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Mig Petes 4 Hi Sues eadatie ples piteereieie/ rhea niet ba aia birt ts se RaAsrol Nerina) ey et mae perry sh atrial A Helena i Pith a ae aeeow hy Dh er Seite as ire seh te Hh ieee fiat Rs ap has Nah ae pe Hehe Vag PNM E AIC) Cis Seay ated We pereaietinr an th fast sonata ce seve 4 Parrceend.teeee fr eeienayt nit f vb Rieu ae aie stewy: rye y at be lipeian ‘ooh ss ee ee ge Meee eiitiy: iN ite nal ipa estat he te i ated of ay depuis by sii edresrns US v 4 5 THE UNIVERSITY OF ILLINOIS LIBRARY, - AUHIGULI UKE Return this book on or before the Latest Date stamped below. University of Illinois Library L161—H41 » ~ ine ARS es > a ee ine. as wy m4 me ~wte * + nm we : hae’ " . eee >t Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/tablesforstatistOOpear TABLES FOR STATISTICIANS AND BIOMETRICIANS CAMBRIDGE UNIVERSITY PRESS Cc. F. CLAY, Manacer ZLondon: FETTER LANE, E£.C. €vinburgh: 100 PRINCES STREET ALSO Zonvdon: H. K. Lewis, Gower Street, W.C. and Wiu1am WesLEy AND Son, 28 Essex Street, W.C. Berlin: A. ASHER AND CO. Leipsig: F. A. BROCKHAUS Bombap and Calcutta: MACMILLAN AND CO., Lrp. Toronto: J. M. DENT AND SONS, Lrp. Tokvo: THE MARUZEN-KABUSHIKI-KAISHA All rights reserved TABLES FOR STATISTICIANS AND BIOMETRICIANS EDITED BY KARL PEARSON, F.R.S. GALTON PROFESSOR, UNIVERSITY OF LONDON ISSUED WITH ASSISTANCE FROM THE GRANT MADE BY THE WORSHIPFUL COMPANY OF DRAPERS TO THE BIOMETRIC LABORATORY UNIVERSITY COLLEGE LONDON Cambridge : at the University Press 1QI4 _ Vie . oS 9 bg 74 2 2» i Tad - ‘a 8 des ARV ha) ' a i > 7 <" ee ; " \ CJ . fe 4 4 ._58 , 4 We: 4 ' 1," 7 " = af : ‘ m b * 4 4\ \ ‘ : . . 4 A! 5 | Cambridge : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS. ‘o ‘ > P2\K BBIVERSITY OF KLLINOIS | x TURE LIBRARY. PREFACE AM very conscious of the delay which has intervened between the announce- ment of the publication of these Tables and their appearance. This delay has been chiefly due to two causes. First the great labour necessary, which largely fell on those otherwise occupied, and secondly the great expense involved (a) in the calculation of the Tables, and () in their publication. This matter of expense is one which my somewhat urgent correspondents, I venture to think, have entirely overlooked. It is perfectly true that only one single Table in this volume has been directly paid for, but a very large part of the labour of calculation has been done by the Staff of the Biometric Laboratory, whose very existence depends on the generous grant made to that laboratory by the Worshipful Company of Drapers. Our staff is not a large one and it has many duties, so that the progress of calculation has of necessity been slow. Even now I am omitting projected * Tables, which I can only hope may be incorporated in a later edition of this ~ work, e.g. Tables of the Incomplete B- and I’-functions, and the Table needed to ‘ complete Everitt’s work on High Values of Tetrachoric r when r lies between _— 80 and —1:00. It would only satisfy my ideal of what these Tables should be, _had I been able to throw into one volume with the present special tables, extensive tables of squares, of square roots, of reciprocals and of the natural trigonometric functions tabled to decimals of a degree. Logarithmic tables are ' relatively little used by the statistician to-day, which is the age of mechanical AA calculators, and he is perfectly ready to throw aside the fiction that there is any gain in the cumbersome notation of minutes and seconds of angle—a system which would have disappeared long ago, but for the appalling ‘scrapping’ of astronomical apparatus it would involve. But the ideal of one handy book for the statistician cannot be realised until we have a body of scientific statisticians far more numerous than at present. Statisticians must for the time being carry ) about with them not only this volume but a copy of Barlow's Tables, and a ~ set of Tables of the Trigonometrical Functions. 345435 vi Tables for Statisticians and Biometricians Beside the cost of calculating these Tables, to which I have referred, must be added the cost of printing them. I had to do this slowly as opportunity offered in my Journal Biometrika, and the Tables as printed were moulded, in order that stereos might be taken for reproduction. Even as it is, there are a number of Tables in this volume, either printed here for the first time (e.g. Tables of the Logarithm of the Factorial and of the Fourth Moment), or published here for the first time (e.g. Tables of the G(r, v) Integrals), the setting up of which has naturally been very expensive. From the beginning of this work in 1901* when the first of these Tables was published and moulded, I have had one end in view, the publication, as funds would permit, of as full a series of Tables as possible. It is needless to say that no anticipation of profit was ever made, the contributors worked for the sake of science, and the aim was to provide what was possible at the lowest rate we could. The issue may appear to many as even now costly; let me assure those inclined to cavil, that to pay its way with our existing public double or treble the present price would not have availed; we are able to publish because of the direct aid provided by initial publication in Biometrika and by direct assistance from the Drapers’ Company Grant. Yet a few years ago when a reprint of these Tables in America was only stopped by the threat to prevent the circulation of the book in which they were to appear entering any country with which we had a reasonable copyright law, I was vigorously charged with checking the progress of science and acting solely from commercial ends! Meanwhile without any leave, large portions of these tables have been reprinted, sometimes without even citing the originals, in American psychological text-books. Two Russian subjects have reissued many of these Tables in Russian and Polish versions, and copies of their works in contravention of copyright are carried into other European countries. It does not seem to have occurred to these men of science that there was any- thing blameworthy in depriving Biometrika of such increased circulation as it obtained from being the sole locus of these Tables, nor did they see in their actions any injury to science as a whole resulting from lessening my power to publish other work of a similar character. It is a singular phase of modern science that it steals with a plagiaristic right hand while it stabs with a critical left. The Introduction gives a brief description of each individual table ; it is by no means intended to replace actual instruction in the use of the tables such as * When issuing their prospectus in the spring of 1901 the Editors of Biometrika promised to provide ‘‘ numerical tables tending to reduce the labour of statistical arithmetic.” : Preface vii is given in a Statistical laboratory, nor does it profess to provide an account of. the innumerable uses to which they may be put, or to warn the reader of the many difficulties which may arise from inept handling of them. Additional aid may be found in the text which usually accompanies the original publication of the tables. In conclusion here I wish to thank the loyal friends and colleagues—Dr W. F. Sheppard, Mr W. Palin Elderton, Dr Alice Lee, Mr P. F. Everitt, Miss Julia Bell, Miss Winifred Gibson, Mr A. Rhind, Mr H. E. Soper and others—whose un- remitting exertions have enabled so much to be accomplished, if that much is indeed not the whole we need. I have further to acknowledge the courtesy of the Council of the British Association, who have permitted the republication of the Tables of the G(r, v) Integrals, originally published in their Transactions. To the Syndics of the Cambridge Press I owe a deep debt of gratitude for allowing me the services of their staff in the preparation of this work. Pages and pages of these Tables were originally set up for Biometrika, or were set up afresh here, without the appearance of a single error. To those who have had experience of numerical tables prepared elsewhere, the excellence of the Cambridge first proof of columns of figures is a joy, which deserves the fullest acknowledgement. Should this work ever reach a second edition I will promise two things, rendered possible by the stereotyping of the tables: it shall not only appear at a much reduced price, but it shall be largely increased in extent. KARL PEARSON. Biometric LABoRATORY, February 7, 1914. Errata The reader is requested to make before using these Tables the following corrections on pp. 82, 83, 84 and 85: For 1:77 N3, and 1:77 Vs, at the top of the Tables read 1177 V3, and 1:177V V3). When you can measure what you are speaking about and express it in numbers, you know something about it, but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatis- factory kind. Lorp KELVIN. La théorie des probabilités n’est au fond que le bon sens réduit au calcul ; elle fait apprécier avec exactitude ce que les esprits justes sentent par une sorte dinstinct, sans qu’ils puissent souvent s’en rendre compte. LAPLACE. CONTENTS PREFACE INTRODUCTION TO THE USE OF THE TABLES. TABLE XI. XII. TABLES * Table of Deviates of the Normal Curve for each Permille of Frequency Tables of the Probability ee Area and Ordinate of the Normal Curve in terms of the Abscissa Tables of the Probability Integral: Abscissa and Ordinate in terms of difference of Areas . Tables of the Probability Integral: Logarithms of Areas for high Values of Deviate Probable Errors of Means and Standard Deviations Probable Errors of Coefficients of Variation Abac for Probable Error of a Coefficient of Correlation r Probable Error of a Coefficient of Correlation: Table to facilitate the calculation of 1— 7° Values of the Incomplete Normal Moment Functions, First to Tenth Moments Numerical Values and Graph of Generalised | Probable Te: Values of the Functions y,, yr, and yy; required to determine the constants of a Normal Frequency Distribution from the Moments of its Truncated Tail : Tables for Testing Goodness of Fit: 3 to 30 frequency groupings : c - : Vv xill PAGE XV 1 XVil 271 xvii 9-10 Xxl iil meat) PAs} XXli 18 XXlil 19 xxill 20-21 xxiv 22-23 XXV 24 XXVIll 25 Xxx1 26-28 * The Roman figures to the pages refer to the Introduction, where the Table is discussed, the Arabic to the Table itself. B. b x Tables for Statisticians and Biometricians TABLE XIII. Tables for Testing Goodness of Fit: Auxiliary Table A, to assist in determining P for high values of ° XIV—XVI. Tables for Testing Goodness of Fit: Auxiliary Tables B—D. Numerical values needed in the calculation and extension of Tables of P. f XVII. Tables for Testing Goodness of Fit: Value of (—log P) for high values of y? when the frequency is in a fourfold Grouping . : , : : : : ; : XVIII. Probability of Association on a Correlation Scale: Values of (—log P) for an observed value of the Tetrachoric Correlation r and the Standard Deviation of r for zero Association ; : : : . XIX. Probability of Association on a Carrelaies Scale: Values of y? corresponding to Values of (— log P) XX, Probability of Association on a Correlation Scale: Values of logy? corresponding to Values of Tetrachoric r and .o; : : : ; XXI. ‘Probability of Association on a Geers Scale: hee to determine ,¢, for a Population of given size, divided into a fourfold Table with given marginal Frequencies XXII. Probability of Association on a Correlation Scale: Abac to determine the Equiprobable Tetrachoric Correla- tion r, from a Knowledge of logy? and ,o, . XXIII. Approximate Values of Probable Error of 7 from a four- fold Correlation Table: Values of y, for values of r XXIV. Approximate Values of Probable Error of 7 from a four- fold Correlation Table: Values of x, for Values of marginal $(1+a) : XXV. Table to determine the Brobabiliey that ine mean Soh a very small sample n (4 to 10), drawn from a normal Population will not exceed (in algebraic sense) the Mean of the Population by more than z times the Standard Deviation of the Sample . : XXVI. Table to assist the Calculation of the Ordinates of the Frequency Curve y=y,e?/"(1 + x/a)? XXVIII. Tables of Powers of Natural Numbers, 1 to 100 XXVIII. Tables of Sums of Powers of Natural Numbers 1 to 100. XXIX. Tables to facilitate the Determination of Tetrachoric Corre- lation: Tables of the Tetrachoric Functions 7, to 7. for given marginal Frequencies PAGE XXXll XXXlll XXXIV XXXV1 XXXVI XXXVI xlil xl xl xl xl xlv xlvi xlvi l PAGE 29 30 31 31 32 32 34 36 37 38-39 40-41 42-51 TABLE XXX. XXXI. XXXII. XXXIIT. XXXIV. XXXV. XXXVI. XXXVII. XXXVITTI. XXXIX. XL. XLI. XLII. XLII. XLIV. Contents Tables to facilitate the Determination of Tetrachoric Correlation: Supplementary Tables for determining high Tetrachoric Correlations (7 ="80 to 1:00) for given positions of the dichotomic lines : Table of the Logarithms of the Gamma Function, log T'(p) from p=1 to p=2 : Table to deduce the Subtense from a neledeest of Are and Chord in the Case of the Common Catenary A, Extension of Table XXXII for very flat Catenaries ; B, Extension of Table XXXII for very narrow Catenaries Diagram to find the Correlation praccenete r from the Mean Positive Contingency on the Hypothesis of a Normal Distribution Diagram to determine the Type of a Frequency Distri- bution from a Knowledge of the Constants 8, and 8.. Customary Values of 8, and 8, Diagram showing Distribution of Frequency Types for High or Unusual Values of 8, and ~,. : Probable Errors of Frequency Constants: Table for the Probable Error of 8, for given Values of 8, and ~, . Probable Errors of Frequency Constants: Table for the Probable Error of 8, for given values of 8, and 8, Probable Errors of Frequency Constants: Values of the Correlation of Deviations in f, and B,(Rg,,) for given values of B, and f, : Probable Errors of Frequency Constants : Eobahle Dee of the Distance between Mean and Mode for given Values of 8, and B, : : : i Probable Errors of Frequency Conceantae Probable Error of the Skewness for given Values of @, and 8, Values of the Frequency Constants f;, ,, 8; and Py for given Values of 6, and @, on thé Assumption that the Frequency can be described by one or other of Pearson’s Frequency Types : : Probable Errors of the Frequency @onstedte Probable Error of the Criterion of Type, «2, for given Values of 8, and Pf, : Probable Error of the ucniuation of F requency iene. Value of Semi-minor Axis of Probability sae for given Values of 8, and 8, ‘ ‘ PAGE lvii lxili Ixiil Ix lxii Ixil Ixil lxii Ixil Ixil lxiv xl PAGE 64 65 72-73 74-75 76-77 78-79 80-81 82-83 b—2 xii TABLE XLV. XLVI. XLVII. XLVIII. XLIX. LI. LIL. LIT. LIV. LV. Tables for Statisticians and Biometricians Probable Error of the Determination of Frequency Type: Value of Semi-major Axis of Probability ae for given Values of 8, and B, ‘ F Probable Error of the Determination of mee Type: Value of Angle between Major Axis of Probability Ellipse and Axis of 8, for given Values of 8, and 8, Probable Error of the Determination of Frequency Type : Diagram to determine for given Values of 8, and 8,, a given Frequency Distribution belongs definitely to a given Type of Frequency : ‘ : Probable Occurrences in Second Small Samples. Per- centage Frequency of each number of Successes in a Second Small Sample of m after p Successes in a First Small Sample of x 3 Logarithm of Factorial |x from n=1 to 1000 . Table of Fourth Moments of Subgroup Frequencies, i.e. n X a, for n=1 to 400, z=1 to 19, for Verification of the Calculation of Raw Moments General Term of Poisson’s Exponential Limit to the Binomial, i.e. of the so-called “Law of Small Numbers.” Value of e™m?/|« for m="1 to 150, and z=0 up to such figure as makes the Function significant in the sixth decimal] Place . : Table of Poisson’s Exponential Limit to the Binomial to be used in the Determination of the Probable Errors of Cell Frequencies n=1 to 30. Percentage of Occurrences in random Samples when the Population proportionately sampled would ae actually n to the Cell . : 5 . Angles, Ares and Decimals of pares Table giving (a) the Arc for each Degree from 1° to 180°, (6) the Are for each Minute and Second of Angle and (c) the Value in Decimals of a Degree of each Are and Second of Angle : : : : : Table of the G(r, v) Integrals. Values of log F(7, v) and H(r, v) for r=1 to 50 and ¢=tan™y from 0° to 45° é : . : : ; : Miscellaneous Constants in Frequent Use by Statisticians and Biometricians . PAGE lxiv lxiv lxx Ixxili PAGE 84-85 86-87 89-97 98-101 Ixxiv 102-112 Ixxvi 113-121 Ixxvii 122-124 Ixxix 125 Ixxxi 126-142 Ixxxili 143 INTRODUCTION TO THE USE OF THE TABLES For this introduction to the use of the Tables I have largely drawn on the prefaces to the original papers in Biometrika, and record here my acknowledge- ments to the authors of the same. INTERPOLATION. (1) A word must first be said as to interpolation. Let a function w be tabled for the argument # proceeding by differences Av=h. Then the scheme of such a table with the differences of wu is: a) lee Au_, | @3 | W» A*u_, Aus Aeu_, | @y | wy A?u_s Atu_, Au_, Aeu_, Lo Up A’u_,; Atu_s Au, Ane, 2, Uy Au, Atu_, etce., ete. Au, Au, Wp Ur Au, Atuy Au, Au, De Us, LU, At Au Aru, %, Uy A*u, Au, Ls Us where : Allg = Ups, — Us, A®u, = Ausy, — Ads, é A®u, = A®u,4, — A®u, ete., ete. Now there are three interpolation formulae which it is desirable to remember. If the function be required for the value «+ @h and this value be termed u, (@), then we have: g(a-@ - - Uy (8) = UW + OA, — ( 21 ) A®uy + ae a 2. Sue iaceeesi (1), — & Nahe Uy (0) =u + OAu, — Se Au, — a NOUS eancnes Gans eG (ii), where ¢=1—6. This is Everett’s formula*. And lastly: e — 2 Up (A) =U, + 84 (Au, + Au) + Spe = ee (Atuy- Atu) 2. (ii), where we work with the differences on or adjacent to the horizontal through a. * Journal of the Institute of Actuaries, Vol. xxxv, p. 452. XIV Tables for Statisticians and Biometricians [INTERPOLATION It is very rarely indeed that we need go beyond second differences, often the first will suffice. Not infrequently the inverse problem arises, namely we are given u,(@) and have to determine @ from it. If we only go as far as second differences, either (i) or (iii) gives us a quadratic to find @ and the root will generally be obvious without ambiguity. Usually it suffices to find = {uy (0) — uw }/Au, and then determine @ from d= (Uo (@) can Uy)/ Aug =F Ss A?u,/ Aug aGAU OMA OOGUODAOCS (iv) 4 or to find A = (uy (A) — w)/d (Aug + Au) and then O = (uy (A) — Uo)/5 (Auy + Au )- a ea (v) 0 0 0 =i 2! 1 (An + Au.) allele ofe(nieisiatete t Very often good results are readily obtained by applying Lagrange’s inter- polation formula which for three values of wu reduces to Up (A) = (1 — &) uy — $0 (1 — 8) wu 4+ $0 (14 8) my... eee (vi). Or, we may use the mean of two such formulae and take uy (A) =(1 — @)(1 — 40) uu + 40 (5 — 8) uy, —4O(1— 0) (uy +m) ... (vii). The resulting quadratics are respectively : A (4 (ty + Uy) — Uy) + OF ( — Wy) $y = H(A) =O eee. (vi), and = @ 4 (uy — uy + U4 + Ue) + 8 4 (Sty — Sly — U_y — Us) + Uy — Uy (A) =O ...( vii). (2) There are some tables in this book which are of double entry, e.g. those for the Tetrachoric Functions and for the G(r, v) Integrals. The eaaplee solid interpolation formula, using second differences, is : Ux, y= Up,o + TAU, 9 + YAU, +4 {a (a — 1) A’u,. + 2ay AAU, + 9 (y — 1) At of... (viii), where A denotes a difference with regard to x, and A’ with regard to y. But if we consider u,z,, to be the ordinate of a surface, and the figure, p. xv, to represent the «y plane of such a surface, then it is clear that, if P be the point z, y, and A, B, C, D, &c. the adjacent points at which the ordinates are known from the table of double entry, only the points A, B, C, D, J, and N are used by the above formula; and of these points, not equal weight is given to the fundamental points A, B, CG, D, for C only appears in a second difference. If another point of the fundamental square other than A be taken as origin, we get a divergent, occasionally a widely divergent result. If we use only four points—A, B, 0, D— to determine the value of the function at P, then we might take the ordinate 1] Introduction XV at P of the plane which (by the method of least squares) most nearly passes through the four points of the surface vertically above A, B, OC, D. We have then Ure, y = $ (Uo,o + a, 9 H Uo,1 + Ua,1) + F (t,o — Moo + Ua,1 — Uo,1) (@ — 5) + 4 (Aor — Uo,o + Uhy1 — M,o) (Y —'5) eee (ix), but by trial it has been found that this formula gives occasionally worse results than that for first differences, using only three points. To find by the methods of simple interpolation (with first or first and second differences) the points a and b, and then interpolate P between them, generally gives a fairly good result; but this result usually differs somewhat from that obtained by first simply F G H I © © O} © (Sly =U) (0, —1) Qk, i) (2, —1) (-1, 9) (0, 0) (1, 0) (2, 0) y A~s+e>a B iO} OQ reevereerees 55 cesanaocs © © Sews iy Cernaccedetess4 (Oe aeareee Oi é © © Spcesacoaccoo: x sastelinsae © © S D b C K (ely 1) (0, 1) (1, 1) (2, 1) © © © © R NV M L (=) 2) (0, 2) (ql, 2) (2, 2) interpolating e and / and then interpolating between e and f*. Various other methods for interpolation in n-dimensioned space will be found discussed by Palin Elderton in Biometrika}. The ideal method can hardly yet be said to be known, and it may well vary from table to table and from one part of the same table to another. One or other of the above methods will, however, suffice in practice for most statistical purposes. I consider now the individual tables. TABLE I (p. 1) Table of Deviates of the Normal Curve for each Permille of Frequency. (Calcu- lated by Sheppard and published by Galton in Biometrika, Vol. v. p. 405.) If NV be the total number in a population, zéz the frequency between « and az+6a, ¢ the standard-deviation, then the frequency curve of the population assuming its distribution to be Gaussian or normal will be: Vi. eee MPL Se Sate scuead weenae ence ratte ed (ix) * B. A. Report, Dover 1899. Tables of G(r, v)-Integrals, Report of the Committee (Drawn up by K. Pearson). : + Vol. vr. p. 94. Zz xvi Tables for Statisticians and Biometricians [I the origin being the mean. Table I. gives the value of a/c for each thousandth of the area of this curve,—each ‘ permille’—reckoned from left to right. In entering the table we enter from the left-hand column and top row if the permille be less than 500. For example, if the frequency below a particular value were 387 per thousand, the corresponding deviate would be — 0:2871, the number placed at the intersection of the °38 row from left and 007 column from top. The negative sign is always to be given when reading permilles below 500, because the deviate will be in defect of the mean, supposing increasing variates to be plotted as usual from left to right. On the other band if the permille be greater than 500 we enter the table from the right-hand column and bottum row. For example, if the permille be 748, the deviate is + 0°6682, the number placed at the intersection of the ‘74 row from right and ‘008 column from bottom of the table. The plus sign must be given, as the deviation is in excess of the mean, if the convention as to plotting variables has been observed. Illustration: The following observations were made on the nature of the degree taken by 1011 Cambridge undergraduates measured at the Anthropological Society’s Laboratory : Poll 487 Second Class 182 Third Class 189 First Class 153 Find the deviates of these on a normal or Gaussian scale. The sums from the lowest to each class top are 487, 676, 858 and 1011 respectively. If we term with Francis Galton the one man in a thousand of surpassing intelligence or special ability a “genius,” we have on multiplying by 0009891197 the reciprocal of 1011, the series for entering Table I. Thus we find: 4817 | “6686 “8487 | and -9990 : | | Hence: (-481) ‘0476 (668) "4344 (848) 10279 (999) 3:0902 (482) ‘0451 (-669)-4372 (849) 1-0322 = A 0025 A -0028 A 0043 — | Ax°7 ‘00175 | Ax 6 00168 Ax 7 -00301 = Deviates : — 0458 +4361 +1:0309 | +3-0902 Supposing with Pearson* that 100 units of intelligence (“mentaces”’) separate the lowest man of the First Class from the highest man of the Poll, we have + 1:0309 — (— 0458) =100/c, where o is the standard deviation of intelligence. Thus «= 100/1:0767 =92'88 mentaces. Hence we conclude that the range of Third Class men is from — 425 (ie. 92°88 x (—‘0458)) below to + 40:50 * Biometrika, Vol. v. p. 109. I—IIT] Introduction xvii (i.e. 92°88 x (+ 4361)) above the average undergraduate. The range of Second Class men is from + 40°50 to + 95°75 mentaces above the average undergraduate, and the range of First Class men all those with more than 95°75 mentaces above the average. The “genius” corresponds to an excess of no less than 287-02 mentaces. If we suppose that one individual in 1000 is completely feeble- minded or practically wanting in all intelligence, we should credit roughly the average man with 300 mentaces, and we should then have our range of intel- ligence on a Gaussian scale: Poll: below 296 mentaces ; Third Class: above 296 and below 340 mentaces ; Second Class: above 340 and below 396 mentaces ; First Class: above 396 mentaces ; “Genius”: above 587 mentaces. In rough numbers: Poll, below 300; Third Class, 300 to 350; Second Class, 350 to 400; First Class, over 400; “Genius,” over 600. Of course there is much that is hypothetical here, but the numbers give us some appreciation of the distribution of ability, and they serve to illustrate the construction of a Gaussian or normal scale. When more than three or four significant figures are needed Tables II and III must be used. Tasies II anp III (pp. 2—10) Tables of the Probability Integral: Area and Ordinate of the Normal Curve in terms of the Abscissa ; and Abscissa and Ordinate in Terms of Difference of Areas. (Caleulated by Dr W. F. Sheppard, and published in Biometrika, Vol. 1. pp. 174—190.) “Sheppard's Tables” were the first to express the Gaussian* or normal probability integral in terms of the standard deviation; they are so familiar to statisticians that it would almost seem a work of supererogation to explain their ‘use, which is further too manifold for full description. We can only give a few sample illustrations. It is most important when using these tables to pay attention to the signs of the differences recorded at the tops of the columns. Tilustration (i). The mean length of cubit in 1063 adult English males is recorded as 18°31 + 019 and of their 1063 adult sons as 18°52 + ‘021. Determine the odds against these two measurements being really identical, ie. random samples from the same population. We assume that the deviation of means and their differences follow the normal law. The difference is 0-21 and the probable error of this difference = V(-019)? + (021)? = 0”.0283. Since the probable error * The term is usual, but inaccurate. Laplace had reached the probability integral and suggested its tabulation several years before Gauss, B. ¢c XViii Tables for Statisticians and Biometricians [I1—II1 = '67449 x standard deviation, we have the standard deviation of the difference = 004196. Hence the deviation in terms of the standard deviation = 0°21/(0:04196) = 50048. Table II, p. 8, gives the area }(1 +) of the normal curve up to the abscissa 2/o. Noting the remark at the foot of the table, we have alo = 5:00, 4 (1 + a) = 999,999,713, a/o =5:01, 4 (1 + a) = -999,999,7278, = 145, A x 48 70, ajo = 50048, $(1+ a) = ‘999,999,7203. Hence 3 (1 — a) = 000,000,2797. Accordingly if we suppose the deviation as likely to be in defect as in excess, the probability that we shall reach the observed deviation, or exceed it, is 2x 4(1—a), and that we shall not is }(1+a)—4(1—a), or the odds against the result on a pure random sampling chance are -999,999,4406 to -000,000,5594, or 1,787,629 to 1, ie. overwhelming odds. Thus we may reasonably argue that sons in the professional classes in 1900 were substantially differentiated from their fathers by a longer forearm of about 1”. Illustration (11). Find the value in mentaces of the mean intelligence of Poll- men, First, Second and Third Class men as given by the numbers in the Illustration to Table I. The equation to the normal or Gaussian curve being za Atel? NV 2aro we easily find that if there be ‘tabled’ ordinates z, and z,* at the abscissae x, and z,, which cut off an area ,., then the mean 2%, of this area is given by Dass = Oye (2 — Za) ] (Maa IN) ace mia tio ome dse ree se eet ane (x). It will be sufficient to take the values of the abscissae already found, ie. a/o=— 0458, a2/o=+ 4361, x,/¢ =+ 10309, x,/o = + 30902. We require the 2’s for these. For example: 2="04, 2='398,6233 05, 2='398,4439 0="58, A, —1793 A, — 397. * The symbol z here used is that of the Tables, i.e. = e— 4 (2/2), - N29 II—IIT] Introduction xix Therefore by formula (i) p. xill: 2, = 398,6283 — 58 [1793] + > ** [397] = -398,6233 — 1040) = 3985241. + 48 \ Or, we might proceed as follows: for the Poll-men $(1—a)=-4817, hence a='0366. But from Table III, p. 9, which gives z for a: a= 03, z ='398,6603 a=04, 2 =398,4408 6=66, A= —2194 As= —627. Hence by formula (i): 2, =°398,6603 — -66 [2194] + 06 x 84 [627] = 398,6603) = =| ='398,5225. +70 We conclude therefore that z would be correct to five figures with second differ- ences, and that for four figures, first differences from either Table II or Table HI will suffice. If we use formula (ii) p. xiii—Everitt’s formula—we find from Table IL: a, = 398,6233 — 58 [1793] + °° 9° rg97) 4 92 * 9° (398) = "398,6233 ~ 1040 + a ee lis and from Table IIT: z, = 98,6603 — -66 [2194] + EGO (earl BA x aoa = 398,6603 = 14s] egy 7 398.6228. + 31) Working with formula (iii), Table II gives us z,=°398,5242 and Table III 2, = °398,5225 with second differences. We shall not therefore without higher differences get from any of our formulae closer than °398,522 with a possible error c2 = 398,5211, [627] XX Tables for Statisticians and Biometricians — {I1—III of 1 or 2 in the last place. This is, of course, amply sufficient for statistical purposes, where four figures as a rule would be sufficient. Using formula (i) p. xii we obtain: 2,=°39852, 2,='23450, Z,='36275, z,= 00337. Whence: = 0 — 39852 : rt. F Boy=+ sive 82730 = — 76°84 mentaces, 39852 — 36275 Bo =+ 39852 z ee o=+ 19250 = + 17°88 mentaces, 1869 = 36275 — ‘23450 F = 5 Lng = ae i o=+°712lo =+ 66:14 mentaces, By =+ 2oueY = DUE o=+1:5378o = + 142°83 mentaces, 1503 Dyn = + —— g o =+33700c = + 313-01 mentaces. Assuming as before the average man to have 300 mentaces of intelligence we find: Average Poll-man has 223 mentaces. Average Third Class man has 318 mentaces. Average Second Class man has 366 mentaces. Average First Class man has 443 mentaces. Average man of “genius” has 613 mentaces. Thus the average First Class Honours man is twice as able as the average Poll- man, and the average “genius” has not quite twice the ability of the average Third Class Honours man. Illustration (iii). It is required to determine normal curve frequencies corre- sponding to the following frequencies of the cephalic index in Bavarian skulls. Here the mean and standard deviation found by moments in the usual way are m= 83'069, o=3482. The deviations from the mean were next expressed in terms of the standard deviation, i.e. these deviations are — 13569, —12569, ...