series we have
|e a;ro <l> s (a;)da; =
n\/2n J
1 + iccoH H +
2! 3!
1 r logs— m l*
~z L » J dx.
The general term in this expansion is of the
form
» 1 rloga;— ml*
"Zl n J
(— co) s co r C
n\/Jn r! J
afe
rfa;
80 Frequency Curves.
which according to the formulas given on page
74 reduces to :
Hence we may write
r = as
]e*°»S> s (x)dx = (-co) 8 V^+WV+DV.,.,
Consequently the relation between the semi-
invariants and the frequency function
Fix) = k %(x) - ^ ^(x)+^ 2 (x) - ^ 3 (x)+ . . .
can be expressed by the following recursion for-
mula
\jO> X 2 (0 2 ^3<D 8
Tr + "2T + ^3T + - ••_ , SjM ^2 SgCQ 3
1! 2! 3!
V =s +^ 1 -+^n-+-^r-+-- :
= \" Sv ^=Y'y l co^ V e m( ' +1)+1/2B2(r+1 V: H
v = » = r =
The constants k are here expressed in terms of
the unadjusted moments or power sums, s. It is
readily seen that the Sheppard corrections for
adjusted moments, M, also apply in this case.
We are, therefore, able to write down the values
Transformed Frequency Series. 81
of the fe's from the above recursion formula in the
following manner
M = k Q e m+1 ' m '
M 1 = J h e m+llm °+k e* m+2n '
M % = k 2 e m+l '* n '+2k 1 e* m+2n '+k e 3m+i - Sn °
M a = k 3 e m+lhn '+M 2 e 2m+2n% +Sk ie Zm+ ^ n2 + k e im+Sn!
M, = k i e m+i, ^+ik 3 e 2m+2n '+Qk 2 e Sm+ ^ + ^k 1 e im+8n '
+fe,e 5m+12,5 " !
It is easy to see that it is not possible to
determine the generating function's parameters m
and n from the observations. These parameters
like M and o* in the case of the Laplacean normal
probability curve must be chosen arbitrarily. If
m and n are selected so as to make k x and k 2
vanish we have
M = k e m+ ''^
M x = k e'
M % = k e
2m+2ri l
Zm+iAn?
the solution of which gives
e
M M 2 2m _ M\
while
82 Frequency Curves.
l^v+'l" = M i -4M 3 e m+1 - 6nl -M e im+9n \e 3n '-4).
This theory requires the computation of a set
of tables of the generating function
i nog x— my
*> / x i ~ si - s - J
wj/2n
and its derivatives. For O (a;) itself we may of
course use the ordinary tables for the normal
curve <p (z) when we consider
log x — m
z = —2 .
n
I have calculated a set of tables of the deriv-
atives of <E> (a;) and hope to be able to publish the
manuscript thereof in the second volume of my
treatise on "The Mathematical Theory of Probab-
ilities".
23. parameters The above development is
Tea^t M squareI based upon the theory of func-
tions and the theory of definite
integrals. We shall now see how the same pro-
blem may be attacked by the method of least
squares after we have determined by the usual
method of moments the values of m and n in the
generating function q> («).
Parameters and Least Squares. 83
Viewed from this point of vantage our problem
may be stated as follows :
Given an arbitrary frequency distribution, of
the variate z with z = (log x — m) : n and where
x is reckoned from a zero point or origin, which
is situated a units below the mean and defined by
the relation
ri 3 A 3 — 3r) 2 Aa = Ajj, where a = \ ± — r\;
to develop F(z) into a frequency series of the
form
F(z) = k y (z) + k 3 y 3 (z) + /c 4 q> 4 (z) + . . . + kn<? n (z) ,
where the fe's must be determined in such a way
that the expression
(r = It,
faipiiz)
gives the best approximation to F(z) in the sense
of the method of least squares.
Stated in this form the frequency function is
reduced to the ordinary series of Gram or the A
type of the Charlier series, already treated in the
earlier chapters.
6*
84 Frequency Curves.
24. application As an illustration of the theory
of a mortality to a practical problem we pre-
sent the following frequency
distribution by 5-year age intervals of the number
of deaths (or Zd s by quinquennial grouping) in
the recently published American-Canadian Mor-
tality of Healthy Males, based on a radix of
100,000 entrants at age 15.
Frequency Distribution of Deaths by Attained
Ages in American-Canadian Mortality Table.
Ages
Zdx
1st Component
2d Comp.
15— 19
1,801
120
1,681
20— 24
1,996
230
1,766
25— 29
2,089
440
1,649
30— 34
2,120
790
1,330
35— 39
2,341
1,370
971
40— 44
2,911
2,270
641
45— 49
3,937
3,570
367
50— 54
5,527
5,400
127
55— 59
7,723
7,722
1
50— 64
10,383
10,383
65— 69
12,987
12,987
70— 74
14,535
14,535
75— 79
13,807
13,807
80— 84
10,328
10,328
85— 89
5,464
5,464
90— 94
1,757
1,757
95— 99
278
278
100—104
16
16
100,000 91,467 8,533
Mortality Tables. 85
The curve represented by the d x column is
evidently a composite frequency function com-
pounded of several series. From a purely mathe-
matical point of view the compound curve may
be considered as being generated in an infinite
number of ways as the summation of separate
component frequency curves. From the point of
view of a practical graduation it is, however, easy
to break this compound death curve up into two
separate components. A mere glance at the d x
curve itself suggests a major skew frequency curve
with a maximum point somewhere in the age
interval from 70 — 75 and minor curve (practically
one-sided) for the younger ages.
Let us therefore break the ~Ld x column up into
the two so far perfectly arbitrary parts as shown
in the above table and then try to fit those two
distributions to logarithmically transformed A
curves.
Starting with the first component the straight-
forward computation of the semi-invariants is
given in the table below with the provisional mean
chosen at age 67.
86 Frequency Curves.
Frequency Distribution of Deaths in American
Mortality Table First Component.
Ages x ?(i) xF(x) x'F(x) z*F(z)
04—100
— 7
16
112
784
5,488
99— 95
— 6
278
1,668
10,008
60,048
94— 90
— 5
1,757
8,785
43,925
219,625
89— 85
— 4
5,464
21,856
87,424
349,696
84— 80
— 3
10,328
30,984
92,952
278,856
79— 75
— 2
13,807
27,614
55,228
110,456
74— 70
— 1
14,535
14,535
14,535
14,535
69— 65
—
12,987
59,172
105,554
304,856
1,038,704
64— 60
+ 1
10,383
10,383
10,383
10,383
59— 55
+ 2
7,723
15,446
30,892
61,784
54— 50
+ 3
5,400
16,200
48,600
145,800
49— 45
+ 4
3,570
14,280
57,120
228,480
44— 40
+ 5
2,270
11,350
56,750
283,750
39— 35
+ 6
1,370
8,220
49,320
295,920
34— 30
+ 7
790
5,530
38,710
270,970
29— 25
+ 8
440
3,520
28460
225,280
24— 20
+ 9
230
2,070
18,630
167,670
19— 15
+ 10
120
1,200
12,000
120,000
32,296
88,199
350,565
1,810,037
Sr 91,468 —17,355 655,421 771,333
Computing the semi-invariants by means of
the usual formulas in paragraph 13, we have :
\ 1 = —17355:91468 = — 0.18974, or mean at
age 67 + 5 (0.19) or at age 67.95
Mortality Tables. 87
X 2 = 655421:91468 — A, 2 = 7.1296
X 3 = 771333:91468 — 3 A^H- 2 A^ = 12.4981.
In order to determine the mathematical zero
or the origin we have to solve the following cubic :
M 3 — 3X 2 2 n 2 = V, or
12.498 n 3 — 152. 511 n 2 = 362.47
the positive root of which is equal to 12.39. The
zero point is therefore found to be situated 12.39
5-year units from the mean or at age 67.95 + 5
(12.39), i. e. very nearly at age 130, which we
henceforth shall select as the origin of the co-
ordinate system of the first component. We have
furthermore
12.39 =e m +i- 5n \ and 7.1296 = e 2m + 3n '(e n '-- 1) =
= (12.39) 2 (e» 9 — 1),
the solution of which gives n 2 = 0.04436, n =
0.2106, m = 2.4504, all on the basis of a 5-year
interval as unit. If we wish to change to a single
calendar year unit we must add the natural
logarithm of 5, or 1.6094, to the above value of m,
which gives us m = 4.0598, while n remains the
same. The above computations furnish us with
the necessary material for the logarithmic trans-
formation of the variate x which now may be
written as
88 Frequency Curves.
z = [log (130 — a:) —4.0598] : 0.2106,
where x is the original variate or the age at death.
Having thus accomplished the logarithmic
transformation we may henceforth write the
generating function as
*o(*) =
1_ pog(130 — z) — 4.0598 -I'
2 L 0.2106 J
.2106|/2jt
= <Po(z) =
271
We express now F (x) by the following
equation.
F{x) = k Q <5> (x) + k s <£> 3 (x) + k^^x) + ....
or in terms of the transformed z :
cp(z) = A: cp (z) + A: 3 cp 3 (2) + A; 4 cp 4 (z) + ,
and proceed to determine the numerical values
of k by the method of least squares.
The numerical calculation required by this
method follows precisely along the same lines as
described in paragraph 17. I shall for this reason
not reproduce these calculations but limit myself
to quote the final results for the various co-
efficients k, which are as follows : — 1
1 Interested readers may consult the detailed com-
putations on pages 246—257 in my Mathematical
Theory of Probabilities (2nd Edition, New York,
1921.
Mortality Tables. 89
ft = 7361.8; /b s = — 212.2; k A = — 9.6.
The final equation of the frequency curve of
the first component F (x) , is therefore : —
Fi(x) = 7361.8q> (*) — 212.2<p,(z) — 9.6<p 4 (z),
where the generating function, y a (z), is of the
form : —
1 Hog (130 — x ) — 4.0598 -I"
<Po(z) = <£„(*) = —7= e~ 2 L °- 210 ^^ ~ J
0.2106)/ 2 jt
The second component, F n (x) , can by means
of a similar process be expressed by the equa-
tion :—
Fn{x) = 947.4cp (z)— 63.4cp 3 (z)— 30.0cp 4 (z),
where
1 Hog (x + 68.8) — 4.532 1'
1 „ 2 L 0.12 J
<PoO) = <J>o(*) =
0.12J/2jt
Addition of these two component curves gives
us the ultimate compound frequently curve,
representing the d x of the mortality table.
A comparison between the observed values of
d x and the values of d x as computed from the
above equation is shown in graphical form in the
attached diagram. Evidently the graduation leaves
but little to be desired in the way of closeness
of fit.
90
Frequency Curves.
ooo
>-"
\
/
\
/
\
""""
/
\
/
\l
/
v
loco
\
l " otJ
\
1000
/
" ou
#
V
-,uo
/
u££— -
"■"— —
— -
-/.
IS So as> 30 35 -*<3 ^S So S5 &o
75 So as <3o <3S loo /KCie-S
Figure 1.
Diagram showing graduation of d x column in the AM (5) table by a
compound frequency curve of the Gram-Charlier types.
25. biological It appears that the Italian
of mortality statisticians were the first to
break up the d x curve into a
system of five or more component frequency
curves, which, however, were all of the normal
Laplacean type. Pearson who in a brillant essay
entitled Chances of Death was the next to attack
the problem, employed a system of five skew
frequency curves. Already as early as 1914 I found
that from ages above 10 the majority of d x
curves in previously constructed mortality tables
could be represented by not more than two skew
Biological Interpretation. 91
frequency curves as shown in the above example
of the AM (5) table.
Although all such investigations may be very
interesting and useful from the point of view of
the actuary, we must, however, not overlook the
fact that the breaking up of the compound d x
curve in the manner just described is merely an
empirical process pure and simple. While such
processes undoubtedly represent very neat methods
of graduation, a quite different and more im-
portant question is whether mathematical work
of this kind allows of a biological interpretation.
It is evident that from a mere mathematical point
of view we may break up the d x curve into various
component parts in an infinite number of ways.
But while such breaking up processes may be
extremely interesting as actuarial graduations and
exercises in pure mathematics, they have evidently
little connection with the underlying biological
facts of a mortality table. This aspect of the
question has been brought out in a very forcible
manner by the eminent American biometrician,
Eaymond Pearl, in his 1920 Lowell Institute
Lectures. The whole subject would appear in a
quite different light if it were possible to give a
biological interpretation of the mathematical
analysis and to show that the component fre-
quency curves as derived from pure mathematics
92 Frequency Curves.
have a counterpart in actual life. This, I think,
would be very difficult, if not impossible to
establish, because it is not mathematics which
determines the conduct or behavior of living
organisms. One might, however, view the whole
problem from the standpoint of the biologist
rather than from the standpoint of the mathema-
tican. The problem then is to ascertain whether
the observed biological facts as shown in the
collected statistical data allow of a mathematical
interpretation, rather than to find a biological
interpretation and counterpart of previously
established empirical formulae.
It is to this important question that I have
devoted the entire discussion of the second chapter
of this book. I have proceeded from certain
observed biological facts (in this particular
instance the statistics on the number of deaths
by sex and attained ages from more than 150
causes of death) which represent the natural
phenomena under investigation. In order to offer
a rational explanation of these facts and to inter-
prete their quantitative relationships, I have
adopted as a working hypothesis the supposition
that the number of deaths according to attained age
and sex among the survivors of a homogeneous
cohort of say 1,000,000 entrants at age 10 tend
to cluster around specific ages in such a manner
Biological Interpretation. 93
that their frequency distribution by attained ages
can be represented by a limited number of sets
of Gram-Charlier or Poisson-Charlier frequency
curves.
On the basis of this hypothesis we can now
by simple mathematical deductions construct a
mortality table from deaths by sex, age and cause
of death and without any information about the
lives exposed to risk at various ages.
Finally we can verify the ultimate results
contained in this final mortality table by working
back from the table to the data originally
observed.
This procedure is in strict conformity with
the model of modern science, which according
to Jevons consists of the four processes of obser-
vation, hypothesis , deduction and verification.
The important factor in this investigation,
and one which most actuaries and statisticians
fail to grasp, is that I have looked at the whole
problem as a biometrician rather than as a
mathematician. Mathematics has been employed
only as a working tool in the whole process, and
the reason that the method has met with success
must be sought for in concrete biological facts
and not in the realm of mathematics.
94 ' Frequency Curves.
26. poisson-s I Q certain statistical series it
P ft?nction Y frequently happens that the
semi-invariants of higher order
than zero all are equal, or that
\ x = X 2 = X 3 = . . . . = X r = X.
We shall for the present limit our discussion
to homograde statistical series where the variates
always are positive and integral, and where there-
fore the definition of the semi-invariants is of the
form : —
Xco Xco 2 Xco s
e Tr + -2T + ^r H "z<p(a;) = ^y( x )e xm =
= cp(0)e 0co + <p(l)e lm + cp(2)e 2co + cp(3)e 3ro + ....,
or
Xco Xco 2 Xco 8 _\ \,co „ , . xca
e
for x = 0, 1, 2, 3, . . .,
which also can be written as
Xe m . X 2 e 2co
e- x (l + — 4
1! ' 2!
= 9(0)1 + 9(l)e ro + cp(2)e 2m +
The coefficient of e TCD gives the relative fre-
quency or the probabitity for the occurence of
x = r, and we find therefore that
Poisson's Function. 95
e- x A r
<f(x) = i|>(r) = -yy
This is the famous Poisson Exponential, so
called after the French mathematician, Poisson,
who first derived this expression in his Recherches
sur la Probabilites des jugesments, but in an
entirely different manner than the one we have
indicated above.
The Poisson Exponential opens a new way
for the treatment of statistical series which poss-
ess the attribute that all their semi-invariants of
higher order than zero are all equal, or nearly
equal. It is readily seen that whereas the Lap-
lacea probability function y (x) contains two
parameters X x and o the probability function of
Poisson contains only one parameter, A.
27. poisson— We have already seen in the
f , fJAJ}T TDD .
frequency previous chapters that the
Gram-Charlier frequency curve
could be written as
F{x) = ~Ld(pi(x) = T.aHi(x)(p (x)
for i=0, 1,2,3,
where cp (^) is the generating Laplacean proba-
bility function.
The idea now immediately suggests itself to
96 Frequency Curves.
use a similar method of expansion in the case of
the Poisson probability function and to employ
this exponential as a generating fuction in the
same manner as the Laplacean function. We are,
however, in the present case of the Poisson
exponential dealing with a generating function
which so far has been defined for positive integral
values only and, therefore, represents a discrete
function. Por this reason it will be impossible to
express the series as the sum-products of the suc-
cessive derivatives of the generating function and
their correlated parameters c. We can, however,
in the case of integral variates express the series
by means of finite differences and write F(x) as
follows :
F{x) = c i\>(x) + c^O) + c 2 A^(» .... (/)
where ty(x) = er m m x :x! for x = 0, 1, 2, 3, .... ,
and
Ai{>0) = t\>(x) — ii>(x — 1),
A 2 i|)(a;) = AiKa:) — A^(a;— l)=i|)(a)— 2\\>(x— 1)
+ $(x— 2).
The series (I) is known as the Poisson-Char-
lier frequency series or Charlier's B type of
frequency curves.
The semi-invariants of these frequency series
are given by the following relation :
Poisson's Function. 97
XjOD + X a CO 2 + X 3 CO 8 + . . .
~2\ 3T
e =
x =
Expanding and equating the co-efficients
of equal powers of co we have :
A = 1 = c S\|) (x) or c = 1
\ t = Zz (i|> (re) + cA$(x) + c t ^(x) + ..-) (II)
\ l z + \. 2 = Zx*{ty(x) + cAi\>(x) + cA 2 Mx) + ---)
We now have
2i))(j) = 1, and
Za;i|) (a;) = Im« _m m x ~ x : (x — 1) ! = mZ\|) (x — 1) = m.
We also find from well-known formulas of the
calculus of finite differences that 1
Za) 2 i|)(a;)
ZxAip(x) =
1 These formulas can also be derived from the de-
finition of the semi-invariants and the well-known rela-
tions between moments and semi-invariants as given on
page 74 when we remember that according to our de-
finition all semi-invariants in the Poisson exponential are
equal to m.
7
98 Frequency Curves.
ZxA 2 ^(x) =
~Lx 2 A^(x) = — (2m + 1)
2,x 2 A 2 i\> (x) = 2
Substituting these values in (77) we obtain
X 1 = m — c x
X x 2 + A 3 = to 2 + m — (2m + 1) c Y + 2c 2
By letting m = A x we can make the coefficient
Cj vanish, which results in
\ ± = m
c 2 = %[>.;, — -to]
where the two semi-invariants X x and A 2 are cal-
culated around the natural zero of the number
scale as origin.
For the above discussion we have limited
ourselves to the determination of the three con-
stants m, c and c 2 . It is easy, however, to find
the higher parameters c 3 , c 4 , c 5 , : . . from the
relations between the moments of the Poisson
function and the semi-invariants of order 3, 4,
5, . . . ect. Charlier usually calls the parameter m
the modulus and c 2 the eccentricity of the B
curve.