—0569, +4 °4381, +1431, +2431, ...+14481, and they are multiplied on a calculator by the reciprocal of the standard deviation, whence the column 2/o is found. Table II gives us $(1+a) knowing a/c; this has been calculated by first differences only. We shall consider as an illustration to Table XII, whether the normal distribution thus reached is to be considered a good fit to the observations. IV] Introduction Xxl Index Observed rio | 4(1+a) Pee oe | | | | 69:°5—70°5 1 | —3°9539 99996 | Under 705 1 70°5—71°5 1 | —3-6625 “99988 2 71:5 —72°5 = | =Syili | eens) 6 725-785 25 | —3-0797 | -99896 1°5 13:5—745 1°5 — 2°7883 ‘99735 3:3 th 5755 35 | —2-4969 99374 67 15:5 —16'5 12°5 —2:2055 | 98629 12:7 16:5 77°5 17 —1:9141 “97219 221 77-5 —78°5 37 —1°6228 | -94768 35°3 718-5—79'5 55 —1:3314 | -90846 519 719°'5—80°5 715 —1-:0400 “85082 70'1 80°5—81°5 82 — -7486 ‘7294 87-0 81°5—82°5 116 — -4572 | -67623 99°4 82:5 —83'5 98 — 1658 | 56584 104-2 83-5—84'5 107 "1256 54997 100°5 8h:5—85°5 82 ‘4170 | “66165 | 89°1 85-5—86'5 74 “7084 ‘76064 726 | 86-5—87'5 58 “9998 "84129 54°3 87 -5—88°5 345 1:2912 | -90167 37-4 88°5—89'5 19 | 15825 “94324 23°7 89°5—90°5 10 1:8739 | -96953 13°8 90°5—91°5 8 21653 | “98482 7-4 91-5—92'°5 3 | 24567 | 99299 3°6 92-5—93'5 15 | 22-7481 | -99700 16 93-5945 2 30395 | 99882 7 9 5—95°-5 1°5 33309 “99957 3 95°5—96°5 = | 36223 | «99985 | Over 95°5 +1 96:5—97'-5 = | 3°9137 | “99995 — 97:5 —98'5 | 4-2050 “99999 = | | Totals 900 ae Tene 900°2 TaBLe IV (p. 11) Extension of the Table of the Probability Integral F=4(1—a). (Calculated by Julia Bell, M.A., Drapers’ Research Memoirs, Biometric Series, vItt, p- 27.) It has been found needful occasionally to determine probabilities for deviations exceeding considerably the limit «/o=6 of Sheppard’s Table II. Illustration. If «/o = 34:31, determine to two significant figures the probability of a deviation occurring as large or larger than this. The table gives us: 33 34 35 36 23839135 a0 25295315 . 48393 a67-y4sey 199973 oao4 15°42967 28337855 Xxil Tables for Statisticians and Biometricians [v—VI Hence using formula (ii) p. xiii: (— log F’) = 25295315 + 31 [1499573] — SE x O08 49393 — 29S 5 43304 ae + 464868} = 9257-55542. ~ acai) Hence log F = — 257°55542 = 2358-44458, F = 2°7834/10, which measures the improbability required. TaBLE V (pp. 12—18) AND TaBLE VI (p. 18) Probable Errors of Means, Standard Deviations and Coefficients of Variation. (Table V calculated by Winifred Gibson, B.Sc.; Table VI by Dr Raymond Pearl and T. Blakeman, M.A. Biometrika, Vol. Iv. pp. 385—393.) If m be a mean, o a standard deviation and V=1000/m a coefficient of variation, for a population of n, we have Probable Error of Mean = "6T448980/V in = YyO oo e-eeneeeeeecceseeeeeeneneees (xi), Probable Error of Standard Deviation . = "67448980 /V In = Yoo .....eeeeseeeseeeeeeeseeees (xii), Probable Error of the Coefficient of Variation = 6744898’ x {! +2 (a) i / nce ee (xiii), = *6744898/V2n x = Ya X Alle is sein sete em gacrelsneteles onleics «eaice ice ag eee eee (xiv). Table V gives y, and y, for each value of n up to 1000, Table VI gives y for each value of V proceeding by units from 0 to 50. When the frequency n is greater than 1000, the tables may still be used iy taking out a square factor, which can be divided out at sight. Illustration (i). n = 2834 = 4 x 7085. n=708, x, ='025385; n=709, x, =°02533. *, N=708'5, y, ='02534, and .°. for n = 2834, we have 1 = 01267. Illustration (ii). In the case of the 900 Bavarian crania of the Illustration (iii) to Table IT the values m = 83:069, o =3'432, V—VIII] Introduction xxiii and therefore V=41315 were found. It is required to find the probable errors of these values. For 900, x, = "02248 and y,= 01590, hence the probable errors of m and o are p.e. of m=yx,0 ='077, pe. of ¢ =y.0 = "055. Next for V = 41315, vr, = 400639 + *1315 [1:00609] — 4 (1315) (8685) x [299] = 413852. p.e. of V=y. x w= '01590 x 413852 = ‘0658. Hence our results should be recorded as m = 83'069 + ‘077, o = 3'432 + 055, V =41315 + 0658. TABLE VII (p. 19) Abac for Probable Errors of r. (Calculated by Dr David Heron, drawn by H. Gertrude Jones, Biometrika, Vol. vit. p. 411.) The probable error of a coefficient of correlation 67449 FE = le (1 — 7°). To ascertain the value of this function approximately, turn the page horizontal, enter with the proper frequency on the scale at base, follow the corresponding vertical until the sloping line with the given correlation is reached, then move along the horizontal to the left until the scale of probable error is reached, which will give the required approximate probable error of the correlation in a population of the given size. Tilustration. r='671 and n=415. Probable error of »=‘018. The actual value is (0182. TABLE VIII (pp. 20—21) Values of 1—7*. (Calculated by H. E. Soper, M.A.) Illustration. As in the last example let | r='671 and n=415. Then probable error of r= y, x (1 — 7°) = 549,759 (from Table VIII) x ‘03311 (from Table V) = 0182. The value of 1 —7* for r to four instead of to three figures can be obtained by interpolation. XXiv Tables for Statisticians and Biometricians [IX TABLE IX (pp. 22—23) Values of the Incomplete Normal Moment Functions. (Calculated by Dr Alice Lee, Biometrika, Vol. v1. p. 59.) The nth incomplete normal moment function is defined to be 1 | x — ha? (2) ee i ies tice Bae Rae rnacaoac tcc L bn (&) Ven ae fy (xv) We take Mp () = Mn (@)/{(r — 1) (wn — 3) (n — 5)... 1} if m be tt (aan = py (a)/{(n — 1) (n — 8)(n—5)...2} if n be odd ) ‘ and mz, (a) is the function tabled. In multiple correlation (supposed normal), the frequency surface is 7 2 °~ On" Tae , PUhd cbt Bh fei 4 ee (xvii), where Ki R (S (Rypap*/ op”) + 2S" (Ryy Xp %q/ ep %q)} and Yr Nigel ce ot aeera cron || creaaincsod cote cauBoAaiod (xviii), Tm Tre Tng ++ 1 while R,, and R,, are the usual minors. x? = constant is the “ellipsoid” of equal frequency in n-dimensional space. The total frequency, i.e. the volume of the surface, inside any ellipsoid y is 1,=[*V V2ar tna(X) se, and I,|/N = WE ene) if n be oven} 2 \ 2, Sree ODT ee ar rey Cea) uae | Thus a knowledge of the incomplete normal moment functions enables us to predict for multiple variables whether an outlying observation consisting of a system of n variate values is or is not reasonably probable. If ,,/N =4, we obtain the ‘ellipsoidal’ contour y, within which half the frequency lies. This y, is the “generalised probable error” of Pearson and Lee. IX—X] Introduction XXV Values of the “generalised probable error” coefficients are given in Table X for n=1 to 11, and by means of a smooth curve the results may probably be extended to n=15. The values found for this extension are: | n=12 n=13 | n=14 | n=15 | | x | 3°367 3513 3°654 | 3°791 | Illustration (i). Let us consider long bone data for Frenchmen. 1='= femur, 2= H =humerus, 3 = 7'= tibia, 4= R=radius*, then by formula (xviii) p. xxiv: R=| 1, ‘8421, -8058, -7439 | | 8421, Ie 8601, +8451 | | 8058, “8601, 1, “7804 | 7439, -8451, 7804, 1 Further in cms: m, = 45°23, o, =2°372, Ma — se Olen mon — Nene: m,—s0'8l, o,=1°799, mM, = 2439, o,=1170. What is the chance that the following individual may be considered French ? Bh =3691, HW’ =26:82, 7 —30:56, RK’ —20:68: The deviations in terms of their standard deviations are : a, =(F’ —m,)/o,=— 3482, 2, =(H’ —m,)/o,=— 4059, a3=(1" —m,)/o,=— 3474, a2,=(R’— m,)/o,=—3:171. Further: 7 = 3°7810, =F = 65496, = = 43406, = = 3'°6508, = = 20231, = = 11404, ae = 02130, oe = 21946, a = 81/5, = = 0°6842. Whence x? = 16'741,035 and y = 40916, n is even, hence: nih — ee OY) =V27r.m,, * For particulars of these length measurements the reader must consult R. S. Proc. Vol. 61, pp. 343 et seq. and Phil. Trans, Vol. 192, A, p. 180, B. XXVi Tables for Statisticians and Biometricians [IX—xX and from the Table, p. 22, we have by formula (i), p. xiii: ms; (40916) = °397,7378 + °916[3650] — 4 (916) (084) [1043] = 398,0682. Hence I,|N = 2x x 398,0682 = “9978. Thus the odds are 9978 to 22, say 454 to 1 against a deviation-complex as great as or greater than this occurring in a French male skeleton, i.e. the bones very improbably were those of a Frenchman. Actually they were those of a male of the Aino race. Illustration (ii). The following are the ordinates of a frequency distribution for the speed of American trotting horses*. It is assumed that they form a truncated normal curve, and we require to determine (i) the mean of the whole population, (ii) its standard deviation, and (iii) what fraction the ‘tail’ is of the whole population. The values of frequency in an arbitrary scale are: q | Seconds Frequency | Seconds | Frequency 29-28 | 92°8 20—19 45:8 28—27 100°4 19—18 38°4 27—26 95:0 o—— lia 27°8 26—25 | 71°2 17—16 19°8 25—24 67°6 16—15 10°7 24—23 61°3 15—14 15°8 23-—22 61°74 14—135 79 22—21 44°8 18—12 5°O 21—20 44°5 12—11 2-1 11—10 5°6 Taking the working origin at 20—19 seconds, we find vy =—3°9214, v/ =32°545,666 for raw moment coefficients. Hence, if d be the distance from 29 seconds, i.e. the stump of the tail from the mean, and = the standard deviation of the tail about its mean: d =95 — 39214 = 55786 secs., 2? = vy — 2? = 17'168,288, and accordingly 27/d?=-5ol7. If this value be compared with those for yy, in Table XI, p. 25, it will be seen that we have got slightly more than the half of a normal curve, Le. not a true tail. We cannot therefore use Table XI, but must fall back on Table IX. * Galton, R. S. Proc. Vol. 62, p. 310. See for another method of fitting, Pearson, Biometrika, Vol. m. p. 3. IX—X] Introduction XXVii Let x be the distance from stump to centre of curve, n equal the area of truncated portion, and V be whole population. Then o a\o ” n/N =f +. = e 2 da! = TARYN G| (08/.G,) econ oem esr (Gx)F ne = No ie is oe alo N 2ar ) = Ne |= i ae aes d. rt \, VQer Vv 2 0 fyel asian ah se +] == (2 a'| I, —alo \ Qar alo a? Sep =No*jh+ A a os aw’ 0 VQar = Io Pose ipa Wale)! GooppSobcocooppaudeoeesauccoactoronsdec (xxl). Now d=2+, and >?=p, —2. 1 x == — m, (a/o) += {4 + m (2/c)} Hence ig - iar ; a nt ae (xxill), o 4+ m (a/c) 1 2 ge +m. (lay) (b+, (a/o)) — | F— = ma (e/a) Se (xxiv) a {E+ my (a/o)}?* i (44+ m,) ($+ m)— (== - m,) =, Var a and aan ea ASL) ee ee (xxv), = —m +a (E+ ms) 7 say, for brevity. Here m, and m, are given by Table IX and 4 + m, is the $+ 4a of Table II. Formula (xxv) has not yet been tabled for different values of a’, as it occurs much more rarely than the corresponding function for a true tail. If we take three values a’ = 0, 0:1 and 0:2, we have, from Tables II and IX, 2=0, $+m,='500,0000, 4+ m,='500,0000, —— — mM, = 398,9423, a ='1, » ='539,8278, » ='500,1325, » = '396,9526, a = "2, » = °579,2597, » = 9010512, » ='391,0427. Whence from formula (xxv) for the three values of S2/d?=-5708, 5528 and °5345. d2 XXViil Tables for Statisticians and Biometricians [IX—XI But our value of =2/d? is ‘5517. Thus we find by interpolation x = ‘1060. It remains to determine m, m, and m, for this value of a’, or simpler $+ m, 0 ’ 0 4+ m, and Fe —m, from the above values for 2’ ="1 and a’ ='2. We find 21 $+ m = 542,194, $4 m= ‘500,184, ave — m, = 396,598. 7 N Qar 396,598 oe Whence d/o — 542,194 + 1060 = 8375. Thus o = 5'5786/('8375) = 6°6610 secs., 2=x2 xo='7061 secs., n/N = 5422. This gives a mean of 28:29 secs. with a variability measured by 6°66 secs. Those actually registered as trotters are 54°/, of the population. TABLE X (p. 24) See under Table IX, p. xxv. TABLE XI (p. 25) Constants of Normal Curve from Moments of Tail about Stump. (Pearson and Lee, Biometrika, Vol. v1. pp. 65 and 68.) This Table may be of service in cases of the following kind: (a) In some cases a record is actually truncated as in the case of the American Trotters dealt with on p. xxvi. Or, again, we may take a record of stature obtained by measuring all men who exceed 69”, or a record of mental capacity found by measuring all persons with low intelligence in a community. (b) When we recognise heterogeneity, e.g. when we have a mixture of male and female bones, or two strains of trypanosomes, we can occasionally get a rough approximate analysis by supposing the tails to represent homogeneous material and then fitting them with normal curves. From the tails we get two components, and if their compound agrees fairly well with the observed total we have performed an analysis far more rapidly than by using the nonic equation*. Difficulties arise owing to deviations from Gaussian frequency being not infrequent; different dichotomic lines may give different results, and owing to paucity of material in the ‘tails’ and corresponding irregularity there will be large probable errors. Cases under (a) and (b) will be treated by exactly the same process, but our Table supposes that the distribution considered is less than half the normal distribu- tion. The rules for determining the mean, standard deviation and total frequency of the untruncated population are given on p. 25 under Table XT itself. * Pearson, Phil. Trans. Vol. 185, A, p. 84. XT] Introduction XX1x Tilustration (i). The following distribution represents the ‘tail’ of a group of 301 mentally defective children measured by G. Jaederholm using Binnet-Simon test methods* : Mental Defect in Years. ey 19 1 1g 1 Yay toy | ese eS aiesee |S ess ress: 8 > > i its © © & | | | | | | | | a 3S Pee Rael iam bpauen te ak) of bakes hs Ss) os | = = 5 iy © oS | oe ] ] ] ] al | - Frequency ... 34 18 | 13 3 4 4 il 1 78 We find, with origin at — 3°45, d=1°8077 ini years and 2?=2°7258. Hence ay, = =3/d? = 834, The Table (p. 25) gives us h’=2186 and w, = 2°833. Hence the mean is at distance from — 3°45 on left = 27186 x c, and for the standard deviation c=, x d=2°833 x 18077 = 51212 4 years = 2°5606 years, whence mean is at distance h = 5597 years from — 3°45. Now h’ = 2'186 corresponds by Table II to $(1 +a) ='98559. *, n/N =:01441, or N=78/(01441) = 5413. Thus the distribution is the tail of a population of 5413 individuals with a mean at 2°15 years of mental excess, and a standard deviation of 2°56 years. The example is merely illustrative and of no importance in itself. Illustration (11). Assuming that the correlation data follow the normal law, determine indirectly the ‘plural’ partial correlationt of habits of mother and health of baby for the limited universe of unclean homes. The correlations between the various characters found by tetrachoric tables are Habits of mother and health of baby: Tp» = 3060, Habits of mother and cleanliness of home: 7, =°7958, Health of baby and cleanliness of home: 3, = ‘2578. There are 947 out of 2931 homes which are not clean. Data from Bradford. Here n/N = 947 /2931 = 3231 =4 (1 —a); hence a= ‘3538, and by Table IIT wv =a/o6 =°459,049. * Mendelism and the Problem of Mental Defect. 11. On the Continuity of Mental Defect. Dulau & Co., 37, Soho Square, W. + Biometrika, Vol. 1x, p. 289. v8.6 Tables for Statisticians and Biometricians [XI Now on the assumption of a normal distribution: 2 N _2 (x2H/g2 NL’ s =| at ——— @ BEI) dy x Varo 8 e /3 —$2” i? = No uve da J a = No® {ps (1 ) — bs («’)}. Here Table IX (p. 22) shows that if s be odd, m, (2% ) = 398,9423, i.e. 1/V 2m, and (p. 28) if s be even, m; (2% )=°500,0000. Hence in obtaining the moment coefficients of the tail, about the mean of the whole population, m,(x’) should be subtracted from °398,9423 or from *500,0000 before the results are multiplied by (s — 1)(s— 3)... 2 or (8 —1)(s— 8)... 1, when s is odd or even respectively. It is convenient to term p.(% )—ps(a’) the com- plementary incomplete moment function of order s*. For s=1 and s=2, we have Np =o {m,(o ) — m, (x’)}, Nps = 0° NV {m.(% ) — mz (x’)}, for in this case the multiplying factors to proceed from m,(a’) to ps (2’) are both unity. Now a’ =2/o can be found when n is known from Tables II or III. Hence we have for the distance of centroid of tail from its stump, and for the square of its standard-deviation about its centroid: d=p/ -—l’c=c E {m, (2% ) — m (a')} — r'] S=c? Oty (20) — mz (a’)) — pw,” Of course m,(% )—m,(a’) is the z of Sheppard’s Tables II and III. Returning to our numerical example, we have from Table IX (p. 22): m, (45905) = °030,6721 + 5905 [162049] — $ (5905) (4095) [26358] = 039,9222, m, (2 ) — mM, (45905) = 359,02. Found directly from Sheppard’s Tables, it equals 35905. Similarly from Table IX (p. 23): Mz ('45905) = 008,1136 + °5905 [73,162] — 4 (5905) (4095) [80661] = ‘012,0630, and = my. (2% ) — my (‘45905) = -487,9370. * It is the function used by Dr Alice Lee and myself, Biometrika, Vol. v1. p. 65 to form Table XI, p. 25, but by an oversight not adequately distinguished in symbol from y, (x’) of p. 60 of the same memoir. XI—XIT] Introduction XXxi We thus reach d = c (:35902/°3231 — 45905) = °§52126, and 3? = o? {'48794/-3231 — (1:11117)} = -274,7860. Or the distance of centroid from stump, and the standard deviation of the tail are respectively We now find N= V1 — S/o? = 85155. The formula for the ‘ plural’ correlation coefficient is* Vio = NT ae ego ane pogo TEED Ape TO ORROCEEE (xxvl1), where Ts = NI%3, Tas = Naz, SONS. | ae Thus isp = POU, Actually taking out the universe of not clean homes, the correlation of habits of mother and health of child is 1615. The difference is considerable, but jt, is deduced from the entire population of 2931 homes, while -1615 depends on only a third of this number. The ‘singular’ partial-correlation is 3r,='1728, ie. the average relation between habits of mother and health of child for each individual grade of cleanliness of home. TaBLe XII (p. 26) Tables for testing Goodness of Fit. (W. Palin Elderton, Biometrika, Vol. 1. pp. 155—163.) The theory of testing frequency distributions for goodness of fit was first given by Pearson} and may be summed up as follows: If a frequency distribution or table contains v’ ‘cells’ and the contents of these cells be m,/, mz’, ms’... Mp/ in number, while m,, m., m;... mM, be the numbers that would occur in these cells on any theory; then calculate ens (rm My square of difference of theoretical and observed frequencies) te =sum ; = = )-Csxviii), theoretical frequency and the probability that random sampling would lead to as large or larger deviation between theory and observation is BP et asf te (hs 4 as wos) a ode DEEN aa Gta tase ens =a) if n’ be even Sees Nie eXG: Xe Mase ) Pe Per e oe ee ate) HE be odd * Pearson, Phil. Trans. Vol. 200, A, p. 25. + Phil. Mag. Vol. u. pp. 157—175, 1900. y XXXil Tables for Statisticians and Biometricians [XII Short provisional tables of P were given in the article referred to and were replaced in the following year by the present standard tables of Palin Elderton. In using the test for goodness of fit, due regard should be paid to the conditions under which it is deduced. It is assumed that the frequencies form a normal system of variates. This is legitimate only when in the binomial (p + q)", q is not very small as compared with p. If gq be not very small as compared with p, even for 7 finite, the binomial approaches closely to the normal curve. Accordingly in using the test it is desirable to club together small frequencies at the tails of curves or margins of surfaces. The difficulty becomes very obvious when theory can go by fractions, but observations only by units. The theory can be extended to cover much ground in all sorts of sampling*. Illustration. The following data for observed frequencies of cephalic index in Bavarian crania and for corresponding frequencies of a fitted Gaussian curve have already been considered on p. xx. Test the goodness of fit. Observed | Gaussian F | (m' — m)? | (m’) (m) m’ —m aes Under 75°5 9°5 12-4 — 2:9 68 75°5—76'5 12°5 12°7 — 02 ‘00 165775 17 221 | = 5:1 1°18 77-5 —78 5 37 35°3 + 17 08 78°5—79'5 55 | files) + 3:1 19 79:°5—80'5 71:5 70°1 + 14 03 | 80°5—81°5 82 870 | - 50 “29 | 81:5—82°5 116 | 99-4 | +16°6 2°77 82°5—83'5 98 104°2 — 62 37 83'5—84'5 | 107 100°5 + 65 “42 8h5—85'5 | 82 89°1 - 71 ‘57 | 85°5—86°5 74 | 72°6 + 14 03 | | 86:°5—87°5 | 58 | 64:3 + 37 25 | 87:5—88°5 |} 345 37°4 — 2°9 "22 88°5—89°5 19 23°7 -— 47 93 | 89:°5—90°5 10 13°8 — 3'8 1:05 | 90°5—91°5 8 T4 + 06 "05 | Over 91°5 | 9 | 63 + 2:7 1°16 | | —— | Totals | 900 | 900-2 18 Groups | x?= 10:27 | * A word of caution must be given about a recent extension by Slutsky (see Journal of Royal Statistical Society, Vol. uxxvu. pp. 73—84) who has applied it to test the goodness of fit of regression curves. In such cases the means and standard deviations of each array should, I think, be deduced from the theoretical surface, and the method would then agree with that illustrated on pp. xxiv—xxvi, i.e. on the probability of a given complex of variates differing from the run of values of a given population significantly. Slutsky after assuming that the observed frequencies and standard deviations of the arrays may replace the theoretical values, deduces his P from Elderton’s Tables instead of from the incomplete normal moment tables. He finds for the fit of a straight regression line, used to predict the probable price of rye at Samara from the price a month previously, y2=22'2, giving P=-02, a bad fit. Had he, however, used the theoretical standard-deviation of an array, i.e. c\/1 -72, instead of the very irregular observed standard deviations of individual arrays, he would have found x?=8-84, leading to P=-64 an excellent fit, which would probably have been still further improved by the use of theoretical total frequencies for the arrays, based, say, on a Gaussian distribution, XITI—X VI] Introduction XXxiil Taking from the column for n’=18 (p. 27) the values for y?=10 and 11, we interpolate P =°891 for y?= 10:27 by first differences, and conclude that in 89 out of 100 trials we should get in random sampling a fit as bad or worse than that observed, if the real distribution were Gaussian. Accordingly we say that a Gaussian curve describes excellently the distribution of Bavarian cephalic indices. TaBLes XITI—XVI (pp. 29—30) Auailiary Tables provided by W. Palin Elderton (Biometrika, Vol. 1. pp. 162— 163), useful for calculating values of P for x? outside the range of the existing table. For such cases we must turn back to the fundamental formulae (xxix) of p. xxx1, and the numerical values of considerable portions of these ‘formulae will be found evaluated in these auxiliary tables. Illustration. Find P for n’=11 and y?=78, iy? =39, hence by formula we have : 29)3 oe 7 ) =e (40 + 760°5 + 98865 + 96393375) = e—® x 107080°375, where the powers of 39 are taken out of Table XX VII (p. 38). Hence using Table XIII, log P = 17-:0625,1520 + 5:0297,0988 = 12:0922,2508, which gives us P =1:23659/10”. As a rule we can select »’ to be odd, but, if it is necessarily even, there is more trouble, not in the determination of the series, but in the evaluation of the integral yy AP peers f= y = I e dx. A table cf the values of #=3/ for y=5 to 500 has been given as Table IV (p. 11). This gives y?=25 to 250000 but the intervals are large. If greater accuracy be required then Schlémilch’s formula* 2 2 _i.2 ) Sine (11 1 1 ay = pacts =,/2 we {T Dees ads, x a XO MY OPH) VOVRFEDOHS 5 9 POE +2 OP +H EF 6) POEFD OP +H OF +6) (+S) P=e*(I +39 +4 (39) + 129 = 3 ; = = . SE eh atone nemererictae NG ye 2) (a 4) Ge 6) G8 £8) (+10) | must be used. Here af = ye x will be found in Table XIII, and the series converges fairly rapidly. * Compendium der hiheren Analysis, Bd. 11, S. 270, Braunschweig, 1879. B. e XXXIV Tables for Statisticians and Biometricians {XVIL Tabte XVII (p. 31) Values of (—log P) corresponding to given values of x? in a fourfold table. (K. Pearson: On a Novel Method of regarding the Association of two Variates classed solely in Alternate Categories. Drapers’ Company Research Memoirs, Biometric Series, vit. Dulau & Co.) If individuals be classed by the characters into A and not-A, B and not-B, we form a tetrachoric table of the form | | A Not-A }| Totals | 4 Bi ee a | b a+b | Not-B ce d e+d | Totals ate | b+d N | For such a table: NV (ab — cd)? X'= G4) (e+ d)b+d)(a+e) gives a measure of the probability of independence, and, if the two attributes are highly associated, x? will be large and P the probability of independence very small and largely outside Palin Elderton’s Table XII. Table XVII provides for such cases. Illustrations. The following tables are given by Mr G. U. Yule in his Theory of Statistics*. His conclusions with regard to them are: 1. Datura: “No Association.” | 2. Eye Colour in Father and Son: “Shows the tendency to resemblance.” 3. Houses in course of erection, Urban and Rural: “Distinct Positive Association.” 4. Imbecility and Deaf-Mutism: “ High Degree of Association.” 5. Developmental Defects and Dullness: “ Very high indeed.” It is required to measure the degree of probability that the variates in these five cases are independent. (1) Datura. (2) Hye-Colour in Father and Son. Colour of Flower. Father. Violet | White | Totals: | Light | Not Light | Totals | | “= | Prickly ... 68 RS | Smooth... 15 | | Totals ... | 83 | * Pp. 37, 34, 62, 33, 34 and 45 respectively. XVII] Introduction XXXV (3) Houses in course of erection im Urban and Rural Districts. (4) Imbecility and Deaf-Mutism. Built | Building } Totals Imbecile | Non-Imbecile | Totals ... | Urban ... | 4960 50 5010 | Deaf Mute 451 14,795 15,246 Rural 1749 12 1761 | | Non-Deaf Mute... | 48,431 32,464,323 | 32,512,754 | Totals ... | 6709 | 62 6771 | | Totals 48,882 | 32,479,118 [32,528,000 | (5) Developmental Defects and Dullness. With Defects Without | Totals 1186 | 2074 | | | Dull 888 Not-Dull 1420 22,793 24,213 Totals 2308 | 23,979 | 26,287 ¢’ the mean square contingency = y’/N and ¢ is the product-moment coefficient on the assumption that the ‘presence of the character’ is to be considered as a concrete unit*. The coefficient of mean square contingency C,=V¢?/(1+ ¢’). The following table gives the values of y*, ¢*, @ and C,, and the values of P deduced. | | Re | ¢° ? Cy P | i ee 3 a So ? == | | | (1) Datura ARO nee 200 “7080 | “085,301 "2921 | -2803 8713 | | (2) EyeColour ... ...... | 138°3265 | -133,327 | -3651 | -3430 | 1-035/10% | | (3) Houses see 200 on | 1°4393 | -000,2125 | -0146 | -0146 6948 (4) Imbecility and Deaf-Mutism 8014-62 00,2464 0157 | -0157 | 3179/1019 | 5 efects an ullness «5 | a206°79 “123,894 "8519 | °332 2°846/10! | (5) Def d Dull 3256-797 | 123.8 3519 | -3320 6/1070 For example, from Table XVIT by Formula (i) we have for y? = 133°3265 : 333265 33'3265) /16°6735) (-log P)=20809 + =="? 10-770} — 3 (“7 %?) (> 01) [026] 50 50 50 = 27-985, and therefore log P = 28015, which leads to P = 1:035/10*. Or again; for y?= 8014°62 : — log P =1785'324 4 a [108-561] — 4 es (a) [-000] =1738:498, * Pearson and Heron, Biometrika, Vol. 1x. p. 167. xxxvi Tables for Statisticians and Biometricians [{XVII—XX and therefore log P = 1739°502, and P=3179/10"™. In the first and third cases a different treatment must be used. For y?= 14393 we use Table XII. We have for n’=4: P ='801253 + -4393 [— 228846] — 4 (4393) (5607) [+ 48064] = 6948. Had we worked from Table XVII by Formula (i), we should have had P='6950. For x?=°7080, we can use Table XII, remembering that for y?=0, P=1. We have P =1-000,000 + 708 [— 198,747] — 4(-708) (292) [— 30,099] = 8624. Had we worked from Table XVII by Formula (i), we should have had P=°865, close enough for practical purposes. The true value of P worked from 2 eee eee Bees Oe re by using Table II is P = "8713. See p. xxxviil. Examining the values of P we see that having regard to the errors of random sampling we can only say that there is no relation between rural and urban districts and houses building or built; there is clearly no ‘distinct association,’ for in 69 out of 100 cases in sampling from independent material we should get more highly associated results. ‘There is likewise no association on the given material in the Datura characters. The other three cases have clearly very marked association, quite independent of any influence of random sampling. If we regard these three tables the order of ascending association judged by either ¢ or C, is (4), (5), (2), as against Mr Yule’s (2), (4), (5). If we disregard the non-significance and take merely intensity of association, without regard to random sampling, the order is (3), (4), (1), (5), (2), as against Mr Yule’s order (1), (2), (3), (4), (5). The best method of inquiry at present for relative association in the case of four-fold tables is, I hold, first to investigate P and throw out as not associated those cases like the ‘Houses, built and building’ above. Then to use either “tetrachoric 7,” or C, according as we are justified in considering the variates as continuous or not. mp (see p. xxxvii) may be used as control. TaBLES XVIII—XX (pp. 31—32) Tables for determining the Equiprobable Tetrachoric Correlation rp. (Pearson and Bell: On a Novel Method of regarding the Association of two Variates classed XVITI—XX | Introduction XXXVIi solely in Alternate Categories. Drapers’ Company Research Memoirs, Biometric Series, vil. Dulau & Co.) We have seen under the discussion of the previous Table how to find a measure of the improbability of two variates being independent, when they are classed in alternate categories. The difficulty in such cases is to appreciate the relative importance of very large inverse powers of 10. The object of the present tables is to enable us to deduce a tetrachoric correlation, 7, of which the improbability is the same as that of the given system supposing it to arise, when the two variates have the same marginal frequencies but are really independent. In order to do this we have to determine ,o, for the given marginal frequencies, i.e. the standard deviation of 7; on the assumption that ris really zero. This may be easily found from Abac Diagram XXI or from Table XXIV (see below). Table XVIII then gives us the value of (—log P) for each value of 7 and ,o,. If we now turn to our original table and calculate its y?, this as we have seen will correspond to a given (—log P). We now make the (—log P) from our y* correspond to the (—log P) from our 7; and ,o,, this gives us a value of 7; which has the same degree of improbability as our observed table. In other words, instead of trying to appreciate the meaning of inverse high powers of 10, we say that a table of the same marginal frequency would be as improbable if it had a tetrachoric correlation r, arising from random sampling of independent variates. Thus we read our improbability on a scale of tetrachoric correlation. We use our correlation merely as a scale to measure probability on. As log x? provides a more satisfactory basis for interpolation, and as many readers use logarithm tables and not calculators, log y? will be the form in which x? will be often presented. Table XX provides the value of 7; corresponding to given a, and given log y”. We will assume for the present that jo, can be readily found from the marginal totals: see p. xli below. Illustration. Obtain the values of rp for the five tables given above on pp. XXXlv—v. The values of log y? and ,o, are as follows: log x° 0%% () Datura ws i Seo ae 1:8500 1941 (2) Eye-Colour, Father and Son S06 2°1249 0514 (3) Houses in Course of Erection ie "1582 0634 | (4) Imbecility and Deaf-Mutism ee 3°9039 ‘0175 | (5) Developmental Defects and Dullness 35128 | -0201 | Of the values here recorded for log y* and ,o,, those for the first and third cases lie beyond the limits of our Table XX. But if the point ,o,=:0634 and log y* = "1582 be noted on the Abac XXII, p. 34, it will be seen that, having due xxxvili Tables for Statisticians and Biometricians {XVII—XX regard to the spacings of the correlation curves, the value of the equiprobable correlation is under ‘03, say ‘027. In other words no significant association can be asserted. In the case of yo, ='1941 we are thrown back on the original formulae*. In the first place we must find P for the given value of yx’, ie. ‘7080 (see p. xxxv). But for n’= 4 from formula (xxix), 1 ers i Aes P=2| | 6 *~ dy + ——en?* V2 J x ao. Xf = 2 {200,057 + -280,0088 x 84142} = '871,3256. To obtain 7 we have to use the formula below, where ,o,='1941, and 1 Z 5 i ( == 3) , the po, 4s, Ms being the normal moment functions of Table IX. ovr 2 2 ue = 1 — A SE crs E (V2m) — po (V2mr)} — a {us (V2m) — py (V2mr7)} P= — ip a ite (Bra) — po (Br) + yes (as (V2) — ps (V2mr)] siege dneedinn (xxxil). Substituting the values of yo, ="1941 and V2m = 4852,107, we have for r= 03, P=-90550, r='04, P=°'86501. Whence for P ='87133, we have r=‘038. We now turn to the three cases which fall inside Table XX, (2) Hye-colour, Father and Son. log yx? = 21249 oo, = 0514, r=05 logy? =2:0942 o> = “O5 Op =06 logy? = 22748 r=06 log y? = 21239 oo, = 06 a e r=07 log y? = 2:2935. Linear differences will suffice ='05 5 S(O weld (Phy SOS 9F, = “VD r=VU'9d T 3806 i af, 0010 oo, = 06 r=06+4 Hence ,o,.= "0514 gives sacl qlee T= 517 T7090 * O84 517 + 012 = 529. ll * Drapers’ Company Research Memoirs. Biometric Series VII. ‘‘A Novel Method,” ete.: see pp. 12, 13. XVITI—XX] Introduction Interpolating for yo, first, “We conclude that the equiprobable correlation is °53. (4) Imbecility and Deaf-nutism. log x? = 3:9039 07 = 0175, XXXIX r=0°95, 9o-= 01, log x2 = 43673; 0, ="02, log y?=3°7660. Hence: r=0°95, .o,='0175, logy? = 39163. Again: r=0:90, »o,= 01, log y2= 42207; ,o,= 02, log y? = 36197. Hence: r=0:90, .o,= "0175, log y?=3'7699. Interpolating log x? = 39039 between 3:9163 and 3°7699, we find rp = 0946. (5) Developmental Defects and Dullness. log y? = 3'5128, ,o, = ‘0201. r=0°8, .o,="02, logy? = 34097; oo, ="03, log x? = 3'0598. Hence: oo, = "0201, log y? = 34062. r=0°9, 97,= 02, log y? = 3°6197 ; »o, ='03, log y? = 3'2690. Hence: log y? = 3'6162, for ,o, = °0201. Thus, by interpolating log y?=3°5128 between 3°4062 and 3°6162, we find ac opile We have accordingly the following results : | ¢, P rp | ", Q (1) Datura Opt Seb 008 | *2803 8713 038 | —*188+°140 | — :282 (2) Eye-Colour ... ... ... | °8430 | 1035/10 | -529 | -5504°027/ ‘581 | (3) Houses nea eS ... | 0146 “6948 027 | —:081+-043 | —-190 (4) Imbecility and Deaf-Mutism | :0157 | 3:179/10179 | -946 *330+°012 | ‘907 (5) Defects and Dullness ..- | °8320 | 2846/1076 851 | °652+-:009 | :846 It will be seen that equiprobable r, confirms generally the results from P, i.e. the tables for ‘Datura’ and ‘ Houses’ give no sensible association. r; also confirms this view and shows that ‘ Houses’ is even lower in the scale than ‘ Datura.’ The order of rp is the same as that of Yule’s coefficient of association Q, but neither rp, Tt, C2, P or Q support the conclusions stated to flow from the percentages on xl Tables for Statisticians and Biometricians {|X XIII—XXIV p. xxxiv. Both rp and Q give very high results for (4) and (5), and this is in accord- ance with the view elsewhere expressed that for extreme dichotomies Q is not to be trusted. It may further be doubted, whether for such dichotomies the theory of the distribution of deviations on which 7p is based can in its turn be accepted. On the whole 7, seems to me the most satisfactory coefficient of association, to be controlled by results for rp in the cases where neither the dichotomies are extreme, nor the numbers so large or so small as to fall outside the moderate range of Tables XVIIJ—XX or Abacs XXI and XXII. Apacs XXI anp XXII (pp. 33—34). See after Tables XXIII and XXIV. Tastes XXIII anp XXIV Tables for determining approximately the probable error of a_ tetrachoric correlation. (Pearson, Biometrika, Vol. 1X. pp. 22—27. Tables calculated by Julia Bell, M.A.) Given a tetrachoric table ate |b+d| WV so arranged thata+c>b+danda+b>c+d, then if 41+a)=(a+b)/N, $(1+m)=(a+c)/N, and 7 be the correlation, we have approximately : Probable error of 7 = X1-X;,- Xa, *Xay? where x1 = 67449/VN, and is tabled in Table V, p. 12, _ Ved +u)$0 =a) H , ay dy ¢ H and K being found from the z column of Table II, p. 2, and ——— sin 7,\? : Xr, = V1l—r2 ve ~ ( 50° t) cave See nee eR eRe (xxxiv), / sin“, being read in degrees. x,, and x,, are tabled in Table XXIV and x, in Table XXIII (p. 35). ’ This value of the probable error is only approximate and may diverge con- siderably from the true value* for extreme dichotomies. In such cases the full formula must be used. * Phil. Trans, Vol. 195, p. 14. xo in formula (1) should of course not be included under the radical. XXIII—XXIV] Introduction xli When 7; is zero in the population and not in the sample, the standard deviation og, of r= 0 is given accurately by Fa Xe Mae Illustration (i). Tetrachoric r, for the Table 22,793 | 1,420 | 24,213 1,186 | 888| 2,074 ear 23,979 | 2,308 | 26,287 is “652. Find approximately its probable error. From Table XXIII: r='65, x, = "6785; r='66, x, = ‘6675. -s Xr="6785.— ‘0110 x ‘2 = 6763. Now (1+) ='9211, $(1+a,)=:9122. Hence from Table XXIII, Xa, = 18249 4-11 [754] = 1:8332, Xa, = 17623 + °22 [626] = 1-7761, Xa, Xay = 3°2559. X cannot be found from Table V in this case as NV is beyond its range. But it equals : 674.49 /\/ 26287 = 67449/162:13 = 00416. Thus finally p.e. of 7, = 00416 x ‘6763 x 3:2559 = 009. Illustration (ii). Find the value of ,o, for the table: 471 | 148 | 619 151 | 230 | 381 622 | 378 | 1000 Here $(1+%)='619 and 4(1+a,)=°622. Xa, = 12712 + 9 [36] = 12744, Nay = 12748 + 2 [89] = 12756. Fr = Xa, Xa,/¥ 1000 = 0514. In a similar manner the values for all the ,o,’s in the table on p. xxxvil were found. B. i xlii Tables for Statisticians and Biometriciuns [XXI, XXII Apac XXI (p. 33) For determination of the standard-deviation of the correlation coefficients obtained by random sampling from a four-fold table in which the correlation is zero. (Drapers’ Company Research Memoirs, Biometric Series, vil. G. H. Soper’s Abac.) Method of use: Enter with the total frequency of the sample on the left-hand scale, and with the first value of 4(1+ a) on the bottom scale. The horizontal through the former and the vertical through the latter meet at a point. At this point pass up the diagonal to the left-hand scale again. Where you meet that scale pass along the horizontal until you meet the vertical through the second value of 4(1+a). Then from this point pass along the diagonal again to the left- hand scale, whence traverse the horizontal to the right-hand scale and there the required value of ,o, may be read off. Illustration (i). Find the value of oo, for the case just given of N=1000, 3$(1+a)=619, £0 + 4,)=622. The vertical through *619 meets the 1000 horizontal in a point whose diagonal reaches the left-hand scale almost exactly in 620. Whence passing horizontally we reach the vertical through ‘622 in a point about midway between two diagonal lines. Passing up midway between these two diagonals, we reach almost exactly the 380 line on the left-hand scale. Passing across to the right-hand scale along this line, we see that we are slightly above the middle of the division between ‘050 and ‘052, say ‘0512. The actual value of oo, is 0514. Illustration (ii). Let N=‘6771, 4(1+4)="7399, 4(1+4,) = "9908. A similar process gives first 450 on left-hand scale and then about 248, whence crossing to right-hand scale we find ,o,=°0635 instead of ‘0634 actual. Apac XXII (p. 34) Abac to determine from log x? and a, the value of the equiprobable correlation rp, for a fourfold table. (Drapers’ Company Research Memoirs, Biometric Series, vit. G. H. Soper’s Abac.) The rule is very simple: Enter the Abac with the proper value of ,o, on the scale at the foot and rise on the vertical till the horizontal through the proper value of log y? on the left-hand scale is reached. Then follow the curve through the meet of these two lines to the right-hand scale, where the requisite correlation will be found inscribed. Illustration. Take the Table for Eye Colour in Father and Son given on p- xxxiv. Here, as just shewn, »,='0514 and (p. xxxvil) log y2=2'1249. If we enter with the vertical through ‘0514 on the scale at the bottom, and the horizontal through 2°1249 on the left-hand scale, the curve through their point of intersection reaches the right-hand scale just below the °53 mark, say 529. This agrees with the correlation found above (p, xxxviii) by interpolation from Table XX. XXV] Introduction xliii TABLE XXV (p. 36) Value of the probability that the mean of a small sample of n, drawn at random rom a population following the normal law, will not exceed (in the algebraic sense) the mean of that population by more than z times the standard deviation of the sample. (“Student”: Biometrika, Vol. vi. p. 19.) When x is greater than 10, it will be sufficient asa rule to use the approximate result PoE IT ie onecn ane Reet eee (xxxv) Mipee Bee, hm etal lee Re | as a measure of the probability. This may be found from Table II. Illustration (i). Experiments of A. R. Cushney and A. R. Peebles on the difference in effect of Dextro-hyoscyamine hydrobromide and Laevo-hyoscyamine hydrobromide*. Additional Hours of Patient Sleep (Laevo — Dextro) 1 +12 2 +2°4 3 +1°3 4 $713) 5 (0) 6 +1:°0 7 +1°8 8 +0°8 9 +4°6 10 +174 Mean es ape +1°58 Standard Deviation Worl + 1:58 % 2= 1:3 Liv Table XXV shows that for z=1°35: P=‘99854, or the odds are 666 to 1 that leavo- is a better soporific than dextro-hyoscyamine hydrobromide. Illustration (11). Difference in weight of crops of potatoes grown by Dr Voelcker with (i) sulphate of potash and (ii) kainite as artificial manure. * Journal of Physiology, 1904. xliv Tables for Statisticians and Biometricians Gain by sulphate of potash. 1904 (a) 10 ewt. 3 qr. 20 lbs. . (b) 1 ton 10 ewt. 1 qr. 26 lbs. 1905 (a) 6 ewt. 0 qr. 3 lbs. “ (b) 13 ewt. 2 qr. 8 lbs. [XXV Average gain=15'25 ewt., and the standard deviation=9 ewt., 2=15:25/9 = 1-694. Here n= 4, and Table XXV gives us P=-9653 + 0:94 x [46] = 9696, or the odds are about 32 to 1 that the sulphate of potash is a better dressing than kainite for potatoes. Illustration (iii). Test whether it is of advantage to kiln-dry barley seed before sowing. The following table gives price of head corn in shillings per quarter for 11 sowings, the first seven in 1899 and the last four in 1900. Not Kiln-dried Kiln-dried A 26°5 26°5 0 28 26°5 15 29°5 28°5 1 1899 ~ 30 29 1 27°5 27 05 26 26 0 29 26 3 29°5 28°5 1 28°5 28 05 bas E 29 1 28°5 28 05 Mean 200 28 “91 Standard Deviation “79 The Gaussian curve gives ‘a %. — S872 Paa/g fe de. De I a Here w=-91/'79 =1:1519, and if a’ =a/N1/8 =1:1519 x 2°8284 = 3258, il B28 ya, SS Bole ) V2Qar ek) which evaluated by Table II, p. 6, gives P='99944, or the odds are 2845 to 1 in favour of not kiln-drying seed barley. XXVI} Introduction xlv If we had actually worked with the non-approximate formula, we should have found P=-9976; or odds of 416 to 1, considerably less than the approximate formula provide, but not enough difference to vitiate any conclusion likely to be drawn in practice*. TABLE XXVI (p. 37) Table for use in plotting Type III Curves, ve. x y=ye ?a(I + =). Beene seweleg oeuineinne os aa (xxxvi) (W. P. Elderton, Biometrika, Vol. 11. p. 270.) Rule: Taking p for the curve, multiply the values in the Table by p in succession on the machine with p on as multiplier. Then subtract the results from the logarithm of y, and we have the logarithms of the ordinates of the curve at the abscissae found by multiplying X in the first column of the Table by a of the curve. The curve can then be plotted. Its origin will be the mode. It is usually quite unnecessary to use the whole series of ordinates, either alternate ordinates will suffice, or we cut off one or both tails at a considerable distance from their tabulated values. Illustration. The frequency curve of barometric heights at Dunrobin Castle is given by the curve — 99-9393 —~ 22.9323 y = 391400 9? r-7661 (1 + 768i) The range X =—°65 to +°90 is easily seen to be sufficient. Column (1) of the accompanying table gives aX for these values, the second gives 22-9323 x (logy (1+ X)— X log, e); * The three illustrations above are drawn from ‘‘Student’s” original paper. He gives (J. c. p. 19) the values for P as drawn from the Gaussian for n=10 to compare with those obtained from the full formula, They are,—corrected for slips : z Full Formula Gaussian F4 Full Formula Gaussian ol *61462 “60411 Teh *99539 | -99819 “2 ‘71846 “70159 1:2 “99713 | 99925 3 "80423 “78641 1°3 “99819 *99971 4 “86970 "85520 14 “99885 “99989 5 91609 90691 11553 *99926 *99996 6 94732 *94375 16 99951 -99999 sf ‘96747 *96799 Ly “99968 — 8 -98007 98285 18 “99978 = 9 ‘98780 99137 19 “99985 — 1:0 99252 *99592 2-0 *99990 = Clearly even for n=10, the Gaussian ascends too rapidly in P, and this must be borne in mind in deducing conclusions for z=1 and upwards when n=11 to 20, say. xlvi Tables for Statisticians and Biometricians [XXVI—XXVIII actually these values are negative and must be subtracted from log y, i.e. 1°592,621; the resulting values are given in the third column. In column (iv) are given the antilogarithms of the numbers in column (111), and these must be plotted to the values in column (i) to obtain the graph of the curve which is a good fit. (i) (ii) (111) (iv) | c=ax pllog,)(1+X) — Xloge]*) log y y | — 977 2°474,991 | — -882,370 13 | — 888 | 1°923,630 331009 ‘AT -— 7:99 | 1-472,368 "120,253 1-325) { = yeni 1°103,755 “488,866 3°08 | | = 6:22 "804,557 "788,064 614 | = ee | 564,456 1-028,165 10°67 | — 4:44 | "375,287 1:217,334 16-49 — 3:55 -230,493 1°362,128 23°02 — 2°67 "124,683 1°467,938 29°37 — 1°78 “053,386 1°539,235 34°61 — 0:89 "012,888 1:579,733 38°00 | 0-00 -000,000 1°592,621 39°14 | 0:89 012,039 1°580,582 38:07 1-78 | 046,713 1°545,908 35°15 2-67 | “101,957 1-490,664 30°95 3°55 | ‘176,074 1°416,547 26°09 4:44 267,482 1°325,139 21°14 5°33 374,828 1:217,793 16°51 6:22 -496,920 1:095,701 12°47 Tale “632,702 959,919 9°12 7:99 | “781,189 "811,432 6:48 8°88 “941,509 651,112 4-48 9°77 | 1°112,905 "479,716 3:02 10-66 1:294,689 *297,932 1:99 11°55 1-486,196 "106,425 1-28 12-44 | 1°686,831 | — -:094,210 “80 13°32 1°896,111 — °303,490 “50 14-21 2°113,510 — *520,889 “30 | 15°10 | 2°338,613 | — °745,992 18 15:99 | 2°570,963 — 978,342 ‘11 Once the reader is used to the process it will be found to work readily, and the same multipliers are kept on the mechanical calculator throughout. TaBLes XXVII anp XXVIII (pp. 38—41) Tables of the Powers and Sums of the Powers of the natural numbers from 1 to 100. (W. Palin Elderton, Biometrika, Vol. 11. p. 474.) These tables can be used in a great variety of ways, for example in finding the roots of equations, or in fitting parabolae of various orders to curves. Illustration (i). Find the positive root of the equation : $ (7) = 002,72677 + 057,1497% + 017,192" + '083,578r4 + 088,3317* + 134,7177° + r — 560,386 = 0. * Actually these values are negative, and are therefore subtracted from log yo to give (iii). XXVITI—XXVIII] Introduction xlvii The positive root is less than ‘56, but the term in 7? shows that it must be less than 52. Take ‘52 and ‘50 as trials. From Table XX VII we have Ist 520,000 and 500,000, 2nd 270,400, ~—--250,000, 3rd 140,608 ,, _~—-125,000, 4th 071,162 , — 062,500, 5th 038,020, ~—-081,250, 6th 019,771 ,, 015,625, ihe = -OLOISi-. -00K813: Multiply out by the coefficients of $(7), retaining the products always on the arithmometer. We find $ (52) = + 016,384. ¢ (50) = — 008,990. Interpolating r= "52 — $8384 x 2= 5071, which is correct to last figure. Illustration (ii). Fit a cubic parabola to the data below, giving the average age of husband to each age of wife in Italy (see Biometrika, Vol. 11. p. 20). We will suppose each observation to be of equal weight,—this is of course not the fact, but it will illustrate the general method of fitting parabolic curves. In the paper just cited illustrations are given up to parabolae of the sixth order. The object here is to show the use of Table XX VII. Age of | Probable Age | Age of | Probable Age | Age of | Probable Age | Bride of Groom Bride of Groom Bride | of Groom | 155 | 25:0 25°5 27°0 35°5 | 36°0 16°5 | 25°2 26°5 27°5 36°5 37°0 icon 25°4 27°5 28°0 37°5 38°5 18°5 25°5 28°5 29°0 38°5 39°5 19°5 25°5 29°5 30°0 39°5 41°5 20°5 25°5 30°5 32°0 40°5 41°5 21°5 25°75 31°5 33°0 41°5 42°5 22°5 26°0 32°5 33°5 42°5 43°5 : 23°5 26°0 33°) 34°0 43°5 43°5 | 24°5 26°8 34:5 34°5 44°5 43°5 = aes = | = 45 ‘5 43:5 The ages of groom have been taken as approximate means. Now we can take our axis of , the age of bride through 30°5, and the age of groom to be measured from 32°0. « will accordingly range from —15 to +15, and the age 32+y of groom will range from y=~7 to y=11'5. We can now re-arrange the above table in a form suitable for working on the following table. Then the squares, cubes, and if necessary, higher powers of x are taken from Table XX VII, p. 38, and are given as Columns (iii) and (iv) below. The entries in Column (i) are then multiplied by those in (ii), (ili) and (iv) by continuous process on the machine, and xlviii - Tables for Statisticians and Biometricians [XXVIII it is not needful to enter separate products, the sums being reached which are placed at the foot. Next from Table XXVIII we read off S(e)=0, S()=2(S8(15), S(@)=0, S() =2(8 (154), S(#)=0, S(a)=2(S (15%). These give us: S (a?) = 2480, S (a) = 356,624, S (a*) = 6096,5840. We have now all the numerical data for a solution. Let the required cubic be : Y= CoH GL + Cok? + Cy". Then we must make w= S(y—¢c,— 2 — G42 — ¢,2°)? a minimum. The resulting equations are S(y) =a@S(1) +a S(z) +¢,8 (2) + 6,8 (2°), S (ay) =o,S (x) + ¢, 8 (2) + oS (2*) + ¢,8 (#4), S (xy) = cS (a) + oS (a) + cS (24) + c,8 (2°), S (ay) = cS (a) + oS (at) + cS (2°) + 58 (2°). (i) (ii) (iii) (iv) (v) (vi) (vii) at 7 y r ae x xy xy wy = ea — 7-0 =—15 995 | —3375 | —_ = a - 68 =a! 196 | —2744 | = | as = — 66 -13 | 169 | —2197 | aa = = — 65 = iz 144 | —1728 = —_ 2* = 65 =ilil iPIL || Sse = a as — 65 —10 100 | —1000 = = = = Ge = 81 | — 729 = = = — 60 = & 64 | — 512 = = a = GD) = 7 49 | — 343 ae ae == = 52 = & 36 | — 216 = | = = — 50 = 5 oy |) = 118} = = att — 45 = 3 16 | — 64 = a = = 2) | =e 3n|) wee a ay = de 2a = Bhi) = 2 Aas = ee a — 2-0 = il it sal = am xt 0 0 0 0 = = = 1:0 1 1 ra = | = = 1°5 2 4 8 = | ae = 2:0 3 9 27 = | = = 2°5 4 16 64 = J8 = 4-0 5 25 125 = | = = 4:5 6 36 216 = = = 6°5 7 49 343 = | a = 75 8 64 512 = = — 9°5 9 81 729 | = pei = 9°5 10 100 1000 = a = 10°5 11 121 1331 = ifs oi, 11°5 12 144 1728 = zh a 115 13 169 2197 = a = 11°5 14 196 2744 = = = 11°5 iy |) 2a 3375 = | = =. al : . BEE (ie ALS eel Sh ze: S (#)=23°65 S (wy) =1833'45 S (x2y)=4560°35 |S (23y) = 248,807°85 | XXVIII] Introduction xlix Write Ci creo — LO cen O21 O0crb; — i O00cs- Then our equations are 23650 = b, x 31000 +b, x *24800, 183345 = b, x 24800 +b, x 35662, 45603 = b, x -24800 +b, x 35662, 248808 = b, x 35662 +b; x 60966 ; giving b, = — 58626, .. Cy = — 58626, b, = 1686453, C, = '016,8645, b, = 959613, ¢, = 959,618, b, = — 1°532,144, es = — 001,532,144, and the required cubic is y = — 58626 + 959.6132 + 016,86452° — -001,532,1442°. 45 SF 40 35 Probable Age of Groom. 30 Fd aN i 15 20 25 30 35 40 45 Age of Bride. The graph of the cubic and the observations are given in the accompanying diagram. If X and Y be the actual ages of bride and groom, then Y =61°30457 — 4344,941.X + °157,0553.X? — -001,53214X°. For higher parabolic curves fitted to the same data, see Biometrika, Vol. I. pp. 21—22. B. g ] Tables for Statisticians and Biometricians [XXTX TABLE XXIX (pp. 42—51) Tables of the Tetrachoric Functions. (P. F. Everitt, Biometrika, Vol. vu. pp. 437—451.) The purpose of these tables is to expedite the calculation of tetrachoric 7, the correlation coefficient from a four-fold table, when we suppose the variates to be Gaussian in the law of their frequency. Let the table be ate, b+d| WN where @ is the quadrant in which the mean falls, then 6 +d and c+d are clearly each less than $V. Let Tm=(b+d)/N=1(1-4@), 7 =(c+d)/N =i(1—a,), then ALN = ToT + Ty Ty 1 TaTe TE wee ETT POP 00 ceecesens (xxxvii) is the equation to determine 7 the tetrachoric correlation, and Table XXIX gives the values for given 7, Le. $(1—a) of the following six tetrachoric functions 7, To... T;, and further of h, the ratio of the abscissa of the dichotomic line to the standard deviation of the corresponding variate. It is occasionally needful to go beyond the first six tetrachoric functions. In this case the following finite difference formula is available: Ti Dn Toe Onn Dae es eee (Xxxvill), where Pr= 1/Vn, In =(n—- 2)/Vvn(n—1 y chi ee (xxxix). The following table gives the values of p, and qg, from n=7 to 24, n i Pn | In n Pn In 7 | +37796 *77152 16 *25000 “90370 8 *BO355 “80178 17 | °24254 “90951 9 33333 “82496 18 | ‘238570 “91466 10 | °31623 *84327 19 *22942 ‘91925 11 “30151 “85812 | 20 *22361 “92338 12 ‘28868 | -87039 | 27 -21822 ‘92711 | 13 27735 “88070 22 | +21320 ‘93048 | U4 “26726 “88950 23 ‘20851 “93356 15 *25820 “89709 24 | -20412 | -93638 XXIX] Introduction li Illustration (i). Find the correlation between dullness and developmental defects as indicated in the following table for 26,287 children. Without Defects | With Defects | Totals Not Dull Dull | Totals Here T. = post = 078,898, 7 = ie = 087,800. Whence by interpolation from Table, p. 43: = 14712, m= 15945, i 14694, T, = 15268, i ‘05977, n= 054.31, T=— 04262, %] =— “Ob5l37, T,=— ‘06702, T, =— ‘06755, T=— ‘00752, Te = ‘00017, h=- 144253, k= 1:35442. Proceeding to apply the difference formula (xxxviii) for four further functions we have tT = ‘04770, Tf = dbpPAl T= 02985, 7/= ‘02486, = — 02530, T, =— 03185, Ty = — 03647, Ty = — 03460. Hence the equation for r is 026,854 = 023,458r + ‘022.4357? + °003,246r° + °002,189r4 + :004,5277° — :000,0017° + 002,49077 + -000,7427% + ‘000,8067° + 001,262, Whence we find 7 = ‘652 + ‘009. Illustration (ii). Find the tetrachoric correlation for the four-fold table given for Houses in course of Erection on p. xxxv. Here 4(1—m) = 7 = 4494 = 260,080; $(1—a)= By simple linear interpolation, T= ‘32442, T, = 02468, T= ‘14758, Ts = "04116, T,=— 07766, T; = 04599, TOs are 08048. 62, = 009,157. ='s lii Tables for Statisticians and Biometricians [X XIX Hence the equation for 7: —‘000,6093 = :008,007r + :006,0727? — 003,57 27° — °003,3577". Whence r= — ‘081 + °043. Or, the association is not definitely significant. Illustration (iii). Find the tetrachoric r for the Table of Bradford Parents : Mother’s Habits. Good Bad Totals... Father’s Habits. Here a brief experience will show the reader that to proceed by tetrachoric functions will require a very large amount of labour. We have 4 (1 —a@) =7 = 543/1696 = 32017; (1 —%)=7)' = 635/1696 = 37441, d/.N = 476/1696 = 28066. We have accordingly the following series of tetrachoric functions—the first 6 from the table, the remaining 18 from the difference formula. d)N 3(1-a) | 71 tT | Ts; T4 75 | T; | E | | | 28066 “32017 “35769 | *11817 | —11415 | —-09489 05674 | -08012 — 37441 | “37901 “08581 | —-*13887 — ‘07178 08288 | 06325 7 78 T) | T10 Ti T12 713 714 lind 4 3 | — ‘02963 | —°06913 01368 | ‘06032 | —°00324 | — 05294 | —-00401 | 04659 , —°05629 | —:05709 | “04034 | °05223 — 02957 — "04819 ‘02176 | :04558 | | | T15 | 716 | Ti Tis T19 | T20 T21 722 “00922 ae *04103 = ‘01304 | *03609 “01586 — ‘03167 — ‘01793 | -02768 —*O01575 | —-04245 ‘01103 | ‘038966 —°00728 | —°03714 “00411 ‘03484 | a | | j | T23 } T24 h =— = — a7 == “01944 — 02407 | 46732 —_ —_ _— —_ — — 00151 — 03272 | 32020 | _— | a | — — ae XXIX—XXX] Introduction liii Considering only the equation as far as Everitt’s Tables extend, we have (7) =— 16079 + 135577 + 0101407? + 015857° + 0068 174 + 00470r* + :00507r* = 0. This leads to 7 =°9365, but the series indicates that the terms are far from converging rapidly. The first 12 tetrachoric functions were then used, the last six being found by the table of p, and q, above, and the value of 7 was found to be 9152. Then 18 functions were used and gave r=-9114. Lastly 24 tetrachoric functions were used, and the equation below obtained, which led to r= "9105. (r) = — "16079 + 135577 + 010147? + -01585r° + 006817 + :004707° + 005077 + 0016777 + 0039578 + 0005579 + -00315r¥ + 000107" + -002557r” — 000097" + 002127" — 000157" + 001749" — 000147" + 00143778 — 000117” + 001187” — 00057r™ + ‘000967 — 000037” + 0007974, It will be seen that even with this very large amount of labour we cannot be sure of having reached a final result*. To obviate this the following table was constructed by Kveritt, and there is no doubt that the extension of this table to the whole range of correlation would much simplify the discovery of tetrachoric 7. At present the calculation of high values of 7, for negative correlations is in hand. TABLE XXX (pp. 52—27) Supplementary Tables for determining High Correlations from Tetrachoric Groupings. (P. F. Everitt, Biometrika, Vol. vi. pp. 385—395.) Using the notation of p. |, : ; ie te = h(a? +9? 2rey) Sa e “1-7 rg] se acroooao00sed ] N A@vV1—rJn Jk ee (=) in the case of a tetrachoric table, or d Ie fe y? 1 Sa ees Vd N WV2a Ji y ( li) ODE ROO SOREHOOCHODOUOS BOE DE XI1). UP eth or h—yr h Y= =| @ue , if ¢=——— a V2 Jt V1 —r? | * Mr H. E. Soper working out this example draws my attention to the fact that convergence is closely given by a form: 7,=7,, (1+a.c”), where n is the number of terms used and a and ¢ are constants. Hence (M — Veo ) (Tr42m ¥, i) = Ont —Te , = Tinton = Tone 4 = Tr +Tntom — 2r n+ In our case take n=6, m=6, and we find or T6718 — 712" = ———_—_ =" 1 a ° 16 +718— 272 as The value 72, is "9105. In this case a=-1567 and c=-7574, but we cannot assert that these would be constants for all tables. If we use ryp, rjg and ry,, we find r,, =-9102. liv Tables for Statisticians and Biometricians [XXX Hence 7, h being known, Y is a tabled integral for each value of Y. Accordingly by aid of Table II we know ao NV Qar be found for each value of h, k and 7. , and using a quadrature formula, d/N can Table XXX gives, for 7 = "80, °85, °90, ‘95 and 1:00, and values of h and k proceeding by ‘1, the values of d/N. For given values of h, k and d/N, we can then find 7 by interpolation from these tables. The process is far shorter than that required by Table XXIX when we have to proceed to many terms. Un- fortunately opportunity has not yet arisen for fully completing similar tables for r negative and over ‘80. Zilustration. Determine the correlation in habits between Mother and Father in Bradford. The data are Mother. | Habits Good | Habits Bad Totals | 8 | Habits Good 1061 | Habits Bad 635 7 | Totals ... 