Numerical Examples. 99
28. numerical Xs an illustration of the appli-
examples cation of the p i sson _charlier
series we select the following
series of observations on alpha particles radiated
from a bar of Polonium as determined by Ruther-
ford and Geiger.
The appended table states the number of
times, F(x), the number of particles given off in
a long series of intervals, each lasting one-eighth
of a minute had a given value x : —
x F(x) x F(x) x F(x)
57
5
408
10
10
1
203
6
273
11
4
2
383
7
130
12
3
525
8
45
13
1
4
532
9
27
14
1
We are here dealing with integral variates
which can assume positive values only and the
observations are therefore eminently adaptable to
the treatment by Poisson-Charlier curves. Select-
ing the natural zero as the origin of the co-
ordinate system we find that tbe first two semi-
invariants are of the form
\ 1 = 3.8754, \ 2 = 3.6257, and we therefore have :
w = \ 1 = 3.86; c 3 = %i[X 2 — to] = —0.125.
The equation for the frequency distribution of
the total N = 2608 elements therefore becomes
7*
100
Frequency Curves.
F(x) = N[T|),. gg (a;) + (—0.125) 2 ^ 3 . ss (x)~].
The table below gives the values as fitted to
the curve, F(x) :
Alpha
Particles
■ Discharged from Film of Polonium
(Rutherford and Geiger).
N = 2608, m = 3.88
i, c 2 =
— 0.125
(i)
(2)
(3)
(4)
(5)
(6)
X
M*)
A 2 i|>M
NX (2)
i^X(3)Xc 2
(*) + (5)
.020668
+ .020668
53.9
— 6.7
47
1
.080156
+ .038820
209.0
—12.7
196
2
.155455
+ .015811
405.4
— 5.2
400
3
.201015
—.029793
524.2
+ 9.7
533
4
.194967
—.051608
508.5
+ 16.8
525
5
.151625
—.037654
394.5
+ 12.3
407
G
.097850
—.009714
254.9
+ 3.2
258
r
.054249
+ .009814
141.2
— 3.2
138
8
.026316
+.015668
68.7
— 5.1
64
9
.011351
+ .012968
29.6
— 4.2
25
10
.004407
+ .008021
11.5
— 2.6
9
11
.001555
+ .004092
4.1
— 1.2
3
12
.000503
+ .001800
1.3
— 0.6
1
13
.000150
+ .000699
0.4
— 0.2
14
.000042
+ .000245
0.1
— 0.1
15
.000010
+ .000076
0.0
— 0.0
16
.000003
+ .000025
17
.000001
+ .000005
As a second example we offer our old friend,
the distribution of flower petals in Ranunculus
Bulbosus. Selecting the zero point at x = 5 and
Transformation of Variates. 101
computing the semi-invariants in the usual
manner we obtain the following equation for the
frequency curve.
F(x) = 222 ^>(x) + 31.5A 2 iMaO, m = 0.631.
A comparison between calculated and observed
values follows : —
x F (x) Obs.
5 134.9 133
6 51.6 55
7 22.5 23
8 9.5 7
9 2.9 2
10 0.6 2
29. trans- For integral variates we have
F thevariat£ shown that the Poisson fre-
quency curve possesses the im-
portant property that all its semi-invariants are
equal. Now while a frequency distribution of a
certain integral variate, x, may perhaps not
possess this property, it may, however, very well
happen after a suitable linear transformation has
been made, that the variate thus transformed will
be subject to the laws of Poisson 's function.
Let z = ax — b represent the linear trans-
formation which is subject to the above laws with
a series of semi-invariants all equal to m.
102 Frequency Curves.
These semi-invariants according to the pro-
perties set forth in paragraph 5 are therefore
m = X x (z) = a\ 1 (x) — b
m = X 2 (z) = a 2 \. 2 (x)
m = X 3 (z) = a?\ 3 (x)
and our problem is to find the unknown para-
meters a, b and m.
Simple algebraic methods, which it will not
be necessary to dwell upon, give the following
results :
a = X 2 :X 3
m = X 2 3 :X 3 2
b = aX 2 — m
As a numerical illustration of this trans-
formation we choose from J0rgensen a series of
observations by Davenport on the frequency
distribution of glands in the right foreleg of 2000
female swine.
No. of Glands.. 01 2 3 4 5 6789 10
Frequency 15 209 365 482 414 277 134 72 22 8 2
The values of the three first semi-invariants are
Transformation of Variates. 103
\ = 3.501, X 2 = 2.825, \ 3 = 2.417,
o = 2.825:2.417 = 1.168,
m = 2.825 3 : 2.417 2 = 3.859,
b = (1.168) (3.501) —3.859 = 0.230.
The new variable then becomes z — az — b
and the transformed Poisson probablity function
takes on the form :
i|)(z) =
A
In general, however, we will find that z is not
a whole number and the expression z ! therefore
has no meaning from the point of view of
factorials at least. This difficulty may, however,
be overcome through the introduction of the well-
known Gamma Function, T(z + 1), which holds
true for any positive or negative real value of z
and which in the case of integral values of z
reduces to Y(z + 1) = z !
Hence we can write the transformed Poisson
probability function as
, . e- m m z
^ = f(^+T)-
Tables to 7 decimal places of the Gamma
Function, or rather for the expression — r (z + 1) ,
have been computed by Jorgensen in his Frekvens-
104 Frequency Curves.
flader and. Korrelation from z = — 5 to z = 15,
progressing by intervals of 0.01.
By means of this table and the tables of
ordinary logarithms it is now easy to find the
values of i|> (z) in the case of the example relating
to the number of glands in female swine. The
detailed computation is shown below. 1
(1) (2) (3)
x z r( z +i)
—.230 .9209
1 +.938 .0108
2 2.106 .6555
3 3.274 .0679
4 4.442 .3216
5 5.610 .4547
6 6.778 .4904
7 7.946 .4446
8 9.114 .3285
9 10.282 .1506
10 11.450 .9177
«
(5)
(6)
(7)
log m?
(3) + (4)
+ loge— m
*W
F(x)
.8651
.1101—2
.0129
30.1
.5500
.8849—2
.0767
179.2
.2350
.2146—1
.1639
382.9
.9199
.3119—1
.2051
479.1
.6048
.2501—1
.1780
415.8
.2897
.0685—1
.1171
273.6
.9746
.7891—2
.0615
143.7
.6595
.4282—2
.0268
62.6
.3444
.9970—3
.0099
23.1
'.0294
.5041—3
.0032
7.5
.7143
.9561—4
.0009
2.1
1 The characteristics of the logarithms have been
omitted in this table (except in column 5) and only the
positive mantissas are shown. Column 7 represents the
2000 individual observations pro rated according to
column 6.
CHAPTER II
(TRANSLATED BY MR. VIGFUSSON)
THE HUMAN DEATH CURVE
In the following paragraphs I
1. INTRODUCTORY & r & Jr-
remarks intend to discuss a method of
constructing mortality tables
from mortuary records by sex, age and cause of
death, but without reference to or knowledge of
the exposed to risk at various ages. This proposed
method is indeed one which has been severely
criticized in certain quarters, and. several critics
flatly deny that it is possible to construct morta-
lity tables from such data without detailed infor-
mation of the exposed to risk. It is, however, a
very dangerous practice to say that a certain thing
is impossible. The true scientist, least of all,
should attempt to set limits for the extension of
human knowledge. It is still remembered how the
great August Comte once denied that it ever
would be possible to determine the chemical con-
stituents of the celestial bodies. Only a few years
after this emphatic denial by the brilliant French-
106 Human Death Curves.
man the spectroscope was discovered, by means of
which we have been able to detect a number of
chemical elements of other worlds than that of
our own little earth. It is but fair to say that the
method which we here shall describe has met with
rather determined opposition in certain actuarial
quarters. Under such circumstances it is natural
that the process will be viewed in a light of scep-
ticism and criticism. I welcome such an attitude
because it has been my purpose to present the
following studies for further investigation and not
to force them upon my readers as authoritative
or as a kind of infallible dogma.
In presenting the outlines of the proposed
method I wish to state that it has never been the
intention to supplant the orthodox methods of
constructing mortality tables where we have ex-
act information of the so-called "exposed to risk"
or number living at various ages. Numerous and
very important examples, however, offer them-
selves in actuarial and statistical practice where
such information is not available. Most of the
greater American Life Insurance Companies,
especially those writing the so-called industrial
insurance, have on hand an enormous amount of
information of deaths by sex, attained age and by
cause of death among their policyholders. Even
the mortuary records of certain occupations, as
for instance metal and coal miners, among the
Introductory Remarks. 107
death claims in the industial class are so numer-
ous, that it would be possible to construct a mor-
tality table for such professions if we know the
exact number exposed to risk at various ages.
Such information is, however, in the majority of
cases wanting, or could only be obtained by means
of a great expenditure of time and labor. Again,
as Mr. P. S. Crum has pointed out in an article
in the "Insurance and Commercial Magazine", a
number of cities and states in United States give
from year to year very detailed information in
regard to mortuary records by sex, age at death
and cause of death. On account of the intense
migration taking place in certain sections of the
United States, especially in those of an industrial
character, it is, however, impossible to know the
exact population at various ages, except in the
particular years in which the federal or state
census has been taken. The fact that for all but
a few states of this country the intercensal period
is no less than ten years, the determination of the
population composition by age and sex for a given
locality and intercensal year, with any degree of
accuracy, becomes a practical impossibility without
a special count. Such a count or census of a
specific locality or a single city is, however, a
costly undertaking at its best, for which the nec-
essary funds are rarely available. In all such
instances the mortuary records are practically
108 Human Death Curves.
worthless in so far as the construction and com-
putation of death rates are concerned, if we are
to rely solely upon the usual method of construct-
ing mortality tables. It will therefore readily be
seen that, apart from purely academic interests,
the possibility of establishing a method of con-
structing mortality tables without knowing the
population exposed to risk at various ages would
be of great practical value, and I deem no apology
necessary to present the following method, which
intends to overcome this very obstacle of having
no information of the exposures.
2. empirical and In order to bring the method
INDUCTIVE ME- ■ , ,1 .• -.
thods of solu- mto the proper perspective it
will be of value to contrast it
with the ordinary methods followed in the con-
struction of mortality tables. Let us therefore
briefly review'those methods and principles com-
monly employed by actuaries and statisticians. A
certain number, say L persons at age x, are kept
under observation for a full calendar year and the
number, D T , who die among the original entrants
during the same year are recorded. The ratio
D x : L x is then considered as the crude probabi-
lity of dying at age x. Similar crude rates are ob-
tained for all other ages and are then subjected to
a more or less empirical process of graduation to
Impirical and' Inductive Methods. 109
smooth out the irregularities arising from what is
considered as random sampling. One then chooses
an arbitrary radix, say for instance 100,000 per-
sons at age 10, which represents a hypothetical
cohort of 10-year old children entering under our
observation. This radix is then multiplied by the
previously constructed value of q and the product
represents the number dying at age 10. This
number, d 10 , is subtracted from l 10 or 100,000 and
the difference is the number living at age 11 or
Z„. This latter number is then multiplied by q xl
and the result is d 117 or the number dying at age
11 out of the original cohort of 100,000. In this
way one continues for all ages up to 105, or so.
It is to be noted that the column of q x in this
process represents the fundamental column while
the columns of l x and d r are purely auxiliary
columns.
Allow us here to ask a simple question. Do
these empirically derived numbers of deaths at
various ages out of an original cohort of 100,000
entrants at age 10 give us any insight or clue as
to the exact nature of the biological phenomenon
known as death, and are we by this method enab-
led to lift the veil and trace the numerous causes
which must have been at work and served to pro-
duce the total effect, the d r curve, of which we
by means of the usual methods have a purely
110 Human Death 'Curves.
empirical representation? I fear that this question
will have to be answered in the negative. The
usual actuarial methods do not give us a single
glance into the relation between cause and effect,
which after all is the ultimate object of investiga-
tion for all real science. Probably some critics
would answer that they are not interested in in-
vestigating causal relations. Such an attitude of
indifference is, however, very dangerous for a sta-
tistician or an actuary whose very work rests upon
the validity of the law of causality. We may,
however, overlook this apparent inconsistency of
the empiricists and turn our attention to the pro-
posed methods of constructing mortality tables-
along inductive lines, or by the process which
Jevons has termed a complete induction.
Such a process we should find diametrically
opposite to the methods of the empiricists, both in
respect to points of attack and deduction. In the
case of the empiricists the q r . is the initial and
fundamental function from which the d x column
is computed as a mere by-product. The rationalistic
method starts with the d column and terminates
with the q x as the by-product.
Being primarily interested in the absolute
number of deaths and not in the relative frequen-
cies of deaths at various ages, our first question
is therefore, "What is the form of the frequency
Property of "Death Curves" m
curve representing the deaths at various ages
among the survivors of the original group of
100,000 entrants at age 10?" Right here we can,
strange to say, apply some purely a priori know-
ledge. We know a priori that the curve must be
finite in extent, because of the very fact that there
is a definite limit to human life, and we also know
that it assumes only positive values. There can be
no negative numbers of deaths unless we were to
regard the reported theological miracles of resur-
rections from the Jewish- Christian religion as
such. This information about the death curve, or
the curve of d , is, however, not sufficient for use
as a basis for our deductions. We must therefore
look about for additional information, whether of
an a priori or an a posteriori nature and of such
a general character that it can be adopted as a
hypothesis.
It was Poincare who once said
3. GENERAL PRO- , . , . .
perties of the that every generalization is a
"DEATH CURVE" / to
hypothesis. Hence we shall
look for some general characteristics which all
mortality tables have in common in the age
interval under consideration (age 10 and up-
wards) . Let us take any mortality table, I do
not care from what part of the world, and
examine the general trend of the curve traced
112 Human Death Curves.
by the values of d for various ages. The curve
rises gradually from the age of ten. The increase
in the number of deaths among the survivors at
various ages will increase, although not uniformly,
until the ages around 70 or 75 are reached. At this
age interval we generally encounter a maximum.
From the ages between 70 and 75 and for higher
ages the number of deaths among the survivors
will decrease at a more rapid rate than at the
earlier stages of life. After the age of 85 only a
small number of the veteran cohort are still alive.
After the age of 90 only a few centenarians
struggle along, keeping up a hopeless fight with
the grim reaper, Death, until eventually all are
carried off between the ages of 110 and 115. We
can much better illustrate this process of the
struggle between the surviving members at va-
rious ages of the cohort and the opposing forces as
marshalled by the ultimate victor, Death, through
a graphical representation. The chart on page 114
shows a mortality graph of the male population
in Denmark (1906-1910) from ages 10 and up-
wards as constructed by the Royal Danish Stati-
stical Bureau. The ordinates of the curve show
the number of deaths at various ages among the
survivors of the original cohort of 100,000 entrants
at agelO. We notice a gradual increase from the
younger ages until the age of 77, where a max-
Property of "Death Curves". ng
imum or high crest is encountered. From that age
a rapid decline takes place until the curve ap-
proaches the abscissa with a strongly marked
asymptotic tendency after the age of 90. At the
age of 110 all the members of the cohort have lost
out and death stands as the undisputed victor, a
victor among a mass of graves. The curve we thus
have traced may properly be called "The Curve of
Death". On the same chart I have also shown
a graphical representation of a comparison between
the Danish death curve and the corresponding
death curves of males for England and Wales in
the period 1909—1911, Norway 1900—1910,
France 1908—1913 and United States period
1909 — 1911, all based upon an original radix of
1,000,000 entrants at age 10.
We will notice quite important variations in
these curves. The curves for the Scandinavian
countries show a relatively heavy clustering around
the maximum point which in the case of Den-
mark is reached at age 75, in England at age 73,
and in France at age 72. The Danish curve is also
more symmetrical and shows a more uniform clu-
stering tendency around the maximum value than
the other curves. The asymmetry or skewness is
most pronounced in the American curve, due to
the comparatively greater number of deaths at
114
Human Death Curves.
s &
"5
3 "
03 t?
< H
g g
is.
younger ages than in the other tables. Tn the
curve for Norwegian males I rnight mention
Property of " Death Curves". 115
another peculiarity which is absent in most other
death curves. I have reference here to a secondary
minor maximum or miniature crest at the age of
21. This maximum point, which is not very pro-
nounced arises from the heavy mortality among
youths in Norway, whose male population always
has consisted of rovers of the sea. A much larger
proportion of young men braves the terrors of the
sea in Norway than in any country in the world.
These sturdy decendents of the Vikings can be
found in all parts of the globe. You are sure to
find a weatherbeaten Norwegian tramp steamer
even in the most deserted and far away harbours
of our continents. But the sea takes its toll. The
result is shown in the little peak in the curve of
death among these sturdy Norwegian youths. 1
Despite all these smaller irregularities all the
curves have, however, certain well defined charac-
teristics , namely :
1) An initial increase with age.
2) A well defined maximum point around the
age period 70 — 80.
2) A more rapid decline from that point until
the ultimate end of the mortality table.
1 Another factor is the high number of deaths from
tuberculosis typical of youth. See in this connexion dis-
cussion in paragraph 12 a under the Japanese Table.
116 Human Death Curves.
The most interesting of these
4. RELATION OF . . . .
frequency c o m m o n characteristics is
CURVES
the encountering or a maxi-
mum point in the neighborhood of 70, and the
subsequent decline toward the higher ages. This
fact has a very important biometric significance,
which we shall discuss in a somewhat detailed
manner. Most of my readers are familiar with the
so-called probability curve, expressed by the
equation :
This Laplacean or normal curve is represented in
graphical form by the beautiful bellshaped curve
so well known to mathematical readers. Various
approximations to this curve are continually en-
countered in numerous instances of observations
relating to certain biological phenomena where
certain measurable attributes of various sample
populations tend to cluster around a certain norm,
such as the measurements of heights of recruits,
fin rays in fish, etc. We also know that where this
tendency to cluster around the mean is asymmetri-
cal or skew, it is in many cases possible to give
a very close representation by the Laplacean-
Charlier frequency curves.
Now let us return to our curves of death. It
Relation to Frequency Curves. 117
will be noted that all these curves for ages above
the crest period 70 to 75 to a very marked degree
approach the form of the normal probability curve
and exhibit a marked clustering tendency around
this particular period. The ages around 70, the
Bible's "three score and ten", can therefore be
looked upon as a norm of life around which the
deaths of the original cohort group themselves
in more or less correspondence with the binomial
probability law. This pronounced grouping ten-
dency is a very significant biological phenomenon,
which it might be of interest to dwell upon.
If all the members of our original cohort were
identical as to physical constitution and characte-
ristics, if they all were exposed to. identically the
same outward influences acting upon their mode
of life, it becomes evident from the law of causa-
lity, which is the basis and justification of every
collection of statistical data, that all members
would die at the same moment. We see, however,
immediately that such hypothetical conditions are
not present in human society. The paramount
feature of our material world is variation. No two
persons are alike in regard to physical constitu-
tion. Certain inherited characteristics, which are
present in the individual in more or less pronoun-
ced form, make themselves felt. No two persons
or group of persons can be said to be exposed to
118 Human Death Curves.
the same outward influences. The clergyman and
college professor living a sort of tranquil and
sheltered life are not exposed to the same dangers
as the working man or the man in business life.