1696 Here (b + d)/N =°32017, (c+d)/N =°37441, and therefore h = °46722, k ="32020 from Table II. Also d/N = 476/1696 = -28066. Inspection of Table XXX shows that 7 will be likely to lie between ‘90 and °95. We extract from the Table for d/N: | | r=*90 h=-4 | 0=o5) | | r="95 h='4 | h='5 | | = | | | | k=3 "2943 2728 k="3 | “3135 +2898 k='4 | -2784 | -2602 | | k=-4 | -2080 | ‘2787 | Hence: | | | r='90 h="4 h="5 r= "95 h=-4 h="5 | | k= -32020 2911 2703 | k= -+32020 | 3104 2876 | Thus: — [ ; ae | 7=°90 | h=*46722 | r="95 | h=46722 | | roe ced ae } | j:= +32020 | 2771 | | k= 32020 “2951 XXXT] Introduction lv We have now the desired h and & and have to interpolate d/N =:28066 between ‘2771 and 2951. There results r=9099. This is in excellent agreement with the value ‘9105 deduced from 24 terms, or from the final value ‘9102, which can be deduced from the 12,18 and 24 term values on the logarithmic rate of decrease hypothesis: see footnote p. lil. TABLE XXXI (pp. 58—61) The V-Function. (J. H. Duffell: Biometrika, Vol. vu. pp. 43—47.) It is well known that ['(2+1)=a2I' (a), and this property enables us to raise or lower the argument of the [-function at will. As a rule in most statistical investigations we require ['(#+1)/a*e-*. The following formula due to Pearson will then be found to give (z+ 1)/x*e~* with great exactness: 25°°623 log (~F) = -0399,0899 + J log w + *080,929 sin == ..(xlii), For values of «+1 less than 6 and often for values less than 10, we find log I'(a@ + 1) or log I'(p) from Table XX XI by reduction to p between 1 and 2. The reader’s attention must be especially drawn as to the rules, given on the Table itself, as to (i) characteristic, (11) change of third figure of mantissa at a bar, and (ii1) the sign of the differences on the facing pages of the tables. The difference tabled under 1:144, say, is the drop from 1:144 to 1145. Illustration (1). Find T' (2346). By the reduction formula I’ (:2346) = [ (1:2346)/:2346. Hence log T (2346) = log T'(1:2346) — 1°370,3280. log I (1:234) = 1-958,9685 =— 1069, log [ (1:235) = 1:958,8616 ‘6A = — [641-4]. . log TP (1:2346) = 1:958,9685 — [641] = 1-958,9044. log 1'(2346)= —1:958,9044 —1:370,3280 _-588,5764 Or T (2346) = 3:87772. lvi Tables for Statisticians and Biometricians [XXXII—XXXIII Illustration (ii). Find T'(8°7614). T (87614) = 7-7614 x 67614 x 57614 x 47614 x 37614 x 27614 x 176141 (17614). log [' (87614) = °889,9401 + log T (17614) -830,0366 ‘760,5280 ‘677,7347 575,3495 4.41 ,1293 245,8580 = 4420,5762 + log T (1'7614). log T (17614) = 1:964,5473 + -4[1113] = 1:964,5918. . log I' (87614) = 4385,1680. Hence T (8:°7614) = 24275°49. TABLE XXXII (pp. 62—63) TABLE XXXII, A and B (p. 64). Subtense from Are and Chord in the case of the Common Catenary. (Julia Bell and H. E. Soper: see Biometrika, Vol. vit. pp. 316, 338, and Vol. 1x. pp. 401—2.) If c be the parameter of the common catenary, then we know that Y HCCOSDU veeeerecccnseeseceeececan essere (xl), where u=2/c is its equation. If the chord be 22, then subtense/chord = (y — c)/(2x) 3 (sinh 50) ee EE (xliv), = — are/chord = = Ue ae Sate ee (xlv), are—chord sinhu-—u £B aan = ae i = T00 (xlvi), subtense (sinh }u)? _ 4 Fr anand in (ie (xlvii). Corresponding values of a and £ are given in the Tables XXXII and XXXII. XXXIII A anv B] Introduction lvii Illustration (i). A cable of 132°5 is suspended over the gap between two towers of the same height, 115 feet apart. What will be the droop of the cable? 132°5 — 115 ee: )1159, Table XXXIII A, gives us a= 21°62 = 100 subtense/chord. *, subtense = "2162 x 115 = 2486. 8=100 Thus the droop is 24°86 ft. Illustration (ii). A catenary arch is to have a rise of 50 ft., centre line measurement, and a span of 200. What is the length of the centre line? a= 100 x 50/200 = 25:0, but a= 25 by Table XXXII gives B=15'1. 100 (are — chord)/chord = 15:1. *, are = 230°2 ft.* Illustration (i). For some races the shape of the nasal bridge is very ap- proximately a catenary. Thus if the nasal chord from dacryon to dacryon be measured and also the tape measure from dacryon to dacryon, we obtain the mesodacryal index 8. The tables enable us to pass to the mesodacryal index a, and thus ascertain the nasal subtense, which is slightly harder of direct measure- ment than the arcual or tape measure. In the skull of a male gorilla the mesodacryal chord was 22°6 mm., and the mesodacryal arc 30 mm. Determine the mesodacryal subtense 3 Se ave WeiGee yen ke 8B=100 Hence, from Table XXXII: a= 38°84 = 100 subtense/22°6. *. subtense = 22°6 x 38884 = 88 mm. The actual value of the mesodacryal subtense measured on the skull was 87 mm. Apac XXXIV (p. 65) Diagram to find the Correlation Coefficient r from Mean Contingency on the Hypothesis of a Normal Frequency Distribution. (Pearson: Drapers’ Company Research Memoirs, No. 1, “On the Theory of Contingency.”) If np, be the frequency in the cell of the pth column and qth row of a correlation or contingency table, and m, be the total frequency in the pth column, n, the * Should there be any use for this table for constructional purposes, which there ought to be when the value of the catenary arch is more fully recognised, I will in a later edition of this work give the value of uw corresponding to each f, so that the parameter c can be at once read off and the form of the arch readily plotted. It might also be desirable to give the values of a and 8 to two decimal places. We have these data in our MS. copies. B. h lviii Tables for Statisticians and Biometricians [XXXIV total frequency in the gth row, and WV the whole population, then if the two variates are independent, the frequency to be expected in the p, gth cell will be N Nq Mp fs Ng Mp NN eee ws NoMp -- . re and the observed excess over this, i.e. Npg——4,”, is termed the ‘contingency in N this cell. The total contingency must be of course zero, i.e. the sum of all the cell contingencies. If, however, we take only the positive excess contingencies and divide them by N, ie. =a (non — “57? ) we obtain the so-called ‘mean contingency.’ On the assumption of normal frequency distribution it is possible to deduce the actual correlation from w, provided that the cells are sufficiently small for summation to replace integration. As in practice our cells are hardly likely to exceed 8 x 8, and may be smaller and unequal in area, we shall generally find a value below that of the true correlation, even if the system be accurately normal. A corrective factor corresponding to the class-index correlation has not yet been theoretically deduced. But experience seems to show that to add half the correction due to class-index correlations gives good results. That is to say, that, if ry be the correlation found from the Abac, p. 65, and r,¢ and rz¢, be the class- index correlations for # and y, we should take for the true correlation: T, r=ryty at - ry | OL MG G =} E a | ee nee ee (xlviii). "rO,"y Cy It is clear that this is the same thing as taking the mean of the crude mean contingency correlation and its value as corrected for the class-index correlations. The following illustrations may indicate the method of procedure. Illustration (i). Find the correlation from the table on p. lix by mean contingency. The first number in each cell is the frequency reduced to 1000, the second number is that to be expected on the basis of independent probability, and the third is the mean contingency of the cell. The sum of the positive contingencies is 94136, hence the mean contingency is 094, Entering the diagram with ‘094 on the base scale, we pass up the vertical to the curve, and then along the horizontal to the left hand scale and find ry, =°285. The class-index correlation for the vertical marginal frequency is 7, ¢, = "9645, and that for the horizontal marginal frequency is ‘9624*. Hence ryl|(rx G,"y C,) = 307, and r=4 (307 + :285) =°296. The table is actually a true Gaussian distribution with correlation equal to °300. * Biometrika, Vol. 1x. p. 218. XXXIV} Introduction First Variate A. 1 2 3 4 | 5+6 | ta 8 Totals | | = S: 4:04 17-16 7:55 3°30 0-91 0:92 0:12 1 | (1:224) | (10-948) | (8-976) | (6-120) | (2-346) | (3-434) | (0-952) | 34 2816 6-212 | —1-426| —2-820| —1-436 | —2-514| —0:832 | | 17-41 123-59 | 79:76 | 44:64 | 14-61 17-67 3°32 2 | (10-836) | (96-922) | (79-464) | (54°180) | (20-769) | (30-401) | (8-428) | 301 | 6-574 | 26-668 | 0:296 | —9-540| —6:159 |—12°731| — 5-108 f | Z | | | 8°86 93:00 | 78-31 52°04 | 19:20 26-40 6-19 _| 2 | (10-224) | (91-448) | (74-976) | (51-120) | (19-596) | (28-684) | (7:°952) | 284 3 —1°364| 1552 | 3-334. | 0-920 | —0°396| —2-284| —1°762 2 pe | ae = 2°83 | 37°73 | 37:24 | 27:51 | 10°95 | 16-31 4-43 S| 4 (4-932) | (44-114) | (36-168) | (24-660) | (9°453) | (13-837) | (8:836) | 137 =| —2:102 | —6°384| 1:072 | 2850 | 1497 | 2-473 | 0-594 | Dp | 162 | 25-21 27°75 | 22:09 9-26 14:64 | 4:43 546 | (3-780) | (33-810) | (27-720) | (18-900) | (7-245) | (10-605) | (2°940) | 105 —2:160 | —8-600| 0-030 | 3:190 | 2:015 | 4:0385 | 1-490 1:02 | 19:50 | 24°47 | 21:39 958 | 16°36 5°68 7 | (8528) | (31-556) | (25-872) | (17-640) | (6-762) | (9°898) | (2-744) 98 — 2-508 |—12:056| —1:402| 3-750 | 2-818 | 6-462 | 2:936 0-22 5°81 8-92 9-03 4:49 8-70 3°83 8 (1476) | (13-202) | (10°824) | (7-380) | (2°829) | (4°141) | (1°148) 41 — 1-256 | —7-892| —1:904| 1:650 | 1:661 | 4559 | 2-682 Totals | 36 322 264 180 | 69 | 101 28 1000 | Illustration (i). Find 7, by mean contingency for the table on p. Ix: lix The sum of the positive contingencies is 169'846, or we have mean contingency v ='170, whence the diagram leads us to ry, ="480. The marginal frequencies are the same as in Illustration (1). Thus we have ry/(r20,7ye,) = 517, r=4$(517 + 480) = 499. The table gives actually a true Gaussian distribution with correlation *500. It will be seen from Illustrations (i) and (11), that if the distribution be Gaussian, even if the marginal frequencies are in fairly irregular groupings, ry will be reasonably close to the true contingency, and corrected as suggested above will give excellent results. h2 Ix Tabies for Statisticians and Biometricians [XXXV—XLVI First Variate A. my SS) i) 4 | 5+6 7 8 | Totals | | | | | 738 | 1985 | 494 | 138 | 026 | O18 | 0-01 1 | (1°224) | (10-948) | (8-976) | (6-120) | (2°346) | (3-434) | (0-952) | 34 | 6156 | 8-902 | — 4-036 | — 4-740 | — 2-086 | — 3-254 | —0°951 | 20°58 | 145-47 | 78:94 35-98 9°72 9:27 1:04 | (96-922) | (79-464) | (54°180) | (20-769) | (30-401) (8-428) | 301 9-744 | 48-548 | —0°524 |—18-200 |—11-049|—21-131| —7-388 | oo = = 2 wn oo 2 6-01 93°63 | 85°41 | 54:34 | 18:59 | 22:33 3°69 _ | 8 | (10-224) | (91-182) | (74-976) | (51-120) | (19-596) | (28-684) | (7-952) | 284 ai —4°214| 2-182 | 10-434 | 3-220 | —1-006| —6-354| —4-262 | = | = 1-26 | 31-81 | 39-49 | 31:03 | 12-29 | 17:36 | 3-76 Ss 4 (4:932) | (44-114) | (36-168) | (24-660) | (9°453) | (13°837)| (8°836) | 137 a ~3°672 |—12:304| 3:322 | 6-370 | 2-837 | 3-523 | —-076 5 | ; D 0:53 18:11 27°79 25°14 11:09 17°62 4:72 5+6 | (3-780) | (33-810) | (27-720) | (18-90) | (7-245) | (10-605)| (2°940) | 105 — 8-250 |-15°700| 0-070 | 6-240 | 3845 | 7-015 | 1-780 0-22 11:02 21:59 23°66 11-86 21°89 7°76 (3°528) | (21:556) | (25-872) | (17-640) | (6°762) | (9°898) | (2-744) 98 — 3°308 |— 20°536 | — 4-282 | 6-020 | 5-098 | 11:992| 5-016 - 0:02 2:11 5°84 8:47 5:19 12°35 7:02 8 | (1-476) | (12-202) | (10-824) | (7-380) | (2°829) | (4°141) | (1-148) 41 ~ 1-456 |—11:092| —4-984| 1:090 | 2-361 | 8-209 | 5-872 Totals | 36 322 264 180 69 101 28 1000 TaBLEs XXXV—XLVI (pp. 66—87) Criteria for Frequency Types and Probable Errors of Frequency Constants. (A. J. Rhind: Biometrika, Vol. vu. pp. 127—147 and pp. 386—397.) It is desirable to consider all these tables under one heading, namely the general investigation of frequency type and of the probable errors of frequency constants. The main lines of Pearson’s theory of frequency are involved in the following statements: XXXV—XLVI] Introduction Ixi If the differential equation to the uni-modal frequency distribution be ldy «-a : y is = f(a) Beene eee eee nent eens eeeeeees (xlix), we may suppose f(x) expanded in a series of powers of «, and so 1 dy L—a = — —= 2 i dudade Heese meets (1). ydx Cot G@+ Coa? +...4+Cpt"+... then a, G, CG, C,... Cy... can be uniquely determined from the ‘moment co- efficients’ of the frequency distribution. These constants are functions of certain other constants B,, 8,—3, 83, 8;—15,... which vanish for the Gaussian curve, and are small for any distribution not widely divergent from the Gaussian. Further Cy, C1, Co+..Cn... converge, if, as usual, these constants are less than unity, the factors of convergence being of the order VB-constant. As a matter of fact ¢, involves the (n+2)th moment coefficient, and thus we obtain values of the c-constants subject to very large errors, if we retain terms beyond c,. If we stop at c, then our differential equation is of the form 1 dy _ xr—a y dt C+ Ca + CX? and we need only 8, = p,?/u.? and 8, = py/ps°, where ps, Ms, ys ave the second, third and fourth moment coefficients about the mean. ldy «x-a ks : If we take the form fom =» We reach the Gaussian, in which each con- 0 tributory cause-group is independent, and if the number of groups be not very large, each cause-group is of equal valency and contributes with equal frequency F rf Sieh: il a —a results in excess and defect of its mean contribution. If we take — “2 = * ; yde Cot+c,x then each contributory cause-group is still of equal valency and independent, but does not give contributions in excess and defect of equal frequency. d. Z— a : Le ., then contributory cause-groups are dz G+ Gqe+ 2 not of equal valency, they are not independent, but their results correlated, and further contributions in excess and defect are not equally probable. The use of this s ay 2 a WAN adopted to allow of this wide generalisation of the ydx M+qu+ Cx” Gaussian hypothesis. Finally if we take 7 form If we adopt it, every @-constant is expressible by means of the formulae: Bn (even) =(n +1) {48, 1+ (1 + 42) Bp }/(1 —$(m—-1) @).. ee. (li), Bn(odd) =(n+1) {$B Brnat+(1+4a) Br}/(1-4$(n—-1)a@) ...... (iii), where G=|(2 873 S16) (Bacto) senceecsccsssaanenrarssane (liv), in terms of lower @-constants. Ixii_ Tables for Statisticians and Biometricians [XXXV—XLVI Table XLII, (a)—(d) gives the values of ,, 8;, 8; and §, in terms of , and 8,. Hence as soon as 8, and £, are calculated we can find the numerical values of Bs= Uspbs/ os’, Bs = pela, Bs= Mr Hs/ oa’, Bo= Mel pat ...eeeeee (lv), theoretically. Although these values will not be those which would be absolutely deduced from the data themselves, they will, considering the large probable errors of fs, Me, M; and ws be reasonable approximations to them. The values of the probable errors of 8, and 8, are determinable by formulae involving £,, By... Bs. From these formulae, Tables XXXVII and XXXVIII, giving the values of VN», and V NXg, have been constructed. Hence multiplying by y, from Table V, we obtain 67449 VN Xe, and the probable errors of 8; and f,. 67449 = Nise: If we add to the standard deviations of 8, and §,, the correlation between deviations in 8, and #,, namely Rg,e,, which correlation is given in Table XXXIX, we can find the probable errors of any functions of 8, and 8,. Two such important functions are the distance d from mean to mode and the skewness sk of the distribution. The probable errors of d and sk can be found from Tables XL and XLI respectively, the former by multiplying the tabulated value VN Xq/o by oxy (from Table V), and the latter by multiplying the tabulated value VVS4 by x1 (from Table V). Thus far we have only been concerned with the constants which describe certain physical characters of the frequency distribution without regard to the type of curve suited to the distribution. We now turn to the latter subject. It is known that the type of frequency depends upon a certain criterion «. Hence near the critical values of «, more than one type of curve may describe the frequency within the limit of the probable error of «,. Table XLIII gives the probable error of «., if the entries in that table be multiplied by the x, of Table V. The following are the series of Type curves which arise according to the value of the criteria Ky = PB pS BiH. a oscise seadaswcte nsec cteceomeeceetee ee (lvi), B, (Bs +3)? (vi), “4 4B, — 38) (28,— 38, — 6) B, is by necessity >28,. Hence for our curves all possible values of f,, 8, lie in the positive quadrant between the lines 8, = $8, and 8, = 158, +4, the latter being if we go to f, the limit of failure of Type IV, for its uw; becomes infinite. Beyond the latter line distributions are heterotypic. XXXV—XLVIT] Introduction Ixiii Criterion Type Equation to Curve Bae on ess 3 VII Y=——— ( =) Me eateisoninester (1vii1) aie m=0 £,=0, B= Normal =ayie | Pt er see vane cetages send (lix). 0 20, <3 im y= (1-5) ed Mn shot car (Ix). a0 f=0, B<18 ST A Oye ee ee (Ixi), (3 a ~v tant = Ky>0<1 IV Y=Yo pene (Ixi1) (+3) a (ay I Ve y= yor vI* eon acne ecieoee (Ixii1). Ky >1l< an VI PST P= CAGES sooosecosootoae (Ixiv). 72 »\P on Le) 26-96; —6=0 {yy = aie Pa(1 +") Seca aes (Ixv). Ky< 0, i.e. negative. Below f=0 I, Y = Yo (1 + =)" (1 _ =\" Soenerk (| bal) . 1 2, te<0. Inside f=0 ee yi (1 + -\" / (1 — 2)". (Ixvii), k.<0. Above f=0 U, Y= Yo ee! ee ...([xvili). For «.< 0, f=0 represents the biquadratic B, (88; — 98; — 12) (482 — 38,) = (1082 — 128; — 18)? (82 + 3)? ...(1xix). Type I is thus divided into three subclasses, limited range curves, J-shaped eurves and U-shaped curves. Diagram XXXV enables the reader at once to find the type appropriate to his distribution, and Diagram XXXVI gives the same figure on a much larger scale to indicate the changes that occur with large values of @, and £3. Knowing the values of 8, and £, the computer can fix his point on the Diagram XXXV, but he may come so near a critical point or line, that one curve may appear as reasonable as another. It is clear, for example, that in the neighbourhood of the Gaussian point G, he might possibly use II, I,, III, VI, V, * The branch of the cubic x.=1 with which we are concerned passes through the Gaussian point, at which p=a, and along this branch p is always >5. Ixiv Tables for Statisticians and Biometricians {[XXXV—XLVI IV or VIII, and as all these types at that point transform into each other, the forms actually deduced will be almost identical, however different their equations. But there will be other occasions when doubt as to the use of the simpler of two curves may arise; for example if 6,=°8, B.=415, are we justified in using Type III as simpler than Type I? Now we have to remember that the variates 8,, 8, form a frequency surface, of which the equation is 1 1 By? ? 2B, B, By Bo uM -Saceep (fet & 8 EPs) =o nS e 2(1-Rs,2,)\2p° =p" =p, Zp. =7 =, =—Bs 1 — Rs, 8, Z= and that the contours of this surface projected onto the 8,, 8, plane of Diagram XXXV form a series of similar and similarly placed ellipses. Within any one of these ellipses a certain amount of the volume of the #,, @.-frequency lies, and therefore if this system of contours were properly placed round the §,, B» point on Diagram XXXV we could tell at once the probability that the given point, owing to random sampling, should fall outside a given elliptic contour. The ellipse which has for principal semi-axes 11772, and 1:1772., where 2, and x, are the principal axes of the ellipse: = 1 : (& a: By bre 2fs, 818s) LS Rs, p, =P Bz Xa, >p. / covers an area on which stands just one half the frequency, i.e. it is the ellipse determined by the generalised probable error of two variates (see Table X, p. 24). The semi-minor axis 1:177>, and the semi-major axis 1177, of this “Probability Ellipse” multiplied by WV are given in Tables XLIV and XLV respectively, and Table XLVI gives the angle in degrees between the major axis of this ellipse and * the axis of ®,. It is thus possible to construct from Tables XLIV—XLVI the “ probability ellipse” round a given point 8, 8, and to test the area within which half the frequency lies. If the probability required be not 4, but much less, then we note that the probability, that a point will lie outside the ellipse with semi- axes AS, and AS, is P=e~™. 67449 Let ASSL VN Xe eee (Ixxii), : VN/q or 2 = q x ‘630,672, ad P = 0-7 * 315,336, Hence log P=—q x 136,949. Accordingly q—10; P=:0427, q=12: P =:0227, q=15: P=-0088, g—20: P=:0018. XXXV—XLVI] Introduction Ixv Hence we select the grade of working probability we require, roughly 1 in 23, 1 in 44, 1 in 114 or 1 in 555, and this determines g. Divide WN the total frequency by qg and look up in Table V, y, for N/g, multiply this by the 1177 VN, of Table XLV, p. 84, and we obtain the semi-major axis of the required ellipse- Multiply the same y, by 1177 VN, of Table XLIV and we have the semi-minor axis. We can then construct round the point £,, 8, this ellipse and ascertain if it cuts critical boundaries on Diagram XXXV, p. 66, the orientation being given by Table XLVI, p. 86. Less accurately, but for practical purposes effectively, we may work on Diagram XLVII, p. 88. We proceed just as before, to select our g and so determine our A>, and AX,. Then we take the ratio of =,/=,. We now pick out of the ellipses on p. 88 the set having the nearest >,/, value and out of this set the ellipse with the nearest XZ, value of its semi-major axis. This ellipse or if necessary an interpolated one is transferred to tracing paper and placed with its centre at the given point (8;, 8), and its major axis touching the dotted curve. If this ellipse does not cut a critical line, we can be certain that to the given degree of probability the curve is of the type into the area of which its 8,, @, point falls. It would be impossible in an Introduction to these tables to give the whole theory of frequency curves*. But one or two formulae may be usefully placed here for reference. Distance d from mode to mean = ee Seopepecea eee (Ixxiii), Skewness sk = 5 ae a a) 1) eR (Ixxiv), Wns = 78. (48) — 248.4 964.98, 8:— 128.4 858.) scereesovesecseueaee. (Ixxv), N3p2= By — 4828, + 482 — B2 +1688, —8Ry+16R,) -se-eseeereeeee (Ixxv bis), Xe, &e,lee,e, = 28; — 3818; — 48382 + 68.28, + 38, 8,—68;+1282+248, (Ixxvi). It is from the above formulae that the Tables now under discussion have been calculated. Illustration. The following percentages of black measured with a colour top are stated to occur with the recorded frequencies in the skin colour of white and negro crosses. Discuss the type of frequency curve suited to the data and determine the chief physical constants of the distribution and their probable errors. * The general theory is given in ‘‘Skew Variation in Homogeneous Material,” Phil. Trans. Vol. 186 (1895), A, pp. 343414: Supplement, Vol. 197 (1901), A, pp. 443459; ‘‘On the Mathematical Theory of Errors of Judgment,” Phil. Trans. Vol. 198 (1902), pp. 274—279 ; ‘‘Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson,” A Rejoinder, Biometrika, Vol. 1v. pp. 169—212. “‘ Skew Frequency Curves,”’ A Rejoinder to Professor Kapteyn, Ibid. Vol. v. pp. 168—171, and “ On the curves which are most suitable for describing the frequency of Random Samples of a Population,” Ibid. Vol. v. pp. 172—175. + Extracted from C. B. Davenport, Heredity of Skin Color in Negro-White Crosses, Carnegie Institution of Washington, 1913. B. t Ixvi Tables for Statisticians and Biometricians [XXXV—XLVI The working origin was taken at 20, the centre of the group 18—22. The centre of the first group at 1:47 / is $(20 — 1:47) = 3°706 on the negative side of the working origin and may be taken to contribute —56, + 206, — 763, + 2880 if | | Percentage Frequency | Percentage Frequency | o0— 2 15 43—fi 45 g5— 7 120 4S—52 24 S—I2 139 5E—57F 14 18—1} 157 58—62 6 18—22 158 63—67 3 23—27 139 68—7T2 3 28—S2 | 117 718—777 2 Soe 92 78-82 2 38—J2 50 Say - | to the first, second, third and fourth moments respectively. The working unit being 5 Wes the raw moment coefficients are: v= “567,2191, vs = 7°708,4990, v3. = 28:982,5042, vg = 253'268,8730. Whence transferring to mean and correcting, we have Mean = 22'8361, o = 270156, fy = 7'298,428, 3 = 16'238,780, f= 198'909,921. These lead to B, = 678,295, B, = 3°734,202, kK, = 28, — 38, —6 = — 566,483, Ky = — 1:052,180, sk = °495,087, Distance from Mean to Mode = d = 1°337,508. These values, except the mean, are all in working units. Therefore in per- centages of black: o =13'5078 and d=6:6875. We can now find the probable errors of these constants. We first want y; from Table V, but 1086 is outside the limit of n. We therefore take y, for 543 and have y, = ‘02047, and we find y, = 014,47. We can repeat our constants with their probable errors Mean = 228361 + 2765, o=13:5078 +1955. Then from Table XX XVII, Bo=3'7: VNSp = 470 + 288 [2] = 4-71, Bo=3'8: VNXp, = 5:05 + 283[2]= 5°06. Hence for Bo=3°7342: VNXpg = 4°71 + 342, [85], VN Xp, = 483. * 4 at 0 and 11 at 2, giving a mean at 1:47 °/,. XXXV—XLVI] Introduction Ixvii Similarly from Table XX XVIII: B.=37: VN, = 12:02 — 283 [66] = 11°65. f.=38: VNXp, = 13°60 — 2s [72] = 13-19. Hence for B.=3°7342: VNSe, = 11-65 + 242, [1°54], VN, = 12:18. Thus we find, multiplying by y,: B= 6783 + :0989, B,=3°7342 + 2493, It is clear that the 8, and @, are significantly different from the Gaussian B,=0 and 6,=3. We next turn to the skewness, using Table XLI: B2.=37: VNSy,=1:98 + 283 [21] = 2:10, B.=38: VNZ~_=1:88 + 283 [16] = 1:97. Hence for Bo= 3°7342: VN Zy = 210 — 342, [13] = 2:06. Thus the skewness = ‘4951 + 0422, or the distribution is significantly skew. Passing to Table XL for the probable error of d, we have B.2=37: VN Salo = 214+ 283 [20] = 2:25, shespuee pe 9:03 + 283 [17] = 2:18. Hence for B,=3°7342: VN 3S q/o = 2:25 — 342 [12] = 2-21. Thus Probable Error of d= xy, x o x 2°21 =°6111, and d = 66875 + ‘6111. The probable error of «, is to be found from the relation : (VS, =4(VN Sp, + 9 (VN Sp)? — 12 (VN Sp.) (WN Se.) x Bp p,- (xxvii) Thus we require Rg,g. Table XX XIX, p. 72, will provide this: B.=3'7: Rep, = 892 + 283 [5] = 895, 8, =3'8: Reg,p, = 893 + 283 [5] = 896. Hence for 8,=3'7342 we may take Rg,g, = *895. Accordingly (VN'%,,)? = 5934096 + 209-9601 — 631-8278 = 17175419. Or, VNX,, = 18-0974. Hence pe. of m=, x VNE,, = 2681, or, «, = — 566,483 + 2681. Ixviii Tables for Statisticians and Biometricians [XXXV—XLVI It would look therefore as if «, were significantly negative, but it is just possible that «, might be zero. Such a big probable error for «, suggests our being in the neighbourhood of a critical limit. This is verified on examining Table XLIII to find the value of VNS,,. We sée that it is over 80, and we thus conclude that the probable error of «, may lie between 1 and 2. Thus we cannot be definitely certain of the sign or magnitude of «., when we are even relatively in the neighbourhood of x, = 0. If we turn to Diagram XXXV (p. 66) we see that the point 8,=°678 and 8,= 37734 is not very close but is approaching the line along which Type III is applicable, and this is the source of the disturbance noted. We accordingly try to measure the probability that Type III would be as satisfactory as Type I within the area of which our f,, B» values actually lie. We must take 1:177/ VX, from Table XLIV and 1:177VNS, from Table XLV. We have B,=°6789, B2=3'7 then 1:177V NS, =23 ”» ” Be = 38 » ” = 2:5. Hence for Bo = 3'7342, 1177V NS, = 23 + 342, [2] = 2:37. Again : Ao=37 LITTVNS,=16 — 283 [1] = 15-48, Bo=38 1177VNX,=18 — 283 [1] =17-43, Bo = 3°7342 1:177V NS, = 15°43 + 842, [2] =1611. Thus: ,/ 2. =2°37/1611=:147 =:15, say. Or, if we turn to Diagram XLVII (p. 88), our system of ellipses is half-way between the 3rd (=,/>,="14) and the 4th (2,/2,.= 16). Now if such a system of ellipses be traced off and centred at the point 8,=678, 8,=3°734 on Diagram XLVII to the right and then the major-axis be brought into parallelism with the dotted lines, we find that the biggest of these ellipses XZ,=°5 fails to reach the critical line III. But the semi-major axis of the probability-ellipse is 11775, =16-11/VN =-493. Hence we must conclude that it is more probable that the curve is of Type I than of Type III. This is readily determined and is usually sufficient guide. Actually the value of AX, must be about ‘6 before we get an ellipse to approximately touch the Type III line. But >, =°493/1:177 =-419, and accordingly \ = °6/:419 = 1-482, which gives P =e~»” ='36 nearly, or the odds are 16 to 9 that the point would not lie outside this contour. But if it did lie outside this contour, the chance of its being on or over the Type III line corresponds to only a very small section of the total frequency outside this contour. If we invert the problem and put the system of ellipses on the nearest point of the Type III line we find that the odds are very much in favour of the point 8,=°678, 8,=3'734 lying outside such a system. On the whole it is reasonable to conclude that Type I is properly used although we should probably not get bad results from a Type III curve. In some respects a suitable fit would be obtained by using Type I, and fixing its XXXV—XLVI] Introduction lxix start at zero*, but the vagueness of what is meant by ‘percentage of black’ as a factor, when the entire pigmentation of the skin probably arises from a single melanin pigment, only varying in concentration in the pigment granules and in the density of granules themselves. We have therefore contented ourselves by fitting a Type I curve, as further illustration of the use of the tables in the present work. The theory of fitting is given in the paper cited belowt. Following the usual notation we find: r =6(8.—B,—1)/(3h, — 28, + 6) = 21:°7755, e =r /{4+ 4B, (r+ 2)'[(r + 1)} = 57-764,468, b? = por? (r + 1)/e = (86'9391). Hence: m, =2:0917, m,:=17'6838, ad, =3°9071, -a, = 330320, and: 1 x ie 1 a ee y=9( + 3-907) ( ~ 330320) To find y, since m, is large, we use the approximation to the formula: PG tm td) » N (m, + m, + 1) e—omtm) (M2, + My)" +2) ee : wees (IXXVII1 w= > Tm +1) * Pim +1) Cy: e7™ my l e7 2 me J : ei Be | 1 il N (m, +. + 1) ee + Ms, 12 \my+my =) : namel Y= — 2 = a= % ek eee (xx mely, p-F Pp Gm, + 1)/(e=™ my) e (Ixxix) Ms the evaluation of the two [-functions for m,+m,+1 and m,+1 following easily by Stirling’s Theorem. If we write Z = ['(3:0917)/{e2™ (2:0917)*"""} we have log Z= log 20917 —2:0917 log 20917, + log 1:0917, + log T (1:0917), + 2:0917 log e. From Table XXXI (p. 58) we find log I (10917) = 1:979,8897 and loge is given by Table LV (p. 143). Hence we determine, log Z7=°576,5176. Evaluating the rest of the expression for log y, we have: log yp = 2'233,3986, Yo = 171157. Thus our curve is £ 2.0917 Z af bess y= 1T145T te sae) (1— s3o300) * For method, see Phil. Trans. Vol. 186, A, pp. 370, 371. + Phil. Trans. Vol. 186, A, pp. 367—370. See also Palin Elderton: Frequency Curves and Corre- lation, Layton Brothers. Ixx Tables for Statisticians and Biometricians {XXXV—XLVI with origin at the mode =1671486 in actual percentages, = 32297 in working units. To calculate y we take the origin at 0 and have log y = 26°134,8705 + 20917 log (w + 6774) + 17-6838 log (36:2617 — 2), where x may be put 0, 1, 2, 3, 4, 5... working units corresponding to 0, 5, 10, 15, 20, 25... actual percentages. The curve is shewn in the accompanying diagram, and considering the nature of the data is a reasonable graduation. 200 ————> 7 — 190 180 170 ; ; 160 150 140 130 / 120 A 110 100 90 Frequencies (per group of five). IN XN F > Mean. 0 5 10 15 20 25 80 85 40 45 50 55 60 65 70 75 80 Per cent. of Black in Skin Colour. TasLte XLVIII (pp. 89—97). Percentage frequency of Occurrences in a Second Sample of m after p Occur- rences in a First Sample n. (M. Greenwood, Biometrika, Vol. 1x. pp. 69—90.) If we assume the truth of Bayes’ Theorem then an event having occurred p times and failed g times in 7 trials, the chance that it will occur s times and fail m —s times in a second series of m trials is: 1 i gprs (1 = iD) tres da ; m 0 coe eres oe 3 0 These results can be evaluated as all the indices are integers and the series C,4+0,+0,+...+C,+... expressed in the usual hypergeometrical form : \q+m jn + 1 mp+l m(m—1) (p+1)(pt+2) _ am (m — 1) (m — 2) lq |pt+m+1 |lLq+m |2 (¢+m)(q +m-—1) [3 (p+1)(p+2)(p+3) ee (@4+m)(q+m—1)(¢+m—2) a seleloetremceplosrepe eas ([xxx). XLVIIT} Introduction Ixxi If x be large and p not widely different from q, then results may be obtained from the Gaussian curve, using as S. D. afm? Z but if either p/n or q/n be very small and m or n are commensurable, this no longer holds*. The case, however, of p and q widely different and m and m commensurable and themselves small numbers frequently arises, especially in laboratory work or in the treatment of rare diseases}. The present table gives the evaluation of the hypergeometrical series, formula (Ixxx) above, for a series of values of m, n, p and q. It is not sufficiently comprehensive to allow of very accurate interpolation in certain of its ranges, but it has involved a large amount of work, and will undoubtedly be of help till a more complete table can be calculated. Meanwhile if the reader feels in doubt as to any interpolation, it is not a very arduous task to calculate the result required from formula (Ixxx) by aid of Table XLIX. Illustration (i). In a batch of 79 recruits for a certain regiment four were found to be syphilitic. What number of syphilitics may be anticipated in a further batch of 40 recruits ? Here n=79, p=4, q=75 and m=40. We must first interpolate in the p=4 column on p. 97 between n=100, m =25, and n= 100, m= 50 for m= 40, ie. we must go $8 towards the m=50 series, or we must add 0°4 times the first series to 0°6 times the second series. We then repeat the same process for the series for p=4 and n = 50, m= 25 and n=50, m=50 on p. 95. There results: Occurrences | n=50, m=40, p=4 | n=100, m=40, p=4 | 0 6°8654 20-7406 1 13°5880 26°7023 2 16°3802 21°5249 3 15-7066 14:1138 4 13-2702 8°2460 5 10°3867 4°4566 6 7°7259 2°2637 7 5°5275 1:0886 | 8 3°8221 “4977 | 9 2°5583 2171 | 10 1°6581 -0906 | 11 | 1:0409 ‘0362 | 12 "6332 0139 8 3734 “0052 Ly ‘2137 0019 | es ‘1188 “0007 | 16 and over 1311 0003 | * For a full discussion of the subject : see Pearson, “ On the Influence of Past Experience on Future Expectation,” Philosophical Magazine, 1907, p. 365. + Tables recently published by Ross and Stott (‘‘Tables of Statistical Error,” Annals of Tropical Medicine and Parasitology, Vol. v. No. 3, 1911), appear to be designed to meet such cases, but being based on the Gaussian curve are, I think, very likely to lead the user to fallacious conclusions. Ixxii Tables for Statisticians and Biometricians [XLVIII We must interpolate between these two series for n =79, that is we must take 0-42 times the first series and 0°58 times the second series. The results are given below, and set against the direct calculation from formula (lxxx), using Table XLIX. By Interpolation. Direet Calculation. n=79, p=4, m=40 | n=79, p=4, m=40 | 0 14-9130 12-6143 | 1 21°1943 21-9379 | | 2 19-3642 22°5152 | 3 14-7828 17-6667 j 10°3562 11-6727 5 6-9472 6°8143 6 45578 | 36137 7 29529 | 17713 8 1:8940 ‘8118 | 9 1-2004 “3507 | 20 7489 “1436 | een 4582 “0560 | 12 2740 | 0208 | > ea "1598 “0074 | uy ‘0908 0025 | 15 0503 0008 16 0271 7 0142 {co | 18 and over 0140 The interpolation does not give a result very close to the actual series. For example, not more than three syphilitics might be anticipated in 70 °/, of samples of 40 by the interpolated series; actually not more than 3 are to be expected in 75°/, of samples. At the same time the result is much better than the normal curve theory provides. In the latter case we have Mean = 40 x 45 = 2025, Standard-Deviation = V40 x 4, x $3 = 1387. Hence (3°5 — 2:025)/1'387 =1:064 and by Table II this value of « corresponds to $(1 +a) =°86,1.e. in 86 °/, per cent. of samples of 40, we should have not more than 3 cases. It will be seen therefore that (i) the values at the latter end of the Table are not close enough to obtain very accurate results by interpolation, but (1i) that the Gaussian gives a still poorer approximation. Illustration (11). Of 10 patients subjected by a first surgeon to a given operation only one dies. A second surgeon in performing the same operation on 7 patients, presumably equally affected, loses 4 cases. Would it be reason- able to assume the second surgeon had inferior operative skill ? XLVITI—XLIX] Introduction Ixxiii On p. 91, we have the series for p=2 when n=10 for the values m=5 and m=10. Taking ‘6 of the first series and ‘4 of the second we have: Interpolation from Actual Value from Table. formula. | m—10)m— 1,1 p— n=10, m=7, p=1 0 37-9762 | 35°9477 1 30°6704 | 31°4542 2 | 17°3366 18°8725 Sane 8-2114 8-9869 Vi) 3°5248 3°4565 i | 1°4446 1:0370 | | 5676 +2200 | The 1996 0251 8 ‘0561 = 9 fon = | 10 [0012 = The chance that if the two surgeons are of equal skill 4 or more patients will die out of the second surgeon’s 7 operations is ‘058 by interpolation and ‘047 actually. Hence the odds against the occurrence are 16 to 1 by the table and 20 to 1 actually. It will be observed that interpolation gives small values at impossible numbers of deaths, but these have to be reckoned in to obtain the total number 100. That all seven patients should die under the second surgeon, if of equal skill, involves odds of 500 to 1 about in the interpolation result, but 4000 to 1 about actually. On the Gaussian hypothesis in the original problem the mean = 7 x j,="7 and the S. D. = V7 x =, x 3, = 7987, and (3°5 —‘7)/‘7937 = 3:52 roughly, or this corresponds to odds of about 4545 to 1—which are wholly un- reasonable. Thus the Table gives by interpolation odds of approximately the right value, which may serve many useful purposes, for those who are unable to work out the values required from formula (lxxx). At the same time it is clear that a much larger Table with closer values of the quantities involved is desirable. TABLE XLIX (pp. 98—101) The Logarithms of Factorials. (Calculated by Julia Bell, published here for the first time.) This table was obtained by adding up in succession consecutive logarithms in a table of logarithms to 12 figures. Not until the work was completed did we realise the existence of the splendid table of C. F. Degen*, which was then used to confirm our own results. De Morgan in his Treatise on the Theory of Probabilities of 1887 published an abridgement to six decimals of Degen’s Table of Factorials. His values cannot, however, be trusted to the sixth figure of the mantissa. The * Tabularum ad faciliorem et breviorem probabilitatis computationem utilum. Havniae, mpcccxxtv. This gives the logarithms of the factorials up to 1200 with 18 figures in the mantissa. B. k Ixxiv Tables for Statisticians and Biometricians [| XLIX—L use of a factorial table is extremely varied, especially in problems in probability involving high numbers. Illustration. In a certain district the number of children born per month is 662 and the chance of a birth being male is ‘51 and of its being female °49. Evaluate the chance that in a given month there should be an equal number of boys and girls born, and compare it with the chance of the most probable numbers (338 boys and 324 girls) being born. The chance of equal numbers of boys and girls being born is: aoe 662 C,. = ('51)* (49)! BI [BBI Therefore . lor C, = 331 | Tene + 1581°714,6156 Seo + 1690,1961) — 1383:941,4114 where the logs of the factorials are found from Table XLIX. Hence log C,= 200-660,6453 + 197°773,2042 or C, = 027155, or once in about 36:8 months, say once in three years the records may be expected to show equal numbers of boys and girls born in the month*. = 9-433,8495 The chance of the most probable number of boys and girls is given by ner : |662 m = (151) (49) 338 [324 log O,,= 338 x 1-707,5702 + 1581:714,6156 +324 x 1690,1961 — 709°645.9652 — 674359,6453 = 200°782,2640 +197°709,0051 Or C,,= 030993, or the most probable numbers will only be born once in 32°3 months, or say once in two years and eight months. | = 2:491,2691. We have C,/C;,=°876, or the chance of equal boys and girls is 88 Y% of the chance of the most probable numbers of boys and girls. TABLE L (pp. 102—112) Tables of Fourth-Moments of Subgroup Frequencies. (Calculated by Alice Lee and P. F. Everitt; published here for the first time.) In the usual method of determining the raw moments of a frequency, we take moments about an arbitrary origin, which is towards the apparent mode and * Actually of course the problem is more complex, because the number of children born per month is not constant. L] Introduction Ixxv multiply by plus and minus abscissae increasing by units—the ‘working unit.’ Thus an error made in an early moment may be carried on to the later moments. To control the results Table L was calculated a number of years ago, and from it the fourth moments for such frequencies as most usually occur can be read off at sight, and the raw fourth moment column thus tested before proceeding further. (i) (ii) (iii) (iv) (v) (vi) (vii) | | ena Frequency | Abscissa | Nyy Nv! | Nv! Nv’ Table L | 171 1 — 20 | — 20 + 400 — 8,000 + 160,000 160,000 | 2 1 19 | 19 361 6,859 130,321 130,321 3 Z 18 | 38 648 11,664 209,952 209,952 J 0 17 we i ae 3 ut 5 3 16 48 768 12,288 | 196,608 | 196,608 6 3 15 45 675 10,125 151,875 151,875 7 5 14 7h) |) 980 13,720 192,080 | 192,080 8 7 Tape SOKeeel *UL1e3 15,379 199,927 | 199,927 9 12 12 | 144 | 1,728 20,736 248,832 248,832 180 1B Th |i 1,573 17,303 190,333 | 190,333 1 17 10 170 1,700 17,000 170,000 | 170,000 2 28 9 252 | 2,268 20,412 183,708 183,708 3 24 8 192 1,536 12,288 98,304 98,304 4 43 7 301 2,107 14,749 103,243 103,243 5 oe leeks 318 1,908 11,448 68,688 68,688 6 57 5 285 1,425 7,125 35,625 35,625 7 55 4 220 880 3,520 14,080 14,080 | 8 68 3 204 612 | 1,836 5,508 5,508 | 9 83 2 166 332 664 | 1,328 1,328 190 85 - 1 — 85 + 83 - 85 + 85 85 1 96 0 = — — —- 2 102 + 1 +102 + 102 + 102 + 102 102 | 3 79 2 158 316 632 1,264 1,264 | 4 83 3 249 7A7 2,241 6,723 6,723 | 5 66 4 264 1,056 4,224 16,896 16,896 i i6 66 5 330 1,650 8,250 | 42,250 | 41,250 | | uf 56 6 336 2,016 12,096 72,576 72,576 8 43 7 301 2,107 14,749 103,243 | 103,243 9 35 8 280 2,240 17,920 143,360 | 143,360 | 200 30 9 270 | 2,430 21,870 196,830 | 196,830 | 1 20 10 200 | 2,000 20,000 200,000 | 200,000 | 2 24 11 264 2,904 | 31,944 351,384 | 351,384 3 14 1 168 2,016 | 24,192 290,304 290,304 4 13 13 169 2,197 | 28,561 371,293 371,293 5 8 14} 112 1,568 21,952 307,328 | 307,328 6 3 15 45 675 10,125 151,875 151,875 7 6 16 96 1,536 24.576 393,216 | 393,216 8 0 17 = OM Ble eee == Lal ¥) 1 18 18 324 | 5,832 104,976 104,976 | 210 1 +19 + 19 + 361 + 6,859 + 130,321 130,321 | | Totals 1306 — + + + + — | k2 Ixxvi Tables for Statisticians and Biometricians [L—LI The multiplication can therefore be done very rapidly and it suffices to re-examine not the whole of the arithmetic but only those rows which do not agree with the table. Illustration. Calculate the first four raw moments of the distribution of head lengths in 1306 non-habitual criminals on the previous page and test whether they are correct. This was an actually worked out case, and it will be seen that in this instance only one slip was made—that of a wrong multiplication by 5 in the contribution to the fourth moment of the frequency of head lengths 196. Often far more serious blunders are found. Correction would be made and the columns then added up on the adding machine. Two points should be noticed. First it is not in practice necessary to copy out the results from Table L,—they are merely compared on the table itself with the items in column (vii) and any divergence noted. Secondly in actual practice, it would be quite sufficient to take 20 instead of 40 sub-groups in this case. Sheppard’s corrections would fully adjust for the difference. TaBLeE LI (pp. 113—121) Tables of the General Term of Poisson’s Exponential Expansion (“Law of Small Numbers”). (H. E. Soper, Biometrika, Vol. x. p. 25.) The limit to the binomial series 1 “AS aes py +np"1q + n( ). nag? 4 n(n —1)(n — 2) n— R : qoe 2 1.2.3 pg +... ..(1Xxx1), when q is very small, but ng = m is finite, was first shewn by Poisson to be m m* \ = T2038 tit ra 4 | aes teee (Ixxxii). The present table provides the value of the terms of this series, Le. e~"m*/x! to six decimals for m = 0:1 to m= 15 by tenths. me can (1 +m+ 1.2 aF A previous table for m=0'1 to m=10 to four decimals has been published by Bortkewitsch*, but his values are not always correct to the fourth decimal. Poisson’s exponential limit to the binomial has been termed the “ Law of Small Numbers” by Bortkewitsch, but there are objections to the term. The approxi- mation depends on the smallness of q (or, of course, p) and the largeness of n, so that the mean m is finite. Thus 100 murders per annum might be quite a “small number,” if they occurred in a population of 40,000,000, for would be large and q would be small. It is therefore space and time which has limited the present table to m = 15, not the idea of m being small of necessity. Illustration (i). The number of monthly births in the Canton Vaud being taken as 662, and one birth in 114 being that of an imbecile, find the chance of 12 or more imbeciles being born in a month. * Das Gesetz der kleinen Zahlen, Leipzig, 1898. LI—LIT] Introduction xxvii F Beene /ALiliss 1 The binomial is Go +74 ng=5'8 nearly. We look out 58 in Table L and sum the terms for 12 and beyond. We find the chance of 12 or more =‘01595. Actually worked from the binomial, it is 01564. Or about once in five years, we might expect in Canton Vand a month with 12 imbecile births*. : a . 7m is accordingly large and q small, while Illustration (11). Bortkewitsch (loc. cit. p. 25) gives the following deaths from kicks of a horse in ten Prussian Army Corps during 20 years, reached after excluding four corps for special reasons: Annual Frequency | Frequency Deaths Observed — Poisson’s Series 0 | 109 108-72 1 65 66°22 2 | 22 20°22 3 2 4°12 J 1 63 5 — ‘08 | | 6 and over — ‘01 | ae | Totals ... 200 | 200 The mean m of the observed frequency is ‘61, whence using Table LI (p. 113) and taking ‘9 the series for 0°6 and ‘1 times the series for 0°7, we reach figures, which multiplied by 200 give us the column headed “ Frequency, Poisson’s Series ” above. Such good agreement, however, is very rare. A good fit to actual data with the Exponential Binomial Limit is not often found. Its chief use lies in theoretical investigations of chance and probable error: see Whitaker, Biometrika, Vol. x. p. 36. TABLE LII (pp. 122—124) Table of Poisson's Exponential for Cell Frequencies 1 to 30. (Lucy Whitaker, Biometrika, Vol. x. pp. 36—71.) Given a cell in which the frequency is n, corresponding to the population NV. Then if x, and WN are very large (or we suppose, without this, the individual to be returned before a second draw), the number in this sth cell will be distributed in M samples of m according to the binomial law M {(1 A a) + i / * See Eugenics Laboratory Memoirs, XIII. ‘‘A Second Study of the Influence of Parental Alcoholism,” p. 22. xxviii Tables for Statisticians and Biometricians HELE The mean will be mmn,/N and the standard deviation fm (1 - mn: If we only have a single sample of m and do not know the distribution in the actual population we are compelled to give n,/N the value m,/m, where m, is the number found in the sth cell of the sample. If n,|N or m;/m be very small and m large: the binomial will approach Poisson’s Exponential Limit, and in such cases the deviations in the samples for the sth cell will be distributed very differently from those following a Gaussian law, and the usual rule for deducing the probability of deviations of a given size by means of the probability integral fails markedly. It is not till we get something like 30 out of 1000 in a cell that we can trust the Gaussian to give us at all a reasonable approach. The present table endeavours to provide material in the case of cell frequencies 1 to 30, which will supply the place of the probability integral. Illustration (i). Suppose the actual number to be expected in a cell is 17, what is the probability that the observed number will deviate by more than 5 from this result? Looking at p. 123 we see that in 8:467 °/, of cases there will be a deviation in defect of 6 or more and in 9°526 °/, of cases a deviation in excess of 6 or more. Hence in 17-993 °/, say 18 °/, of cases we should get values less than 12 or greater than 22. Thus once in every 5 or 6 trials we should get values which differ as widely as 6 or more from the true value. Now look at the matter from the Gaussian standpoint. The standard deviation is Vm (i My. oe m m m Here m is supposed large compared with 17, so that the S. D. =V17 = 4123 nearly. But suppose m = 800, we should have S. D. = V17 (1 — 02125) =V17 x ‘97875 = 4079. Now we want deviations in excess of 5, i.e. we must take 5:5/4079 = 1348, If we turn to Table IT we find for this argument 1(1+a)="9102 or $(1—a) = 0898. Hence we should conclude that in not more than 17:96 °/, of cases would deviations exceed +5. Actually such occur in 17-99°/, of cases. Thus the actual per- centages are very close, but the Poisson series tells us that 8:47 °/, of cases will be in defect and 9°53 °/, in excess, while the Gaussian gives 8°98 */, in both excess and defect. We may further ask the percentage of times that 17 itself would occur; according to the Gaussian it will occur in 9°76 °/, of trials, actually it will occur in 9°63°/,. With values of cell-frequency less than 17, say in the single digits, far greater divergences will be encountered. LII—LU1} Introduction lxxix Illustration (11). Consider the fourfold Table below and discuss the relative probabilities that it has arisen from a population which shews 0, 1, 2, 3, ete. indi- viduals for this size of sample in the cell B, not-A. On the assumption that 0 is really the population of this cell, the probability is unity. Hence we have the following result. Population iy hoe Begre a ie | ame ae | | 13 Berean a | 4 | 5 (ST steal erst 9 | 10 | wu 2 | ove ae a | | i} Ss << i == Probability | | | | | | of 0 788 | ° — 04979| 01832) :00674| -00248 -00091 | -00034 | 00012 "00005 | 00002 | 00001 | ‘00000 | occurring " | ial Sc. Sum = 1°58200. Whence taking the & priori probabilities proportional to the probability of 0 occurring on the separate possibilities we have: Probabilities that the Table arose from a population with x in the B, not-A cell. i} x Probability x | Probability 7) 632,110 7 | 000,575 1 232,541 8 000,215 2 085,550 9 | 000,076 3 031,473 10 000,032 4 011,580 11 000,013 5 004,260 12 000,006 6 001,568 13 and over “000,000 The “association” of such a Table cannot therefore be considered “ perfect,” for in 37 °/, of cases it would arise from a Table with a unit or more in the B, not-A cell. The above is actually a Table of the correlation of stature in father and son. Grave caution is therefore needful in discussing such “perfect association” tables. TABLE LIII (p. 125) Angles, Arcs and Decimals of Degrees. (Based on Hutton’s Mathematical Tables.) This Table gives degrees in radians for the first two quadrants; it then gives minutes and seconds from 1 to 60 in fractions of a degree and in radians. The lxxx Tables for Statisticians and Biometricians [LIT need of such a table is very obvious, and arises in too great a variety of cireum- stances to be specified. Illustration. It is required to plot the curve *: we = 149917 tan 8, y = 235°323 cos Oe-*95% 8, Here log y = log 235°323 + 32°8028 log cos 0 — 45696 log e x 0. To cover the whole range of observations we must proceed from @=— 45° to 6=+45° roughly. It will be found sufficient to take @ by steps of 3° and ultimately perhaps of 4°. Hence 149917 is put on the arithmometer and multi- plied in succession by the natural tangents of 3°, 6°, 9°... etc. Plus and minus signs are given to these values of «. The corresponding values of y are found in three columns. The first is obtained by putting 32°8023 on the arithmometer and multiplying by the logarithmic cosines of 3°, 6°, 9°, etc. The second is obtained by multiplying (taking the third factor from Table LITI) 45696 x log e x ‘017,4533 = :216,7955 on the machine and multiplying the result in succession by 3, 6, 9, etc. The first column is added to log 235°323 = 2°371,6644 and the second column first subtracted and then added to the result to obtain the value of logy for positive and negative abscissae respectively. The antilogarithms give the ordinates. Another problem sometimes arises given x to find y. For example: In the above curve the mode is at 117°9998 cms. of stature and the origin at 113°8228, thus the distance between them =41770 cms. or since the working unit of x=2 cms. and the positive direction of # is towards dwarfs, the mode is at x =—2:0885. Required to find the maximum ordinate y,,,. We have tan @ = — 2:0885/14-9917 = — 139,3104, whence by a table of natural tangents @=—7° 55’ 851265, =—- 7° 55’ 51”. The log cosine of this value of @ is 1:995,8962. Table LITII gives us: = 122,1730 in are ay = 015,9989 ,, ,, 851,265’ = 851,265 x 000.2909 = -000,2476 ,, ,, Hence 06=—'138,4195 ,, ,, * See Phil. Trans. Vol. 186, A, p. 387. Pearson’s Type IV frequency curve fitted to the stature of 2192 St Louis School Girls aged 8. LITI—LIV] Introduction Ixxxi Hence log Ymo = 2°371,6644 + 32°8023 (— 004,1038) + 1:984,5521 x 188,4195 2°511,7510. 324901. Hence Ymo TaBLE LIV (pp. 126—142) Tables of the G(r,v) Integrals. (Calculated by Alice Lee, D.Sc. Transactions British Association Report, Dover, 1899, pp. 65—120.) The purpose of this table is to obtain the value of the integral CGE | maint Goede) 2. eat S (8s), 0 In order to obtain small differences in tabulated values two additional functions F'(r, v) and H(r, v) are introduced. The relations between the three functions are then expressed by the following series of equations: TONED) SCALE CD (ee (Ixxxiv), r+ F(r, v)= men gr" Jal GRAD) encconcoscen SeCeeeece (Ixxxv), Vr — CGA Dien A (esp adhe, of eee ete. Heke (Ixxxvi), vp+hur r+ G(r, v) = ——eer Ja! Gs) woosoeneonascd (Ixxxvii), Ha — Vr —1e- H(r, v)= a EGE v)\ Bacestsitnetcorbastenie: (Ixxxviii), al —vp—tvr H (r, v) = ea GGA) aeronaemansaeecss (lxxxix) where tan ¢ = p/r. Pearson’s Type IV Skew scontnes Curve is of the form -ytan-! = a eailaseanecandasenadtaseutrnice (xc). Sey a Hence if NV be its total area, i.e. the entire population under discussion, N = yae~ Aer cE sin” Ge” dé, 0 ee i i iy = = @ FG,9) Ixxxil Tables for Statisticians and Biometricians [LIV The function H (r, v) is introduced because, as a rule, its logarithms have far smaller differences and it is thus capable of more exact determination from a table of double entry. Its physical relation to the curve may be expressed as follows ; let the origin be transferred to the mean, then if y, be the ordinate at the mean, N 1 ;\bis n= = H(r, v) wininininie MUtelwia/ujainlrintelsinivisisie(ale/nie(olely (xci)? 