All these and other factors, almost infinite in
number, tend to produce a decided variation in
the actual duration of life. Of these influencing
factors those relating to purely inherited or na-
tural characteristics are without doubt the most
powerful. If it were possible to eliminate certain
forms of deaths due to infectious diseases, tuber-
culosis and accidents, causes more or less due to
outward influences, we should have left a number
of causes due to a gradual wearing out of the
human system, similar in many respects to the
deterioration of the mechanism in ordinary ma-
chinery. The death curve from such causes of death
would be more related to the normal curve than
the death curve which includes causes of death
from non-inherent or anterior causes as menti-
oned above. This statement is borne out in the
shape of the Danish death curve. In Denmark
where a very determined and largely successful
fight has been carried on against tuberculosis, and
where the accident rate is very low we also find
that the curve is more symmetrical than for in-
stance in this country or in England.
This tendency to an approach towards the bi-
Relation to Frequency Curves. H9
normal probability curve was already noted by
Lexis, who from such considerations tried to de-
termine what he called a "Normalalter" or normal
age for various countries and sample populations.
Speaking of this attempt the eminent Danish sta-
tistician, Harald Westergaard, says in his „Sta-
tistikens Teori i Grundrids" (Copenhagen 1916)
"An unsually interesting attempt has been made
by Lexis to determine the normal age of man.
A mortality table will, as a rule, have two
strongly dominant maximum points for the num-
ber of deaths. During the first year of life there
dies a comparatively large number. From the age
of 1 the number of deaths decreases and reaches
its lowest point in early youth. It then again
begins to increase, at times in wavelike motions,
until the maximum point is reached at the old
age period".
"The clustering around the latter point has
now a great likeness with the normal or Gaussian
curve, and we might for this reason call this
specific age the normal life age. For the cal-
culation of such a normal age the argument may
be put forth that experience shows that the great
variations in mortality tend to disappear in old
age. Let the rate of mortality in a certain gene-
ration at age .r be \x x and the number of the cor-
responding survivors be l x . The quantity \x x l x will
120 Human Death Curves.
then increase from a certain point, while l x de-
creases, in the beginning slowly, but later on at a
more rapid pace. "During a long period of life the
quantity \i x l x — the number of deaths at a certain
age — -will increase with age. Later on a reversed
motion takes place. But when this reversion will
occur depends on many conditions, the successful
fight against certain diseases, progress in econo-
mic conditions, or change in the mode of living.
All this exercises an important influence, and the
maximum point occurs therefore sometimes sooner
and sometimes later. It is also important to in-
vestigate the natural selection in old age, which
so to say divides the population in different strata,
each with its own state of health. The healthiest
of such groups will with the increase in age play
a greater role. Here as everywhere it is the more
important problem to study the clustering around
the mean inside the special groups rather than to
attempt to find a derived expression for the morta-
lity. On the other hand, the correspondence be-
tween the normal curve as established by Lexis
is another testimony to the fact that this curve
or formula very often can be applied, even in
complicated expressions".
Compound Curves. 121
s. the "death Lexis was satisfied to deter-
CVRVE" AS A ,, , a
compound mine the normal age. A more
ambitious attempt to investi-
gate the mortality by means of frequency curves
throughout the whole period of life was made by
the eminent English biometrician, Pearson, in a
brilliant essay in his "Chances of Death". Pear-
son took the number of deaths in the English
Life Table No. 4 (males) and succeeded in break-
ing up the compound curve into five component
curves typical of old age, middle age, youth, child-
hood and infancy. I want to advise my readers to
study this brilliant and illuminating essay, especi-
ally on account of its beautiful form of exposition
which makes the whole subject appear in a most
interesting light.
Speaking of this attempt by Pearson, the
American actuary, Henderson, is of the opinion
that „the method has not, however, been applied
to other tables and it is difficult to lay a firm
foundation for it, because no analysis of the deaths
into natural divisions by causes or otherwise has
yet been made such that the totals in the various
groups would conform to these (the Pearson)
frequency curves". We shall later on come back
to this statement by Henderson, which we feel
is a partial truth only. On the other hand, it must
be admitted that the system of Pearson's types of
122 Human Death Curves.
skew frequency curves (by this time twelve in
number) are by no means easy to handle in
practical work and often require a large amount
of arithmetical calculation. Moreover, there seems
to be no rigorous philosophical foundation for the
Pearsonian types of curves, and they can at their
best only be said to be exceedingly powerful and
neat instruments of graduation or interpolation.
On the other hand, I am of the opinion that
the goal can be reached more easily if we, instead
of the Pearsonian curve types, make use of the
Laplacean-Charlier andPoisson-Charlier frequency
curves, which are expressed in infinite series of
the form :
F(x) = q ,(ar) + p 8 q,in( a: ) + p 4 q,iv( a . )+ ..: ( 2 )
or2f(s)=iKaj) + Y I A»iMs) + Y,A»iMa!) + ....(3)
These two curve types have been treated
elsewhere by Gram, Charlier, Thiele, Bdgeworth
J0rgensen, Guldberg and other investigators, and
it is therefore not necessary to dwell further upon
their analytical properties, which were discussed
in Chapter I.
Eeturning now to the general form of our d x
curve of the mortality table which we discussed
above, it is readily seen that this curve has all the
properties of a compound frequency curve, that
Compound Curves. 123
is, a curve which is composed of several minor or
subsidiary frequency curves, generally skew in
appearance. As proven both by Charlier and by
J0rgensen, any single valued and positive comp-
ound frequency curve vanishing at both -\- oo and
— cc can be represented as the sum of Laplacean-
Charlier and Poisson-Charlier frequency curves.
We know thus a priori that the d x curve is comp-
ounded of the two types of frequency curves. But
how are we to determine the separate component
curves? It is readily admitted that no a priori
reason will guide us here. The purely empirical
observer might therefore abandon the project
right here, because to all appearances it would
seem hopeless to attempt a solution by purely
empirical means. The positive rationalist does
not despair so easily. "Very well", he says, "if
we can not make further progress by purely
empirical means, we are at least permitted to try
deductive reasoning and attempt to bridge the gap
by means of an hypothesis". The hypothesis I
shall adopt is the following :
The frequency distribution of deaths ac-
cording to age from certain groups of causes
of death among the survivors in a mortality
table tend to cluster around certain ages in
such a manner that the frequency distribution
can be represented by either a Laplacean-
124 Human Death Curves.
Charlier or a Poisson- Charlier frequency
curve.
A study of mortuary records by age and cause
of death immediately supports this hypothesis.
We notice, for instance, that diseases such as
scarlet fever, .measles, whooping cough and diphr
theria often cause death among children, but
rarely seem to affect older people. We know, for
instance, that there is a much greater probability
that a 5-year old boy will die from scarlet fever
than a man at the age of 40 wiill die from the
same disease. On the other hand, there is quite
a large probability that an old man at age 85
will die from diseases of the prostate gland, while
such an occurrance is almost unheard of among
boys. Similarly deaths from cancer and Bright's
disease are very rare in youth, but quite frequent
in early old age. Tuberculosis, on the other hand,
causes its greatest ravages in middle life, and has
but little effect upon older ages.
6. mathematical Leaving, however, the ques-
PROPERTIES OF ° ^
nIntfreq P uen- tl0n 0f the 8 T0U P in g of causes
cy curves of death into a limited num-
ber of typical groups to a later discussion, we shall
in the meantime see how the hypothesis can carry
us over the difficulties. Let us for the moment
Mathematical Properties. 125
assume that we are able to group the causes of
death into say 7 or 8 groups. We shall also as-
sume that we know the percentage frequency
distribution of deaths according to age in each
of the groups. This means in other words that
we know the equation of the frequency curves
giving the percentage distribution. Let the ana-
lytical expression for these frequency curves be
denoted by the symbols :
Fj(x), F a {x), F m {x), ..., .Fviii(z). (4)
Again, let the total number of deaths among the
survivors in the mortality table from causes of
death according to the above grouping be denoted
by
N u Nu, Niu, Nix, . . ., Nviu respectively. (5)
The number of deaths in a certain age interval,
say between 50-54 can then be expressed as
follows :
x = bi
^d x =^N Fi (x) +^N U F n {z)-\-..
X = 50 60
54
+ y,^
vmFy\ii{x).
(6)
In this relation the only known quantities are
the equations for the frequency curves Fi{x),
126 Human Death Carves.
Fa(x), . . ., Fvm(x, of the percentage frequency
distribution according to age in each of the eight
groups. Neither d x nor any of the various N's are
known. The only relation we know a priori among
the quantities N is the following :
JV, + N u + N m + ■ ■ • JVvm = 1 ,000,000. (7)
The latter equation is simply a mathematical
expression for the simple fact that the sum total
of the sub-totals of the various groups of causes
of death, in other words the deaths from all
causes among the survivors in the mortality table,
must equal the radix of the entrants of our orig-
inal cohort of 1,000,000 lives at age 10. Viewed
strictly from the standpoint of frequency curves,
we might express the same fact by saying that
the sum of the areas of the various component
curves must equal 1,000,000.
It is readily seen that on the assumption that
the expressions of the different F(x) conform to
the above hypothesis it is possible to find d for
any age or age interval if we can determine the
values of the different N's. It is in this possibility
that the importance of the proposed method lies,
and we shall now show how it is possible to deter-
mine the N's without knowing the exposed to
risk.
Observation Equations. 127
r. observation Consider for the moment the
EQUATIONS
following expression :
50 III
JTiVm Fm (a)
£Ni Ft (x) +£Fn (x) Kn +
50 51)
54 54
-^Vin Fm (x) + . . . +^^111 jPViii (a)
(8)
What does this equation represent? Simply the
proportionate ratio of deaths in group III to the
total number of deaths in all type groups (in
other Words the deaths from all causes) in the age
interval 50-54. Such ratios are usually known as
proportional death ratios. It is readily seen that
these proportionate death ratios are dependent on
the deaths alone and absolutely independent of
the number exposed to risk, provided tne total
number of deaths from all causes in a certain age
group is large enough to eliminate variations due
to random sampling. 1 In other words, we can find
1 Strictly speaking this statement is only true for an
age interval of one year or less and may in the case of
large perturbing influences in the population exposed to-
risk be subject to appreciable errors when we use large
age intervals of 10 or more in our grouping for the com-
puting of R{x). When the age interval for the grouping
of causes of deaths by attained ages is 5 years or less
the error committed in assuming R(x) as being indepen-
128 Human Death Curves.
a numerical value for the term JB rir (x) on the left
side of the equation from our death records alone
without reference to the exposed to risk in this
interval. Similar proportionate death ratios can
of course without difficulty be determined for the
other groups of causes of death and for arbitrary
ages or age intervals. In this manner we can
determine a system of observation equations with
known numerical values of .R. (#)(& = I, II, III, . . .)
The fact that the number of observation equations
in this system is much larger than the number of
the unknown N's makes it possible to determine
these unknowns by the method of least squares.
Probably the simplest manner is first to deter-
mine by simple approximation methods, or by
mere inspection, approximate values for the
various N's and then make final adjustments by
the method of least squares.
Let, for instance,
'JVi, 'N n , 'N }
nil
dent of the number exposed to risk is in most cases
negligible. One of the difficulties encountered in the
construction of a mortality table for Massachusetts Males
was that the age interval used for the grouping was 10
years instead of 5 years or less. See in this connection
the remarks at the beginning of paragraph 11 and at
the conclusion of paragraph 16 of the present chapter.
Observation Equations. 129
be the first approximations of the areas of the
various groups of frequency curves so that
#, = <#!, N n =a,'N n ,
-tfvm = a 8 'JVvm.
-}
(9)
Let us furthermore introduce the following
symbols :
1
(10)
'JVi Fj (x) = <&! (x) , 'N n F a (x) = <D 2 (fc) ,
'N Y inF vm {x) = <£> a (x).
The different values of
®i(«). ® 2 (*). *s(*). ••-, ^ 8 (*)
may then be regarded as a system of component
frequency curves to which we now must apply the
different correction factors c^, a 2 , a 3 , . . . , a 8 in order
to fit the curves to the observed proportional death
ratios, R(x), for the various groups of typical
causes of death. Let us for example assume that
the observed death ratio of a certain age (or age
group), x, under a certain group of causes of
death, say group No. Ill, is Rm(x). We have
then the following observation equation :
B m (x) = a s ® 3 (x): [a^W+a^W-U
+ a t <J> 4 (z)+. • .+a a ® 8 (x) + a 2 <P 2 (x)} } (U)
130 Human Death Curves.
Since the sum of the areas of the different comp-
onent curves necessarily must equal 1,000,000 it
is easy to see that we may write the factor a 2
in the last term of the denominator in the follow-
ing form :
a, Y O 2 0) = 1,000,000
or
1,000,000- [a^X^+ag^ 7 <D 3 (z)-
... + a 8 ^ '<D 8 (x)]) : JT<D, (x) =
= h — [h « x + /j a ;i + . . . + /> 8 a 8 J
where
1 ,000,000 _ Z p! (x)
~ ! I$ 2 (l) ' J ~~ I$ s (l)'
1 " s$,(i)' '•■' 8 KD 2 (i)'
(12)
The expression for i?m (a;) can then be put in the
following form :
Bm {x) = a 3 O s (a;) : [c^ ^ (x) + a ;} <J> 3 (x) + '
+ a i * i (x) + ....+a 8 <l> s (x)+ /(IS)
+ (*t> — *! a x — . . . — /.•„ a 8 ) <D 2 (a;)] . .
Classification of Deaths. 131
Similar observation equations for the other
groups are derived without difficulty.
Once having formed the observation equations
it is simply a matter of routine work to compute
the normal equations from which the values- of
the unknown N's can be found. We shall, how-
ever, not go into detail with the derivation of the
necessary formulas, since this is a process which
belongs wholly to the domain of the theory of
least squares and which has received adequate
treatment elsewhere. (See for instance Brunt's
Combination of Observations.)
s. classifica- We think it more advantage-
TI °oF°DEATi ES ous to illustrate the method by
a concrete example. As an
illustration we may take the case of Michi-
gan Males in the period 1909—1915. The
mortuary records of Males in Michigan are
for that period given in the reports issued
annually by the Secretary of State on "Begistrat-
ion of Births and Deaths, Marriages and Divorces
in Michigan". The deaths by sex, age and cause
of death are given in quinquennial age groups. A
very serious drawback is the grouping of all ages
above 80 into a single age group instead of in at
least 4 or 5 quinquennial age groups. This makes
it impossible to obtain good observation equations
9*
132 Human Death Curves.
for ages above 80. When we consider that about
one fifth of the original entrants at age 10 in the
mortality table die after the age of 80, it is readily
seen that this defect in the Michigan data is of a
very serious character, which makes it out of the
question to determine correctly the areas of the
curves for middle old age and extreme old age.
For ages below 70 these curves do not play so
important a role, and the method ought therefore
in these ages yield satisfactory results. We now
make the assertion that the deaths among the
survivors in the final life table can be grouped in
the following typical groups.
Causes of Death typical of : —
Group I Extreme Old Age.
II Middle Old Age.
— Ill Early Old Age.
— IV Middle Life.
V Early Middle Life.
— VI Pulmonary Tuberculosis, Etc.
— Vila Early Life Occupational Hazard.
— Vllb Middle Life Occupational Hazard.
— Villa Childhood.
The classification of causes of death according
to this scheme is given in the following table, mar-
ked Table A.
Classification of Deaths. 133
Table A. Michigan Males 1909—1915
Classification of causes of death according to the
chosen system of curves.
No. in Inter-
national Class
fication.
81.
i_ GROUP I
Diseases of the arteries.
124.
Diseases of the bladder.
125—133.
Other diseases of the genito-urinary
142.
154.
126.
system.
Gangrene.
Old age.
Diseases of the prostate.
GROUP II
10.
Influenza.
47—48.
Rheumatism.
64.
65.
66.
79.
Apoplexy.
Softening of the brain.
Paralysis.
Heart disease.
82.
Embolism.
89.
Acute bronchitis.
90.
Chronic bronchitis.
91.
94.
Broncho-pneumonia .
Congestion of the lungs.
96—97.
Asthma and emphysema.
103.
Other diseases of the stomach.
134 Human Death Curves.
No. in Inter-
national Classi-
fication.
105. Diarrhea and enteritis, (over 2 years)
14. Dysentery.
GROUP III
39. Cancer of the mouth.
40. Cancer of the stomach and liver.
41. Cancer of the intestines.
44. Cancer of the skin.
45. Cancer af other organs.
46. Tumors.
50. Diabetes.
53 — 54. Leukemia and anemia.
63. Other diseases of the spinal cord.
68. Other forms of mental diseases.
80. Angina pectoris.
109 — 110. Hernia, intestinal obstruction, and
other diseases of the intestines.
120. Bright's disease.
121. Other diseases of the kidneys
123. Calculi of urinary passages.
GROUP IV
56. Alcoholism.
18. Erysipelas.
62. Locomotor ataxia.
73 — 76. Other diseases of the nervous system,
77. Pericarditis.
Classification of Deaths. 135
No. in Inter-
national Class
fication.
i-
78.
Endocarditis.
83.
Diseases of the veins.
84.
Diseases of the lymphatics.
85
—86.
Other diseases of the circulatory sy-
stem.
87.
Diseases of the larynx.
88.
Diseases of the thyroid body.
92.
Pneumonia.
93.
Pleurisy.
95.
Gangrene of the lungs.
98.
Other diseases of the respiratory sy-
stem.
99-
-101.
Diseases of the mouth, pharynx, and
oesophagus.
111.
Acute yellow atrophy of the liver.
113.
Cirrhosis of the liver.
114.
Biliary calculi.
115-
-116.
Diseases of the liver and spleen.
118.
Other diseases of the digestive system.
143-
-145.
Furuncle, abscess, and other diseases
of the skin.
147-
-149.
Diseases of the joints, and locomotor
system.
GROUP V
4.
Malarial fever.
13.
Cholera nostras.
136 Human Death Curves.
No. in Inter-
national Classi-
fication.
20.
Septicemia.
24.
Tetanus.
32.
Pott's disease.
33.
White swellings.
34.
Tuberculosis of other organs.
35.
Disseminated tuberculosis.
55.
Other general diseases.
60.
Encephalitis.
70—71.
Convulsions.
102.
Ulcer of the stomach.
117.
Peritonitis.
119.
Acute Nephritis.
164.
Diseases of the bones.
155.
Suicide by poison.
156.
Suicide by asphyxia.
157.
Suicide by hanging.
158.
Suicide by drowning.
159.
Suicide by firearms.
160.
Suicide by cutting instruments.
161.
Suicide by jumping from hight places
163.
Suicide by other or unspecified means
164—165.
Accidental poisonings.
166.
Conflagration.
167.
Burns (conflagration excepted).
168.
Inhalation of noxious gases.
172.
Traumatism by fall.
Classification of Deaths. 137
No. in Inter-
national Classi-
fication.
175 — (2). Traumatism by electric railway.
175 — (3). Traumatism by automobiles.
175 — (4). Traumatism by other vehicles.
176. Traumatism by animals.
178. Cold and freezing.
179. Effects of heat.
185. Fractures and dislocations (cause not
specified.
GROUP VI
28. Tuberculosis of the lungs.
29. Miliary tuberculosis.
37 — 38. Venereal diseases.
186. Other accidental traumatism.
57 — 59. Chronic poisoning.
67. General paralysis of the insane.
31. Abdominal tuberculosis.
GROUP VII
1. Typhoid fever.
69. Epilepsy.
108. Appendicitis.
182. Homicide.
169. Accidental drowning.
170. Traumatism by firearms.
171. Traumatism by cutting instruments.
138 Human Death Curves.
No. iD Inter-
national Classi-
fication.
173. Traumatism by mines and quarries.
174. Traumatism by machinery.
175 — (1). Traumatism by railroads.
180. Ligthning.
61. Meningitis.
GROUP VIII
5. Smallpox.
6. Measles.
7. Scarlet fever.
8. Whooping cough.
9. Diphtheria and croup.
30. Tubercular meningitis.
150. Congenital malformations.
9. outline of com- ^ e numDer of deaths in the
put in g scheme various groups according to the
above classification and ar-
ranged according to age during the period 1909 —
1915 is given in the table B on page 140.
From that table it is a simple matter to com-
pute the proportionate death ratios of the separate
groups of causes of death. Such a computation is
shown in table C on page 141.
It is readily seen that these death ratios are
independent of the number exposed to risk. More-
Computing Scheme. 139
over, the number of observations seem to be suffi-
ciently large to eliminate serious variations due
to random sampling. This might perhaps not hold
true for the age intervals 10 to 14 and 15 to 19
where not alone random sampling is present, but
a somewhat modified classification seems neces-
sary. I have, however, not used the observed pro-
portionate death ratios for the two younger age
intervals in my computations which only took into
account the ratios above 20. For this reason I do
not deem it necessary to go into a closer investiga-
tion of a re-classification of causes of death for
these younger age groups. A more serious defect
which cannot be overcome is presented in the
ages above 80 where, as mentioned before, a clas-
sification according to age is absent in the original
records for the state of Michigan. The fact that
the highest number of deaths (12,473) occurred
in ages above 80 makes this defect more serious
than the omission of a re-classification of causes
of death below 20.
So far we have only been concerned with the
first step in the complete induction according to
the model of Jevons, namely that of simple observ-
ation. The next step in the induction is the hypoth-
esis. We present now the following working
hypothesis.
The frequency distribution of deaths according
140 Human Death Curves.
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142 Human Death Curves.
to age of the above groups of causes of death
among the survivors of an original cohort of
1,000,000 entrants at age 10 can be represented by
a system of frequency curves determined by the
following characteristic parameters:
Parameters
Group
Mean
Dispersion
Skewness
Excess
I
79.6 years
9.5730 years
+ .1066
+ .0546
II
70.5 -
12.8000 -
+ .0967
+ .0126
III
65.5 -
13.6870 -
+ .1248
+ .0650
IV
59.5 -
17.0890 -
+ .1790
- .0106
V
65.5 -
19.9411 -
+ .0555
- .0367
VI
44.5 -
16.0352 -
- .0124
- .0272
Vllb
57.5 -
12.1552 -
+ .0008
- .0005
Vila
Poisson-Charlier Curve: Modulus = 28.5 years,
Eccentricity =
1.0001
Villa
Poisson-Charlier Curve: Modulus
= 13.5 ye;
irs.
From these parameters and from well-known
tables of the probability or normal frequency curve
and its various derivatives it is easy to determine
the frequency distribution for any desired interval.
For this system of frequency curves we now
shall try to find the various areas of N v iV n ,
iV In , , N YUI so as to conform to
the observed values of R x in Table C. As a first
approach to the final values of N , we may by an
inspection (which of course is improved upon by
Computing Scheme. 143
a long practice in curve fitting) choose the follow-
ing approximations. 1
Group Approximate Value of 'N.
I
123000
II
366000
III
183000
IV
105000
V
75000
VI
70000
Vila & Vllb
61000
VIII
17000
1000000
These preliminary numerical values represent
the first approximations of the areas of the various
frequency curves. The sequence represented by
'NjFJz), 'N n F n (x),'N m F m (x),. ■ -'N^F^x^U)
gives the number of deaths at age x. We notice
thus that by multiplying the various equations of
frequency curves for arbitrary age intervals with
1 These numbers represent as a matter of fact a first
rough approximation of the areas of the different com-
ponent curves by means of the method of point contours.
Hence it is to be expected that the final adjustments
will be comparatively small. This fact has, however, no
influence upon the application of the method.
144
Human Death Curves.
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Computing Scheme. 145
their respective 'ATs we can get a first approxima-
tion of the final death curve. I give on page 144 an
approximate table arranged in 5 year intervals.
We might now first compute the various factors
k n , k„ /c„ which will be common for all
observation equations. We have, referring to the
above formulas (llandl2) for the various k's (15).
_ 1000000 _ 123089 . = 183045 )
°~ 365995 ' 1_ 365995' 3__ 365995'
_ 104888 75030 69996 .
365995 5 365995 " 365995
61003 17002
(15)
365995 8 365995
Or
& = 2,732, ^ = 0,336, &3 = 0,500,& 4 =0,287,
k b = 0,205, ft, = 0,191, k 7 = 0,167, k 8 = 0,046.
To illustrate the further process of the compu-
tation of the observation equations, let us take a
certain age interval, say the interval between
50-54. The value of <1> 2 taken from the above table
is 163.39. The value of R m (x) for this interval is
0.234 (see table page 141) . Hence we have the
following observation equation (16).
10
146 Human Death Curves.
0.234 = 104.53a 3 : [15.76(^ + 104.530,+
84.16a 4 + 64.52a g + 73.55a 6 + 35.01 a ? +
0.00a 8 + (2.732 — 0.336a 1 — 0.500a 3 — (16)
0.287 a 4 — 0.205a 6 — 0.191a 6 — 0,167 a g -
— 0.046 a g ) 163.39]-
After a few simple reductions this may be
brought to the following form :
9.16cl + 99.19a, — 8.72a, — 7.26a. — )
13 4 5 (17)
9.91 a 6 — 1-81 a, + 1.76 a g — 104.45 = 0. j
In the routine work I usually use a system of
computing the various equations which is out-
lined in detail in the accompanying tabular scheme
referring to all the groups in the age interval
50-54 and shown on pages 148-154.
Similar observation equations are arrived at in
exactly the same manner for other groups and
other age intervals. For the whole interval from
age 20 and upwards we get, in this way 96 obser-
vation equations from which to determine the cor-
rection factors. The coefficients of theae obser-
vational equations are then written down, and
Computing Scheme. 147
their various products formed in turn. We deem
it not necessary to give all these observational
equations and their coefficients for all the 96
observations, but shall limit ourselves to give all
the necessary computations for the interval from
50-54 as previously considered. With the usual
system of notation employed in the method of
least squares we get the scheme on pages 148-154.
Normal Equations, Michigan Males 1909 — 1915.
723763 400750 218930 150776 135184 115318 30325 1801152
877847 253187 176242 149858 129697 34600 2053941
237159 90440 72317 62110 16246 964843
105346 47022 39939 10576 628608
76774 28909 8668 525295
53378 7012 437390
2391 111625
The addition of the various columns of the sum
products of the coefficients gives us finally the
above set of normal equations of which we only
submit the coefficients in the usual scheme em-
ployed in the method of least squares.
Solving the above system of normal equations
by means of the well-known method devised by
Gauss, we obtain finally the values on page 154 for
the various a's by which the approximate values
'N must be multiplied in order to yield the prob-
able values- of N.
10*
148 Human Death Curves.
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Human Death Curves.
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Human Death Curves.
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154 Human Death Curves.
gg gh S s
50-54
50-54
0.0
0.8
0.5
3.2
188.1
39.6
1.4
88.3
6.1
0.8
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0.3
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1.5
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1740.4
69.7
n: 2391.0
- 111625.0
-1807.0
hh
hs
— — — — —
— — —
72.3
45.9
10920.3
2299.0
5416.9
375.4
2819.6
15.9
2631.7
584.8
979.7
93.9
103877.3
4157.7
Sum: 6630212.0 107358.0
Correction Factors, a.
Group I 1.03284
II 1.00017
— Ill 1.03635
IV 1.03731
V 1.00956
VI .0.97334
— Vila 0.90332
— Vllb 0.60565
— VIII 1.13743
Goodness of Fit. 155
Applying the above correction factors to the
respective values of 'IV, we get finally as the total
areas of the respective component curves :
Group
I
127,131
II
366,059
III
189,699
IV
108,750
V
75,747
VI
68,130
Vila
33,032
Vllb
12,133
VIII
19,339
1,000,000
Multiplying the equations of the various frequency
curves, F(x), of the percentage distribution in
each group with the above values of N we ob-
tain finally the complete mortality table as will
be given in the Appendix. The final graphical
representation of the frequency curves is shown
in Figure 2.
io. goodness of This completes the third step
FIT in the inductive process. The
fourth and final step is the
verification of the results thus arrived at by a mere
deductive process. Here it must be remembered
156
Human Death Curves.
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Goodness of Fit. 157
that the condition which the final component fre-
quency curves shall fulfill is the one that observed
proportionate death ratios shall agree as closely
as possible with the expected or theoretical pro-
portionate death ratios as computed from the final
table. In this connection it must be borne in
mind that the observed proportionate death ratios
are given in quinquennial age groups. Thus the
observed proportionate death ratios in a certain
age interval, as for example between 50 — 54 are
really the average or "central' ' proportionate death
ratios at age 52. From the complete table it is,
however, possible to compute the proportionate
death ratios for each specific age. Graphically the
expected proportionate death ratios will therefore
represent a continuous curve, while the observed
ratios will be represented by a rectangular shaped
column diagram. Such a graphical representation
is shown in Pig. 3 which simply represents the
figures in Table C and Table E in graphical form.
The "goodness of fit" of the "expected" or theore-
tical values to the ''actual" or observed values is
seen to be very close, especially in the largest and
most important groups. It is only in the combined
groups Vila and Vllb that the "fit" might prob-
ably be open to criticism for higher ages, but even
here the deviation is small between the actual and
theoretical values. A very small increase in the
158
Human Death Curves.
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Goodness of Fit. 159
area of the Vllb curve would easily adjust this
difference. It is, however, doubtful if such a cor-
rection or adjustment would have any noteworthy
effect upon the ultimate mortality rates q x , and I
do not consider it worth while to go to the addi-
tional trouble of recomputing the areas, especially
in view of the fact that the observation data above
the age of 80 are not exact and detailed enough to
be used in this method of curve fitting. For ages
up to 70 or 75 I consider, however, the table as
thus constructed as sufficiently accurate for all
practical purposes.
u Massachusetts ^ s an °th er example of the me-
1914^917 "hod I take the construction
of a mortality table for the
State of Massachusetts from the mortuary records
for the three years 1914, 1915 and 1916. The
records as given by the Registration reports are
better than the records for Michigan, in as much
as they have avoided the deplorable practice of
grouping all deaths above the age of 80 into a
single age group. On the other hand, the classifi-
cations of cause of death in Massachusetts by at-
tained age are given in ten year age groups only.
Hence it is readily seen that we will only be able
to secure half as many observation equations as
in the case of the five year interval in Michigan.
160
Human Death Curves.
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severe test. In spite of this drawback I shall for
Massachusetts Males. 161
the benefit of the readers briefly outline the results
I have obtained from an analysis of the Massachu-
setts data.
While for the Michigan data I employed a sy-
stem of frequency curves previously used with
success for certain Scandinavian data, I found it
was easier to fit the Massachusetts data to a sy-
stem of frequency curves used in the construction
of a mortality table for England and Wales for
the years 1911 and 1912 from the mortuary records
of deaths by age and cause among male lives. The
classification by age of the causes of death in 8
groups is also different from that of Michigan,
especially for middle life and younger ages. The
parameters of the system of component frequency
curves to which I fitted the Massachusetts data are
shown in the following table F :
Table F.
Parameters of the System of Frequency Curves
for Massachusetts Males 1914—1916.
Group
Mean
Dispersion
Skewness Excess
I
78.70 years
7,9775 years
+ .0920 + .0331
II
68.00 -
12,2051 -
+ .1151 + .0234
III
63.05 -
13,0532 -
+ .1210 + .0471
IV
60.45 -
17,8552 -
+ .0983 - .0091
V
49.60 -
18,6100 -
+ .0328 - .0309
VI
43.80 -
14,6750 -
- .0091 - .0272
Vllb
57.40 -
12,1550 -
+ .0021 - .0026
Vila and Villa constructed from Poisson-Charlier Curves.
11
162 Human Death Curves.
The observed number of deaths according to the
8 groups of causes of death, and their correspond-
ing proportionate death ratios are given in the fol-
lowing tables G and H.
By finding first approximate values and then by
a further correction of these approximation areas
by means of the factors a. determined by the
method of least squares in exactly the same man-
ner as demonstrated in the case of Michigan, we
finally arrive at the following areas of the various
groups.
Areas of the component fre
quency curves in the
Life Table for Massachusetts Males t 1914 — 1916.
Areas
Group I
90064
— II
281470
— Ill
207854
— IV
151316
— V
99543
— VI
107718
Vila & Vllb
40719
— Villa
21316
1000000
Forming the products N F (x) for the various
groups and integral ages we obtain finally the
life table as shown in the appendix. In order
Massachusetts Males. 163
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164
Humaxi Death Curves.
Massachusetts Males. 165
to test the "goodness of fit" of the curves it is
necessary to compute the expected or theoretical
proportional death ratios from this latter table and
compare such ratios with the observed or actual
proportionate death ratios as shown in Table H.
The theoretical values are shown in Table I, and
a graphical representation illustrating the "good-
ness of fit" between the observed and theoretical
ratios is given in Fig. 5. I think it will be generally
admitted that the fit is satisfactory for all practical
purposes.
The State of Massachusetts has always been the
foremost state in the union for reliable and trust-
worthy statistical records, and in all probability it
would be possible to secure the deaths by causes in
5-year age groups instead of ten-year groups. By
taking the above table as a first approximation one
should then obtain a very accurate table. On the
other hand, it is possible to verify the final results
in the above Life Table for Massachusetts by an
entirely different process. It happens that the
State of Massachusetts took a census in April 1915.
This census for living males by attained ages could
then be used as an approximation for the exposed
to risk, while the deaths for the three years could
be used as a basis for the number of deaths in a
single year. A Life Table could then be con-
structed by means of the orthodox methods usually
166
Human Death Curves.
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168 Human Death Curves.
employed by actuaries and statisticians in the con-
struction of mortality tables from census returns.
12 ' coMmfvB en. As a third illustration, I shall
TABLiFitnt—ir cons truct a table for American
other tables Locomotive Engineers for the
period 1913—1917. The statistical data forming
the basic table are the mortuary records by at-
tained age and cause of death among the members
of The Locomotive Engineers' Life and Accident
Insurance Association, a large fraternal order of
the American Locomotive Engineers. The total
number of deaths in the five year period amounted
to more than 4,000. Distributed into separate
groups of causes of death, it was found that it
was possible to use a system of frequency curves
similar to that employed in the State of Massachu-
setts, except for Group No. IV, for which it was
found exceedingly difficult to find a single curve
which would fit the data, and much points towards
the actual presence of a compound curve of that
group of causes of death among the Locomotive
Engineers. The grouping of causes of death is, also
slightly, different from that of Michigan and Mas-
sachusetts. I shall not go into further details as
to the actual construction of this table, except to
mention the areas of the various component fre-
Locomotive Engineers. 169
y/i
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ill !
V
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170 Humaii Death Curves.
quency curves of which I present the following
table.
Areas
Group I 44,857
— II 342,645
— Ill 226,022
— IV 147,420
V 47,650
— VI 31,260
— Vila 79,005
— Vllb 77,713
— VIII 3,428
1,000,000
It must also be remembered that the radix of
this table is taken at age 20, instead of at age 10
as is the case in the preceding tables. The final
graph is shown on the preceding page. A num-
ber of diagrams illustrating the "goodness of
fit" are also attached and need no further com-
ment. It might, however, be of interest to men-
tion the fact that the American actuary, Moir,
has recently constructed a mortality table for
American Locomotive Engineers along the ortho-
dox lines from the data contained in the Medico-
Actuarial Mortality investigation. Moir's table --
or at least the great bulk of the material from
Locomotive Engineers.
171
which it was derived — falls in the interval be-
tween 1900 and 1913. Owing to the energetic
'safety first" movement which since 1912 has been
actively pursued by most of the leading American
joia
.016
.014
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.006
MH
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Fig. 7.
railroads, it is, however, to be expected that the
period 1913 — 1917 indicates a reduced mortality as
compared with that of Moir's period. This fact
is also shown in the diagrams in Fig. 7. 1 On the
other hand, the almost parallel movements of
Moir's table with that of the table of the fre-
quency curve method of 1913 — 1917, seems to
indicate the soundness of the proposed method.
1 Curves I, II and V are Locomotive Engineers' Mor-
tality Tables for various periods.
172 Human Death Curves.
„x,„T~T„„r, T A similar table showing mor-
12 a. ADDITIONAL °
mortality tality conditions among a de-
TABLES
cidedly industrial or occupational
group has been constructed for coal miners in the
United States. The original data of the deaths by
ages and specific causes were obtained from the
records of several fraternal orders and a large indus-
trial life assurance company and comprised nearly
1600 deaths. The number of deaths above the age of
sixty were, however, too few in number to determine
with any degree of exactitude the area of component
curves for the older age groups. For ages below
sixty-five the table should on the other hand give a
true representation of the mortality among coal
miners in American collieries during the period under
consideration 1 ). A particular feature of this table is
the comparatively low mortality in group VI, which
contains primarily deaths from tuberculosis. Coal
miners present in this respect different conditions
than those usually prevailing in dusty trades where
the death rate from tuberculosis is unusually high.
The same feature is also borne out in previous in-
vestigations on the death rate of coal miners in Eng-
1 It was not possible to seperate anthracite and bituminous coal miners.
The data indicate, that anthracite mine workers have a higher accident
rate than workers in bituminous mines.
Coal Miners.
173
jff, \
*_/,•
/ ;i
/
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-'[
ii
/
J\/.
i
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174 Human Death Curves.
land, and by the recent investigations by Mr. F. L.
Hoffman on dusty trades in America.
In order to have a measure of the mortality pre-
vailing among industrial workers in America, we
submit a table derived from a very detailed collection
of mortuary records by age, sex and cause of death
as published by the Metropolitan Life Insurance Com-
pany of New York. A deplorable defect in this splen-
did collection of data is the grouping together of all
ages above seventy in a single age group, which
makes it almost impossible to determine the com-
ponent curves for higher ages with any degree of
trustworthiness.
The defect in the original Metropolitan data for
older age groups made it neccessary to modify the
earlier sets or families of curves which were used
on the Michigan and Massachusetts data and to
combine several of the subsidiary component curves,
especially those for the older age groups. Such
modifications were, however, easily performed by
means of simple logarithmic transformations.
I give below my grouping scheme for the Metro-
politan data designated by the code numbers of the
international list of causes of death. The actual
cause of death corresponding to each code number
is found under paragraph 8 of the present chapter.