5 where o is the standard-deviation of the curve a &: — Verena A ey ara otaveravele alaleis aise ieie:e'nia fe belwimints fabs (xen) The distance of the mean from the origin is given by [OY SSO) Casdonccossdoanesoconogodsoote (xciil). When r is fairly large: cos*d@ 1 a : 1 igh earn tee ; F(r, v) = A/ on (cos oy RB aquicteteletaie «isiaiataintatats (xciv) 1 Tr 1 eu Hence ———— oe ge eee xcv), a H (7, v) Nope oe Ce (a 2 where A= vy, —— and thus the evaluation if ¢ be > 60° may be made by aid of Table II*. Illustration. In the curve fitted to the statures of St Louis School Girls, aged 8 (p. lxxx), we have N= 2192, a= 149917, r = 30°8023, v= 456967. Find y. We have : tan @ = v/r = '148,3548. Hence b=8° 2631315 = 8°-43855. Turning to the Tables, p. 136, we see the large differences of log F'(r, v) at this value of ¢, and accordingly settle to work with log H(r, v). We have for log H (r, v), r= 30 r=l o=8: 388, 2032 3885583, o=9° 388,2278 *388,5822, = 87-4386, r= 30: log H (r, v) ="388,2032 + (4386) [246] — 4 (4386) x (5614) [28] =‘388,2137. * For a fuller discussion of these integrals see Phil. Trans. Vol. 186, A, pp. 376—381, B. A. Trans. Report, Liverpool, 1896, Preliminary Report of Committee..., and the B.A. Trans. Report, Dover, 1899, already cited, LIV—LV] Introduction Ixxxiii $ = 8°-4386, r=31: log H(r, v)= °388,5583 + (4386) [238] — 3 (4386) (5614) [27] = 388,5684. $ = 8°-4386, r=32: log H (r, v) ="388,8910 + (4386) [231] — 3 (4386) (5614) [26] = '388,9008. Hence ¢ = 8°4386, 7 = 30°8023: log H (r, v) ="388,2137 + ‘8023 [3547 | — $ (8023) (1977) [— 223] = 3885001. Hence by formula (lxxxv) : log F(r, v) = v¢ loge + r + 1 log cos $ — 4 log (r —1) + log H(r, v). Or, using Tables LIII and LV, we have log F(r, v)= °-292,2901 — -737,1249 + 1:849,6578 + 388,5001 — °737,1249 y log F(r, v)= 1-793,3231 Finally from formula (xci) : log y = log NW — log a — 1:793,3231 = 3°340,8405 = — 1:175,8509 — 1793,3231 — -969,1740 = 2°371,76665. Or Yo = 235°324*, TABLE LV. This table contains some miscellaneous constants in frequent statistical or biometric use and requires no illustration, It has already been used in the illustrations to previous tables. I have had the generous assistance of my colleagues Miss E. M. Elderton and Mr H. E. Soper in the preparation of the Illustrations to these Tables. I can hardly hope that arithmetical slips have wholly escaped us in a first edition, and I shall be grateful for the communication of any corrections that my readers may discover are necessary. * The value 235-323 obtained in Phil. Trans. Vol. 186, A, p. 387, was found by the approximate formula (xciv) before tables were calculated. Every reader may now see in what way the higher branches of mathematics are concerned in our present subject. They are the abbreviators of long and tedious operations, and it would be perfectly possible, with sufficient time and industry, to do without their use....... When both the ordinary and the mathe- matical result are derived from the same hypothesis, the latter must be the more correct : and in those numerous cases in which the difficulty lies in reducing the original circumstances to a mathematical form, there is nothing to show that we are less liable to error in deducing a common sense result from principles too indefinite for calculation, than we should be in attempting to define more closely, and to apply numerical reasoning —DrE Morean. Tables of the Probability Integral TABLE I, Tuble of Deviates of the Normal Curve for each Permille of Frequency. Permille] -000 | -001 | -002 ‘09 «© | 30902 | 2:8782 ‘01 | 2°3263 | 2-2904 | 2-2571 02 | 2:0537 | 2:0335 | 2:0141 ‘03 | 1-8808 | 1:8663 | 1-8522 ‘04 | 1°7507 | 1:7392 | 1°7279 05 | 16449 | 1-6352 | 1-6258 06 | 1:5548 | 1°5464 | 1-5382 ‘07 | 1-4758 | 14684 | 1-4611 ‘08 | 1-4051 | 1°3984 | 1-3917 09 | 13408 | 1°3346 | 1-3285 10 | 1-2816 | 1:2759 | 1-2702 11 ‘| 1:2265 | 1:2212 | 1:2160 12 | 11750 | 1:1700 | 1-1650 13 | 1°1264] 1:1217 | 1°1170 14 | 1-0803 | 1:0758 | 1-0714 15 | 1:0364| 1-0322 | 1-0279 16 | 0-9945 | 0-9904 | 0:9863 ‘17 | 0-9542 | 0-9502 | 0:9463 18 | 0°9154 | 0-9116 | 0-9078 19 | 0°8779 | 0°8742 | 0°8705 -20 | 0°8416 | 0-8381 | 0-8345 21 | 0:8064 | 0-8030 | 0-7995 22 | 0-7722 | 0:7688 | 0-7655 -23 | 0°7388 | 0:7356 | 0°7323 -24 | 0-7063 | 0:7031 | 0:6999 25 | 0:6745 | 06713 | 0°6682 26 | 06433 | 0-6403 | 0-6372 ‘27 | 0-6128 | 0-6098 | 0-6068 -28 | 0°5828 | 0°5799 | 0:5769 29 | 0:5534 | 0:5508 | 05476 ‘30 | 0°5244 | 0-5215 | 0°5187 “31 | 0:4959 | 0-4930 | 0-4902 “32 | 0:4677 | 0-4649 | 0-4621 33 | 0:4399 | 0-4372 | 0-4344 “34, | 0°4125 | 0:4097 | 0-4070 ‘35 | 0°3853 | 0-3826 | 03799 -36 | 0°3585 | 0-3558 | 0°3531 ‘37 | 03319 | 0-3292 | 03266 ‘38 | 0°3055 | 0-3029 | 0-3002 39 | 0-2793 | 0:2767 | 0:2741 40 | 0°2533 | 0-2508 | 0-2482 41 | 02275 | 0-2250 | 0-2224 ‘42 | 0-2019 | 0-1993 | 0-1968 +43 | 0-1764 | 01738 | 0:1713 071510 | 0:1484 | 0:1459 071257 | 0:1231 | 0-1206 01004 | 0:0979 | 0:0954 0-0753 | 0:0728 | 0:0702 0:0502 | 0:0476 | 0:0451 0:0251 | 0:0226 | 0:0201 010 | :009 | -008 “003 2°7478 2°2262 1°9954 1°8384 1°7169 1°6164 1°5301 1°4538 1°3852 1°3225 1°2646 1°2107 171601 171123 1:0669 1:0237 0°9822 0°9424 0°9040 0°8669 0°8310 0°7961 0°7621 0°7290 0°6967 0°6651 0°6341 0°6038 0°5740 0°5446 05158 0°4874 0°4593 0°4316 04043 0°3772 0°3505 0°3239 0:2976 0°2715 0°2456 0:2198 0°1942 0°1687 071434 0-1181 00929 0:0677 00426 00175 ‘007 004 2°6521 2°1973 1:9774 1°8250 1°7060 1°6072 1°5220 1°4466 1°3787 1°3165 1°2591 1°2055 171552 1°1077 1°0625 1:0194 0°9782 0°9385 0°9002 0°8633 0°8274 0°7926 0°7588 0°7257 0°6935 06620 0°6311 0°6008 0°5719 0°5417 0°5129 074845 0°4565 0-4289 0°4016 0°3745 0°3478 0°3213 0°2950 0°2689 0°2430 0°2173 0°1917 071662 0°1408 071156 00904 0°0652 0°0401 00150 006 005 006 2:5758 | 2°5121 21701 | 2°1444 1-9600 | 1:9431 1°8119 | 17991 1-6954 | 1°6849 1°5982 | 1°5893 1°5141 | 1°5063 1°4395 | 1°4325 1°3722 | 1°3658 1°3106 | 1°3047 12536 | 1°2481 1°2004 | 171952 171503 | 171455 171031 | 1°0985 1°0581 | 1°0537 1:0152 | 1:0110 0°9741 | 0°9701 0°9346 | 0°9307 0°8965 | 0°8927 0°8596 | 0°8560 0°8239 | 0°8204 0°7892 | 0°7858 0°7554 | 0°7521 0°7225 | 0°7192 0°6903 | 0°6871 0°6588 | 0°6557 0°6280 0°5978 0°5681 0°5388 0°5101 0°4817 0°4538 0°4261 0°3989 0°3719 0°3451 0°3186 02924 0°2663 0°2404 0:2147 0°1891 0°1637 0°1383 0°1130 0:0878 0:0627 0:0376 0°0125 0°6250 0°5948 0°5651 0°5359 0°5072 0°4789 0°4510 074234 0°3961 03692 0°3425 0°3160 0:2898 0°2637 0°2378 0°2121 01866 01611 071358 071105 00853 00602 0°0351 0:0100 “007 ‘008 “009 -010 2°4573 | 2°4089 | 2°3656 | 2°3263 2°1201 | 2:0969 | 2:0749 | 2:0537 1°9268 | 1°9110 | 1°8957 | 1°8808 1°7866 | 1°7744 | 1°7624 | 1°7507 1°6747 | 1°6646 | 1:6546 | 1°6449 1°5805 | 1°5718 | 1°5632 | 1°5548 1°4985 | 1°4909 | 1°4833 | 1°4758 1°4255 | 1°4187 | 1°4118 | 1°4051 9 1°3595 | 1°3532 | 1°3469 | 1°3408 “91 1°2988 | 1:2930 | 1:2873 | 1:2816 ‘90 1°2426 | 1:2372 | 1:2319 | 1°2265 89 11901 | 171850 | 171800 | 1°1750 88 1°1407 | 1°1359 | 1°1311 | 1°1264 “87 1:0939 | 10893 | 1:0848 | 1:0803 86 1:0494 | 10450 | 1°0407 | 1°0364 85 1:0069 | 1°0027 | 0°9986 | 0:°9945 “8h 0:9661 | 0°9621 | 0:9581 | 0°9542 “83 0°9269 | 0:9230 | 0°9192 | 0:9154 "82 0°8890 | 0°8853 | 0°8816 | 0°8779 “81 0°8524 | 0°8488 | 0°8452 | 0°8416 “SO 0°8169 | 0°8134 | 08099 | 0°8064 TO 0°7824 | 0°7790 | 0°7756 | 0°7722 ‘78 0°7488 | 0°7454 | 0°7421 | 0°7388 fi 0°7160 | 0°7128 | 0°7095 | 0°70638 ‘76 0°6840 | 0:6808 | 0°6776 | 0°6745 Ces 0°6526 | 0°6495 | 0°6464 | 0°6433 Th 0°6219 | 06189 | 0°6158 | 0°6128 ‘73 0°5918 | 0°5888 | 0°5858 | 0°5828 ‘72 0°5622 | 0°5592 | 0°5568 | 0°5534 chil 0°5330 | 0°53802 | 0°52738 | 0°5244 Cr 0°5044 | 0:5015 | 0-4987 | 0:4959| -69 04761 | 0°4733 | 0°4705 | 0°4677 ‘68 074482 | 0°4454 | 0°4427 | 0°4399 sin 0°4207 | 0°4179 | 0°4152 | 0°4125 66 0°3934 | 0°3907 | 0°3880 | 0°3853 65 0°3665 | 0°3638 | 0°3611 | 0°3585 “64 0°3398 | 0°3372 | 0°3345 | 0°3319 63 0°3134 | 0°3107 | 0°3081 | 0°3055 62 0°2871 | 0°2845 | 0°2819 | 02793 61 02611 | 0°2585 | 0°2559 | 0°2533 60 02353 | 0:2327 | 0:2301 | 0-2275| -59 02096 | 0°2070 | 0°2045 | 0°2019 58 0°1840 | 0°1815 | 0°1789 | 0°1764 ron 0°1586 | 0°1560 | 0°1535 | 0°1510 ‘56 071332 | 0:1307 | 071282 | 0°1257 a) 0:1080 | 0:1055 | 0:1030 | 0°1004 Sh 00828 | 0:0803 | 0°0778 | 0:0753 53 0°0577 | 0°0552 | 0°0527 | 0:0502 be 0:0326 | 0°0301 | 0°0276 | 0:0251 “Di 0:0075 | 00050 | 0:0025 | 00000 “50 003 “002 001 ‘000 | Permille 005 | ‘004 2 Tables for Statisticians and Biometricians TABLE II. Area and Ordinate in terms of Abscissa. 2 2 Foe aay et Be z aah ps ‘00 } 5000000 o | -3989423 399 : 39894 Reeees 199 or | “5030804 | 3555, 4 | 3089223 Pee Nae8 o2 | -so797e3 | 320% s | -3988625 ae, 399 os | -siige6s | 39802 | 12 | -3987628 soos | ies “04 D159534 39854 16 3986233 1793 398 05 |} -sigg3ss | 3°84 | 90 | 3984439 Bee egy 06 | 5239222 24 | -3982248 397 06 | -5239222 | 39810 uob ae 2588 09 | -535s564 | 29750 | 36 | -3973298 | 3379 | 304 eREDG 39714 eens 3773 10 | -5398278 | 33708 40 | 3969525 | 3773 | 393 11 | 5437953 i 44 | -3965360 : 392 Be | aoe 39631 ( 4558 ie | Samael | set | ae | ome | gs | 14 | 5556700 | 39°82 55 | -3950517 5337 | 387 ia eS ECT Pg eet 5724 15 | 3506177 | 22077 | 59 | -sodavo3 | Bitg | 386 16 | ~:5635595 £ 63 | 3938684 384 46 | ~263 39355 o> | |S 6493 : ie | are | Sat |S | ibe | gee | i 19 | ‘5753454 | 30217 | 74 | 3918060 eal ri 20 | -5792597 | Zog2 | 78 | -ao1ozav | £693 | 375 21 | 5831662 _ s2 | -3902419 : 373 ; 38983 J) |) Seg) 8381 7 ee ee BB sen n[ eu lea -2, | -5948349 | 38898 | 3 | “3e7e166 | 2120 | 365 Be ew ee ree 4) Fess Sew ree 9485 25 | -5987063 | 3°12 | 97 | 3866681 Pie eae? 26 * -6025681 es 100 | -3856834 360 oe re 38518 PAS MR esi || Oe 2 | mato | gee | IN | Rees | ar | oe -29 | -e14ogi9 | 28°06 | aia | -3895146 | 20917 | 350 BA pene | S819 Bees 11268 3 i7piia || 28he, | 114) seaave: |Pateven| a7 31 | 6217195 ~ 11s | -3802264 344 32 | -e255158 | Srey | 121 | 3790309 | j3993 | 340 | gen | etl | Gea ‘Bres7a | 12635 ae is nag | | 8288 Ser oGa nl” Lecoanalmese 8 -eseaso7 | 2/28? | 131 | -aveeso3s | iaeee || 329 36 | -6405764 z 135 | -3739106 325 ‘3 6405764 | 37303 | 2 BISS106 "|" Nagas yl) poe | te | Se | BS | SS | ae | 39 | -esi73i7 | 37044 | yaa | -segzo77 | 14262 | 33 eH pase 36900 = nen 14575 . 40 | 6554217 | 36900 | 147 | 3682701 | ‘ene | 309 ‘Ht | 6590970 | ggeq9 | 150 | 3667817 | yey9q | 305 b | fers | gens | ie | Sear | tei | i 6700314 | 26293 | 359 | -3ea1349 | 1787 | 93 55 \ -e73z6iis | 26133 | 169 | -3605270 | 16079 | ogg 4 36448 | 35971 16367 46 | 6772419 | asaog | 165 | 3588903 | jean | 283 47 | -6808225 | 308 68 | -357295: 4s eBes808 oe ries OReEReaee ioe a ‘9 | 6879331 | g5oqq | 178 | “3588124 | arazo | 262 50 | 6914625 176 | 3520653 264 2 zs | 3@+e) | | aie 50 | -6914625 176 51 | 6949743 bi 179 52 | -egsacs2 | 34989 | 181 53 | -roisss0 | 3478 | 184 54 | “7054015 | 343%" | 186 55 | “7088403 | 3130" | 189 56 | -7122603 191 57 | -7ise612 | 3300? | 193 58 | “7190427 | 38857 | 196 59 | 7224047 | 33620 | 198 oA . i= Af co | -7257469 | 32555 | 200 ‘61 | -7290691 202 62 | °7323711 ae 204 63 | -7356527 | 32816 | 906 64 | -7390137 | 22650 | 908 “65 ‘ 65 | “7421539 | 22508 | 210 66 | 7453731 212 ‘67 | 7485711 aoe 214 8 | -rei7a7s | 31/87 | ois 69 | -7549029 | 31991 | 917 7o | -7580363 | 31382 | 19 71 | 7611479 220 72 | 7649375 pe 222 73 | -7673049 | 300/t | 993 75 | 7703500 | 30838 *| 285 75 | -733736 | 30°06 | 996 76 | -7763797 297 “77 | -rrgss01 | 22042 | 338 73 | -7823046 | 32202 | 930 49 °|) -ra52361 |) Soe geaiiaeall so | -73si4s6 | 32082 | ope ‘31 | -7910299 | . 233 82 | -7938919 Bk 234 33 | -7967306 | 38387 | 935 84 | -7995458 | 28092 | 935 35 | 3028375 | S/al? | 936 86 | 8051055 237 -87 | -8078498 sete 238 38 | 8105703 | 3720 | 938 39 | -8132671 | 368% | 939 ‘90 | 8159399 | 38738 | 939 91 | -8185887 240 ‘92 | 8212136 RS 240 3. | -sazaias | 26008 | oat 04 | -s2ezo12 | 22/68 | 941 “O5 *! 95 | -sago439 | 20087 | oat 96 | -8314724 242 ‘oy | -8339768 Fert 242 93 | -8364569 | 2180 | o4p 99 | -s3sgi29 | 34360 | 949 1-00 242 8413447 : | A 3520653 | yay 3502919 2 | 17994 3agag25 | 12380 3466677 | ieigy 3448180 | j802) 3429439 | T6981 3410458 | 9915 3301243 | 1oi44 3371799 | Ioeey 352132 | toon 3332246 | donoc 3312147 3291840 | 20307 20510 eopisan | 30) 3250623 30899 229724 | 2080 3208638 igisya71 | 21267 165929 | 31802 3144317 | Siac °3122539 31936 3100603 3078513 a 3056274 | 33230 3033803 | 3030) eplien, | 2° -2988724 2965948 | Sonne 2943050 | 22807 2920038 | 32033 asg616 | 3355 2873689 “ie 2850364 a 2526945 | 3380 9803438 | 23007 2779849 | 330% 2756182 ‘2732444 rath 2708640 | 2380? 2684774 | 23866 2660852 | 33073 -2636880 2619863 | 34017 3588805 | 24058 s5eaTia | 24093 540591 | Salis 2516443 mag9977, | are! -2a6g095 | 34te° 2443904 | 3410 2419707 Tables of the Probability Integral TABLE II.—(continued). 2 2 az Pecan te ao. = 264 1-00 | 8413447 | giorg | 242 259 101 | -3437524 | 2anl8 | 94s 254 102 | -saeiass | 33°35 | 249 249 103 } -g494950 | 32002 | 942 244 104 } 8503300 | 33351 | 249 239 105 | -s631409 | 33009 | ai 234 106 | -8554977 oe | 24 229 ror | -ss7e903 | 32626 | 241 994 108 | -s599289 | 33388 | 240 219 roy | -se21434 | Seat? | 240 213 110 | 8643339 | 312? | 240 208 111 | 8665005 | grog | 239 203 112 | -sescas1 | 21326 | 239 197 113 | -8707619 | Sooeq | 238 192 114 | 8728568 | Sore | 237 187 115 } 8749281 | Soy75 | 237 181 116 | 8769756 | gooq | 236 176 117 | -8789995 | 30%) | 935 170 118 | -ssog999 | 20004 | 935 165 119 | -8s29768 | 12769 | 934 159 120 } -9849303 | j2n40 | 933 154 121 | 8868606 | joo-9 | 232 148 1:22 | -8887676 | jeuan | 231 143 1:23 8906514 18609 230 137 1:24 | 8925123 | ies, | 229 132 1%5 8943502 18151 228 126 126 | 8961653 | jrooq | 227 121 127 | 8979577 | jie, | 226 115 128 | -soo7a7a | A7i7! | 995 110 129 | 9014747 | {/oj, | 224 104 130 | -9031995 | joo | 293 99 131 | -9049021 299 93 132 | -gocssas | 16804 | 990 88 133 | -9082409 | jane | 219 83 134 | 9098773 | ipa, | 218 77 135 | 9114920 | 18037 | a7 72 1:36 | -9130850 men Marit 66 137 | 9146565 | 12/0" | 214 61 138 | 9162067 | }223, | 212 56 ra MES com | mee 51 140 | 9192433 | Vers | 210 45 141 | 9207302 208 40 142 | 9221962 | 1350 | 207 35 143 | 9236415 | tioaq | 205 30 rj4 | 9250663 | Tia, | 204 25 145 | 9264707 | j3e49 | 202 20 146 | -9278550 201 15 147 | -9292191 ee 199 10 1-48 | 9305634 | 13543 | 197 5 1-49 | -9318879 | j351, | 196 0 150 | -9331928 2) “y04 - A ne — + ‘2419707 0 9395511 | 24196 5 2371820) || Zajes. | 10 2347138 ae 14 2322970 hon 219 2298821 | oa195 24 2274696 & 28 24097 29 2202508 | 22027 | 41 2178522 | 23986 | 46 * | 93940 2154582 | 5 50 Bare 23890 Sac |) eee ee aossors | 278 | a 2059363 | da649 66 2035714 | oacn 70 2oigiss | 23578 | 74 agssGet || Geer cl ae "1965205 ea 82 1941861 23259 85 sme | saxo | “igyaasa |) 28077 96 1849373 | 2298) | 99 1826491 pav7g | 103 us03012 ol) ogee ide" 1781038 ea 109 Boe 22564 : 1758474 92459 112 1736022 9337 115 "1713686 22218 118 1691468 | 9. 121 1669370 een 124 1647397 91847 127 T6255 pean etiie lgleo 1603833 | 2/717 | 39 ee 21585 ‘1582248 yi 134 1560797 | 21374 | 137 1539483 | S375 | 139 -1518308 is 142 1497975 | 210383 | j44 2 20890 1476385 = 146 1455641 Ur 148 2 20596 e "1435046 20446 150 1414600 | Soo94 | 152 1394306 | Soj4g | 154 1374165 155 1354181 vee 157 1334353 | joggq | 159 1314684 19508 160 *1295176 162 1—2 Sed TS wR RRR SanND ~% ~2 ~~ S WHOS GOS TABLE II , . Area and Ordinate in terms of Absci SCUSSAL. 3 (1+a) 4 A? + Ss Z a a2 9331928 7 2 e 7 (1 - 9314783 | 12855 194 } -1295176 pe s -9357445 12662 193 “1275830 19346 162 2-0) “9369916 12471 191 "1256646 19183 163 a1 pe .9389198 12282 189 1237628 19018 165 2-02 ‘9777844 5345 ey || pieeoe. | Sec Wane 1gs53 | 166 geo || Goh sige ree | 186 Fee ta000BD alll ere 167 we || need 5031 -9406201— 18517 | 168 | opel yee 9417924 11724 184 ‘1181573 : ee Pes 9429466 | 11541 183 | -1163225 | 18348 ate 2-06 Bee icons, | FeO: “|. coal orate isi77 | 1 cor t -onoeeseal aes 9459007 | 11181 179 | 1127049 | 18006 171 pa ee) ae jn eTeT Al are Wi “1t0s208= |e 12 4) ae 4559 ‘9463011 17661 173 2-1 ‘es16911 | 4445 SECT) me pe | ee 240 | 9821356 | 359 “9484493 10654 174 1074061 17487 174 2-11 berg a Pe a lar Tyg be Ary \|ecae 9505985 | 10311 170 | -los9611 | 17187 ne 2-13 9820970 | fa79 10142 | 169 | °1022649 16962 | 176 21) osa4142 | Aone ‘9515428 ie7ag | 1%6 pune 3098 ‘9525403 9975 167 "1005864 : ‘92224 | Sor 9535213 gsio | 165 | 0989255 16609 | 1%? 2-16 ae 9554345 9485 | 162 | 0956568 16255 | 177 2:18 9849966 | 3r47 oece | teo) 4) 0940401 ||| Geaad 178 4 |) ae 3666 9563671 iss99 | 178 J) ee eee ae ‘9572838 9167 | 158 } °0924591 220 | 9860966 | 3509 9581849 9011 156 0908870 15722 178 2-24 som *9590705 8856 155 "0893326 15544 178 2-22 Reset ‘9599408 g7oa | 153 | 0877961 is366 | 178 2-23 9867906 | 3357 9553 | 151 0862773 | 15188 Lie 2-2) 9871263 | 3083 Dee seni pe soe 9874545 | 3283 9616364 8403 149 ‘0847764 3 Bi a a8 spo IW BenG arabe: aoeeee 1agazen) Ae 2:26 iy aEaeeaD 8256 | 146 | -0818278 14654 | 178 es “9880894 9640697 7966 144 0803801 14477 177 9:98 9883962 3068 7824 142 0789502 14300 177 9-29 “9886962 2999 ae Taree |: a ze -ggggsg3 | 2982 ‘9656205 7es4 | 140 J 0775379 0 | 9892759 | 3500 ‘9663750 7545 | 189 | 0761433 1s946 | 176 23. ee gees: || adopt Ler ah ea tees 13770 | 476 ae | sean eesu |) gaya, | 182 Rie Cees 13594 | 176 ose | snd 074 ae ie ures EE tye 2 = 9900969 2674 eee Ieee alt as Be | eee 2612 pe ses 1c Aes 2 9906133 2552 “9699460 6879 130 0694333 13071 174 2:36 aaa See apis ea eee (et) es bere Aiog Rates ee 9712834 6624 126 0668711 12725 173 2-38 Bones pie 6500 125 0656158 12553 172 2:39 ‘9913437 2377 eS iin eee |. Wl 4 9915758 | 221 eee | ery ely eee yo | 9918025 | S13 ‘9731966 6255 121 0631566 12211 170 OHI = ‘9738102 613g | 120 | 0619524 12041 | 170 Ae) Meee! ‘9744119 oase. |. 118 | ~oaoyenat | Tae re ue rs (see ae oe 116 ie fees 168 oe "9924506 2108 ‘9750021 11538 | 167 en | ween ates ‘9755808 5787 115 0584409 Ee Benoa as ‘9761482 5674 | 11s | 0578038 ame -|° tee 246 pi hie WR Me ape er 11306 | 165 Hide (eetRa rroiGAt Ut B4b8 Ame cl cose gary 11042 | 164 ous | po3ie08 1865 oava > ee oy 9934309 | 1865 162 249 | 9936128 1820 250 | 9937903 | 1775 108 0539910 Tables of the Probability Integral 5 TABLE II.—(continued). A 2 2 z a £ 3(1+a) . = z a i 0539910 250 | -9937903 44 | -017528: 0529192 ee 251 | -g939034 | 1731 | 4g ie 4336 | 3 0518636 2-52 | -9941323 | 1688 2 carol | 2228 || eg 05186: ee : 42 | 0166701 ZD |) Gs 0508239 | jog 253 | -go49969 | 1646 | 41 | -o1e2545 | 4157 0498001 ee 2-54 | -9944574 | 1605 | Oe 0498001 | 0081 2 o44574 | 1605 | 40 | -o1sea76. | Son, | 86 2 5084 255 | -9946139 | 126? | 39 | -o1ssa93 | 3207 | 85 0477996 256 | -9947 05 . ‘047791 ate 5 9947664 _ | 39 | 0150596 0468226 aeie 257 | -goagis1 | 1487 | 33 | -o1ae7s2 | 3814 es 0458611 2-58 | -9950600 | 1449 7 See | cere ee } 9463 ae 600 > 37 0143051 ae 81 0449148 : 259 | -g99sa012 | 1412 | 36 | -oisgac BAU | Gn 0439836 | Oigo 260 | -gooazes | 1876 | 35 | ‘oisses0 | 2571 | 7 9162 BM san lice) 2882890. Nesio3: || 278 0430674 149 261 | - ¢ 5 |. 0132337 ‘0430674 sos | M9 2: 9954729 | jag | 35 |. 0192887 | gag | 77 eel gees | 147 262 | 9956035 | jo79 | 34 | 0128921 | 35 5 | HG 011279 se 6 68 | 9957308 | 1555 | 33 | 0120581 | 356, | 74 0404076 aie 145 264 | 9958547 | jon, | 32 | 0122315 3266 | a3 meee Wy Sarees 207 = ee 3193 | & oe 65 | -g9s9754 | 1°07 | 3a | coisa. | 337 | 72 0387069 Ee 266 | - 0387069 gag | 2 206 | 9960030 | q44, | 31 | 0116001 | 495, | 70 ae 267 | -9962074 30 | 0112951 0370629 9 | 139 268 | -996318 1115 | 43¢ sage || eel |e “037 enc 268 189 5 | 99 | -o109969 1 | 68 0362619 cay |p iss 2-69 | -9964974 | 1985 | 99 | -o1ozose | 2213 | 67 0354746 S Be 3 = 056 ae eee 2847 854746 eee. |. 136 27 9965330 | 1002 | 28 | 0104209 ae 66 0347009 aay es 271 | -996 01014: 0339408 te 133 2-79 Eecaro 0 = eave. | 2 iS 0331929 468 | i390 273 | -996833: 974 | 9 cps lt eee tees ‘OBB1924 Lees 2 68333 26 | -0096058 6 62 os24603 | 138 130 274 | -9969280 | 248 | 2 gage | 2092 mee 7206 es a I 9966 999 26 ‘0093466 2731 61 7077 ‘ 275 | -9970202 g97 | 25 | 0090936 ea7i | 80 0310319 197 276 | -997 relies Pee rae a | jet) gee ln |e | eee |e | 0296546 Panes | 188 273 | -9972821 BASe || aso) guescey |e ee alleee 0289847 699 | 193 279 | -9973646 825 | 3 0083697 | 5303 | 56 ‘oze9e47 | org | 128 Sot 23 | -ooeiae8 |vce,) || 359 vee 2 2:30-| 9974449 Bos | 92 | ovate | Fia5 | 54 0276816 . | 190 2:81 | 9975229 Fy) Lagat 0270481 | 639) | 119 ge | goveoss | 79 | ai | oorasa | 2136 | 50 Pee ens |) Lee 283 | 9976726 | 728 | 21 ee peme 0258166 116 23, | 9977443 | 717 | 9 0707 2038 | 5 10258166 posi | 16 2 : AE «20 | o0rowat 20 50 aar6 2:85 | -9978140 ee BS 49 0246313 113 2:36 | 997 00667 0240556 oe 111 co See ae Jeol Sceeiora 128 re 02: 110 2:38 | -9980116 0 8 | -oos3067 | 1839 ‘0234910 sag | 110 501 610 | 18 | 0063067 Seen ll 46 Miea 2:89 | -9980738 22 | ig | -oosia74 | 1793 | a5 0223945 Faso. (4 10% 2-90 | -9981342 ote 17 | 0059525 ey 44 i 0218624 105 291 | - 0057 0213407 are 104 2-92 Bee 570 Z ee 1661 | 75 ‘oans294 | Synz | 102 293 | -9983052 553 | 16 | -oos4541 | 1819 | a7 101 2-9), ‘9983589 537 = ERR 1578 Giga 4910 a es. 9835 529 16 0052963 1b 40 ni 2:95 | -9984111 oe | is | 0051496") vee 40 0193563 98 2: f : ona 96 | -9984618 15 | -0049929 "0188850 pote 96 2-97 | -9985110 492 | 14 | -oo4sa70 | 1459 & oisaea3 | 4607 95 2-98 | -9985588 Ae ie Neato |e! allege ‘overt Fee 93 2-99 | -9986051 464 | 14 | oossecs | 1384 36 283 92 3:00 | -9986501 450 | 13 | -oovisis | 147 | 35 a 6 Tables for Statisticians and Biometricians TABLE II. Area and Ordinate in terms of Abscissa. A A? A a2 i A A? x 3 (1+a) a <4 z Be i Ey 4(1+a) = 7% 00 | 9986501 | 4» | 13 | o0143is | ,,,,'| 35 350 | 9997674 | g, | 3 301 | -9986938 | 45, | 13 ] 0043007 | 55.7 | 35 351 | 9997759 | 92 | 3 3:02 | -9987361 | 43) 13 | 0041729 | jo45 | 34 352 | -9997842 | oo | 3 $03 | 9987772 | So, | 12 | 0040486 | 3555 | 33 58 | 9997922 | 7 | 3 3-04 | 9988171 | 33- | 12 | 0039276 | 3373 | 32 $54 | 9997999 | 21 | 3 305 | 9988558 | 5-4 | 12 } 0038098 | jyi_ | 32 3:58 | -g998074 | 45 | 3 pala . \2 * 5 oh pele ¢ so6 | 9988933 | 5,, | 11 | -0036951 | 444, | 31 56 | 9998146 | gy | 3 307 | 9989997 | S25 | 11 | 0035836 | 3592 | 30 $57 | -so08215 | go | 2 3-08 | 9989650 | 335 | 11 | 0034751 | 3558 | 29 $58 | -oogses2 | Qf | 2 3:09 | -9980992 | 355 | 10 | -0033695 | 3539 | 29 359 | 9998347 | go | 2 310 | -9990324 | 355 | 10 | -oosace8 | “G54 | 28 360 | 9998409 | go | 2 311 | 9990646 | 5,5 | 10 | ‘oosi6c9 | 9, | 27 361 | gooss69 | ,. | 2 312 | 9990957 | 355 | 10 | ‘003098 | G1, | 27 362 | -9098527 | 2. | 2 313 | 9991260 | So, | 9 | 0029754 | G7, | 26 3:63 | -9098083 | PY | 2 314 | 9991553 | 55, | 9 | 0028835 | 24, | 26 364 | 9998637 | 25 | 2 315 | 9901836 | 57, | 9 } 0027943 | fee | 25 865 | 9998689 | 5 | 2 316 | -9992112 _ | 9 | -0027075 24 3:66 | -9998739 2 17 | -g9gaa78 | 282 | 8 | -oozeaz1 | 85? | 94 367 | -go9s787 | 48 | 2 318 | -9992636 | $25 | 8 ]| 0025412 | ~o | 23 368 | -oggsss4 | 47 | 2 319 | -g992886 | 395 | 8 | ‘oozes | foi | 23 sé9 | -gooss79 | 45 | 2 2.7) “ 216 = “005 9° 2. . 2999 = 3-20 | -go93129 | $55 | 8 | ‘cozsss1 | 7/5 | 22 so | -gogsoa2 | 45 | 2 g21 | -9993363 | oo | 7 | 0023089 | .., | 21 371 | -9998064 | 4. | 2 3:22 | 9993590 | 55, | 7 | ‘0022858 | 5) | 21 v2 | -9999004 | 35 | 1 323 | -9993810 | 55 | 7 | oozles9 | fo, | 20 373 | -g999043 | 35 | 1 324 | -g9o4024 | 55% | 7 | -0020960 | Gea | 20 37h | -o999080 | 32 | 1 3:25 | 9994230 | 55, | 7 | 0020290 | gs5 | 19 g75 | g9o9116 | 32 | 1 326 | -9994429 6 | 0019641 19 376 | -9999150 1 3-27 | -go0i623 | 192 | 6 | -ooi9010 | 62) | 18 377 | -googis4 | 33 | 1 3-23 | -9994810 | 33, | 6 | 0018397 | Zo | 18 7s | -so9o216 | 37 | 1 3-29 | g9o4991 | j-, | 6 | ‘Ool7803 | fo | 17 sro | gogo2d7 | 39 1 3:30 | -9995166 6 | -0017226 Ci ily 380 | -99995 ae 169 560 29 331 | 9995335 | jo, | 6 | oo1esss | .,. | 17 381 | 9999305 | 5, | 1 332 | -9995499 5 | -0016122 oe ipo de 382 | 9999333 | Se | 1 A 159 : 527 : in . Bs 27 1 3:33 | -9995658 | je, | 5 | ‘oo1s595 | 735 | 16 3:83 | 9999359 |. 56 3:34 | 9995811 | j43 | 5 | 015084 | fo, | 15 $84 | 9099885 | 95 | 1 335 | -9995959 5 | -0014587 15 385 | 9999409 143 481 24 3:36 | -9996103 | j59 | 5 | ‘0014106 | 4— | 15 ss6 | 9999433 | 5, | 1 337 | -9996242 | 13, | 5 | ‘0013639 | 423 | 14 ssv | 9900456 | 59 | 1 33. ‘9996376 130 4 "0013187 439 14 3°88 ae a1 4 3:39 | -9996505 | 355 | 4 | oo1g74s | 456 | 18 ss | 9999499 | 55 | 1 2. 5 = p 2099 13 3 . 5 340 | 9996631 | j5, | 4 | 0012322 | 435 Fi s4y1 | 9996752 | 44, | 4 | oo11910 | 4) | 18 3:91 | 9999539 | 9 | 1 42 | -g996869 | 335 | 4 | ‘0011510 | 35, | 12 3-92 | -9999557 | 43 | 1 FAS “9996982 109 4 0011122 2376 12 BIS 9999575 17 1 344 | 9997091 | joe | 4 | ‘Oolo747 | Sey | 12 3-94 | 9999503 | 34 | 1 345 | 9997197 | jo, | 4 | ‘oolosss | 325 | 11 3:95 | 9999609 | 54 | 1 3:46 | 9997299 | oo | 3 | 0010030 | 4,, | 11 396 | 9900625 | 4, | 1 s47 | 9997398 | 9, | 3 | ooogss9 | S37 | s-o7 | 9909641 | jp | 1 348 | 9997493 | go | 38 | 0009358 | 355 | 10 sos | 9909655 | yy | 1 ayo | 9997085 | go | 3 | -oo090a7 | 379 | 10 399 | -999967 4 | e Z (24 ‘9997674 10 400 | -9999683 Tables of the Probability Integral TABLE IL—(continued). a 1 A A? . A? | z * % xv a (1 + a) ic = 2 re 0008727 | 45, | 10 400 | -9999683 | 45 1 “0001338 2 0008426 | 54, | 10 4-01 9999696 | 33 1 ‘0001286 2 0008135 | S256 | 9 402 } -9999709 | 55 0 ‘0001235 2 0007853 | 5-3 | 9 403 | -9999721 5 ‘0001186 2 -0007581 2a 9 LO4 ‘9999733 “= ‘0001140 2 0007317 | o5_ | 8 40s | 9999744 | 55 “0001094 2 0007061 | 5, | 8 406 | 9999755 | 45 ‘0001051 2 0006814 | 525 | 8 407 | 9999765 | 55 ‘0001009 2 0006575 | 53, | 8 408 | -9999775 A “0000969 2 0006343 | 55, | 8 409 | -9999784 5 “0000930 1 0006119 | 317 | 7 410 | -9999793 3 ‘0000893 1 9 0005902 | 94, | 7 Hit “9999802 = “0000857 1 0005693 | do, | 7 412 | 9999811 3 ‘0000822 1 0005490 | jog | 7 418 | -9999819 2 0000789 1 0005294 | 124 | 6 414 | 9999826 e ‘0000757 1 0005105 | 325 | 6 415 | -9999834 7 ‘0000726 1 “0004921 wy | & 416 | 9999841 = ‘0000697 1 0004744 a 6 417 | 9999848 d ‘0000668 1 0004573 ie 6 18 9999854 e “0000641 1 “0004408 i a 6 419 ‘9999861 2 ‘0000615 1 0004248 | 57. | 5 420 | -9999867 6 ‘0000589 1 0004093 | jy, | 5 yt ‘9999872 6 0000565 1 0003944 | 44, | 5 4:22 | 9999878 2 “0000542 1 0003800 | 3, | % 423 | 9999883 3 “0000519 1 0003661 | 45. | 5 424 | -9999888 F ‘0000498 1 0003526 *|_ 130 5 425 9999893 5 ‘0000477 1 0003396 | jo. | 4 4:26 | -9999898 i “0000457 1 0003271 ei 4 42t ‘9999902 4 ‘0000438 1 0003149 | 47, | 4 428 | 9999907 i ‘0000420 1 0003032 | 43, | 4 429 | 9999911 i ‘0000402 1 0002919 | 355 | 4 430 | 9999915 yi ‘0000385 1 “0002810 en 4 + 431 ‘9999918 ‘0000369 1 002705 | 109 | 4 432 | 9999922 | 3 0000354 1 “0002604 98 d 433 "9999925 3 ‘0000339 1 “0002506 seals 43h ‘9999929 : “0000324 1 0002411 Bietes 435 | -9999932 3 “0000310 1 “0002320 ag oe: 436 | -9999935 5 ‘0000297 1 0002232 Peale 437 | -9999938 5 “0000284 0002147 ee alee 438 | -9999941 = ‘0000272 0002065 9 | 3 4389 | 9999943 5 ‘0000261 “0001987 7 | 3 440 | -9999946 9 “0000249 0001910 oe 44 ‘9999948 = “0000239 0001837 las 442 | -9999951 : “0000228 ‘0001766 as 3 443 | 9999953 6 ‘0000218 ‘0001698 pale 444 | 9999955 5 “0000209 0001633 es 2 LAS 9999957 9 “0000200 ‘0001569 avsita® 446 | -9999959 65 ‘0000191 ‘0001508 59 fee 447 | -9999961 A ‘000183 “0001449 eae Ie 448 | -9999963 A “0000175 | -0001393 ale 449 | -9999964 2 ‘0000167 0001338 2 450 } -9999966 “0000160 Tables for Statisticians and Biometricians TABLE II. Area and Ordinate in terms of Abscissa*. z $(1+a) Fe 4 450 66023 159837 451 67586 152797 52 69080 146051 5S 70508 139590 54 71873 133401 455 73177 127473 456 74423 121797 LT 75614 116362 458 76751 111159 SS 77838 106177 460 78875 101409 461 79867 96845 462 80813 92477 468 81717 88297 464 82580 84298 465 83403 80472 4-66 84190 76812 467 84940 733811 68 85656 69962 469 86340 66760 470 86992 63698 471 87614 60771 472 88208 57972 ATS 8877 55296 T 74 89314 52739 475 89829 50295 476 90320 47960 477 90789 45728 478 91235 43596 479 91661 41559 480 92067 39613 481 92453 37755 482 92822 35980 483 93173 34285 SJ 93508 32667 485 93827 31122 86 94131 29647 487 94420 28239 88 94696 26895 489 94958 25613 490 95208 24390 491 95446 23222 492 95673 22108 OS 95889 21046 LOY 96094 20033 495 96289 19066 46 96475 18144 LO 96652 17265 498 96821 16428 499 96981 15629 Zz $(1+a) 300 97133 o01 97278 502 97416 503 97548 FO 97672 305 97791 506 97904 5:07 98011 DUS 98113 509 98210 5:10 98302 5-11 98389 5:12 98472 ols 98551 a1 98626 515 98698 516 98765 517 98830 518 98891 5-19 98949 5-20 99004 521 99056 a2 99105 F223 99152 oe. 99197 O25 99240 5°26 99280 527 99318 228 99354 529 $9388 5°30 99421 531 99452 D2 99481 DSS 99509 Tosh 99535 589 99560 ‘36 99584 37 99606 538 99628 589 99648 540 99667 S41 99685 TAQ 99702 543 99718 a4 99734 545 99748 546 99762 FL? 99775 FAS 99787 T4I 99799 * Prefix ‘99999 to each entry. 14867 14141 13450 12791 12162 11564 10994 10451 9934 9441 8972 8526 8101 7696 7311 6944 6595 6263 5947 5647 5361 5089 4831 4585 4351 4128 3917 3716 3525 3344 3171 3007 2852 2704 2563 2430 2303 2183 2069 1960 1857 1760 1667 1579 1495 1416 1341 1270 1202 1138 99810 99821 99831 99840 99549 99857 99865 99873 99880 99886 99893 99899 99905 99910 99915 99920 99924 99929 99933 99936 99940 5-71 99944 572 99947 OTS 99950 TTL 99953 anes 99955 576 99958 5-77 99960 578 99963 579 99965 5°80 99967 581 99969 5°82 99971 58S 99972 584 99974 5:85 99975 586 99977 5ST 99978 5°88 99979 589 99981 5-90 99982 591 99983 5:92 99984 FOS 99985 OJ 99986 5:95 99987 5:96 99987 597 99988 5:98 99989 5:99 99990 6:00 99990 n 1077 1019 965 913 864 817 773 731 691 654 618 585 553 522 494 467 441 417 394 372 351 332 313 296 280 264 249 235 222, 210 198 187 176 166 157 148 139 131 124 117 110 104 98 92 87 82 77 73 68 65 61 Tables of the Probability Integral TABLE IIL Abscissa and Ordinate in terms of difference of Areas. a 0000000 0125335 0250689 0376083 0501536 0627068 0752699 0878448 1004337 1130385 "1256613 "1383042 1509692 1636585 1763742 1891184 2018935 2147016 2275450 2404260 2533471 2663106 2793190 2923749 3054808 “3186394 3318533 3451255 3584588 *3718561 3853205 3988551 4124631 4261480 ~ 4399132 4537622 4676988 4817268 4958503 5100735 5244005 5388360 5538847 “5680515 5828415 5977601 6128130 6280060 6433454 6588377 6744898 A 2 Ab # Ss £ 125335 os 20 195354 suai 125394 ete oe 125453 | 20 125532 | 20 125631 20 125750 ie 20 125889 20 126048 ao | 30 126228 an 21 126429 21 Wogan | = seen ea 126893 ae ae 197157 area ee eae 128081 eH 23 128434 pee | Res 128811 aoe a \aead 129211 il 2 129635 2 25 130084 alee 130559 Bap oh 28 131059 pee weed 131586 Pe ear 132140 28 132722 Be eo 133333 ie 30 133973 ee these 134644 von | 2 135346 CS reny 136081 fe | 34 136849 ate 35 137652 oe) a6 138490 | $83 | 37 139366 39 140281 Ae 40 141235 eee || 149231 | yha9 | 43 azar | 1028" | 45 144355 47 ‘145487 | ioe | 49 146668 | Tjs9 | 51 147900 | 1332 | 54 14on8e~| jase | ‘56 150529 59 151930 ied 62 153304 | 148) | 65 154923 {aa 69 ies | 1Pe | ve 3989423 *3989109 *3988169 *3986603 *3984408 3981587 3978138 3974060 3969353 3964016 “3958049 3951450 3944218 "3936352 "3927852 3918715 "3908939 3898525 "3887469 3875769 3863425 “3850434 3836794 3822501 3807555 3791952 *3775690 *3758766 3741177 3722919 *3703990 3684386 *3664103 3643138 3621487 3599146 3576109 3552374 “3527935 3502788 3476926 3450346 "3423041 3395005 3366233 3336719 3306455 3275435 3243652 3211098 “3177766 313 940 1567 2194 2821 3449 4078 4707 5337 5967 6599 7232 7866 8501 9137 9775 10415 11056 11699 12344 12991 13641 14292 14946 15603 16262 16924 17589 18258 18929 19604 20283 20965 21651 22342 23036 23735 24439 25148 25861 26580 27305 28035 28772 29514 30264 31020 31783 32554 33333 Dee Se ey mpnwwwry He Wo Oo Wo WO Ww wh b te wmomo-~I~n WTIDADH Drover an 10 Tables for Statisticians and Biometricians TABLE III. Abscissa and Ordinate in terms of difference of Areas. “6744898 6903088 "7063026 “7224791 *7388468 7554150 3177766 3143646 3108732 3073013 3036481 2999125 158191 159937 161765 163678 165682 167782 34119 34915 35719 36532 37356 38189 “7721932 “7891917 8064212 "8238936 8416212 2960936 2921902 2882013 2841256 "2799619 169984 172296 174724 177276 179961 39034 39889 40757 41637 42530 *8596174 8778963 8964734 9153651 9345893 2757089 2713653 2669295 2624000 2577753 182789 185771 188917 192242 195760 43437 44358 45295 46247 47217 "9541653 ‘9741139 "9944579 1:0152220 10364334 2530535 2482330 2433117 2382877 2331588 199486 203440 207641 212114 216882 48205 49213 50240 51289 52362 1:0581216 1:0803193 11030626 11263911 11503494 *2279226 +*2225767 2171185 2115451 2058535 221977 227432 233286 239583 246374 53459 54582 55734 56916 58130 1:1749868 1:2003589 1:2265281 1:2535654 1:2815516 2000405 1941024 1880356 1818357 1754983 253721 261693 270373 279861 59380 60669 61999 63374 Lables of the Probability Integral TABLE IV, Extension of Table of the Probability Integral F=4(1—-a). f= =| e-* dx, The table gives (— log F) for x. VQ i -log F Eo —log F x -logF 5 6°54265 30 197 -30921 50 544°96634 6 9-00586 31 210°56940 60 783°90743 it 11°89285 52 224°26344 70 1066-26576 8 15°20614 3S 238°39135 80 1392-04459 9 18°94746 BL 252°95315 90 1761°24604 10 23°11805 35 267°94888 100 2173°87154 11 27°71882 36 283°37855 150 4888-38812 12 32°75044 37 299°24218 200 8688-58977 13 38°21345 3s 315°53979 250 13574-49960 1h 44°10827 39 332°27139 300 19546 °12790 15 50°43522 40 349°43701 350 26603°48018 16 57°19458 41 367 03664 400 34746 °55970 17 64°38658 z 385°07032 450 4397536860 18 72°01140 43 403 53804 500 54289°40830 19 80°06919 4A 422°43983 20 88 °56010 45 441°77568 N.B. Toobtain anything 21 97°48422 46 461°54561 but a rough apprecia- 22 106°84167 Av 481°74964 tion after c<=50, the 23 116°63253 48 502°38776 table would require 24 126°85686 49 523°45999 much extension, but for many practical 25) 137°51475 50 54496634 problems it suffices to 26 14860624 take after 7=50: 27 160°13139 28 | 172-09024 pa i,-i 29 | 184-48283 V2 o : 30 197 °30921 From each of the values in this table -30103 must be subtracted, if we wish to obtain the probability 2F, then given by (—log 2F’), that the value is greater than x, without regard to sign. 2—2 12 TABLE V. Probable Errors of Means and Standard Deviations. Tables for Statisticians and Biometricians 1 | -67449 | -47694 2 | -47694 | :33724 3 | -38942 | -27536 4 | 33724 | -23847 5 | -30164 | -21329 6 | 27536 | -19471 7 | -25493 |- -18026 8 | -23847 16862 9 | -22483 | -15898 10 | -21329 | -15082 11 | -20337 14380 12 | -19471 ‘13768 13 | -18707 | 13228 14 | 18026 | -12747 15 | 17415 | 19314 16 | -16862 | -11923 17 | -16359 | 11567 18 | 15898 | -11241 19 | 15474 | -10942 20 } -15082 | 10665 21 | 14719 | -10408 22 4 -14380 | 10168 23 | -14064 | -09945 24 | 13768 | -09735 25 | -13490 | -09539 26 | -13228 | -09353 ay | -12981 | -09179 28 | 12747 | 09013 29 | -12525 | -08856 30 | 12314 | -08708 31 | 12114 | -08566 32 | 11923 | -08431 33 | 11741 | *08302 3 11567 | 08179 35 | 11401 | -08062 36 | 11241 | -07939 7 | -11088 | -07841 3s | 10942 | -07737 39 | 10800 | -07637 4o | "10665 | :07541 41 | 10534 | -07448 42 | 10408 | -07359 48 | 10286 | -07273 44 | 10168 | 07190 45 | 10055 | :07110 46 | :09945 | 07032 4y | -09838 | :06957 48 | -09735 | 06884 49 | :09636 | -06813 50 | -09539 | 06745 n X Xe 51 09445 ‘06678 52 09353 06614 53 09265 06551 54 09179 06490 59 “09095 06431 56 09013 06373 57 08934 06317 58 08856 06262 59 08781 “06209 60 “08708 06157 61 08636 06107 62 08566 06057 63 08498 06009 64 ‘08431 05962 65 08366 05916 66 08302 05871 67 08240 05827 6S 08179 05784 69 708120 | °05742 70 08062 05700 71 08005 05660 12 07949 “05621 73 07894 05582 7. 