"Metropolitan" Life Table. 175
GROUP I
10, 39 to 46, 48, 50, 54, 63 b, 64 to 66, 68, 79, 81,
82, 89 to 91, 94, 96, 97, 103, 105, 109 a, 120, 123, 124,
126, 127, 142, 154.
GROUP II
4, 13, 14, 18, 26, 27,, 32 to 35, 47 (over age 20), 49,
51 to 53, 55, 60, 62, 70 to 72, 77, 78, 80, 83 to 88, 92,
95, 98 to 102, 106, 107, 109 b, 110 to 119, 122, 125, 143
to 145, 148, 149, 155 to 163.
GROUP III
28, 29, 31, 37, 38, 56 to 59, 67.
GROUP IV a AND IV b
1, 5 to 9, 17, 19, 20 to 25, 30, 61, 63 a, 73 to 76, 108,
146, 147, 150, 164 to 186, 47 (under age 20).
It will be noted that under this scheme Group I
includes practically Groups I to III of the Michigan
classification, Group II corresponds partly to IV and
V for Michigan, Group III is practically Michigan's
Group VI, while Group IV a and IV b takes in partly
V, VII, and VIII in the Michigan experience. As a
further correction I found it also advisable to transfer
some of the deaths in the age intervals 10 — 14, 15 — 19,
20—24, and 25—29 in Groups I and II to Group IV a
so as to avoid the long left tail ends in these older
age curves.
176
Human Death Curves.
After grouping the deaths (more than 200,000) of
the Metropolitan experience according to the above
scheme, it is a simple matter to compute the various
PER
CENT
~1\
80
' "k GROUP ^k
GROUP
"Si 1 •
U^-.
K
\
\ 1
\
60
\|
. \
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X-
GROUP
III.
'
\
40
'>
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\]
V
20
GROUP \|
IVA ANO IV37I
\ x
\_
"^■OW*
^^^t}--=
20 30 40 SO 50 A3ES .
Fig. 9.
values of R(x) of the four groups for quinquennial
age intervals and use these values (altogether 52 in
number) for finding the observation equations and in
the subsequent determination of the component curves
as shown in the final mortality table in the appendix
Japanese Life Tahle. 177
to this chapter. A comparison between the observed
values of R(x) by quinquennial ages and the con-
tinuous values of R(x) (indicated by dotted curves)
as computed from the final mortality table is shown
in Fig. 9. The "fit" between calculated and observed
values is evidently satisfactory.
A most instructive and unique experience is of-
fered in the table of Japanese Assured Males for the
four year period 1914-1917 and based upon the death
records of more than a dozen of the leading Japanese
Life Assurance Companies. About 35,000 deaths by
cause and arranged in quinquennial age groups were
available for this construction. The component curves
for the older age groups were determined by a simple
logarithmic transformation of the variates and offered
no particular obstacles in the a priori determination
of the parameters. The curves for middle and younger
life were more difficult to handle, especially the
curves typical of tuberculosis, spinal meningitis and
the peculiar Oriental disease known as Kakke, aris-
ing from an excessive rice diet. A first attempt to
use the same curve types as employed in some of the
European and American data did result in a very
poor fit between the observed and calculated values
of R(x) for the younger age intervals clearly indica-
ting that the clustering tendencies were different in
the case of the Japanese data than in the other experi-
ences I had previously dealt with.
12
178 Human Death Curves.
The peculiar form of the observed values of R(x)
for the tuberculosis group indicated beyond doubt
that the frequency curve for this group itself was a
compound curve. I therefore decided to include both
spinal meningitis and kakke with the tuberculosis
group, and treat this new group as a compound fre-
quency curve with two components. By successive
trials I finally succeeded in establishing a complete
curve system which satisfied the ultimate require-
ment of the fit between the observed and calculated
values of R{x) for the various groups. 1
Grouping of Causes of Death in Japanese Assured
Males 1914—1917.
GROUP I
Diseases of Arteries, Senility, Influenza, Cerebral
Hemorrhage, Acute and Chronic Bronchitis, Broncho-
pneumonia.
GROUP II
Asthma and Pulmonary Emphysema, Cancer (all
forms), Tumor, Diabetes, Other Diseases of Body,
Paralytic Dementia, Tabes Dorsalis, Diseases of other
organs for circulation of Blood, Chronic Nephritis,
Other Diseases of Urinary Organs.
GROUP III
Mental Diseases, Other diseases of Spine and
Medulla Oblongata, Other Diseases of Nervous
1 See Addenda for the final table.
Japanese Life Table. 179
System, Diseases of Cardiac Valves, Pneumonia,
Pleurisy, Other Respiratory Diseases, Gastric Catarrh,
Ulcer of Stomach, Hernia, Other Diseases of Stomach,
Diseases of Liver, Acute Nephritis, Diseases of Skin
and Diseases of Motor Organs.
GROUP IV a AND IV b
Typhoid Fever, Malaria, Cholera, Acute Infectious
Diseases, Peritonitis, Suicide, Dysentery, Tuberculosis
(all forms), Syphilis, Kakke, Menengitis, Inflamma-
tion of the Caesum, Death by external causes (acci-
dents, etc.).
Arranging the collected Japanese statistics on
causes of death among assured males by attained
age at death in accordance with the above scheme
of grouping, using a 5 year interval as the unit, we
obtain the following double entry table for the 35207
deaths as used in my computation for the various
values ofR(x).
Ages Group I Group II Group III Group IV Total
10—14
3
4
37
79
123
15—19
17
23
216
714
970
20—24
37
65
181
1640
1923
25—29
62
109
324
1975
2470
30—34
124
257
800
1993
3174
35—39
278
480
1147
2065
3970
40—44
449
662
1299
1674
4084
45—49
701
957
1352
1482
4491
50—54
742
959
1115
990
3806
12*
180 Human Death Curves.
Ages
Group I
Group II
Group III
Group IV
Total
55—59
864
1045
1041
728
3678
60—64
865
847
874
482
3068
65—69
626
571
612
186
1995
70—74
399
268
347
80
1094
75—79
123
76
100
20
319
80—84
16
13
10
3
42
The observed values of R(x) as derived from the
above table are shown in the staircase shaped histo-
graph in Fig. 10. The correlated values of R(x) as
calculated from the final mortality table are shown
as dotted curves on the same diagram. The "fit"
between observed and calculated values of R(x) is
evidently satisfactory except for the youngest age
intervals.
The construction of the present Japanese table con-
stitutes probably the most severe trial to which the
proposed method has hitherto been put. We are here
dealing with an entirely different race living under
different economic conditions than the nations of
Europe and America and afflicted with certain forms
of diseases which are comparatively rare or unknown
among the Western nations.
It is therefore gratifying to note that the eminent
Japanese actuary, Mr. T. Yano, in comparing the
above mentioned table with an investigation he made
on the aggregate mortality in 1913-1917 of all the
Japanese life assurance companies (about 45 in num-
ber) from the actual number of lives exposed to risk
Japanese Life Table.
181
So
6s.
Zo
V^
s -
%
^
V
GffOuP X
\
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X
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i
.
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GrOoof
li *v
'n.
t
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X
V
N
N
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Gtvow
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Qooupt)
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2Q 3o •^o 5o bo Jo Ayat Peo^Vi.
Fig. 10.
182 Human Death Curves.
at various ages has been able to test independently
the validity of the proposed method to complete
satisfaction. (See remarks in preface).
13. criticisms and With these remarks I shall
summary close the mere technical dis-
cussion of the proposed method
and turn my attention to the arguments advanced
by certain American critics against the possibility
of constructing mortality tables from records of
death alone. I deem no apology necessary to meet
those critics and give a brief historical sketch of
the origin of the proposed method, because re-
marks along this line will tend to accentuate the
difficulties the mathematically trained biometrician
has to contend with in obtaining a hearing among
the present day school of actuaries and stati-
sticians.
A good many critics, among whom I may men-
tion Mr. John S. Thompson and Mr. J. P. Little,
apparently have received an erroneous impression
of the fundamental processes of the proposed me-
thod and its evident departure from the conven-
tional methods. Mr. Thompson states "If we un-
derstand the process, the result is simply a gradua-
tion of "d " the "actual" deaths, and it is not
apparent why a mortality table should not be
formed from the unadjusted deaths and some other
Criticism and Summary. 183
function of graduation with equally good re-
sults" 1 . From this it would appear that Mr.
Thompson is of the opinion that I have graduated
the deaths as actually observed. As any one who
will take the trouble to read the above article can
see this is not the case. The actually observed
numbers of deaths have only been used to con-
struct the observed proportionate death ratios 2 .
The whole process may be summarized -as fol-
lows :
1) The choice (a priori) of a system of fre-
quency curves based upon the hypothesis that the
distribution of deaths according to age from typi-
cal causes of death can be made to conform to
those postulated frequency curves whose para-
meters are known or chosen beforehand.
2) The grouping of causes of death so as to
conform with the above mentioned system of fre-
quency curves.
3) The computation for each age or age group
of the proportionate death ratios of such groups
1 Proceedings of the Casualty Actuarial Statistical
Society of America, Vol. IV, Pages 399—400.
2 These objections by Thompson and Little are shown
in their full obscurity in the case of the tables for Lo-
comotive Engineers, Coal Miners and Japanese Assured
Males where the greatest number of observed deaths fell
between ages 35 — 49.
184 Human Death Curves.
from the oollected statistical data of deaths by age
and by cause of death.
4) The choice of approximate values of the
areas of the various component frequency curves.
Such approximate values can be determined by
inspection or by simple linear correlation methods.
5) The determination by means of the theory
of least squares of the various correction factors a
with which the approximate values of the areas
must be multiplied in order that we may obtain
the probable values of the areas of the component
curves. The observation equations necessary for
this computation are obtained from the observed
proportionate death ratios, which are indepen-
dent of the exposed to risk.
6) The subsequent calculation of the products
NF(x) for all groups and for all integral ages.
This gives us again the total number dying from
all causes at integral ages among the original
cohort of 1,000,000 entrants at age 10. In other
words the d x column from which the final morta-
lity table can be constructed.
7) The computation of the "expected" or
theoretical proportionate death ratios from the
final table and their subsequent comparison with
the "actual" or observed proportionate death ra-
tios to illustrate the "goodness of fit".
It is this last step which constitutes the verifica-
Criticism and Summary. 185
tion of the results derived by means of a purely
deductive or mathematical process, and is a test
of very stringent requirements. It is namely re-
quired that there must be a simultaneous "fit",
not alone for all groups of causes of death, but
for all age intervals as well.
The sole justification of the proposed method
hinges indeed upon the validity of the hypothesis.
Is it indeed possible to choose a priori a system
of frequency curves to which to fit our observed
data? Theoretically speaking each population or
sample population, as for instance certain occupa-
tional groups such as locomotive engineers, far-
mers, textile workers, miners, etc. will in all pro-
bability have its own particular system of fre-
quency curves. From a purely practical point of
view — and this is the one in which we are chiefly
interested — we may, however, easily get along
with a limited system af frequency curves for the
various groups of causes of death and limit our-
selves to a comparatively few sets of frequency
curves to which to fit our statistical data. The
case is analogous to that confronting a manufac-
turer of shoes. Undoubtedly the foot of one indi-
vidual is different in form from that of any other
individual, and in order to get an absolutely fault-
lessly fitting boot we would all have to go to a
custom boot maker. Practical experience shows,
186 Human Death Curves.
however, that it is possible to manufacture a few
sizes of boots, say 6's, 7's, 8's and intermediate
sizes in quarters and half s, so as to fit to com-
plete satisfaction the footwear of millions of
people. Exactly in the same manner I have found
from a long and varied experience in practical
curve fitting that it is possible to fit the mortuary
records of male deaths by attained age and cause
of death to a comparatively limited number of sets
of component curves, say not more than 5 or 6
sets. Moreover, if in a certain sample population
a certain curve should not exhibit a satisfactory
fit it is indeed a simple matter to change its para-
meters so as to improve the fit.
14 additional ^- n re g ar d *° * ne classification
PIUNCIPLES OF 0f the CaUSeS 0f death int0 a
method limited number of groups it
seems that some of the critics of the method are
of the opinion that this classification is ironclad
and fixed. This, however, is not the case. While
in a specific sample population a certain cause of
death might fall in group II, it is quite likely
that the same cause of death would come under
another group in another sample population. For
instance, the deaths from asthma are in Michigan
grouped under Group II. In the case of Coal
Miners such deaths would, however, go into group
Additional Remarks. 187
IV or group V. If the classification of causes of
death were fixed, the frequency curves for separate
population would show great variations, and it
would be out of the question to limit ourselves to
a small set of systems of component curves. Mak-
ing the classification flexible, we are, on the other
hand, in a better position to proceed with a fewer
number of curves. For instance, in order to use
the postulated frequency curve for Group VI for
Michigan it was necessary to place the cause of
death listed as No. 186 (other accidental trau-
matism) of the International Classification of
Causes of Death in that group instead of in group
V or VII, where most deaths of this type are or-
dinarily classed.
It would be interesting to see to what extent
the proposed classification and the chosen system
of frequency curves in Michigan deviates from
the theoretically exact system of frequency curves.
In the case of Michigan it would be impossible to
test this. An approximate test might be obtained
from the Michigan mortality data for the three
year period 1909 — 1911. Professor Glover has con-
structed a mortality table for males in the State
of Michigan in this three-year period by means
of the usual methods employed by actuaries by
resorting to the exposed to risk. Starting with a
radix af 1,000,000 at age 10 it is possible to break
188 Human Death Curves.
up the deaths or the d x column of the Glover
table into a set of subsidiary columns of death
from groups of causes of death in the same order
as given in Table A on page 133 by means of a
simple application of the observed proportionate
mortality ratios as derived from the 1909 — 1911
period. On the basis of a radix of 1,000,000 sur-
vivors at age 10 we find that according to the
Glover Table, 5016 will die in the interval from
50 — 54. Let us also suppose that the proportionate
mortality ratios in group III for ages 50 — 54
amounted to 0.23, then the number of deaths from
group III in that particular interval in the Glover
table would be 5016 x 0.23 = 1154. Similar num-
bers could be found for the other groups and for
arbitrary age intervals, and we would in this man-
ner have an empirical representation of the fre-
quency curves. This aspect of the matter is treated
in brief form on another page.
Keturning now to our original discussion, it will
readily be admitted that the method of construc-
ting mortality tables by means of compound fre-
quency curves cannot be considered as absolutely
rigorous from the standpoint of pure mathematics.
But neither can the usual methods of constructing
mortality tables by graduation processes either by
analytical formulas, mechanical interpolation for-
mulas or a simple graphical process be considered
Additional Remarks. 189
as mathematically exact. All statistical methods
are, in fact, approximation processes. In the
greater part of the realm of applied mathematics
we have to resort to such approximation processes.
It is thus absolutely impossible to solve correctly
by ordinary algebraic processes simple equations
of higher degree than the fourth. We encounter,
however, in every day practice innumerable in-
stances in which an approximation process, as for
instance Newton's or Horner's methods or the
method of finite differences, is sufficiently close to
determine the roots of any equation so as to satisfy
all practical requirements.
From this point of view I claim that the pro-
posed method in the hands of adequately trained
statisticians will yield satisfactory results, and I
am inclined to think that the results are probably
as true as the ones obtained by means of the usual
methods, which especially in the case of gradua-
tion by interpolation formulas often are affected
with serious systematic errors. Moreover, there
are sound philosophical and biological principles
underlying the proposed method, which is perhaps
more than can be said about the usual methods,
purely empirical in scope and principle. On the
other hand, I will readily admit that the proposed
method is by no means a simple rule of the thumb
and it can under no circumstances be entrusted to
J 90 Human Death Curves.
the hands of amateurs. The whole process can in
my opinion only be employed when placed in the
hands of the adequately trained statistician who is
thoroughly familiar with his mathematical tools,
as provided in the formulas from the probability
calculus. Such adequate training is not acquired
over night, but only through a long and patient
study. Meticulous and patient work is often re-
quired before one is finally brought upon the right
track, especially in the classification of the causes
of death. Failure upon failure is oftentimes en-
countered by the beginner in this work, and it is
probably only through such failures that the in-
vestigator is enabled to avoid the pitfalls of the
often treacherous facts as disclosed by statistical
data and steer a clear course. Mathematical skill
is only acquired through a long and careful study.
The illustrious saying of the Greek geometer,
Euclid, who once told the Ptolemaian emperor
that "there is no royal road in mathematics" holds
true to-day as it did in the days of antiquity.
The fact that the method is no simple mechani-
cal rule, but one which can be entrusted into skill-
ful hands only, is, moreover, in my opinion, one
of its strong points, because it eliminates all at-
tempts of dilletantes to make use of it. A large
manufacturing plant would not, for instance, put
an ordinary blacksmith or horseshoer to work on
Additional Remarks. • 191
making the fine tools for certain parts of automa-
tic machinery employed in the manufacture of
staple articles. Only the most skilled and highly
trained tool makers are able to produce machine
parts, which often require precision measurements
running into one thousandth part of an inch. Nor
would a large contracting firm dream of putting
a backwoods carpenter in charge of the construc-
tion of a skyscraper. Yet, this case is absolutely
analogous to that of letting the mere collector of
crude statistical data make an analysis and draw
conclusions from certain collected facts as ex-
pressed in statistical series of various sorts.
While some American critics to all appearances
have misunderstood the principles underlying the
method, several European reviewers of the short
summary of the method as originally published in
the "Proceedings of the Casualty Actuarial and
Statistical Society of America" evidently have un-
derstood its fundamental principles completely.
The European critics seem, however, to be of the
opinion that there is a rather prohibitive amount
of arithmetical work involved in the actual con-
struction of the mortality table. Thus a review in
the Journal of the Royal Statistical Society for
May 1918 has this to say :
"Mr. Fisher's object is to construct a life
table, being given only the deaths at ages and
192 Human Death Curves.
not the population at risk. The hypothesis
employed is that the total frequency of deaths
can be resolved into specific groups of deaths,
the frequencies of which cluster around cer-
tain ages. The parameters of these sub-fre-
quencies having been determined, the areas
are deduced from a system of frequency cur-
ves of the form :
R (x) = N * F *&
■ BK ' ~ N B F B {x) + N c F c {x) + N D F D (x). . .
where Rb(x) , the proportional mortality at
age x of deaths due to causes in group B and
F B (x), is obtained from the equation of the
sub-frequency curve for cause B , while Nb +
N c + N D + . + N E = 1,000,000. The
values of R(x) provide a system of observa-
tional equations from which (by least squares)
the values of N B , &c., can be obtained.
"Since particularly in industrial statistics,
or in general statistical inquiries under war
conditions it is easier to obtain accurate data
of deaths at ages than of exposed to risk, the
success of the method is encouraging. It is,
however, to be noted that the amount of arith-
metical work envolved is considerable. Quite
apart from the determination of the para-
meters of the frequency curves, the formation
and solution of the normal equations needed
to compute the areas is a heavy piece of work.
It would be of interest to see whether the re-
solution into but three components effected by
Professor Karl Pearson in his well-known
Additional Remarks. 193
essay published in the "Chances of Death"
could be made to describe with sufficient ac-
curacy an ordinary tabulation of deaths from
age 10 onwards to lead to approximately cor-
rect results for life table purposes. The test
should, of course, be made with mortality
data derived from a population very far from
being stationary and the deductions compared
with the results of standard methods. The
subject is one of peculiar interest at the pre-
sent time."
From the above quotation it is evident that this
English reviewer has a clear conception of the
fundamental principles upon which the method is
based. His criticism is mainly directed against
the heavy piece of arithmetical work involved.
This work can, however, not be compared with
the much more difficult task of obtaining the ex-
posed to risk at various ages, which under all cir-
cumstances would take much greater time and be
infinitely more costly, in fact be absolutely pro-
hibitive from a financial point of view. I wish in
this connection to state that the whole arithmeti-
cal work involved in the construction of the Michi-
gan table was done by two computers in less than
70 hours, while the corresponding table for Mas-
sachusetts took about 75 hours. I do not know if
this can be called exactly prohibitive.
In regard to the remarks of my British critic
13
194 Human Death Curves.
concerning the Pearsonian method I might add
that in my first attempt of an analysis of mortality
conditions along the lines as described above I
tried to subdivide the causes of death into four
groups. It was, however, found that this was not
always sufficient to describe the frequency dis-
tribution of the number of deaths around certain
ages. 1 doubt whether it is at all possible to des-
cribe the frequency distribution in the various sub-
groups by a system of normal curves, which, of
course, would somewhat lessen the work. I have
made attempts to do this, but so far I have not
been successful except in a few cases. 1 It might
be possible that we should succeed in this if we
first set up a hypothetically determined curve of
the numbers exposed to risk. Such a curve might,
for instance, be a normal curve. Personally, I be-
lieve that little would be gained by such a proce-
dure. More fruitful appears an analysis by means
of correlation surfaces. The mortality table con-
structed by the process as I have described it con-
stitutes in its final form a correlation surface,
wherein the age at death and the group of causes
of death are the independent variables, and the
number of deaths at a certain age and from a
1 See Addenda for the Metropolitan Table and the
Japanese Table.
Another Application. 195
certain group af causes of death is the numerical
value of the correlation function of the two va-
riates. Provided one could obtain an exact equa-
tion of such a correlation surface, it would be a
simple matter to construct a mortality table, and
I hope that some statistician may in the future be
induced to attempt a solution of the problem in
this lieht.
15. another ap- Before closing the discussion of
PLICATION OF &
thefreqven- this subject we shall, however,
CY CURVE ME- J .
thod give a brief description of an-
other application of compound frequency curves in
the construction of mortality tables. We have here
reference to the use of skew frequency curves in
the graduation of crude mortality rates as com-
puted in the usual empirical manner as the ratio
of deaths to the number of lives exposed to risk
at various ages. On page 165 it was mentioned
that the State of Massachusetts took a census in
April 1915. This census together with the deaths
for the triennial period from 1914 — 1916 makes
it an easy matter to construct a mortality table in
the conventional manner. Moreover, such a table
can be compared with the previously constructed
table from mortuary records by sex , age and cause
of death only and shown in the appendix.
In this connection it might be worth mention-
13*
196
Human Death Curves.
ing that my first table for Massachusetts as con-
structed by compound frequency curves was pre-
pared during the summer of 1918 and first pre-
sented in a series of lectures delivered at the
University of Michigan during the month of
March 1919, while the final official report of the
1915 Massachusetts census did not come in the
hands of the present writer before May 1919.
Another Application. 197
The official census of the population of Mas-
sachuetts by sex and single ages is given on page
478 in Vol. Ill of the Massachusetts report from
which Fig. 11 has been constructed. It is seen
from a mere glance of this graph that there is an
unduly high tendency among the figures to cluster
around ages being multiples of 5. This tendency
is especially marked in the age interval 30 — 60
and presents a defect which is of no small im-
portance in the construction of a mortality table
by means of the conventional methods. It is in-
deed doubtful if a table constructed from data
so greatly influenced by observation errors and
misstatements of ages can be considered as ab-
solutely trustworthy. On the other hand the data
ought to be sufficiently exact to test the results
arrived at by the proposed method of compound
frequency curves.
We give below the male population in 5 year
age groups for the middle census year of 1915
and the corresponding deaths from all causes
durirg the triennial period 1914 — 1916.
MASSACHUSETTS
1915 Male Population and Number of Deaths
among Males from 1914 — 1916.
Ages Population, L x . Deaths 1914— 10. D x .
5— 9 169010 1715
10—14 152419 1004
198 Human Death Curves.
Ages J
15—19
J opulation, L % . 1
154773
Jeaths iyi4—
1537
20—24
171961
2353
25—29
171017
2726
30—34
149294
2979
35—39
142617
3535
40—44
125462
4007
45—49
107909
4393
50—54
89490
5026
55—59
65133
5459
60—64
49079
5679
65—69
34790
6027
70—74
23638
5946
75—79
13724
4752
80—84
6494
3166
85—89
2479
1751
90—94
530
540
95—99
124
133
100 & over
12
23
A few small discrepancies will be found to exist
between this table and the table printed on page
163, giving the observed deaths from various
causes in ten year age intervals. This arises solely
from the fact that a number of deaths were re-
corded where the contributing cause was unknown
and could, therefore, not be distributed in their
proper groups. But this defect is of no influence
in the construction of mortality table by means
Another Application. 199
of the method of compound frequency curves, un-
less all the causes reported as unknown should
happen to belong to the same group, which hardly
can be assumed to be the case. At any rate the
proportionate death ratios which are the keystone
in this method of construction are for practical
purposes left unaltered whether we include or ex-
clude these few numbers of unknown causes. In
the usual way of constructing tables from ex-
posures and number of deaths it is on the other
hand absolutely essential to include all deaths as
otherwise the death rate will be underestimated.
Bearing these facts in mind we therefore refer
to the above figures of L x and D x for Massachu-
setts Males from which we without further diffi-
culty can construct an empirical mortality table,
either by graphic methods or by simple summa-
tion or interpolation formulas. There is indeed no
dearth of such formulas, of which a large number
have been devised by Milne, Wittstein, Woolhouse,
Higham, Sprague, Hardy, King, Spencer, Hen-
derson, Westergaard, Gram, Karup and several
other investigators. In the following computation
I have used a formula originally devised by the
Italian statistician, Novalis, and later on some-
what modified by the English actuary, King.
The following schedule shows the actual process
in detail.
200
Human Death Curves.
MASSACHUSETTS MALES.
A. Population.
Graduated Quinquennial Pivotal Values.
Graduated
Ages Population L x A L x A 2 L X Age
Population
12 29332
17 30836
34537
34369
22
5— 9 169010 — 16591
10—14 152419 + 2354 + 18945
15—19 154773 + 17188 + 14834
20—24 171961— 944 — 18132
25—29 171017 — 21723 — 20779 27
30—34 149294— 6677 + 15047 32 29739
35—39 142617 — 17155 — 10478 37 28607
40—44 125462 — 17553— 398 42 25095
45—49 107909 — 18419— 866 47
50—54 89490 — 24357— 5938
55—59 65133 — 16054+ 8293
60—64 49079 — 14289+ 1765
65—69 34790 — 11152+3137
70—74 23638— 9914+ 1238
75—79 13724— 8130 + 1884
80—84 6494— 4015 + 4115
85—89 2479— 1949 + 2066 87
90—94 530— 406 + 1543 92
95—99 124— 112 +
100—104 12
52
57
62
67
72
77
82
294 97
102
21587
17946
12961
9802
6933
4717
2731
1265
480
104
23
1
Graduated Population = u x+7 = 0.2 L x+5 —
0.008A 2 L, +5
Another Application. 201
B. Deaths 1914—1916.
Graduated Quinquennial Pivotal Values.
Ages
5— 9
10—14
15—19
20—24
25—29
30—34
35—39
40—44
45—49
50—54
55—59
60—64
65—69
70—74
75—79
80—84
85—89
90—94
95—99
100—104
In this manner we obtain the graduated quin-
quennial pivotal values of the population and of
the deaths for ages 12, 17, 22, 27, ... . etc. Then
No. of
Deaths D x A * '
(\ 2 n*
Age
Graduated
Deaths
1715— 711
1004 + 533 +
1244
12
200.8
1537 + 816 +
283
17
307.4
2353+ 373 —
443
22
470.6
2726+ 253 —
120
27
545.2
2979 + 556 +
303
32
595.8
3535 + 472 —
84
37
707.0
4007 + 386 —
86
42
801.4
4393 + 633 +
247
47
878.6
5026 + 433 —
200
52
1005.2
5459+ 220 —
213
57
1091.8
5679 + 348 +
128
62
1125.8
6027 — 81 —
429
67
1205.4
5946 — 1194 —
1113
72
1189.2
4752 — 1586 —
392
77
950.4
3166 — 1415 +
171
82
633.2
1751 — 1211 +
204
87
350.2
540— 407 +
804
92
108.0
133— 110 +
297
97
26.6
23
102
4.6
202 Human Death Curves.
by dividing one third of the graduated deaths by
the population we have the graduated pivotal
values of the so-called "central death rates", or
m x for quinquennial ages from age 12 and up.
From these values of m, we easily find the corre-
sponding values of q x by means of the formula :
1*- 2 + m x
We give below the results of this computation
Massachusetts Males 1914—1916.
Age 1000 q x from Novalis' Formula
12 2.21
17 3.33
22 4.64
27 5.29
32 6.68
37 8.25
42 10.65
47 13.53
52 18.67
57 26.38
62 38.29
67 58.12
72 81.90
77 109.91
82 165.02
87 240.18
92 325.64
Graduation of d x Column. 203
The intervening values of q x are without diffi-
culty derived by interpolation formulas or by a
graphical process. Once having all the values of
q x for separate ages from age 10 and up it is a
simple matter to form tables of l x and d x commen-
cing with a radix of 1,000,000 at age 10. Without
going into tedious details we present the following
values of l x for decimal ages.
Massachusetts Males 1914—1916.
kge
h
Ages
1,d x
10
1,000,000
10—19
27,700
20
972,300
20—29
47,330
30
924,970
30—39
66,750
40
858,220
40—49
98,650
50
759,570
50—59
153,900
60
605,670
60—69
233,150
70
372,520
70—79
237,130
80
135,390
80—89
124,760
90
10,640
90 & over
10,640
100 32
16. graduation It is to this table that we now
BY FLUENCY sha11 **& » P™* 88 ° f re "
curves graduation by means of the
method of compound frequency curves.. Here we
have already an empirical representation of the
total compound curve of death or the d x curve.
204 Human Death Curves.
This compound curve can now by simple and
straightforward processes be broken up into its
various component parts as to causes of deaths by
means of the various observed proportionate mor-
tality ratios, R x shown in Table H on page 163.
Let us for the sake of illustration take the age
interval 40 — 49. According to our empirically con-
structed table as derived from the Massachusetts
1915 census we find that the number of deaths
among the survivors in this age interval amounts
to 98,650.
Applying to this number the observed propor-
tionate death ratios, B , in table H we are able to
break this number up into its various component
parts according to the groups of causes of death
from which the numerical values of R x were de-
rived. These component parts are as follows :
Group Nc
i. of Deaths
I
1180
II
18050
III
17170
IV
17170
V
14300
VI
23970
VII a & b
5820
VIII
990
Total :
98650
Graduation of d r Column.
205
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206 Human Death Curves.
In the same manner we can break up the com-
pound curve (the d x curve) in its eight component
parts for all other age intervals, which finally gives
us the following table of component groups,
printed on the preceeding page, and graphically this
table will represent a series of frequency diagrams
of the various groups of causes of deaths. It is an
easy matter to fit such diagrams to a system of
Laplacean-Charlier or Poisson-Charlier frequency
curves, which symbolically may be represented as
follows :
N^x), N u F u (x). .N^F^x)
where F(x) is the frequency function of the per-
centage distribution according to age of the va-
rious component groups or curves, while N stands
for the areas of such curves.
These curve areas are simply the sub-totals of
the respective groups in the above table. The pa-
rameters giving the equations of the curves F t (x),
F n (x), F UI (x), .... are easily computed by the
methods of moments and are shown in the follow-
ing table on page 207.
Once having determined the parameters of the
various frequency curves it is a simple matter to
construct the final mortality table which is shown
in the addenda.
Graduation of d x Column. 207
Values of Parameters of Component Curves,
Massachusetts, 1914—1916 Males. 1
Group Mean Dispersion Skewness Excess
I 75.0 9.78 +0.080 —0.005
II 67.5 13.65 +0.117 +0.017
III 64.0 14.12 +0.124 +0.030
IV 60.5 16.51 +0.089 —0.006
V 50.0 18.61 +0.026 —0.034
VI 43.5 15.57 —0.036 —0.023
Vllb 57.5 16.33 —0.027 —0.028
It now remains for us to compare the final values
of q x which we obtain from the three tables :
A) The values of q x as computed in the usual
1 In this grouping I have combined Vila and VIII
into a single group and roughly fitted this group to a
truncated Poisson-Charlier curve. This, of course, is not
exact and introduces evidently errors in the younger
age interval from 10 — 19. For ages above 20 this curve
plays no importance and the other curves should for
the ages above 20 give a satisfactory fit. If absolutely
exactitude was required for younger ages it would
indeed offer no difficulties to compute curves Vila and
VIII separately and thus obtain a much closer fit in
the youngest age interval. In view of the fact that
the present calculation is a test case only, it has not
been thought necessary to go to these refinements.
This defect will af course also effect to a slight extent
group VII b.
208 Human Death Curves.
way from the number of lives exposed to risk and
the corresponding deaths at various ages.
B ) The values of q x as obtained by a re-gradua-
tion of the mortality table under A by means of
compound frequency curves.
G) The values of q x constructed from mortuary
records by sex, age and cause of death, but with-
out knowing the numbers of lives exposed to risk.
Massachusetts Males. 1914—1916.
Values of 3000 q by various methods.
Age
A
B
C
17
3.33
3.15
3.27
22
4.64
3.99
4.28
27
5.29
5.04
5.46
32
6.68
6.72
7.03
37
8.25
8.63
8.88
42
10.65
10.83
11.05
47
13.53
13.86
14.05
52
18.67
18.83
19.13
57
26.38
26.88
27.66
62
38.29
38.79
40.26
67
58.12
59.04
56.54
72
81.90
76.50
77.61
77
109.91
103.69
107.51
82
165.02
137.97
148.79
I think that every unbiased investigator will
admit that there exists a close agreement be-
Comparison between Methods. 209
tween the three series. It is indeed difficult to
say which one of the three is the most probable.
We know that on account of the great perturba-
tions due to misstatements of ages the values
under A are effected with considerable errors. The
usual interpolation or summation formulas do not
suffice to remove these errors and tend often to
increase them. A re-graduation by means of fre-
quency curves as shown in series B will in all
probability give better results, although on ac-
count of the large age interval (10 years) in which
the causes of deaths are grouped in the Massa-
chusetts reports this method does not come to its
full right 1 . The values of q x under A and B are
naturally closely related to each other, and those
in series B cannot be derived unless the values
in series A are known beforehand. Series C on
the other hand is independent of either A or B,
having been derived by means of entirely different
methods of construction.
17. comparison A comparison between the pa-
B §£WJi?£'£l F ~ rameters in the seperate com-
thods ponent curves in B and C
gives us, however, a way of testing the validity
of the hypothesis upon which the method of
See footnote on page 127.
14
210 Human Death Curves.
series G rests. In the case of the series G we star-
ted with the hypothesis of the existence of a set
of frequency curves of the percentage distribution
of the number of deaths according to age among
the various groups. On the basis of this hypothesis
and from the observed values of the proportionate
death ratios, R , we determined by the method
of least squares the areas of this postulated set of
frequency curves. In the case of the B series we
broke up the empirically constructed compound
death curve (the d curve) into its various com-
ponent parts according to a similar classification
of causes of deaths as under C. We have therefore
in this case an empirical determination of the
areas of the component curves and all that we
need to do is to graduate the rough frequency
diagrams as represented by such areas to a system
of frequency curves.
Let us now briefly examine how far the various
skew frequency curves in series B and C differ
from each other. In regard to the various statis-
tical parameters of the separate groups we have
the following results :
Means.
Group
Series G
Series B
I
78.5
75.0
II
68.0
67.5
Comparison between Methods.
211
Group
Series G
Series B
III
63.0
64.0
IV
60.5
60.5
V
49.5
50.0
VI
44.0
43.5
Vllb
57.5
Dispersions
57.5
Group
Series C
Series B
I
7.98
9.78
II
12.21
13.65
III
13.05
14.12
IV
17.86
16.51
V
18.51
18.61
VI
14.68
15.57
Vllb
12.16
Skewness.
16.33
Group
Series C
Series B
I
+ 0.092
+ 0.080
II
+ 0.115
+ 0.117
III
+ 0.121
+ 0.124
IV
+ 0.098
+ 0.089
V
+ 0.033
+ 0.026
VI
—0.010
—0.036
Vllb
—0.002
—0.027
14*
212 Human Death Curves.
Excess.
Group
Series G
Series B
I
—0.033
—0.005
11
+ 0.023
+ 0.017
III
+ 0.047
+ 0.030
IV
—0.009
—0.006
V
—0.031
—0.034
VI
—0.027
—0.023
Vllb
—0.003
—0.028
Taken all in all there is found to exist a satis-
factory agreement between the hypothetical va-
lues in series C and the values derived by empiri-
cal methods. It is only in group I that we find
some important discrepancies. This group contains
causes of death typical of extreme old age where
we naturally may expect great perturbations
owing to large errors from random sampling,
especially in series B. In this same connection
we may also mention that the empirically deter-
mined values under series B are subject to a slight
correction by means of the Sheperd formulas,
which were not employed in my computations.
We have already mentioned that the system
of frequency curves which we choose a priori
for Massachusetts (Series C) was the same system
which we had used on a previous occasion
in the construction of a mortality table for Eng-
Comparison between Methods. 213
lish Males for the period 1911— 1912 1 ). This is a
fact of no small importance. It will in general be
found that the percentage distribution according
to age in the various component curves differs
little in different sample populations. Even in the
case of American Locomotive Engineers it was
found possible to use the same set of curves as in
the case of Massachusetts and England and Wales.
In the same way I have found that the set of
curves used in the construction of the table of
Michigan Males also can be used in the case of
males in the urban population of Denmark. With
a very few exceptions I have found it possible
to get along with a limited number of sets of
curves, say four or five sets. Should it never-
theless prove impossible to fit the original data to
any one of these particular curve systems, it will
in most cases be found possible by means of suc-
cessive approximations to reach a system of cur-
ves which may be made the a priori basis for the
construction of the final table as was the case in
the table for Japanese assured males.
Finally we come to the comparison of the vari-
ous areas of the component curves. We have
here :
1 See " Proceedings of the Casualty Actuarial Society
of America", Vol. IV, page 409.
214 Human Death Curves.
Areas.
G
B
I
90064
105000
II
281470
296190
III
207854
213010
IV
151316
144200
V
99543
87850
VI
107718
106260
VII & VIII
62035
47410
Total 1000000 1000000
Evidently the agreement is not so close in this
case. But it would indeed be rather rash to assert
that the values in series G are faulty. One must
here bear in mind the diametrically opposite
principles employed in the determination of these
areas. In series B we have a direct determination
by empirical methods. In this determination we
shall, however, find reflected all the original sy-
stematic and observational errors originally pre-
sent in series A from which the curves under B
were computed. Every error due to misstatements
of ages and systematic errors introduced by the
summation or interpolation formulas will be di-
rectly reflected in the areas under series B, and
such areas can therefore in a sense only be con-
sidered as a first approximation to the true or
presumptive areas.