07841 “05544 75 07788 05507 76 07737 05471 77 ‘07687 “05435 ve 07687 *05400 79 07589 05366 80 07541 05332 81- 07494 05299 82 07448 05267 83 07403 05235 84 | -07359 “05204 85 07316 05173 86 07273. 05143 s7 07231 05113 8&8 07190 05084 89 07150 05056 90 07110 05027 91 ‘07071 “05000 92 07032 04972 93 06994 04946 D4 06957 04919 95 06920 “04893 96 06884 04868 97 06848 04843 98 “06813 04818 99 ‘06779 04793 100 06745 04769 xX Xe “06711 ‘06678 06646 “06614 06582 06551 06521 “06490 “06460 06431 “06402 06373 "06345 06317 “06290 06262 06236 06209 06183 06157 06132 06107 06082 06057 06033 “06009 “05985 “05962 05939 05916 05893 05871 05849 “05827 05805 05784 05763 "05742 05721 05700 05680 05660 05640 05621 05601 05582, ‘05563 05544 05526 05507 04746 04722 “04699 “04677 “04654 “04632 04611 “04589 “04568 “04547 04527 “04507 *O4487 “04467 “04447 “04428 “04409 “04391 04372 04354 04336 04318 04300 04283 04266 "04249 04232 04216 04199 04183 04167 04151 04136 04120 04105 04090 “04075 “04060 “04045 04031 “04017 “04002 03988 03974 ‘03961 08947 08934 03920 03907 “03894 Tables for Facilitating the Computation of Probable Errors 13 TABLE V. Probable Errors of Means and Standard Deviations. 05489 05471 05453 05435 “05418 “05400 *05383 05366 ‘05349 05332 05316 “05299 05283 “05267 “05251 "05235 ‘05219 “05204 ‘05188 ‘05173 “05158 05143 05128 “05118 “05099 05084 “05070 “05056 05041 ‘05027 “05013 “05000 04986 "04972 04959 “04946 04932 “04919 04906 “04893 04880 “04868 04855 04843 04830 “04818 04806 “04793 04781 04769 03881 03868 “03856 03843 03831 08819 “03806 ‘O3794 “03782 O3771 03759 03747 ‘03736 “08724 03713 03702 03691 03680 “03669 03658 03647 08637 “03626 03616 “08605 03595 03585 “03575 03565 03555 03545 03535 03526 03516 “03507 03497 “03488 03478 03469 03460 03451 03442 08433 03424 * 03415 03407 03398 03389 03381 03372 n ».¢ Xo 201 04757 “03364 251, 202 ‘04746 “03356 252 203 04734 ‘03347 253 204 04722 03339 254 205 ‘04711 03331 255 206 “04699 03323 a. 207 “04688 03315 2, 208 ‘04677 03307 C2, 209 “04666 “03299 2! 210 “04654 03291 26 211 “04643 03283 She “04632 ‘03276 213 “04622 ‘03268 214 ‘04611, | *03260 215 “04600 08253 216 "04589 “03245 217 "04579 03238 218 “04568 03230 219 “04558 - 03223 220 04547 “03216 221 04537 03208 222 04527 “03201 223 “04517 “03194 224 “04507 ‘03187 225 ‘04497 “03180 226 “04487 03173 27 ‘04477 “03166 228 04467 “03159 229 "04457 03152 23. “04447 03145 280 231 04438 ‘03138 281 282 ~ "04428 03131 282 233 “04419 “03125 283 234 “04409 ‘03118 284 235 “04400 “03111 285 236 04391 03105 286 237 04381 03098 287 288 04372 “03092 288 239 04363 “03085 289 240 "04354 03079 290 241 “04345 ‘03172 291 242 04336 “03066 292 243 04327 “03060 293 244 *04318 "03053 294 245 “04309 ‘03047 295 246 “04300 03041 296 247 04292 03035 297 248 “04283 03029 298 249 04274 03022 299 250 “04266 03016 800 X 04257 “04249 “04240 04232 "04224 04216 ‘04207 ‘04199 “04191 “04183 04175 ‘04167 “04159 “04151 04143 04136 “04128 04120 “04112 04105 04097 ~ -04090 “04082 “04075 ‘04067 “04060 *04053 “04045 “04038 “04031 "04024 04017 04009 “04002 03995 03988 03981 03974 03968 03961 03954 03947 03940 03934 03927 03920 03913 03907 03901 03894 “02919 02913 “02892 ‘02887 ‘02881 ‘02876 02871 02866 “02860 02855 02850 02845 “02840 *02835 “02830 ‘02825 “02820 “02815 ‘02810 ‘02806 ‘02801 02796 ‘O2791 ‘02786 02782 ‘02777 02772 ‘02767 02763 “02758 02754 Xp ‘03010 “03004 02998 02993 ‘02987 02981 02975 “02969 02964 02958 02952 02947 “02941 02935 02930 02924 02908 02903 ‘02897 alii a 14 Tables for Statisticians and Biometricians TABLE V. Probable Errors of Means and Standard Deviations. DG | Xe a XG i¢ 301 | -03888 | -02749 03600 | -02546 -03368 | -02382 302 | -03881 | -02744 ‘03595 | -02542 03364 | -02379 303 | -03875 | -02740 ‘03590 | 02538 -03360 | -02376 304 | -03868 | -02735 03585 | ‘02535 -03356 | -02373 305 | -03862 | -02731 ‘03580 | -02531 -03352 | -02370 306 | -03856 | -02726 ‘03575 | 02528 -03347 | -02367 30% | -03850 | -02722 03570 | 02524 ‘03343 | -02364 308 | -03843 | -02718 03565 | -02521 -03339 | -02361 309 | -03837 | -02713 “03560 | -02517 -03335 | 02358 310 | -03831 | -02709 03555 | 02514 -03331 | 02355 311 | -03825 | -02704 -03550 | -02510 03327 | -02353 312 | -03819 | -02700 -03545 | :02507 -03323 | -02350 313 | -03812 | -02696 03540 | -02503 03319 | -02347 314 | -03806 | -02692 ‘03535 | -02500 03315 | 02344 315 | -03800° | 02687 -03530 | 02496 03311 | -02341 316 | -03794 | -02688 03526 | 02493 -03307 | 02338 giy | -03788 | -02679 “03521 | 02490 -03303 | 02336 318 | 03782 | -02675 -03516 | 02486 -03299 | 02333 319 | -03776 | -02670 03511 | 02483 -03295 | -02330 320 | -03771 | -02666 03507 | 02479 ‘03291 | -02327 321 | -03765 | -02662 ‘03502 | -02476 -03287 | -09324 322 |) -03759 | -02658 03497 | -02473 ! -03283 | -02322 323 | -03753 | -02654 “03492 | 02469 2 03279 | -02319 32 ‘03747 | 02650 -03488 | 02466 g 03276 | -02316- 925 | -03741 02646 -03483 | -02463 g “03272 | 02313 526 | -03736 | -02642 -03478 | 02460 03268 | -02311 327 | -03730 | -02637 03474 | 02456 -03264 | -02308 928 | -03724 | -02633 03469 | -02453 -03260 | -02305 329 | -03719 | -02629 03465 | -02450 -03256 | 02303 330 | 03713 | -02625 03460 | -02447 -03253 | -02300 331 \ -03707 | -02621 03456 | 02443 / 03249 | -02297 332 | -03702 | -02618 03451 | 02440 03245 | -02295 333 | -03696 | -02614 -03446 | 02437 -03241 | -02292 334 | -03691 | -02610 03442 | 02434 434 | -03238 | -02289 335 | -03685 | -02606 -03438 | -02431 ‘03234 | -02287 33 ‘03680 | -02602 ‘03433 | 02428 3 | -038230 | -02284 337 | -03674 | -02598 03429 | -02424 ‘03227 | 02281 338 | 03669 | -02594 03424 | -02421 “03223 | -02279 339 | -03663 | 02590 -03420 | 02418 ‘03219 | 02276 340 | -03658 | -02587 03415 | -02415 03216 | 02274 341 | -03653 | :02583 03411 | 02412 ‘03212 | -02271 342 | 03647 | :02579 03407 | -02409 -03208 | -02269 343 | 03642 | -02575 03402 | 02406 -03205 | 02266 344 | °03637 | -02571 ‘03398 | 02403 -03201 | 02263 345 4 03631 | 02568 -03394 } 02400 ‘03197 | -02261 346 | -03626 | -02564 ‘03389 | 02397 03194 | 02258 ayy | -03621 | -02560 -03385 | 02394 ‘03190 | 02256 348 | -03616 | 02557 03381 | 02391 03187 | 02253 349 | 03610 | 02553 -03377 | 02388 03183 | -02251 850 03605 "02549 03372 *02385 ‘03180 “02248 Tables for Facilitating the Computation of Probable Errors 15 TABLE V. Probable Errors of Means and Standard Deviations. xX Xe te x Xe 451 03176 "02246 03013 02131 ‘ 551 02873 02032 452 03173 "02243 “03010 °02129 552 °02871 “02030 453 03169 *02241 *03007 °02127 553 “02868 *02028 4S 03166 "02238 “03004 02124 554 °02866 "02026 455 03162 02236 “03001 "02122 555 02863 “02024 03159 02233 “02998 "02120 556 "02860 02023 703155 02231 02996 02118 557 02858 “02021 03152 "02229 “02993 “02116 558 "02855 "02019 “03148 "02226 é “02990 “02114 559 "02853 02017 03145 "02224 ‘02987 02112 560 “02850 “02015 03141 02221 “02984 02110 561 02848 “02014 03138 02219 “02981 02108 562 “02845 02012 03135 02217 “02978 “02106 563 02843 “02010 03131 “02214 02975 “02104 oG4 "02840 “02008 03128 02212 ‘02972 *02102 565 02838 02006 03125 02209 02969 “02100 566 *02005 “03121 *02207 "02966 “02098 567 “02003 703118 02205 02964 02096 568 q “02001 “03115 02202 “02961 “02094 569 “02828 “01999 ‘03111 “02200 “02958 “02092 570 02825 “01998 ‘03108 “02198 02955 “02089 571 “02823 01996 “03105 "02195 02952 “02087 572 “02820 “01994 “03101 02193 “02949 *02085 573 02818 *01992 “03098 02191 ; “02084 574 02815 ‘01991 03095 02188 4 “02082 575 02813 01990 03092 02186 g “02080 576 02810 01987 “03088 02184 6 “02935 “02078 57 “02808 “01986 ‘03085 02181 02935 ‘02076 578 “02806 01984 03082 02179 ‘0293 “02074 579 “02803 “01982 ‘03079 02177 "02 02072 580 02801 “01980 03075 02175 i +0292 02070- 581 02798 ‘01978 03072 02172 32 "0292 02068 582 “02796 01977 ‘03069 “02170 °02922 02066 583 02793 01975 “03066 ‘02168 ‘ 02064 584 02791 “01974 03063 “02166 ‘ 02062 585 02789 01972 “03060 02163 “02060 586 ‘02786 “01970 “03056 “02161 : “02058 587 02784 “01969 03053 *02159 4 "02056 588 02782 “01967 “03050 *02157 q 5 02054 589 ‘02779 01965 03047 “02155 ‘| 02052 590 02777 “01964 03044 02152 02051 591 02774 01962 ‘03041 *02150 *02897 02049 592 02772 “01960 03038 *02148 “02895 02047 593 ‘02770 “01959 03035 02146 “02892 "02045 594 ‘02767 ‘01957 03032 "02144 02889 02043 595 ‘02765 01955 -03029 "02142 02887 “02041 596 02763 “01954 “03026 02139 “02884 02039 597 02761 01952 *03022 02137 “02881 ‘02037 598 02758 “01950 “03019 02135 “02879 02036 599 02756 “01949 03016 02133 02876 020384 600 02754 01947 16 Tables for Statisticians and Biometricians TABLE V. Probable Errors of Means and Standard Deviations. n XxX Xg Xe Xx, Xo 601 02751 01945 651 "02644 01869 02548 01801 602 "02749 701944 652 “02642 01868 02546 “01800 603 “02747 “01942 653 02639 *01866 02544 01799 GO4 “02744 *O1941 694 02637 01865 02542 01798 605 “02742 01939 655 02635 01864 02540 01796 606 02740 01937 656 02633 01862 “02538 01795 607 02738 701936 657 02631 01861 02537 01794 608 02735 01934 658 02629 01859 02535 ‘01792 609 02733 01933 659 02627 *01858 g 02533 01791 610 02731 “01931 660 02625 01856 02531 01790 611 “02729 701929 661 “02623 01855 02530 01789 612 02726 701928 662 02621 01854 g 02528 ‘01787 613 02724 “01926 663 02620 01852 02526 “01786 614 02722 “01925 664 02618 01851 "02524 01785 615 02720 01923 665 02616 “01849 "02522 ‘01784 616 02718 01922 666 02614 01848 01782 617 02715 01920 667 "02612 01847 ‘O1781L 618 02713 ‘01919 668 02610 “01845 01780 619 02711 ‘01917 669 “02608 01844 ‘01779 620 02709 701915 670 “02606 01843 ‘01777 621 02707 01914 671 02604 01841 ‘01776 622 "02704 01912 672 “02602 “01840 *O1775 623 02702 “O1L911 673 “02600 ‘01838 723 02508 ‘O1774 624 02700 01909 674 02598 ‘01837 Uk: 02507 ‘01773 625 02698 “01908 675 02596 01836 " 02505 ‘01771 626 02696 “01906 676 "02594 01834 Z 02503 ‘01770 627 02694 “01905 677 02592 01833 Z 02502 ‘01769 628 “02692 “01903 678 02590 01832 Z 02500 ‘O1L768 629 02689 “01902 679 02588 01830 2 "02498 ‘01766 630 02687 “01900 680 02587 01829 > 02496 01765 631 02685 “01899 681 02585 01828 é "02495 01764 632 02683 01897 682 02583 “01826 ¢ "02493 01763 633 02681 “01896 683 02581 01825 5 02491 01762 634 02679 01894 684 02579 01824 3 02490 01760 G35 02677 01893 685 02577 01822 02488 01759 636 02675 01891 686 “02575 701821 3 "02486 01758 637 “02672 01890 687 02573 01820 “02485 *O1757 638 “02670 01888 6388 02571 01818 73. “02483 01756 639 02668 01887 689 02570 01817 “02481 01754 640 “02666 01885 690 "02568 01816 th 02479 01753 GAL 02664 01884 691 02566 01814 “02478 01752 642 02662 01822 692 02564 01813 : 02476 ‘O1751 643 02660 01881 693 02562 01812 02474 “01750 644 | 02658 | -01879 694 | -02560 | -01810 02473 | -01749 645 02656 01878 695 02558 “01809 02471 01747 646 | 02654 | -01876 647 | 02652 | -01875 696 697 02557 01808 02555 ‘01807 02469 01746 02468 01745 648 02650 01874 698 02553 “01805 02466 01744 649 02648 ‘01872 699 02551 “01804 02465 01743 650 “02646 ‘O1871 700 02549 01803 02463 01742 Tables for Facilitating the Computation of Probable Errors 17 TABLE V. Probable Errors of Means and Standard Deviations. xX, Xo Xo ee n Xx, Xo n 02461 -01740 801 02383 “01685 851 02460 ‘01739 802 *02382 01684 852 "02458 *01738 803 *02380 “01683 853 “02456 ‘01737 S804 "02379 “01682 S54 02455 ‘01736 895 02377 ‘01681 855 02312 01635 02311 01634 02309 01633 02308 01632 02307 “01631 02453 01735 806 02376 “01680 856 “02451 01733 S07 02374 “01679 S857 02450 01732 S808 02373 01678 858 "02448 O1731 809 02371 01677 859 “02447 01730 810 02370 01676 560 02305 “01630 “02304 01629 02303 01628 02301 01627 02300 01626 02445 01729 811 02368 01675 861 02443 01728 812 02367 “01674 562 “02442 01727 813 02366 ‘01673 863 02440 *01725 S14 02364 01672 S64 *02439 01724 $15 02363 01671 865 02299 “01625 02297 “01624 02296 01624 "02295 01623 02293 01622 02292 01621 02291 01620 "02289 01619 02288 ‘01618 02287 01617 *02437 01723 $16 02361 ‘01670 866 "02435 01722 &17 02360 “01669 867 02434 ‘01721 $18 02358 “01668 868 02432 ‘01720 819 02357 ‘01667 869 02431 01719 820 02355 “01666 870 "02429 01718 821 02354 01665 871 “02285 01616 02428 ‘OL717 822 02353 01664 872 “02284 “01615 02426 ‘O1715 823 02351 01662 373 02283 ‘01614 "02424 ‘OL714 824 02350 ‘01661 874 “02281 “01613 "02423 017138 825 02348 01660 875 “02280 01612 02421 01712 826 02347 01659 876 02279 01611 “02420 ‘O1711 827 02345 01658 877 02278 “01610 “02418 ‘01710 828 02344 01657 878 02276 “01610 “02417 “01709 829 02343 01656 879 02275 “01609 “02415 01708 8350 “02341 01655 880 “02274 “01608 02414 ‘OL707 831 02340 01654 881 02272 ‘01607 02412 ‘O1L706 S32 02338 01653 882 “02271 01606 “02410 “01704 833 02337 01652 883 02270 01605 02409 01703 S34 02336 “01651 884 02269 01604 02407 01702 835 02334 01651 885 *02267 01603 “02406 *O1701 836 02333 01650 886 02266 “01602 02404 “01700 837 02331 01649 887 02265 ‘01601 02403, “01699 835 01648 888 02263 01600 02401 01698 859 01647 889 02262 “01600 “02400 01697 840 01646 890 02261 01599 02398 01696 841 “01645 891 02260 01598 02397 01695 852 01644 892 02258 01597 02395 01694 S43 “01643 893 02257 01596 “0: ‘01693 S44 01642 894 02256 01595 01692 845 “01641 895 02255 “01594 02391 01699 S846 ‘01640 896 02253 01593, 02389 ‘01659 847 01639 897 02252 *01592 02388 ‘01688 848 01638 898 “02251 “01592 02386 ‘01687 549 *01637 899 02250 “01591 02385 “01686 850 02313 01636 900 “02248 01590 18 Tables for Statisticians and Biometricians TABLE V. TABLE VI. Probable Errors of Means and Standard Deviations, Probable Errors of Coefficient of Varration. Xx X2 n Xy X2 Vi 02247 | -01589 951 | 02187 | -01547 a Bee melee es 02246 | -01588 952 | 02186 | -01546 t 1 acgeeae || 2000705 | aeoemnnen 02245 | -01587 953 | -02185 | -01545 2b Roose || 2700180 Uae cameron 02243 | -01586 954 | -02184 | -01544 } | @ooeap.| 4700870 pea -60 “19942 "01585 la “05 q é ” fe 4 02242 | -01585 955 | -02183 | -01543 4b oreae. |: 1700609 ih eeeerect 02241. | 01585 a56 | -02181 | -01543 oars 1-00908 pepe I Meerees o5y | -02180 | -01542 ee Ree senna 01583 958 | -02179 | -01541 7 | eee (| aeoness Pail BB 01582 959 | -02178 | -01540 BT ae eaa: ow Oaiby #10 ade a| bs ry 58 9 > “02 . 2 ty, * | 01581 160 | -02177 | -01539 1) VL cappere: 2026900)» penal by 01580 oer | -o2176 | -01589 ; 103280 02233 | -01579 962 | -02175 | -01538 tT | Fees ceoagay | | Pet be 02232 | -01578 963 | -02174 | “01537 Te Deeper te oseno | 208 ag 02231 | -01578 964 | 02172 | -01536 18 | eee ieeoeses |. 9) os +022! 0157 965 02 5 . + eecblhe x Paleo 02230 | -01577 65 | -02171 | 01535 ie | isasara | 106202 | Soy | 54 ‘02229 | -01576 966 | -02170 | -01535 Brace: (|= 2b:07070 02227 | 01575 967 | -02169 | -01534 16 | ease, (|) 07991 Senn mes 02226 | -o1574 968 | -02168 | -01533 UT | Gee 1 m8985 | ake | 5a 02225 | -01573 969 | -02167 | -01532 Te | foetege feog00 | 1028 | 51 . 229, “f) i= lod * “f . 5? Le A - . i} 4 02224 | 01572 970 | 02166 | -01531 co | aovreaci | 121066 | q195 | 50 02293 | -01572 971 | -02165 | -o1531 Feo tf E19192 p 02221 | “01571 972 | -02163 | -01530 ot ghee Ht euaaoe | |e On ag 02220 | 01570 973 | -02162 | -01529 oo 1 gaseme cael eee eee 02219 | -01569 974 | 02161 | -01528 oy | d5-saazs | ABBE ie | ee 02218 | -01568 975 | -0216 0152 2 aot 1717 : 0 01568 975 | -02160 | -01527 oe higeetegs (oevete s,s |e 02217 | -01567 ove | 02159 | -01527 Bee 1°18539 02215 | -01566 ovr | 02158 | -01596 26 | 2770190 |) 149945 | 1298 | ag 02214 | -01566 978 | -02157 | -01595 27 || 2B S085 | 5 apagb) «| tae? al ag 02213 | -01565 979 | -02156 | -01524 pe tee fee 02212 | -0156 so | 02155 | - 01564 980 | -02155 | -01524 oo lt aeeasee || Deeliole ee alee 02211 | -01563 9st | 02153 | -01593 = 125991 02209 | -01562 982 | -02152 | -01522 Bt op: O8 See) | eye0be ae mene 02208 | -O1561 983 | 02151 | 01521 ped Peep ei fe 02207 | -01561 984 | 02150 | -01520 ie Br one 130940 os 36 "022 ‘O156 985 “02 0152 OUp ave . 06 | -01560 985 | 02149 | -01520 Se L. Secbubay (| i oaeeanat ce ee enue 02205 | -01559 986 | 02148 | -01519 : ; 134422 02203 | -01558 987 | 02147 | -01518 86 | a0 80707 1) 3621s fae eee "02202 01557 ISS 02146 01517 at lee 138041 | . 1826 32 02201 | -01556 939 | 02145 | “01517 oe ecgeege || 1°30890 |) teen) 02200 | 01556 99 02 “01516 me AIT 01556 0 02144 — 01516 40 45°95650 1:41789 1920 30 02199 | -01555 oot | -02143 | -o1515 roe 143709 02198 | -01554 992 | -02142 | ‘O15i4 4L | 4739359 | 7.45658 | 1950 | 99 02196 | -01553 993 | 02140 | “O1D14 42 | 48°85017 | J .4ngag | 1978 | 96 02195 | -O1552 994 | 02139 | -o1si3 2 a eeene 149642 oe = 02194 | -O1551 95 | -o2ts 01512 51675 | 20! 15 995 | -02138 | -O15I 4e° | Ssenory || S167 gn ae 02193 | “01551 996 | 02137 | ‘1511 a f: 1-D3734 02192 | -01550 997 | -02136 | -01510 46 | 54°87706 | ysseig | 2084 | o4 ae ee A Were Oto yy | 5643524 | 159818 | o109 p2191 | -01549 998 01510 1:57927 24 ‘02189 | 01548 999 OF 01509 8 Be 451 | 160059 eis ‘02188 | 01547 100 0213: ‘01508 : rt “625 2 g 1547 | 0 02133 150 50 61:23724 1°62214 2177 22 ol s Probable Error of a Coefficient of Correlation TABLE VII. Abac for Probable Errors of r. 19 Scale of Correlation oan ~ Fo G6 OSSSSErRRL LoS sg 8 Scale of Probable Errors yy YY iil i, ”, // rsh = = fre} o Bee eno. oo. oe 8 3 8 8 = = fo} 8 mp ma 7 7 >) ~ 3 S 7 7 7 ° S f=) oO / [ / © = i 7 / / / fo} rot 7 r 7 o 2 RTS, AV / / 7 iol y S ° lin She Thi y 7 7 7 5 9 Ly / / / 1 : = / L 5 z : 7: , iff te 7 é bs - - 8 i Z : -—= g wo [_ f vA / wo | eles 7 femme / OE Ve — ine £ TH - fo} / vA f 4 = / / f ¢ i 7 7/V/ 01 FAT 7 7 / 7 / 7 = ff So e | a thet 8 7 j 7 ca 17 Z £ / = 7 ] 7 7. 7 Z 2 eee - g L Oo ° L & 2 w : 38 ; : 7 = ; as : 2 4 L IS) L yi al d 3” # awe “ “8 ah 5 7 Z / me) — — oD / o” = g - s io} Fi 8 : 8 8 ° tow > mos & ae = ) 3 2 a od 2 f 2 i=) bs 8 bs * 1 t a fo] g To) je = 7 v ” a 4 y U f 6 é 8 5 Q ee a h. | 7 | 2 & f : i rs. iG j Ww w/o wD oO as oD rr) a a = } My i ite Wi, TT, Os a : ; g odeget ers f Eee = 8 : Abac for determining the Probable Errors of Correlation Coefficients. 3—2 20 P 000 Tables for Statisticians and Biometricians TABLE VIII. “O01 “000 | 1:000 000 ‘010, “999 900 020 | +999 600 ‘030 | +999 100 ‘040 | +998 400 -050| -997 500 ‘060 | -996 400 ‘070 | -995 100 -080 | -993 600 | 090 991 900 -100| -990 000 -110| ‘987 900 {120 | +985 600 ‘130 | -983 100 -140| -980 400 -150| ‘977 500 -160| ‘974 400 ‘170 | +971 100 ‘180 | -967 600 190 | -963 900 -200| -960 000 “210 | *955 900 “220 | *951 600 “230 | ‘947 100 "240 | ‘942 400 250 | +937 500 260 | “932 400 270 | -927 100 *280 | 921 600 “290 | *915 900 300 | +910 000 “310 | +903 900 820 | ‘897 600 ‘330 | +891 100 “340 | *884'400 ‘850 | +877 500 300 | *870 400 “87 863 100 "380 | *855 600 390 | +847 900 ‘400 | *840 000 ‘410 | *831 900 420 | -823 600 430} 815 100 ‘440 | 806 400 450 | “797 500 ‘460 | “788 400 ‘470 | “779 100 ‘480 | “769 600 ‘490 | *759 900 | 989 799 ‘987 679 985 359 982 839 980 119 ‘977 199 974 079 ‘970 759 ‘967 239 963 519 959 599 955 479 951 159 946 639 941 919 936 999 ‘931 879 926 559 921 039 915 319 909 399 903 279 896 959 890 439 883 719 876 799 869 679 862 359 854 839 847 119 “839 199 -831 079 *822 759 814 239 “805 519 796 599 “787 479 ‘778 159 ‘768 639 ‘758 919 999-999 999 879 999 559 999 039 998 319 ‘997 399 996 279 994 959 993 439 ‘991 719 002 Values of 1—7°- Values of 1-7? for r=-001 to ‘999, 003 004 005 ‘006 007 008 “009 "999 996 999 856 999 516 998 976 998 236 ‘997 296 996 156 “994 816 993 276 991 536 989 596 987 456 985 116 ‘982 576 ‘979 836 ‘976 896 ‘973 756 ‘970 416 ‘966 876 963 136 959 196 955 056 950 716 946 176 941 436 936 496 931 356 926 016 920 476 ‘914 736 908 796 902 656 896 316 889 776 883 036 876 096 868 956 861 616 854 076 846 336 838 396 830 256 821 916 813 376 “804 636 “795 696 “786 556 777 216 “767 676 “757 936 989 391 ‘987 231 984 871 982 311 ‘979 551 ‘976 591 973 431 ‘970 O71 966 511 962 751 958 791 954 631 950 271 945 711 940 951 935 991 930 831 925 471 919 911 914 151 908 191 902 031 895 671 *889 111 882 351 875 391 868 231 860 871 853 311 845 551 837 591 829 431 821 071 812 511 803 751 794 791 “785 631 “776 271 “766 711 ‘756 951 999/091} 999 831 999 471 "998 911 998 151 ‘997 191 996 031 994 671 993 111 991 351 "999 984 999 804 999 424 998 844 998 064 ‘997 084 995 904 994 524 992 944 991 164 “989 184 ‘987 004 984 624 982 044 ‘979 264 ‘976 284 ‘973 104 969 724 966 144 962 364 958 384 954 204 949 824 945 244 940 464 “935 484 930 304 924 924 919 344 913 564 907 584 901 404 895 024 888 444 881 664 874 684 867 504 860 124 852 544 844 764 836 784 828 604 820 224 811 644 “802 864 “793 884 “784 704 “775 324 "765 744 755 964 754 975 ‘999 975 999 775 ‘999 375 998 775 ‘997 975 996 975 ‘995 775 994 375 992 775 ‘990 975 988 975 986 775 984 375 ‘981 775 ‘978 975 ‘975 975 ‘972 775 969 375 965 775 961 975 ‘957 975 953 775 949 375 944 775 939 975 934 975 929 775 924 375 918 775 912 975 906 975 900 775 "894 375 887 775 880 975 873 975 866.775 859 375 851 775 843 975 835 975 827 775 819 375 810 775 801 975 “792 975 "783 775 “774 375 764 775 999 964 ‘999 744 999 324 998 704 ‘997 884 ‘996 864 995 644 994 224 “992 604 ‘990 784 988 764 986 544 “984 124 “981 504 ‘978 684 975 664 972 444 “969 024 965 404 961 584 957 564 953 344 948 924 944 304 939 484 934 464 929 244 923 824 918 204 ‘912 384 906 364 “900 144 893 724 887 104 *880 284 873 264 866 044 858 624 851 004 843 184 835 164 826 944 818 524 809 904 *801 084 “792 064 “782 844 "773 424 “763 804 753 984 999 951 “999 711 999 271 998 631 ‘997 791 996 751 995 511 994 O71 992 431 ‘990 591 988 551 ‘986 311 983 871 981 231 ‘978 391 975 351 972 111 968 671 965 031 ‘961 191 957 151 952 911 948 471 943 831 938 991 933 951 928 711 923 271 ‘917 631 ‘911 791 905 751 899 511 893 O71 886 431 879 591 872 551 865 311 857 871 850 231 842 391 834 351 826 111 ‘S17 671 509 031 800 191 “791 151 “781 911 772 471 762 831 752 991 999 936 ‘999 676 “999 216 998 556 ‘997 696 ‘996 636 “995 376 ‘993 916 992 256 ‘990 396 988 336 ‘986 076 ‘983 616 980 956 ‘978 096 ‘975 036 ‘971 776 968 316 964 656 960 796 956 736 952 476 948 016 943 356 938 496 933 436 928 176 922 716 ‘917 056 911 196 905 136 898 876 892 416 885 756 878 896 871 836 864 576 857 116 849 456 841 596 833 536 825 276 816 816 *808 156 "799 296 “790 236 *780 976 “771 516 “761 856 751 996 999 919 999 639 999 159 998 479 ‘997 599 996 519 995 289 993 759 992 O79 990 199 988 119 985 839 983 359 980 679 ‘977 799 974 719 ‘971 4389 *967 959 964 279 ‘960 399 956 319 952 039 947 559 942 879 ‘937 999 932 919 927 639 922 159 916 479 ‘910 599 904 519 898 239 891 759 885 079 878 199 871 119 863 839 856 359 848 679 840 799 832 719 824 439 815 959 807 279 “798 399 “789 319 “780 039 “770 559 ‘760 879 “750 999 Probable Error of a Coefficient of Correlation TABLE VIII. Values of 1-7". Values of 1-1? for r=-001 to ‘999. 21 000 ‘001 002 003 “750 000 “739 900 ‘729 60) “719 100 “708 400 “697 500 “686 400 “675 100 663 600 “651 900 “640 000 “627 900 615 600 “603 100 “590 400 “577 500 564 400 *551 100 “537 600 523 900 510 000 495 900 481 600 467 100 “452 400 437 500 422 400 ‘407 100 391 600 375 900 “360 000 343 900 327 600 311 100 294 400 277 500 260 400 243 100 225 600 ‘207 900 "190 000 ‘171 900 “153 600 “135 100 ‘116,400 ‘097 500 '| ‘078 400 059 100 039 600 019 900 “748 999 “738 879 728 559 ‘718 039 “707 319 “696 399 685 279 673 959 662 439 “650 719 ‘638 799 626 679 614 359 ‘601 839 589 119 “576 199 363 079 549 759 636 239 622 519 508 599 494 479 480 159 465 639 450 919 435 999 420 879 “405 559 390 039 374 319 “358 399 342 279 325 959 309 439 292 719 275 799 258 679 241 359 223 839 206 119 188 199 170 079 151 759 133 239 114 519 095 599 076 479 057 159 037 639 017 919 “747 996 “737 856 727 516 ‘716 976 ‘706 236 695 296 684 156 672 816 661 276 649 536 637 596 625 456 613 116 600 576 387 836 ‘O74 896 561 756 548 416 534 876 521 136 ‘507 196 493 056 478 716 “464 176 449 436 434 496 “419 356 404 016 “388 476 372 736 “356 796 340 656 324 316 307 776 291 036 274 096 256 956 "239 616 222 O76 "204 336 "186 396 ‘168 256 149 916 ‘131 376 112 636 -093 696 074 556 ‘055 216 035 676 015 936 746 991 736 831 ‘726 471 “715 911 ‘705 151 “694 191 683 031 ‘671 671 “660 111 648 351 636 391 624 231 ‘611 871 599 311 586 551 573 591 560 431 ‘547 O71 533 511 519 751 505 791 491 631 ‘477 271 462 711 447 951 432 991 ‘417 831 402 471 386 911 371 151 355 191 339 031 322 671 306 111 289 351 272 391 255 231 237 871 220 311 "202 551 184 591 166 431 148 071 129 511 ‘110 751 091 791 072 631 053 271 033 711 013 951 | ° 004 745 984 "735 804 | "725 424 “714 844 “704 064 693 084 681 904 670 524 “658 944 647 164 635 184 “623 004 “610 624 598 044 ‘585 264 ‘572 284 ‘559 104 ‘545 724 ‘532 144 | ‘518 364 | “5 504 384 "490 204 475 824 461 244 446 464 431 484 ‘416 304 400 924 385 344 369 564 353 584 337 404 321 024 304 444 287 664 270 684 253 504 236 124 218 544 200 764 "182 784 | 164 604 146 224 127 644 “108 864 “089 884 ‘070 704 051 324 031 744 O11 964 | 006 008 ‘009 502 975 488 775 474 375 459 775 444 975 429 975 414 775 399 375 383 775 “B67 975 | 351 975 | *335 775 319 375 302.775 285 975 268 975 ‘251 775 | ‘234 375 -216 75 | ‘198 975 | “180 975 | "162 775 ‘144 375 125 775 "106 975 087 975 068 775 049 375 029 775 009 975 “743 964 733 744 “723 324 | 7 712 704 701 884 | | 690 864 ‘679 644 | 668 224 656 604 | | °644 784 | | -632 764 | 620 544 608 124 595 504 “582 684 569 664 556 444 "543 024 529 404 ‘515 584 ‘501 564 487 344 | - “472 924 “458 304 443 484 “428 464 “413 244 397 824 382 204 | 366 384 | 350 364 334 144 “B17 724 | “301 104 | ‘284 284 | 267 264 250 044 232. 624 215 004 ‘197 184 179 164 160 944 142 524 123 904 105 084 086 064 066 844 047 424 027 804 007 984 “700 791 689 751 678 511 667 O71 655 431 643 591 631 551 619 311 606 871 094 23 581 391 ‘441 991 426 951 ‘411 711 396 271 380 631 364 791 348 751 *332 511 316 O71 299 431 282 591 265 551 248 311 230 871 213 231 195 391 177 351 159 111 140 671 122 031 103 191 084 151 064 911 045 471 025 831 005 991 ‘741 936 “731 676 “721 216 “710 556 “699 696 688 636 | 677 376 665 916 654 256 642 396 630 336 “618 O76 “605 616 *D92 956 “580 O96 567 036 553 776 340 316 | 526 656 | ‘512 796 498 736 484 476 470 016 455 356 440 496 425 436 410 176 394 716 "379 056 "363 196 | 347 136 *330 876 314 416 297 756 280 896 263 836 246 576 229 116 ‘211 456 193 596 “175 536 “157 276 138 816 120 156 “101 296 082 236 ‘062 976 043 516 023 856 003 996 “740 919 | “730 63% “720 159 “709 479 “698 599 “687 519 | 676 239 664 759 653 O79 641 199 629 119 616 839 “604 359 “591 679 578 799 565 719 552 439 538 959 525 279 *d11 399 497 319 483 039 468 559 453 879 438 999 “423 919 ‘408 639 393 159 377 479 *361 599 B45 519 329 239 312 759 296 O79 279 199 "262 119 244 839 "227 359 209 679 “191 799 173 719 "155 439 136 959 ‘118 279 ‘099 399 ‘080 319 “061 039 “041 559 ‘021 879 ‘001 999 | 22 Tables for Statisticians and Biometricians TABLE IX. Values of the Incomplete Normal Moment Function py (2). A. Odd Moments m, (#) = pun (w)/\(n—1) (vn — 3) (n — 5)... 