Comparison between Methods. 215
Another point well worth remembering is the
one that no conditions are imposed upon the areas
in series B. In series G where we work with mor-
tuary records only we have on the other hand the
very important condition or restriction requiring
that the areas of the component curves must be
so determined that their ratios to the compound
curve for various age intervals will conform as
closely as possible with the observed proportionate
death ratios, R x , for those same age intervals.
In order to test the influence of this additional
requirement in respect to conformity to observed
proportionate death ratios we might use the values
of the component curves under series B as a first
approximation and then afterwards determine the
correction factors a for the areas in exactly the
same way as in the case of series G. No doubt
such a calculation would tend to improve the
table.
A difficulty occurs, however, in the case of
the Massachusetts data owing to the large interval
of 10 years into which the causes of death by
attained ages are grouped. As pointed out in the
footnote on page 127 the quantity R B (x), (x =
30, 11, 12, 100 ; B =1, II, III, ) ,
can only be considered as being independent of
the "exposed to risk" if the age interval into which
the deaths fall is sufficiently small. If this is not
216 Human Death Curves.
the case, the "central" values of Rb (%) are
subject to certain corrections. In the case of the
groups of causes of death typical of younger ages
the observed "central" values of Ryii (%) and
Iivm (x) for the age intervals 10 — 19, 20 — 29,
30 — 39 are evidently too high, while on the other
hand the values of Rj (x) and JJ n (z) in the case
of the age intervals 60—69, 70—79, 80—89,
90 — 100 are too low as compared with the true
values of R(x) at these "central" ages. I have,
however, tacitly ignored this fact in my computa-
tions. The subsequent result is that the final
values of q x for the younger ages in column C as
shown on page 208 are in all probability a little
too high, and the values of q x above 65 too low.
In the case of the other tables as shown in the
present book the age interval into which the causes
of death were arranged was 5 years or less, and
the error was thus reduced to such an extent that
further corrections may be disregarded for all
practical purposes.
ADDENDA I
Showing Detailed Mortality Tables and Death
Curves for
1) Japanese Assured Males (1914 — 1917)
2) Metropolitan Life. White Males (1911—1916)
3) American Coal Miners (1913—1917)
4) American Locomotive Engineers (1913 — 1917)
5) Massachusetts Males (Series C) (1914—1916)
6) Michigan Males (1909—1915)
7) Massachusetts Males (Series B) (1914—1916).
218
Addenda.
Mortality Table — Japanese Assured Males
1914—1917 (Aggregate Table)
Age
I
II
III
IVa
IVb
dx
lx
lOOOqx
15
24
65
343
2379
2811
1000000
2.81
18
39
74
360
3645
4118
997189
4.13
17
43
84
388
4888
5403
993071
5.44
18
48
93
415
5981
6557
987668
6.64
19
54
107
446
6826
7433
981111
7.58
20
60
120
478
7447
8105
973678
8.32
21
68
135
513
7716
8432
965573
8.73
22
77
153
550
12
7734
8526
957141
8.91
23
87
171
591
27
7581
8457
948615
8.92
24
101
195
633
50
7274
8253
940158
8.86
25
111
218
678
77
6864
7948
931905
8.53
26
126
246
729
112
6384
7597
923957
8.22
27
140
278
780
153
5860
7211
916360
7.87
28
160
315
838
206
5341
6860
909149
7.54
29
178
353
899
26S
4821
6519
902289
7.22
30
198
305
963
341
4323
6220
895770
6.94
31
227
446
1033
425
3853
5984
889550
6.73
32
252
501
1109
521
3421
5804
883566
6.59
33
286
557
1185
629
3021
5678
877762
6.46
34
319
626
1273
751
2665
5633
872084
6.46
35
358
700
1364
885
2336
5643
866451
6.51
36
401
779
1460
1031
2048
5719
860808
6.64
37
450
872
1564
1186
1797
5869
855089
6.86
38
502
970
1671
1350
1566
6059
849220
7.13
39
570
1081
1791
1524
1366
6332
843161
7.51
40
638
1197
1916
1701
1191
6643
836829
7.94
41
716
1332
2049
1883
1037
7017
830186
8.45
42
802
1475
2193
2066
903
7439
823169
9.04
43
899
1632
2341
2249
783
7904
815730
9.69
44
1005
1799
2501
2428
680
8413
807826
10.41
45
1126
1985
2671
2599
598
8979
799413
11.23
46
1261
2180
2852
2764
514
9571
790434
12.10
47
1406
2393
3042
2917
447
10205
780863
13.07
48
1575
2611
3236
3061
395
10878
770658
14.12
49
1764
2867
3459
3187
339
11606
759780
15.27
50
1957
3122
3666
3298
295
12338
748174
16.49
51
2180
3395
3892
3389
257
13113
735836
17.82
52
2426
3679
4136
3473
224
13938
722723
19.29
53
2692
3984
4380
3532
195
14783
708785
20.86
54
2987
4285
4638
3576
172
15658
694002
22.56
55
3306
4610
4922
3611
147
16596
678344
24.47
56
3654
4940
5177
3612
130
17513
661748
26.46
57
4026
5274
5456
3605
113
18474
644235
28.68
58
4432
5603
6742
3581
97
19455
625761
31.09
69
4857
5937
6025
3544
84
20447
606306
33.72
60
5316
6257
6316
3498
74
21461
585859
36.63
61
5795
6568
6604
3424
69
22460
564398
39.79
62
6293
6860
6890
3345
59
23447
541938
43.27
63
6805
7129
7162
3255
51
24402
518491
47.15
64
7332
7361
7423
3150
43
25309
494089
51.22
65
7854
7570
7672
3042
38
26176
468780
55.84
Addenda.
219
Ige
I
II
III
IVa
IVb
dx
Ix
lOOOqx
66
8366
7727
7896
2919
36
26944
442604
60.88
67
8863
7838
8089
2791
31
27612
415660
66.43
68
9313
7894
8257
2655
28
28147
888048
72.53
69
9719
7894
8385
2511
23
28532
359901
79.27
70
10053
7829
8468
2362
20
28732
331369
86.71
71
10294
7700
8503
2212
18
28727
302637
94.92
72
10424
7496
8477
2067
15
28479
273910
103.97
73
10424
7227
8389
1901
13
27954
245431
110.69
74
10280
6897
8230
1746
13
27166
217477
124.91
75
9970
6503
8002
1593
10
26078
190311
137.02
76
9492
6057
7695
1444
10
24698
164233
150.38
77
8834
5571
7313
1298
8
23024
139535
165.00
78
8037
5047
6853
1159
7
21103
116511
181.12
79
7086
4499
6314
1026
6
18931
95408
198.42
80
6046
3943
5733
900
5
16621
76477
217.33
81
4953
3400
5091
784
4
14232
59856
237.77
82
3871
2862
4421
676
3
11833
45624
259.35
83
2813
2365
3730
577
2
9487
33791
280.75
84
1957
1907
3046
489
1
7400
24304
304.48
85
1232
1498
2396
412
5538
16904
327.61
86
701
1141
1797
340
3979
11366
350.08
87
343
844
1275
277
2739
7387
370.76
S8
140
603
844
225
1812
4648
389.78
89
48
40S
516
179
1151
2836
405.85
90
14
269
283
141
707
1685
419.58
91
5
171
134
110
420
978
429.44
92
111
53
83
247
558
442.65
93
56
14
63
133
311
452.10
94
28
4
44
76
178
457.05
95
14
2
31
47
102
460.78
96
5
1
22
28
55
509.01
97
14
14
27
518.50
98
99
4
9
4
13
4
692.30
1000.00
Mortality Table
Metropolitan White Males 1911-
-1916
Age
I
II
III
IVb
IVa
dx
lx
lOOOqx
10
SO
153
205
47
1720
2205
1000000
2.21
11
95
179
274
01
1776
2385
997795
2.39
12
118
210
350
77
1812
2567
995410
2.58
13
141
244
444
96
1832
2757
992843
2.78
14
168
282
550
116
1834
2950
990086
2.98
15
202
327
671
140
1825
3165
987136
3.21
16
240
373
810
171
1803
3397
983971
3.45
17
282
427
960
199
1772
3640
980574
3.71
18
336
483
1130
233
1733
3915
976934
4.01
19
393
545
1315
274
1680
4207
973019
4.32
20
454
611
1514
311
1612
4502
968812
4.65
21
527
685
1728
358
1539
4837
964310
5.02
22
599
765
1951
407
1449
5169
959473
5.39
23
6S7
845
2184
459
1363
5538
954304
5.80
220
Addenda.
Age
I
II
III
IVb
IVa
dx
iX
lOOOqx
24
775
932
2428
515
1279
5929
948766
6.25
25
874
1024
2674
575
1190
6337
942837
6.72
26
977
1120
2924
638
1107
6766
936500
7.32
27
1088
1223
3173
703
1012
7199
929734
7.74
28
1202
1328
3414
770
923
7637
922535
8.28
29
1324
1436
3648
839
840
8087
914898
8.84
30
1473
1549
3879
909
757
8567
906811
9.45
31
1584
1662
4089
985
684
9004
898244
10.02
32
1702
1779
4283
1052
614
9430
889240
10.60
33
1863
1899
4459
1125
545
9891
879810
11.24
34
2012
2015
4604
1196
485
10312
869919
11.85
35
2160
2139
4740
1266
427
10732
859607
12.48
36
2324
2259
4842
1332
378
11135
848875
13.12
37
2485
2379
4919
1399
335
11517
837740
13.75
38
2664
2501
4968
1462
296
11891
826223
14.39
39
2847
2617
4989
1520
25S
12231
814332
15.02
40
3057
2734
4988
1577
226
12578
802101
15.68
41
3272
2848
4953
1628
192
12893
789523
16.33
42
3508
2960
4898
1675
163
13204
776630
17.00
43
3767
3066
4821
1719
143
13516
763426
17.70
44
4057
3170
4719
1757
120
13823
749910
18.43
45
4389
3267
4604
1789
100
14149
736087
19.22
46
4748
3358
4471
1816
90
14483
721938
20.06
47
5153
3447
4320
1839
75
14834
707455
20.97
48
5599
3526
4160
1855
61
15201
692621
21.95
49
6064
3598
3991
1867
50
15590
677420
23.01
50
6631
3663
3810
1872
42
16018
661830
24.20
31
7198
3721
3630
1872
35
16456
645812
25.48
52
7820
3769
3443
1867
30
16929
629356
26.90
53
8492
3809
3254
1857
22
17434
612427
28.47
54
9168
3839
3069
1840
10
17926
594993
30.13
55
9897
3858
2876
1820
1
18452
577067
31.98
56
10637
3868
2696
1793
18994
558615
34.00
57
11378
3867
2519
1762
19526
539621
36.18
58
12114
3853
2340
1726
20033
520095
38.52
59
12847
3830
2169
1687
20533
500062
41.06
60
13555
3794
2004
1640
20591
479529
43.77
61
14217
3746
1844
1591
21396
358538
46.67
62
14817
3685
1692
1541
21735
437140
49.72
63
15359
3615
1547
1484
22005
415405
52.97
64
15820
3535
1408
1425
22188
393400
56.40
65
16179
3443
1277
1364
22263
371212
59.97
66
16450
3340
1153
1299
22242
348949
63.74
67
16610
3229
1037
1235
22111
326707
67.68
68
16691
3109
930
1166
21896
304596
71.89
69
16591
2981
828
1098
21498
282700
76.05
70
16412
2851
736
1030
21029
261202
80.51
71
16107
2711
649
955
20422
240173
85.03
72
15721
2568
571
892
19752
219751
89.88
73
15225
2423
500
825
18973
199999
94.87
74
14629
2271
434
759
18093
181026
99.95
75
13946
2126
377
695
17144
162933
105.22
76
13225
1976
325
632
16158
145789
110.83
77
12423
1828
278
572
15101
129631
116.49
78
11580
1684
237
515
14016
114530
122.38
Addenda.
221
Age
I
II
III
IVU IVa
dx
lx
lOOOqx
79
10729
1543
200
461
12933
100514
128.67
80
9840
1406
167
411
11824
87581
135.01
81
8950
1272
138
363
10723
75757
141.54
82
8092
1144
115
318
9669
65034
148.68
83
7237
1024
98
282
8641
55365
156.07
84
6420
911
79
247
7657
46724
163.88
86
5645
806
65
208
6724
39067
172.11
86
4920
707
53
1S1
5861
32343
181.21
87
4240
615
43
150
5048
26482
190.62
88
3622
531
34
126
4313
21434
201.22
89
3065
457
27
106
3655
17121
213.48
90
2550
387
22
87
3046
13466
226.20
91
2099
327
16
70
2512
10420
241.07
92
1698
270
14
56
2038
7908
257.71
93
1355
222
11
45
1633
5870
278.19
94
1053
179
8
35
1275
4237
300.92
95
805
143
6
27
981
2962
331.20
96
595
112
5
20
732
1981
369.51
97
412
85
1
14
512
1249
409.93
98
286
62
10
358
737
485.75
99
198
27
6
231
379
609.50
100
95
15
4
114
148
770.27
101
27
5
2
34
34
1000.00
Mortality Table — American Coal Miners
(1913—1917)
Age
I n
III
IV
Va
Vb
VI
dx
be
lOOOqx
18
99
124
142
4566
7
366
5304
1000000
5.30
19
114
144
164
4702
10
408
5542
994696
5.57
20
140
168
187
4954
14
452
5915
989154
5.98
21
162
194
214
5196
19
498
6283
983239
6.39
22
190
223
243
5234
27
546
6463
976956
6.62
23
223
250
272
5151
38
597
6531
970493
6.73
24
256
282
307
5067
50
646
6608
963962
6.86
25
298
315
341
4952
69
697
6672
957354
6.97
26
341
349
379
4846
91
749
6755
950682
7.11
27
390
386
421
4748
120
802
6867
943927
7.27
28
440
424
465
4683
156
853
7021
937060
7.49
29
498
461
508
4569
202
903
7141
930039
7.68
30
557
500
560
4413
257
953
7240
922898
7.84
31
622
538
609
4220
326
1002
7317
915658
7.99
32
688
579
663
4000
408
1048
7386
908341
8.13
33
761
618
718
3757
505
1093
7452
900955
8.27
34
837
654
777
3500
618
1133
7519
893503
8.42
35
915
693
840
3233
749
1175
7605
885984
8.58
36
994
732
905
2963
898
1212
7704
878379
8.77
37
1084
775
973
2697
1064
1246
7839
870675
9.00
38
1171
818
1045
2435
1251
1277
7997
862836
9.27
39
1267
867
1124
2184
1452
1305
8199
854839
9.59
40
1364
920
1206
1946
1667
1329
8432
846640
9.96
41
1471
978
1293
1723
1894
1352
8711
838208
10.39
42
1581
1045
1386
1515
2131
1369
9027
829497
10.88
222
Addenda.
Age
I
II
III
IV
Va
Vb
VI
dx
lx
lOOOqx
43
1705
1125
1489
1325
2372
1383
9399
820470
11.46
44
1835
1222
1585
1106
2609
1395
9752
811071
12.02
45
1
1976
1322
1712
883
2841
1403
10133
801319
12.65
46
6
2132
1444
1837
853
3063
1408
10743
791186
13.58
47
10
2302
1584
1971
729
3265
1410
11271
780443
14.44
48
21
2492
1741
2114
619
3443
1408
11838
769172
15.39
49
32
2705
1918
2265
524
3595
1402
12441
757334
16.43
50
42
2934
2118
2423
442
3706
1395
13060
744893
17.53
51
54
3190
2337
2589
368
3790
1383
13711
731833
18.74
52
73
3470
2567
2764
307
3832
1368
14380
718122
20.02
53
94
3775
2820
2945
255
3832
1352
15073
703742
21.42
54
123
4104
3086
3130
210
3790
1331
15774
688669
22.91
55
153
4437
3355
3313
173
3706
1308
16445
672895
24.44
56
185
4843
3637
3501
141
3595
1281
17183
656450
26.18
57
225
5246
3922
3689
115
3443
1252
17892
639267
27.99
58
268
5656
4192
3872
93
3265
1220
18566
621375
29.88
59
310
6085
4454
4047
76
3063
1186
19221
602809
31.89
60
354
6530
4703
4209
61
2841
1148
19846
583588
34.01
61
402
6970
4936
4364
48
2609
1109
20438
563742
36.25
62
450
7403
5133
4500
39
2372
1076
20964
543304
38.59
63
508
7832
5305
4618
30
2131
1023
21447
522340
41.05
64
573
8230
5438
4718
24
1894
978
21855
500893
43.63
65
648
8615
5533
4795
19
1667
931
22208
479038
46.36
66
746
8954
5581
4846
15
1452
884
22478
456830
49.20
67
875
9255
5596
4871
13
1251
834
22695
434352
52.25
68
1015
9507
5563
4871
9
1064
785
22814
411657
55.41
69
1207
9704
5479
4841
6
898
736
22871
388843
58.81
70
1437
9846
5358
4786
6
749
686
22868
365972
62.49
71
1702
9917
5196
4701
4
618
637
22775
343104
66.38
72
2008
9931
4999
4592
4
505
588
22627
320329
70.64
73
2334
9871
4771
4460
2
408
540
22386
297702
75.20
74
2677
9747
4513
4302
2
326
494
22061
275316
80.10
75
3028
9557
4233
4125
2
257
449
21651
253255
85.49
76
3332
9307
3941
3929
1
202
408
21120
231604
91.19
77
3610
9001
3638
3722
1
156
366
20494
210484
97.37
78
3827
8643
3322
3496
120
329
19737
189990
103.88
79
3967
8237
3012
3267
91
293
18867
170253
110.82
80
4020
7799
2704
3029
69
258
17879
151386
118.10
SI
3980
7327
2411
2788
50
226
16782
133507
125.70
82
3916
6803
2123
2552
38
198
15630
116725
133.90
83
3658
6315
1846
2313
27
171
14330
101095
141.75
84
3370
5801
1596
2085
19
147
13018
86765
150.04
85
3040
5286
1366
1862
14
125
11693
73747
15856
86
2684
4776
1151
1650
10
105
10376
62054
167.21
87
2305
4281
957
1448
7
88
9086
51678
175.82
88
1937
3809
789
1261
5
71
7872
42592
184.82
89
1584
3353
640
1085
3
60
6725
34720
193.69
90
1269
2924
513
927
2
48
5683
27995
203,00
91
985
2535
404
784
2
38
4748
22312
212.80
92
747
2168
310
650
1
29
3905
17564
222.33
94
551
1845
231
531
22
3180
13659
232.81
94
396
1545
170
428
17
2556
10479
243.92
95
278
1279
119
338
12
2026
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255.71
96
198
1050
79
261
7
1594
5897
270.31
97
126
845
48
195
5
1219
4303
283.29
Addenda.
223
Age
98
99
100
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85
70
35
II
672
525
401
III
26
9
IV
140
96
59
29
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217
351
963
364.48
103
14
149
240
612
392.16
104
10
97
163
372
488.17
105
8
55
107
209
511.96
106
6
25
63
102
727.65
107
3
2
37
39
794.87
108
2
5
S
625.00
109
1
1
3
1
666.67
1000.00
224 Addenda.
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Addenda.
231
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232
Addenda.
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Addenda. 233
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ADDENDA II
In order to show a rapid application of frequency
curve methods to the graduation of mortality tables
when the number of lives exposed to risk at various
ages is known, the following data, relating to appli-
cants who had been rejected for life assurance on
account of impaired health, by Scandinavian assur-
ance companies is instructive. The original stati-
stics as collected by a committee of the insurance
companies were first published in the quinquennial
report (1910—1915) of the Danish Government life
Assurance Institution (The Statsanstalt) for 1917.
The material related to Scandinavian and Finnish
applicants who previously to 1893 (and in the case
of two Danish companies before 1899) had been re-
jected for life assurance. By a special investigation,
the committee followed up these rejections and sought
to establish whether the applicants were alive at July
1, 1899, or were previously deceased. Detailed re-
ports for the full period during which the risks were
under observation were available for 8,208 individual
applicants. For 2,023 applicants complete data were
not available.
The final statistical results of the Statsanstalt's in-
vestigation are shown in the following summary
table:
Addenda. 035
TABLE I.
Mortuary Experience of Rejected Risks of
navian
Life Companies.
Attained
No. Exposed Number
Age
to Risk of Deaths
15-19
434 6
20-24
3,831 28
25-29
11,405 145
30-34
17,644 233
35-39
19,442 318
40-44
17,600 324
45-49
13,971 296
50-54
10,179 295
55-59
6,640 264
60-64
3,927 194
65-69
1,995 96
70-74
836 71
75-79
306 32
80-84
98 20
85-89
12 3
The exposed to risk by separate ages and the
correlated deaths are shown in Table II in Columns
2 and 3, from which we, without difficult}', obtain the
crude or ungraduated mortality rates, as shown
Column 4.
We next assume a purely hypothetical frequency
distribution of the exposed to risk, according to age,
represented by a Laplacean normal probability curve
with its mean or origin at age fifty and a dispersion
equal to 12.5 years, as shown in Column 5. The fre-
quency distribution of the number of deaths on the
basis of the ungraduated mortality rates in Column 4
236
Addenda.
and the above-mentioned normal probability curve is
shown in Column 6, which may be considered as an
ungraduated compound frequency curve. *
Arranged in quinquennial age intervals this latter
frequency distribution is shown in the following sum-
man,- table:
Ages
No. of Deaths
13-17
51
18-22
75
23-27
329
28-32
711
33-37
1,464
38-42
2,498
43-47
3,649
48-52
5,377
53-57
6,238
58-62
6,232
63-67
5,254
68-72
3,605 1
73-77
2,536
78-82
1,425
83-87
1,169
88-92
351
93 or over
95
Total . . . 41,059
The above frequency distribution is now subjected
to a graduation by means of the Laplacean — Charlier
or Gram — Charlier frequency function. The mathe-
matical calculations give the following parameters:
1 A slight adjustment was made in the figures in column (6) corres-
ponding to age 70, and in the age groups above the age oi 88.
Addenda. 237
Mean Age 57.75 years
Dispersion 13.32 years
Skewness —0.0031
Excess —0.0037
Applying these parameters to standard probability
tables we obtain the usual Laplacean — Charlier fre-
quency curve. Distributing the 41,059 individual
deaths according to this frequency curve we obtain
column (7) which is the graduated death curve cor-
responding to the hypothetical exposure as- given by
column (5). The final mortality rates per 1,000 of
exposed to risk are then found by dividing (7) with
(5) and are shown in column (8).
In order to show how close the graduation by
means of frequency curves agrees with the actual
observations, I have made a calculation of the
" actual" to the " expected" deaths by quinquennial
age intervals as shown in the following table:
TABLE III.
Comparison between "Actual" and "Expected"
Deaths on the Basis of the Graduated Mortality
Rates of the Scandinavian Mortality Table for
Rejected Lives
No. Exposed
A S es to Kisk
15-19 434
20-24 3,831
25-29 11,4-05
30-34 17,644
35-39 19,442
40-44 17,600
Actual
Expectec
Deaths
Deaths
6
3.4
28
37.6
145
133.4
233
242.2
318
314.3
324
336.8
238
Addenda.
Ages
No. Exposed Actual
to Risk Deaths
Expected
Deaths
45-49
13,971 296
321.8
50-54
10,179 295
287.2
55-59
6,640 264
234.8
60-64
3,927 194
178.6
65-69
1,995 96
119.5
70-74
836 71
67.4
75-79
306 32
33.8
80-84
98 20
15.1
85-89
12 3
2.5
Total 108,320 2,325 2,328.4
Considering the somewhat meager experience on
which the graduation was based, I think it must be
admitted that the method of frequency curves comes
surprisingly close to the actual facts. In this connec-
tion it is of interest to note that the actuaries of the
Danish Statsanstalt made a graduation of the above
data on the basis of Makeham's method and obtained
from least square methods the following values for
the constants. x
A = 0.006
log B = 7.0566 — 10
log C = 0.025
The " expected" deaths according to this latter
graduation, and on the basis of the above experience,
amount in total to 2,317 as against 2,325 "actual"
deaths and 2,328 " expected" deaths according to the
frequency curve method. "Viewed from the stand-
1 See formula (6) page 192 of Institute of Actuaries Text Book. Life
Contingencies by E. E. Spurgeon, London, 1922.
Addenda.
239
point of the principle of least squares it is also found
that the sum of the squares of the deviations is smal-
ler under the frequency curve method than under the
method of Makeham, which seems to be pretty good
evidence of the soundness of the method in spite of
the fact that I throughout have worked with un-
weighted observations. If properly chosen weights
were applied to the observations even closer results
could be obtained.
TABLE II.
Mortality Experience of Rejected Scandinavian Risks
(Male).
(5)
(6)
(7)
G t d, du&t 6 d
f\ \
(2)
(3)
(3) : (2)
Hypo-
(5) X (4)
(8)
(1)
Exposed
No. ol
thetical
Crude
Death
(7) : (5)
ige
to Risk
Deaths
Expo-
Death
Curve
lOOOqx
sure
Curve
15
11
0.00000
792
5.6
7.07
16
31
1
0.03226
987
32
7.1
7.07
17
64
1
0.01562
1223
19
9.2
7.52
18
121
0.00000
1506
11.7
7.77
19
207
4
0.01932
1842
3
15.4
8.36
20
340
1
0.00294
2239
7
19.7
8.80
21
501
1
0.00200
2705
5
25.0
9.24
22
719
6
0.00834
3246
27
30.8
9.49
23
982
6
0.00611
3871
24
38.8
10.02
24
1289
14
0.01086
4586
50
47.8
10.42
25
1619
22
0.01359
5399
73
58.2
10.78
26
1986
23
0.01158
6316
73
70.6
11.18
27
2287
34
0.01487
7341
109
85.0
11.58
28
2597
29
0.01117
8478
95
101.7
12.00
29
2916
37
0.01269
9728
123
120.5
12.39
30
3180
38
0.01195
11092
133
142.0
12.80
31
3395
50
0.01473
12566
185
166.4
13.24
32
3564
44
0.01235
14146
175
193.5
13.68
33
3700
46
0.01243
15822
197
223.4
14.12
34
3806
55
0.01445
17585
254
257.0
14.61
35
3882
48
0.01236
19419
240
293.3
15.10
36
3943
64
0.01623
21307
346
332.8
15.62
37
3921
72
0.01836
23230
427
375.3
16.16
38
3880
66
0.01701
25164
428
420.0
16.69
39
3816
68
0.01782
27086
483
467.7
17.27
40
3737
66
0.01766
28969
512
517.6
17.87
41
3637
63
0.01732
30785
533
566.9
18.41
240
Addenda.
(i)
Age
(2)i
Exposed
(3)
No. of
(*)
(3) : (2)
(5)
Hypo-
thetical
(6)
(5) * (4)
Crude
(7)
Graduated
Death
Curve
(8)
(7) : (5)
to Risk
Deaths
Expo-
Death
1000 qx
sure
Curve
42
3539
59
0.01667
32506
542
623.3
19.17
43
3426
62
0.01810
34105
617
678.2
19.89
44
3261
74
0.02269
35553
807
732.7
20.61
45
3079
67
0.02176
36827
801
787.8
21.39
46
2941
61
0.02074
37903
786
842.4
22.23
47
2793
46
0.01647
38762
638
895.1
22.97
48
2653
61
0.02299
39387
906
945.9
24.02
49
2505
61
0.02435
39767
968
994.3
25.00
50
2348
61
0.02598
39894
1036
1039.0
26.04
51
2184
65
0.02976
39767
1183
1079.9
27.16
52
2024
66
0.03261
39387
1284
1116.0
28.33
53
1882
59
0.03135
38762
1215
1147.4
29.53
54
1741
44
0.02527
37903
958
1173.3
30.96
55
1610
62
0.03851
36827
1418
1193.0
32.39
56
1447
60
0.04147
35553
1474,
1206.9
33.95
57
1308
45
0.03440
34105
1173
1214.3
35.60
68
1189
47
0.03953
32506
1285
1214.9
37.37
59
1086
50
0.04604
30785
1417
1209.0
39.27
60
966
44
0.04555
28969
1320
1197.0
41.32
61
871
35
0.04019
27186
1089
1178.8
43.52
62
786
35
0.04453
25164
1121
1154.2
45.87
63
701
44
0.06277
23230
1458
1124.6
48.41
64
603
36
0.05970
21307
1272
1090.1
51.16
65
518
22
0.04247
19419
825
1050.7
54.11
66
453
24
0.05298
17585
932
1006.3
57.22
67
392
19
0.04847
15822
767
960.1
60.68
68
340
16
0.04706
14146
666
909.6
64.30
69
291
15
0.05155
12566
648
858.4
68.31
70
244
25
0.10246
11092
1136
804.2
72.50
71
193
17
0.08808
9728
857
750.9
77.19
72
158
13
0.08228
8478
698
695.7
82.06
73
132
9
0.06818
7341
501
642.4
87.51
74
109
7
0.06422
6316
406
589.1
93.27
75
91
8
0.08791
5399
475
537.7
99.59
76
74
10
0.13514
4586
620
486.8
106.15
77
58
8
0.13793
3871
534
440.3
113.74
78
45
4
0.08889
3246
289
393.8
121.32
79
37
2
0.05405
2705
146
351.9
130.09
80
31
5
0.16129
2239
361
311.8
139.26
81
24
6
0.25000
1842
461
274.5
149.02
82
18
2
0.11112
1506
168
241.6
160.42
83
15
4
0.26667
1223
326
209.5
171.30
84
9
3
0.33334
987
329
181.5
183.89
85
6
2
0.33334
792
264
155.9
196.84
86
3
0.00000
631
000
133.4
211.41
87
2
1
0.50000
499
250
113.4
227.26
88
2
1
0.50000
393
197
95.5
243.00
89
0.5
0.50000
307
154
79.2
257.98
Note: — The observations above age 87 are not reliable.
TABLE OF CONTENTS
CHAPTER I.
Introduction to the Theory of Frequency Curves.
Page
1. Introduction 1
2. Frequency Distributions 6
3. Property of Parameters 8
4. Parameters as Symmetric Functions 11
5. Thiele's Semi-Invariants 12
6. Fourier's Integrals 16
7. Solution by Integral Equations 19
8. First Approximation 21
9. Hermite's Polynomials 26
10. Gram's Series 33
11. Co-efficients and Semi-Invariants 41
12. Linear Transformation 45
13. Charlier's Scheme of Computing 47
14. Observed and Theoretical Values 51
15. Principle of Least Squares 53
16. Gauss' Normal Equations 57
17. Application of Methods 60
18. Transformation of Variate 69
19. General Theory of Transformation 70
20. Logarithmic Transformation 72
21. The Mathematical Zero 75
22. Logarithmically Transformed Frequency Curves. 77
23. Parameters Determined by Least Squares 82
24. Application to Graduation of a Mortality Table. 84 ,
25. Biological Interpretation 90
26. Poisson's Probability Function 94
27. Poisson — Charlier Curves 95
28. Numerical Examples 99
29. Transformation of Variate 101
CHAPTER II.
The Human Death Curve.
1. Introductory Remarks 105
2. Empirical and Inductive Methods 108
3. General Properties of Death Curves Ill
4. Relation of Frequency Curves 116
5. Death Curves as Compound Curves 121
Page
6. Mathematical Properties 124
7. Observation Equations 127
8. Classification of Causes of Death 131
9. Outline of Computing Scheme 138
10. Goodness of Fit 155
11. Massachusetts Life Table 159
12. American Locomotive Engineers 168
12a, Additional Mortality Tables 172
13. Criticism and Summary 182
14. Additional Remarks 186
15. Another Application of Method 195
16. Graduation of dx. Column 203
17. Comparisons of Methods 209
ADDENDA I.
Mortality Tables for:
Japanese Assured Males 218
Metropolitan White Males 219
American Coal Miners 221
American Locomotive Engineers ; . . . 224
Massachusetts Males 1914—1916 226
Michigan Males 229
Massachusetts Males (Series B) 231
ADDENDA II.
Mortality Experiences of Rejected Risks of Scan-
dinavian Life Companies 234
INDEX
Archimedes, 4.
Biological Interpretation of
mortality, 90 — 91.
Bi-orthogonal functions, 28.
30, 32.
Broggi, U., 53.
Bruhns, 72.
Brunt, 131.
Cauchy, Theorem of, 17.
Causality, Law of. 110. 117.
Charlier, 1, 2, 17, 19, 48, 51.
98. 122.
Charlier's A type series, 83.
B type series, 96.
Scheme of Com-
putation, 47.
Charlier — Gram series, 60,
64. 90, 93, 95.
Charlier — Laplace series,
53, 70, 116. 122. 123. 206.
Charlier — Poisson series, 93
96, 99. 122, 123. 206.
Coal Miners, American, 172,
174.
Component frequency cur-
ves. Mathematical pro-
perties of, 124.
Comte, August, 105.
Crum, F. S., 107.
Davenport, 102.
Davis, M., 60, 61.
da Vinci, Leonardo, 5.
Death curve, general pro-
perties of. 111 — 115.
Death curve as a com-
pound curve, 121.
de Vries, 60. 63, 100.
Dispersion, or Standard De-
viation, 40, 4 5.
Eccentricity, 98.
Edg-eworth. 122.
Empirical and Inductive
Method, 109. 110.
Error Laws of precision
measurements, 53.
Euclid, 190.
Eulerean relation for com-
plex quantities, 22.
Exposed to risk. 105. 106.
Fechner, 72.
Fourier, function, conju-
gated. 19, 21.
Fourier, integrals, 16.
Fourier, integral theorem
17.
Fourier series, 32.
Fredholm, 4, 69.
Fredholm determinants, 4.
5.
Fredholm integral equa-
tions, 32.
Frequency Distribution, de-
finition of, 6.
Gamma function, 103.
Gauss, 56, 147.
Gaussian algorithms of suc-
cessive elimination, 57.
Gaussian curve, 119.
Gaussian solution of nor-
mal equations. 57.
Geigrer, 99 — 100.
Glover. 187, 188.
Gram, 4, 5, 122.
Gram series, 33, 41, 53. 70,
83.
Gram-Charlier series, 60,
62. 64. 90, 93, 95.
Guldberg. 122.
Heiberg, 4.
Henderson, 121.
Hermite polynominals, 27,
33. 36, 38. 69.
Hoffman, F. L., 174.
Homogeneous Sum Pro-
ducts, 56.
Homograde Statistical Se-
ries, 94.
Horner. 189.
Integral Calculus. Founda-
tion of, 4.
Integral equations, 4, 5.
Frequency curve as so-
lution of, 19.
Fourier's 17.
Fredholmian. 32.
Japanese Assured Males,
176, 182.
Jevons, 93, 110, 139.
Jorgensen, 27, 51, 63, 70. 72,
102. 103. 122. 123.
King-, 199.
Laplace, 1, 69.
Laplacean probability func-
tion, 24, 73, 77, 95, 96,
Laplacean Normal frequen-
cy curve, 24. 26. 71, 81,
90.
Laplacean-Charlier series,
53. 70, 116, 122, 123. 206.
Least squares, principles of
the methods of. 53, 57.
Least squares. Parameters
determined by, 82, 85.
Lexis, 119, 120.
Little. J. P., 182.
Locomotive Engineers,
American, 168, 171.
Lowell Institute, 91.
MacLaurin Series. 2.
Massachusetts Males, 128,
159 — 167, 195 — 209.
Mathematical Zero, 75 — 76,
87.
Metropolitan Life Insurance
Co., 174 — 176.
Michigan Males. 131 — 158.
Modulus, 98.
Moir. 170 — 171.
Moments, of a frequency
function, 38, 47, 73.
Sheppard corrections for
adjusted moments, 80.
Mortality, biological inter-
pretation of, 90 — 93.
Mortality Tables:
American-Canadian, 84,
86.
American Coal Miners,
172 — 174.
American Locomotive
Engineers, 168 — 171.
Massachusetts Males,
159 — 167, 195 — 209.
Metropolitan Life Ins.,
Co., 174 — 176.
Michig-an Males, 131 —
158.
-Japanese Assured Males
176 — 182.
Myller-Lebedeff, Vera, 32.
Newton. 189.
"Normalalter", 119.
Normal equations, 56 — 59,
67,
Gauss solution of, 57.
Novalis. 199.
Nucleus of an equation. 19
Observation equations, 54.
127 — 131.
Orthogonal functions. 5, 36.
Orthog-onal substitution, 69.
Parameters, determined by
least squares, 82, 83'.
Parameters, .properties of,
8 — 10.
Parameters viewed as sym-
metric functions, 11.
Pearl, Raymond,, 91.
Pearson. Karl, 1, 2, 25, 38,
60, 90, 121, .192.
Percentag-e frequency dis-
tribution. 125—126.
Poincare, 111.
Poisson, Exponential Bino-
mial Limit, 95 — 97.
Poisson, Probability func-
tion, 94, 98. 101, 103.
Poisson-Charlier series, 93,
96, 99, 122, 123. 206.
Power Sums, 11, 47. 73.
Probability function, oTefl-
nition of, 6.
Reduction equations, 67.
Relative frequency func-
tion, 6.
Rutherford, 99, 100.
Semi-invariants. definition
of, 12.
Computation of, 4 6 — 4 8.
General properties of, 16.
Sheppard corrections for
adjusted moments, 80.
Standard deviation, 40.
Statistical series, homo-
grade, 94.
Sum products, homogene-
ous, 56.
Symmetric functions, 38.
Parameters viewed as, 11.
Taylor series, 2, 3.
Thiele, 1, 12, 19, 38. 61, 69.
72, 122.
Thompson, John S„ 182.
Transformation,
General theory of, 70.
Linear. 4 5, 6 2, 101.
Logarithmic. 72, 74, 77,
82, 87.
Of variates. 101 — 104.
Westere-aard, 119.
Wicksell, 72.
Yano, T„ 180.
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