2}, m, (x) Mey (x) | ms (x) my; (x) Mg (x) 0:0 0000000 *0000000 0000000 “0000000 “0000000 O-1 0019897 “0000050 “0000000 “0000000 0000000 0-2 0078996 0000787 “0000005 “0000000 0000000 Os *0175545 0003920 0000059 0000001 0000000 O-4 0306721 “0012105 *0000321 0000006 “0000000 0-5 "0468770 0028688 0001183 0000037 0000001 0-6 0657177 0057372 0003390 “0000151 0000005 O7 “0866883 0101861 “0008146 0000493 “0000024 0s 1092507 0165494 0017172 “0001350 0000086 0-9 1328570 0250925 0032702 0003242 0000259 1-0 “1569716 “0359862 0057399 0006988 0000687 1. “1810901 0492895 “0994199 0013795 0001634 1-2 “2047562 "0649423 0146092 0025293 “0003549 13 2275737 0827672 0215865 0043539 0007135 1-4 2492148 1024819 0305828 “0070957 0013414 15 2694247 1237174 0417570 0110219 0023776 16 "2880214 1460428 0551764 0164068 0040005 17 3045932 “1689923 0708039 “0235098 “0064248 18 3199921 1920929 0884945 0325513 0098944 19 3333265 2148899 1080009 0436894 0146688 2:0 3449513 2369694 “1289874 0569995 0210055 2-1 *3549587 “2579749 1510502 0724606 0291380 22 3634677 2776192 1737425 “0899486 0392533 23 *3706152 "2956902 1966019 1092390 “0514703 24 3765478 3120515 2191769 “1300173 0658224 25 “3814140 “3266380 “2410506 1518971 0822459 26 *3853593 "3394489 2618602 1744437 “1005767 27 "3885213 *3505370 2813106 “1972006 1205553 28 *3910268 3599983 2991823 *2197160 "1418391 29 “3929897 3679593 3153329 2415682 1640231 3:0 3945104 3745671 “3296946 2623860 “1866637 31 “3956755 “3799784 3422662 2818638 2093055 52 “3965582 3843517 "3531029 2997718 2315079 BS *3972197 “3878403 “3623049 3159582 “2528687 34 “3977101 *3905878 “3700046 3303476 *2730432 35 “3980696 “3927244 3763548 “3429335 2917571 36 “3983304 3943653 3815183 3537687 3088145 37 “3985175 *3956099 3856585 3629529 “3240979 38 “3986503 3965425 “3889331 “3706199 3375646 39 3987436 *3972329 3914881 *3769253 3492376 40 “3988085 3977378 *3934552 3820351 "3591947 41 *3988530 *3981028 3949499 3861165 3675554 42 "3988833 *3983635 3960708 3893304 “3744677 4E *3989037 3985475 3969007 3918253 3800964 44 “3989173 “3986759 *3975073 "3937367 3846117 45 “3989263 3987645 "3979452 *3951801 3881809 46 3989321 “3988248 *3982573 3962557 3909614 47 3989359 “3988656 3984770 3970466 "3930967 4°8 “3989383 3988927 3986298 3976205 “3947135 49 “3989398 “3989106 3987348 *3980315 3959207 50 3989408 3989222 3988061 3983221 “3968097 3989423 “3989423 3989423 *3989423 3989423 8 Incomplete Normal Moment Functions TABLE IX. Values of the Incomplete Normal Moment Function. B, Even Moments mp (x) = pn (x)/{(n —1)(n— 8) (n — Dy inalle mg (x) ms (x) mg (x) mg (x) m9 (x) 0:0 “0000000 “0000000 -0000000 “0000000 ‘0000000 0-1 ‘0001325 70000002 “0000000 “0000000 “0000000 0-2 “0010512 “0000084 “0000000 “0000000 “0000000 03 *0034951 -0000626 “0000008 “0000000 “0000000 O4 -0081136 "0002572 “0000058 “0000001 -0000000 0:5 "0154298 *0007604 “0000270 “0000008 “0000001 0-6 "0258121 “0018200 -0000925 ‘0000037 “0000001 0-7 "0394585 0087575 “0002588 “0000139 *0000006 0°8 "0563914 *0069507 -0006223 -0000437 “0000025 0-9 ‘0764632 "0118045 *0013297 ‘0001177 “0000086 1:0 0993740 ‘0187171 ‘0025857 “0002812 “0000251 LEST, "1246965 ‘0280428 "0046525 ‘0006094 “0000658 HI) “1519070 *0400559 "0078427 “0012160 “0001558 LS “1804203 "0549214 *0125028 "0022617 ‘0003386 14 *2096248 ‘0726741 "0189894 *0039577 “0006842 wich *2389164 ‘0932091 -0276408 "0065653 “0012964 16 *2677274 *1162835 *0387442 “0103869 ‘0023209 Leh *2955511 *1415300 *0525059 "0157516 “0039494 1°8 *3219594 “1684803 *0690258 “0229926 "0064207 19 "3466134 °1965937 ‘0882796 *0324204 "0100147 2:0 "3692680 "2252921 "1101113 70442938 “0150415 2 *3897700 "2539927 *1342371 “0587910 *0218224 2:2 "4080525 2821413 "1602593 ‘0759866 “0306667 2:3 "4241237 “3092387 *1876903 "0958345 0418437 24 “4380556 “3348616 *2159821 "1181613 "0555560 25 "4499695 "3586763 *2445598 "1426700 0719132 26 "4600231 *3804450 *2728554 "1689546 *0909136 a7 "4683965 *4000247 *3003387 *1965228 “1124320 2:8 *4752816 "4173616 *3265431 "2248963 *1362197 29 *4808719 "4324798 °3510842 *2532933 “1619132 3:0 "4853546 "4454679 °3736720 "2813629 “1890538 31 "4889053 "4564647 “3941138 “8085150 °2171145 82 “4916838 "4656432 “4123121 "3342962 "2455315 33 "4938321 *4731975 "4282552 *B583379 ‘2737379 OL "4954736 °4793298 “4420056 *38038672 *3011962 35 *4967130 "4842409 "4536843 “4002102 *3274261 36 “4976381 "4881218 *4634555 *4177877 *3520261 37 "4983205 "4911484 “4715111 "4331061 “3746880 38 “4988183 "4934784 “4780568 "4462441 *3952025 39 ‘4991771 "4952491 "4833001 *4573366 “4134583 4:0 "4994330 "4965779 “4874418 *4665592 "4294345 ol 4996133 "4975627 “4906683 *4741120 *4431886 42 “4997391 "4982835 "4931479 *4802063 "4548407 43 "4998258 “4988045 “4950279 "4850521 "4645574 4h “4998849 *4991766 “4964343 "4888500 "4725352 4d "4999247 °4994392 "4974729 "4917846 “4789861 46 "4999512 "4996222 "4982298 "4940207 4841246 4? “4999688 *4997483 4987744 “4957010 “4881574 4s “4999802 *4998342 *4991613 *4969464 “4912765 49 "4999876 “4998919 4994326 “4978572 "4936544 50 *4999923 “4999303 “4996206 “4985144 "4954417 8 5000000 *5000000 5000000 -5000000 “5000000 24 Tables for Statisticians and Biometricians TABLE X. Diagram of Generalised ‘ Probable Error. Table of Generalised ‘ Probable Errors. ee ay | Probable Error | if | 0:674,4898 2 1:177,4062 3 | 1°538,1667 4 1:832,1239 5 2-086,0146 6 2°312,5982 5 2°519,0869 8 2°710,0022 9 2-888,3962 10 3°056,4366 | 11 3°215,7402 | | | Diagram for Value of Probable Error for n Variables. Number of Variables Determination of Normal Curve from Tail 25 TABLE XI. Constants of Normal Curve from Moments of Tail about Stump. Values of the Functions vr, and wv, required to determine the Constants of a Normal Frequency Distribution from the Moments of its Truncated Tail. nv "1 Yo 3 n ys 0°00 571 1°253 2-000 ITs one 0°01 573 1°259 2°016 12 8°690 0:02 574 1°265 2°032 1°3 10°331 0:03 576 1271 2°049 14 12°383 0-04 578 1-276 2-066 15 14°968 0°05 *580 1°282 2-083 16 18°248 0°06 “581 1-288 2-100 HIG 22°439 0-07 583 1-294 2°118 18 27°832 0:08 585 1°300 2°136 1°9 34°3823 0°09 ‘587 1°305 2°155 2:0 43°956 Ol 588 1°31) 2°173 21 55°977 02 “605 1°371 2°377 22 71°925 0-3 622 1°432 2°617 23 93°248 O-4 *638 1°495 2°902 24 121-988 05 "653 1°560 3°241 a5 161-038 0-6 668 1°626 3°646 26 214°537 o'7 "682 1°693 4°133 2°7 288°434 0s “696 1:762 4°720 2°8 391°374 0-9 ‘709 1°833 5°433 29 535°963 1:0 "722 1°904 6°303 3°0 740°796 ha) "734 1/977 7371 85 [4299-226] Let d equal distance of centroid of tail from stump, = = standard deviation of the tail about its mean, and n=its area, (i) Find y, from y,=>?/d*. Hence from table determine h’. (ii) From this value of h’ find y., then c=d x yy gives the standard deviation of the uncurtailed normal curve. (il) h=h' xo gives the origin of the uncurtailed normal curve. (iv) Knowing h’, Table II gives }(1+ a) and therefore the ratio $(1—a) of tail to total area of curve NV, or N=n/}(1-—a). For many purposes it is sufficient to use V=n x py. 26 Sa NVA AY & S&S Mm cS 2 ea rr SSSRCSRRKRSRSRESSRVSASBRES Y QaWD 9s So 93 ©& 606581 367879 223130 135335 082085 049787 030197 018316 011109 006738 004087 002479 001503 000912 000553 000335 000203 000123 000075 000045 “000028 000017 “000010 “000006 000004 000002 000001 “000001 “000001 “000000 000000 “000000 000000 000000 Tables for Statisticians and Biometricians 801253 572407 391625 261464 ‘171797 111610 071897 046012 "029291 018566 011726 007383 004637 002905 001817 001134 000707 000440 0002738 000170 000105 000065 000040 000025 000016 ‘000010 000006 000004 000002 000001 000000 7000000 000000 “000000 909796 *735759 557825 406006 ‘287298 “199148 ‘135888 ‘091578 -061099 040428 026564 ‘017351 ‘011276 007295 004701 003019 001933 ‘001234 -000786 000499 000317 ‘000200 -000127 ‘000080 -000050 000032 -000020 -000012 -000008 000005 000000 000000 000000 000000 962566 849146 699986 549416 415880 306219 220640 156236 "109064 075235 051380 034787 023379 015609 010363 006844 004500 002947 001922 001250 000810 000524 000338 000217 000139 000090 000057 000037 000028 000015 ‘000000 000000 000000 000000 985612 *919699 "808847 76076 543813 "423190 320847 238103 173578 "124652 088376 061969 043036 029636 020256 013754 009283 006232 004164 002769 001835 001211 000796 000522 000341 000223 000145 000094 000061 000039 000001 000000 000000 000000 TABLE XII. Test for Goodness of Fit. 994829 959840 885002 ‘779778 “659963 5639750 428880 382594 ‘252656 188573 138619 100558 072109 051181 036000 025116 017396 011970 008187 005570 003770 002541 001705 001139 ‘000759 000504 000333 000220 000145 000095 000001 “000000 000000 000000 Values of P. 998249 ‘981012 934357 857123 “757576 647232 536632 “433470 *342296 265026 201699 151204 111850 081765 059145 042380 030109 021226 014860 010336 007147 004916 003364 002292 001554 001050 000707 000474 ‘000317 000211 000003 000000 000000 000000 999438 991468 964295 911413 834308 “739919 637119 534146 437274 350485 275709 213308 *162607 122325 090937 066881 048716 035174 025193 017913 012650 008880 006197 004801 002971 002043 ‘001399 000954 000648 000439 000008 000000 000000 000000 999828 996340 981424 ‘947347 891178 815263 °725444 628837 532104 440493 357518 "285057 ‘223672 “172992 132061 099632 074364 054964 040263 029253 021093 015105 010747 007600 005345 003740 ‘002604 001805 001246 000857 000017 000000 000000 000000 Tables for Testing Goodness of Fit oi TABLE XII.—(continued). x n= 12 n’ =13 | n=14 n=15 n’=16 n=17 n'=18 n’=19 n' =20 1 | ‘999950 | -999986 | -999997 | :999999 Ti I ie 110 ue 2 | 998496 | -999406 | -999774 | :999917 | ‘999970 | *999990 | 999997 | :999999 ike 8 | 990726 | 995544 | -997984 | -999074 | 999598 | :999830 | ‘999931 | 999972 | ‘999989 4 | (969917 | 983436 | -991191 | :995466 | 997737 | :998903 | :999483 | 999763 | 999894 5 | 931167 | 957979 | 975193 | -985813 | -992127 | 995754 | 997771 | ‘998860 | ‘999431 6 | -878365 | 916082 | 946153 | -966491 | 979749 | ‘988095 | ‘993187 | 996197 | ‘997929 7 | ‘799078 | 857613 | -902151 | ‘934711 | 957650 | -973260 | ‘983549 | -990125 | 994213 8 | 713304 | 785131 | -843601 | -889327 | 923783 | :948867 | ‘966547 | -978637 | -986671 9 | 621892 | 702931 | -772943 | ‘831051 | ‘877517 | ‘918414 | -940261 | -959743 | ‘973479 10 | -530387 | ‘615960 | -693934 | °762183 | -819739 | 866628 | ‘903610 | 931906 | 952946 11 | -443263 | 528919 | -610817 | ‘686036 | °752594 | 809485 | 856564 | °894357 | 923839 12 | 362642 | 445680 | 527643 | *606303 | ‘679028 | °748980 | *800136 | ‘847237 | 885624 18 | -293326 | °369041 | -447812 | 526524 | -602298 | ‘672758 | °736186 | ‘791573 | 838571 14 | -232993 | 800708 | 373844 | -449711 | -525529 | 598714 | -667102 | *729091 | “783691 15 | *182498 | -241436 | 307354 | 378154 | -451418 | 524638 | *695482 | ‘661967 | °722598 16 | 141180 | *191236 | -249129 | -318374 | -882051 | *452961 | ‘523834 | :592547 | -657277 Lad ~ 107876 081581 061094 149597 | 199304 | -256178 | °818864 | 385597 | 454366 | -523105 | ‘589868 "115691 | 157520 | ‘206781 | -262666 | 323897 | 388841 | °455653 | 522438 088529 | 123104 | 164949 | -213734 | 268663 | 328532 | *391823 | *456836 18 20 | -045341 | ‘067086 | 095210 | -180141 | *171932 | -220220 | -274229 | -332819 | ‘394578 21 | :033371 | 050380 | 072929 | *101632 | 136830 | 178510 | :226291 | 279413 | 336801 22 | 024374 | -037520 | ‘055362 | 078614 | -107804 | *143191 | *184719 | -231985 | -284256 23 | 017676 | 027726 | ‘041677 | ‘060270 | 084140 | *113735 | *149251 | -190590 | 237342 24 | 012733 | 020841 | -031130 | 045822 | 065093 | ‘089504 | *119435 | 155028 | -196152 25 | 009117 | 014822 | 023084 | ‘034566 | 049943 | 069824 | ‘094710 | *124915 | -160542 006490 004595 003238 002270 001585 010734 | 017001 | 025887 | 088023 | 054028 | 074461 | ‘099758 | °130189 007727 | 012441 | 019254 | 028736 | 041483 | 058068 | 078995 | -104653 005532 | 009050 | 014228 | ‘021569 | 031620 | 044938 | -062055 | 083428 003940 | 006546 | 010450 | ‘016085 | 023936 | 034526 | 048379 | 065985 002792 | 004710 | ‘007632 | ‘011921 | 018002 | 026345 | 037446 | 051798 000036 000001 000000 000000 000072 | 000138 | ‘000255 | ‘000453 | ‘000778 | 001294 | ‘002087 | 003273 000001 | 000003 | ‘000006 | ‘000012 | 000023 | 000042 | 000075 | 000131 :000000 | 000000 | 000000 | -000000 | -000001 | -000001 | 000002 | 000004 000000 | 000000 | 000000 | ‘000000 | ‘000000 | -000000 | ‘000000 | -000000 4—2 BC mHNRAAY Ss BH recy ee ee ee ee ee ee rr ee SPS see yeNSPeaLgceseasoaos & Ro avr is) (Sy te) 1S 28 1° 1 999996 999954 999722 “998898 996685 991868 982907 968171 946223 916076 877384 830496 “776408 "716624 652974 587408 521826 457930 397132 340511 *288795 242392 201431 165812 135264 109399 087759 069854 004995 000221 000007 000000 Tables for Statisticians and Biometricians TABLE XII. Test for Goodness of Fit. n' =22 “999998 “999980 999868 999427 998142 995143 "989214 ‘978912 ‘962787 939617 908624 869599 822952 “769650 "711106 649004 585140 521261 458944 399510 343979 293058 247164 206449 ‘170853 140151 114002 091988 007437 000365 000013 000000 n’=23 | 1 1: “999999 999992 "999939 999708 “998980 997160 993331 986304 974749 957379 933161 901479 862238 815886 “763362 “705988 645328 583040 ‘520738 459889 ‘401730 347229 297075 251682 "211226 175681 144861 118464 010812 000586 000022 000001 Values of P. *999997 999972 999855 999452 998371 995957 991277 983189 ‘970470 *951990 ‘926871 *894634 855268 "809251 "757489 "701224 “641912 581087 "520252 460771 “403808 *350285 300866 "255967 215781 "180310 149402 015369 000921 000038 “000001 999999 999987 999929 ‘999711 999085 “997595 994547 989012 979908 966121 946650 920759 ‘888076 848662 803008 *751990 696776 “638725 579267 519798 -461597 405760 *353165 304453 260040 +220131 184752 021387 001416 “000064 “000002 1° 999994 999966 999851 999494 998596 996653 992946 986567 ‘976501 961732 941383 914828 881793 842390 “797120 "746825 “692609 "635744 577564 5619373 462373 407598 355884 307853 263916 22.4289 029164 “002131 000104 000004 n' =27 1 1: 1 fo ile 999998 | -999999 999984 | -999998 | :999997 | -999999 999924 | -999962 | -999981 | 999991 *999726 | -999853 | -999924 | -999960 *999194 | -999546 | 999748 | -999863 997981 | -998808 | -999302 | -999599 “995549 | -997239 | 998315 | -998988 -991173 | -994294 | -996372 | -997728 983974 | :989247 | -992900 | 995384 973000 | -981254 | -987189 | 991377 957384 | -969432 | :978436 | -985015 936208 | -952947 | -965819 | 975536 909088 | -931122 | -948589 | 962181 875773 | ‘903519 | ‘926149 | -944272 836430 | °870001 | -898136 | ‘921288 "791556 | -8380756 | *864464 | 892927 741964 | °786288 | -825349 | ‘859149 688697 | °737377 | 781291 | -820189 632947 | *685013 | -733041 | °776543 575965 | -630316 | °681535 | 728932 5618975 | 574462 | 627835 | -678248 -463105 | °518600 | 573045 | 625491 409383 | -463794 | *518247 | 571705 358458 | :410973 | -464447 | 517913 *311082 | -360899 | °412528 | -465066 ‘267611 | °314154 | -363218 | -414004 039012 | 051237 | :066128 | -083937 003144 | :004551 | -006467 | -009032 “000168 | :000264 | ‘000407 | :000618 000007 | ‘000011 | 000019 | -000030 © Com™™D NW So TON Tables for Testing Goodness of Fit TABLE XIII. Ausiliary Table A. log {x 2 en | 68479282 61816058 48905897 33438109 Ih }h— M4 _ ra) fon} or for) @ oo oO on] bo ive} [ee] (ee) = SS) SS) N) rs ! 280445839 261630713 2°42473615 223046765 2:03401675 3°83576379 3°63599760 3°43494971 3:23277708 302964420 482566143 462092598 441551928 420951024 400295765 5°79591210 558841744 5-38051190 517222904 6:96359847 6°75464644 654539633 6°33586907 6'12608346 791605644 7-70580334 7-49533808 7-28467333 707382065 8°86279064 865159301 8 44023670 8-22872997 801708042 9°80529511 9:59338058 9°38134293 _ 916918780 10-95692047 10°74454589 10-53206866 1031949311 10°10682329 1189406301 | log e 2x" “78285276 66570552 34855828 13141104 91426380 69711655 47996931 26282207 04567483 3°82852759 3°61138035 39423311 3°17708587 95993863 74279139 52564414 30849690 09134966 87420242 65705518 43990794 “22276070 00561346 “78846622 57131898 1 HD) HD oe) oe) Oe] Ot) Oe ST HST ST | C9] C9] Co! 2] BO} ne DO] BO] BO! et] et eT for) No) ou ns I ~I e ~ i) 6 13702449 791987725 770273001 748558277 7-26843553 7-05128829 8°83414105 861699381 8-39984657 8°18269932 9:96555208 9°74840484 9°53125760 9°31411036 _ 909696312 10°87981588 1066266864 10°44552140 10-22837416 1001122691 1179407967 1157693243 1135978519 11°14263795 2 x2 | log {x af 2e-ve} §1 11-68121586 52 1146828520 53 1125527422 5h 11-04218593 55 1282902315 56 12°61578858 57 1240248475 58 12°18911408 59 13-97567885 60 13:76218123 61 13°54862328 62 13-33500696 63 13°12133415 64 14-90760662 65 14:69382607 66 14-48099412 67 14-26611232 68 1405218213 69 1583820498 70 15°62418221 71 1541011512 72 15°19600496 3 1698185290 Th 16°76766009 7 16°55342762 "6 16°33915654 17 16°12484787 v4 1791050256 19 17°69612157 80 17°48170578 81 1726725605 82 17:05277323 3 1883825810 8h 1862371146 85 18°40913404 86 18°19452656 87 19:97988972 88 19°76522419 89 1955053062 90 19°33580963 91 1912106183 92 20:90628780 93 20°69148812 ey 20°47666333 95 2026181397 96 20:04694054 97 21°83204355 98 21-61712348 99 2140218080 100 21-18721596 -29 log e~ 3X” 12-92549071 12°70834347 12°49119623 12:27404899 12:05690175 13:83975450 13-62260726 13-40546002 13:18831278 1497116554 1475401830 14:53687106 1431972382 1410257658 15°88542934 15°66828209 15°45113485 15°23398761 15:01684037 1679969313 16°58254589 16-36539865 16714825141 1793110417 17°71395693 17:49680968 1727966244 17-06251520 18°84536796 18°62822072 18°41107348 18°19392624 1997677900 19°75963176 19°54248452 1932533727 1910819003 20-89104279 20°67389555 20°45674831 20°23960107 20°02245383 21-80530659 21°58815935 21°37101211 21°15386486 9293671762 22°71957038 22°50242314 2° 28527590 30 Tables for Statisticians and Biometricians TABLES XIV—XVI. Ausiliary Tables B,C and D. TABLE XIV (B). Table of colog [n]:—[nJ=n(n —2)(n— eae n n odd nos. colog'[n] even nos, 1 “00000000 2 8 152287875 4 5 2:82390874 6 7 397881070 8 9 3:02456819 10 11 598317551 12 13 686923215 Lh 15 7-69314089 16 17 846269197 18 19 _9:18393837 20 21 1186171908 22 23 12-49999124 24 25 73'10205123 26 27 15°67068747 28 29 16-20828947 30 31 18-71692778 32 3: 19:19841384 34 85 2165434575 36 87 22-08614407 38 8 24-49507946 40 41 26°88229561 42 3 27-248827 15 dA 45 29°59561464 46 4 31-92351678 48 49 32-23332070 50 51 3452575052 52 53 36-80147465 54 55 3706111196 56 57 39°30523711 58 59 41-53438510 60 61 43°74905526 62 3 45:94971471 64 65 4613680135 66 67 48-31072655 68 69 50:47187746 70 71 52-62061911 72 7 54-75729625 Th 75 56°88223499 76 7 58-99574426 78 7 59:09811717 80 81 61-18963215 82 83 63-27055406 8h 85 65°341138514 86 87 67-40161588 88 89 69-45222588 90 91 71°49318448 92 93 73°52470154 a4 95 75°54697793 96 97 7756020620 98 99 79°56457100 100 colog [n] 169897000 109691001 2:31875876 3:41566878 4-41566878 5'33648753 619035949 898623951 _9°73096701 10°42993701 1108751433 13-70730309 14-29232974 1684517171 17°36805045 19°86290048 20°33142156 22°77511906 23°19533546 25°59327547 27-97002618 28°32657350 30°66381567 32-98257443 33°28360443 35°56760109 3783520733 38-08701930 40°32359131 4254544006 44-75304837 46-94686839 47-12782446 49°29481554 5144971750 53°59238501 55°72315329 5784233970 59-95024509 60°04715511 62°13334125 6420906197 66-27456352 6833008084 70°37583833 72°41205051 74-43892265 7645665142 7846542534 80:46542534 TABLE XV (C). > [2 Ne | e-4x? dx Tax 3173106 1572992 0832646 0455003 0253474 0143060 0081506 0046776 0026998 0015654 0009112 0005321 0003115 0001828 0001076 0000634 0000374 0000221 0000132 0000078 0000046 0000027 0000016 0000011 0000007 “0000004 0000003 “0000002 0000001 0000000 TABLE XVI (D). Function Log. Function 1-7828527590 1:9019400615 Values of 7. Probability of Association on Correlation-Scale TABLE XVII. 31 Values of (—log P) corresponding to given values of yx? in a fourfold table. (Extension of Table XII for n’ = 4.) -log P -logP —-log P x? ~ SSaArD Np osten RN ON S mw ee 5021 5230 5-440 5-650 5°860 6071 6281 6-492 6°703 6914 77126 7337 7549 7-761 7-972 8184 8397 8609 8'821 9°034 9°246 9-459 9672 9°885 10°07 10-097 12°231 14°370 16513 18°659 20°809 31-579 42°375 537184 64:002 74826 85°655 96-487 107°321 118158 128-997 139°837 150°678 161°520 172°364 183-208 194:053 204'899 215°745 226°592 237°439 237°439 248°287 259°135 269-983 280°832 291-681 302°531 313381 324-231 339°081 345931 356°782 367°633 378484 389°335 400°187 411-038 421-890 432-742 443°594 454-446 4767151 497-856 519°561 541267 562'973 562973 584680 606°387 628°094 649°801 75834 866°886 975°434 1083995 11927538 1301°092 1409°649 1518206 1626°765 1735°324 1843°885 1952-446 2061°008 2169°570 2278°133 2386°697 2495261 2603°825 2712°390 2820°955 2929°521 13500 14000 14500 15000 15500 16000 16500 17000 17500 18000 18500 19500 20000 20500 21000 21500 22000 22500 23000 23500 24000 24500 25000 19000 | —logP 2929°521 3038°086 3146°652 3255°219 3363°785 3472°352 3580°919 3689°486 3798°053 3906621 40157188 4123°756 4232-324 4340°892 4449-461 4558-029 4666°597 4775°166 4883°735 4992°304 5100°873 5209°442 5318011 TABLE XVIII. Values of (—log P), entering with r and vo; Values of c,. ‘01 02 03 “OL 05 ‘06 “07 0:05 6:248 1:907 1:020 0675 0-498 0°392 0°322 0075 13°228 3°760 1:908 1-217 0874 0674 0°545 Ol 22924 6267 3076 1:910 1:343 1-019 0814 015 50°687 13°329 6:298 3°784 2-586 1:916 1-498 0-2 90-035 23°254 10°771 6343 4:259 3100 2384 08 206348 52°453 23°836 | 13°758 9:057 6478 4:906 O4 380°266 96013 43°254 | 24°726 | 167112 11°407 8°552 05 626428 157:607 70°669,| 40177} 26-025 18312 | 13-642 06 970879 243°753 108:980 | 61°747 | 39°845 | 27-922] 20-713 07 1463:946 367:033 | 163-781 92°579 | 59584 | 41°634] 30-792 0s 2220-267 556100 | 247-801 | 139°832 | 89°819] 62-625 | 46-209 0-9 3607:924 902949 | 401-907 | 226:479 | 145-241 | 101-085 | 74-442 0:95 | 5056547 | 1265°013 | 562-757 | 316-904 | 203-069 | 141-207 | 103-886 5426°580 08 0:273 0°456 0-675 1:218 1924 3°903 6686 10°597 16:020 23°740 35°539 57°134 79671 32 Tables for Statisticians and Biometricians TABLE XIX. Values of x? corresponding to the values of (—log P) in Table X VIII. Values of ,o;. ‘01 02 05 0:05 0:075 OL 015 0-2 03 O4 O85 06 ovr Os 0-9 0:95 Values of 7 31:84 64:66 109°82 238°45 429-29 956°68 1758°21 2892°33 4479-02 6750°09 10233°49 16624°37 23295°86 10°88 19°95 31:93 65°13 111°35 246°62 447°81 731-95 1129°10 1697°24 2568°34 4166712 5833°82 3°52 5°58 8:03 14:24 22°35 45-11 78:13 12422 188°28 27958 41922 674:92 94156 03 ‘04 6:36 451 10:89 7:38 16:64 | 10:90 32:08 | 20:07 53:16 | 32-29 11405 | 67714 204-07 | 11818 330:80 | 189-82 507°65 | 289:58 760°43 | 431-96 1147-76 | 649-98 1857°93 | 1049-48 2599-00 | 1466-24 TABLE XX. Values of log x? corresponding to values of r and yo, in Tables XVIT and XVIII, Values of vr. ‘01 02 0:05 | 15030 | 1-0366 0-075 | 18106 | 1:2999 20407 | 1:5042 015 | 23774 | 1:8138 0-2 2°6256 | 2:0467 3 2-9808 | 2°3920 0-4 32451 | 26511 05 34612 | 2°8645 06 36512 | 3:0527 Or 3:8293 | 3:2297 08 40100 | 3:4097 0:9 42207 | 36197 0:95 | 43673 | 3°7660 Values of .c;. 03 0°8035 1:0370 1:2212 15062 1°7256 2:057 1 23098 25196 27056 28811 30598 32690 34148 “O4 06542 08681 10374 13025 15091 1:8270 2°0725 2°2783 24618 26354 2°8129 30210 31662 05 0°5465 0:°7466 0:9047 11535 1°3498 16543 1:8928 20942 2°2748 24465 26224 2°8293 29738 06 ‘07 08 2°91 2°48 2°19 451 3-78 3:28 6°35 5:26 4°51 10:93 8°82 7°39 16°75 13°25 10°97 32°93 25°45 20°64 56:14 42°73 33°92 88°38 66°60 52°34 133-02 99°55 77°70 19657 146°35 11361 293°64 | 21774 | 16834 47122 | 348:23 | 268-26 65632 | 48415 | 37237 06 ‘07 08 0:4639 0°3945 03404 0°6542 05775 05159 0:8028 0°7210 0°6522 1-0386 09455 0°8686 1:2240 1:1222 1°0400 15176 14057 13148 17493 16307 15305 19464 18235 17188 271239 19980 18904 2°2935 21654 20554 2°4678 2°3379 2°2262 26732 25419 24286 28171 2°6850 2°5710 Abacs for Determining the Equiprobable Tetrachoric r. 33 XXI. Abac to determine yo. NSSSUV USS SESS RM AGQRARAYRK DRpK#OOHe S SNA — WANS NNN SR SS NSS SEEN S RAH SSS SSDS SISSS SERS SSSI TS = SSIES WSS ROO SAN SS =< SS SE En an SeneEeE EERE oa - SI SS = <5 SS = SSS Sie c= See ae es ee eee S= —— PS SSESSVVSSWNNN NNN iS SENNNESSS SSS SASSES = =s SS KN oa Ss SSS BSS SESTSSSESSSPSSS] SSS ES ESS SS 2 eS ee * : 9 993 *° ga 90 92 98 94 95 96 97 980 985 990 loge "994 -995 -996 997 9986 “999 Scale of 4(1+a) and 4(1+a4)). Scale of oop+ Tables for Statisticians and Biometricians XXII. = im S > | 3S EE Ir rele = A 5 cot S st im ; ~ {tf SS ee S ia < =| - | =— > . > Ss Ss S 3S ‘woynquusigg pouwoyy v fo sisayjodhigZ ay2 uo hovahuyuoy wunayy wolf 4 woynjatuy puf op wouliog "AIXXX or Cr os Oo *L quaiagaog u014072aLL09 66 Tables for Statisticians and Biometricians XXXV. Diagram to determine the type of a Frequency Distribution from a. knowledge of the Constants B, and B,. Customary Values of B, and By. [J | HH ACU HEE EREEOREECS tik CLG / fetal eal can — fee fae fsb = ites ALLL fi ae va : EER rr : HIEEy A Frequency Type from B, and B, 67 XXXVI. Diagram showing Distribution of Frequency Types for High Values for B, and Bf. B 7 aN u 3x5, | AAs layne ea, | 7 = anaes 140] \V agian eee ee FS & 3 3 zy ~ u =| rs ——_—_—<—$_——————— cy wy ° 160 Asymptote of Cubic —-- 180] ; \ Asymptote of Biquadratic 220); \ —_— Tables for Statisticians and Biometricians 68 To jind the Probable Error of PB. TABLE XXXVII. Values of VN Sp. By 0-75 54 18 80 64 29 18 | 14°18 43 10 | 15°11 05 | 16°07 et) paciUely qui \sey pers dae! Nein ce ae ela Rex temerce. (a! Nel Get wc) sect sine Uelyrent lat’ (ww sat! sme .erales leh us licuell: oe 0°65 | 0-70 0°60 HN SMO BOLT ODOW M10 D NAPSDHMODEMrDDSOAHNMMOI- OY 60 OD AAG AAG GI 99 60 60 0 oo H = 05 | 15 15°98 | 16 “08 4 4°37 4°70| 4° | 15 76 Asta HAOAMOAMAOADADM-OCNOAMDODDAMINBDOO rs 1 SQLEGSOOCOEERBSNOrSPOSOSPHNGCAONALSD 14°91 15°90 | 4°87 0-45 | 0°50 0-40 9°54 HOUMANMOVAAAHEeOMODHRAAHe aD Hoe QI =H igiegite ol ICO Uae Vice uC ea ial ye Meare (TT i re) 7 23 7 31 88 47 0 76 86 87 737 8°46 9°05 49 | 1°67 37 | 1°57 | 1°77 30 | 1°50 | 1°7 30 | 1°48 | 1°67 45/1 32 | 1 37 ONDLE-ANDH coe gta ave eT | ee | oD on OD =H AMNMONOODOMDADOMDI~DN19 O ID SO I~ DO NMINI- AN 15} 1 12/1 13/1 15/1 20] 1 1 05 | 1:26 | 1 10 | 1°34] 1°54) 1 42 02) 1 15) 1 2 i) 0 3 7 2, i) 6 3 2°50 | 2 215 3 2°86 3 ‘07 "29 53 “78 B) “33 58 | 0°93} 1 59 | 0°95 | 1 €0 | 0-97 62 | 0:99} 1 64/1 0:00 | 0:05 | 0°10 | 0:15 | 0°20 | 0°25 | 0°30 | O 00 | 0-66 | 1 00 | 0-69} 1 ‘00 0-73 | 1 ooo 00 |0 00 | 0 00 | 0 "00 | 0 00 | 0 e200000 SOO OC OO OOOO OOOO OOOOOS S$3S838883S88 POE TIT IE C SL els ay Go Se sete oolelg oar eaten Sas Sass yeN Ga eae ay Sk ae Ca eae Sete EAMG ota eg es pity es es este Gane a al Probable Errors of Frequency Constants 69 TABLE XXXVII.—(continued). Values of V N%g,. By 0:80 | 0°85 | 0-90 | 0-95 | 1:00 | 1-05 | 1:10 | 1-15 | 1-20 | 1:25 | 1°80 | 1-85 | 1-40 | 1-45 | 1:50 3°96] 4-21] 4-47] 4:73] 5°00] 5-27| 5:55] 5°83| 6-12] 6-41] 6-71] 7-01] 7-31] 7-62] 7-94| 2-0 3°80 | 4:03] 4:27] 4:53| 4:80] 5-07| 5-34] 5°62] 5°90! 618] 6-48] 6-77| 7-07] 7:37] 7:69] 2-2 3°66| 3°88] 4:11] 4:36] 4-63] 4-88] 5-15] 5-42] 5°69| 5-96] 6-25] 6-54] 6:84] 714] 7-45] 2-2 3°52| 3°74] 3°96] 4:20| 4-46] 4-71| 4:96] 5-22] 5-48| 5°75] 6:02] 6-31] 6-61] 6-91| 7-21] 23 3-41| 3-62] 3-83] 4:05] 4:29] 4:54] 4-78] 5°03| 5:28] 5°55] 5-82] 6-10] 6-38] 6°68] 6-97] 2-4 3°32| 3°51] 3°71] 3-92] 4:15] 4:38] 461] 4°85] 5:10| 5°36] 5°62] 5°89| 6-16) 6-45| 6-74] 2-5 3-25 | 3:42| 3-60] 3-80| 4:01] 4:23] 4-45| 4:68] 4:92] 5°17] 5°43] 5°68| 5-94] 6-22] 6-51] 2-6 3:20] 3-35] 3°51| 3°69| 3°89] 410] 4:32] 4-54] 4-76] 5:00] 5-24] 5-48] 5-73] 6:00] 6-28] 2-7 3'18| 3:32] 3-47] 3-63| 3:80] 4-00] 4-21] 4-41] 4-62] 484] 5-07] 5°30] 5-53) 5-79] 6-06] 2-8 3-19| 3:32| 3-45] 3-60| 3-75] 3-92] 4-11] 4:30] 4-49] 4:70] 4:91] 5:12] 5°34] 5:59] 5-85| 2-9 3°27| 3°38| 3-49| 3-61| 3-74] 3°87] 4:03] 4:21] 4°39] 4:58] 4:78] 4:98] 5-19] 5-42} 5°68] 3-0 3°38| 3°47] 3:57| 3°67| 3-77| 3°89] 4:02] 4:16] 4:31| 448] 4:66] 4:85] 5-05) 5-28] 5:53] 3-1 3°53] 3°60] 3-68| 3-76] 3:84] 3-93] 4:03| 4:15] 4-28] 4-43/ 4:59] 4-75] 4:92] 5-15] 5-40] 3-2 3°70| 3°75| 3-81| 3°88] 3-95] 4-02] 4:10] 4:19] 4-28] 4-42] 4:56] 4-69] 4:84] 5-04] 5-28] 3-3 3-90] 3-93] 3-97| 4-03| 4:08] 4-14] 420] 4:26] 4:34] 4-45| 4:56] 4:66] 4-78| 4:97] 5:18] 3-4 4:14] 4:17| 4:19] 4:22| 4-26] 4:30] 4:34] 4:39] 4-45] 4:52] 4:60] 4:68] 4:79] 4:93] 5-12] 3-5 4:42| 4:44] 4-45| 4-47| 4:49] 4-51] 4:54] 4:57| 4°62] 4-67| 4:74] 4:78] 4:81] 4:95] 5-09] 3-6 474| 4°75] 4-76| 4-76] 4-77] 4°78] 4-79| 4:81] 4:84] 4:87| 4:90] 4°92] 4:95] 5-04] 5°13] 3-7 510] 5°10] 5-09] 5-08] 5:08] 5-07] 5-07| 5:06] 5°06] 5-08| 5-09] 5-11] 5:14] 5-18] 5-22] 3-8 5:49| 5°48] 5:46) 5-44] 5°42] 5:40] 5°37| 5°35] 5°33] 5-32| 5°32] 5°33] 5°34] 5:°35| 5°37 | 3-9 5°89] 5°88] 5-86| 5°83] 5°80] 5°76] 5°72| 5°69] 5°65] 5°62) 5°60] 5°58] 5°57) 5:57] 5°59] 4-0 6°33| 6:32] 6-30] 6-26] 6-21] 6-16] 611] 6-06] 6:02] 5-98] 5-94] 5-91] 5:88) 5-86} 5°86] 4-7 6°80| 6°79] 6-76| 6-71| 6°65] 6-60| 6:54| 6-48] 6-42] 6-36] 631] 6-27| 6-24] 6-21] 618] 4-2 730) 7:28] 7:25] 7-19] 7-13] 7-07] 7-01] 6-93] 6:87] 6-80] 6-74] 6-67] 6-62| 6:57] 653] 4-3 783| 7-80] 7°76| 7-71| 7-65] 7°58] 7°51| 7-44] 7:37] 7-28| 7-20] 7-12] 7-:05| 6-98] 6-92] 44 838| 8:36] 8:32] 8-28| 8-21] 814] 8-07| 7:99] 7:90] 7°81] 7-71] 7°61| 7-51] 7:42] 7°34] 45 8:96| 8:95] 8-91| 8-86| 8-79] 8-72] 8-64] 8°55| 8-45] 8-35| 8-24] 8-13] 8-00] 7:90] 7°80} 46 9°57] 9°57| 9°53| 9-47] 9-40] 9:33] 9-24] 9:14] 9:04] 8-93] 8°82] 8-69] 855] 8-42] 8-31} 4-7 10-20 | 10-23 | 10°16 | 1010 | 10-05 | '9:97| 9:88] 9°77] 9°67] 9°55| 9-42] 9:28] 9-14] 9-00} 8°87 | 4-8 10-91 | 10-92 | 10°87 | 10-80 | 10°74 | 10-66 | 10°57 | 10-44 | 10°32 | 10-18 | 10°04] 9:90] 9-76] 9°63} 9:50] 4-9 11°66 | 11°65 | 11-61 | 11:55 | 11-48 | 11-39 | 11-29 | 11-17 | 11-04 | 10-90 | 10-77 | 10-62 | 10-46 | 10-30 | 10°14 | 5-0 19°45 | 12-43 | 12°38 | 12-32 | 12-24 | 12-14 | 12-03 | 11-91 | 11-78 | 11-64 | 11-50 | 11-34 | 11-15 | 10-96 | 10°82 | 5-2 13°28 | 13-25 | 13-20 | 13-13 | 13-04 | 12-92 | 12-80 | 12°67 | 12-54 | 12-40 | 12-24 | 12-07 | 11-88 | 11-72 | 11°54 | 5:2 14°16 | 14-12 | 14-07 | 13-98 | 13°87 | 13°76 | 13°63 | 13-47 | 13-35 | 13-20 | 13-02 | 12-84 | 12°66 | 12°48 | 12°30 | 5-8 15:09 | 15:06 | 15-00 | 14-90 | 14°78 | 14-65 | 14°51 | 14-36 | 14-22 | 14-05 | 13-81 | 13-67 | 13-48 | 13-29 | 13-11 | 5-4 1606 | 16-02 | 15-96 | 15°87 | 15-76 | 15-63 | 15-49 | 15°33 | 15-17 | 15-00 | 14°81 | 14-61 | 14-40 | 14-18 | 13-97 | 5-5 — | — | 17-02] 16-91 | 16-79 | 16-67 | 16-51 | 16-34 | 16-18 | 15-95 | 15-70 | 15-50 | 15-30 | 15-07 | 14°84 | 5-6 — | — | 18-14] 17-99 | 17-88 | 17-75 | 17-58 | 17-40 | 17-23 | 16-94 | 16-70 | 16-47 | 16-26 | 16-04 | 15-77 | 5-7 — | — | 19-34} 19-13 | 19-02 | 18-87 | 18-69 | 18-48 | 18-26 | 17-98 | 17-74 | 17°50 | 17-26 | 17-01 | 16-76 | 5-8 — | — | 20-57 | 20:36 | 20-20 | 20-03 | 19-84 | 19-62 | 19-39 | 19-11 | 18-84 | 18°59 | 18-32 | 18-05 | 17-78 | 5-9 — | — | 21-86 | 21-65 | 21-45 | 21-25 | 21-03 | 20-79 | 20-54 | 20-29 | 20-02 | 19-76 | 19-47 | 19°18 | 18°90 | 6-0 — | — | — | — | — | — | 22°36 | 22-18 | 21-92 | 21-61 | 21-31 | 20-97 | 20-61 | 20-30 | 20-13 | 6:1 — | — | — | — | — | — | 23-77 | 23-61 | 23-32 | 23-00 | 22-63 | 22-22 | 21-82 | 21-50 | 21:29 | 6-2 = — | 25-33 | 25-09 | 24°74 | 24-38 | 24-00 | 23°55 | 23-13 | 22-78 | 22-50 | 6-3 —|- 26-95 | 26-64 | 26-27 | 25-86 | 25-43 | 26-00 | 24-52 | 24-12 | 23-82 | 6-4 — | — | — | — | — | — | 28-61 | 28-18 | 27-73 | 27-30 | 26-89 | 26-46 | 26-06 | 25-65 | 25°24] 6-5 pe = = = — = — — = = — — 27°67 | 27°21 | 26°75 | 6-6 —— — — — — = — = a — — — 29°40 | 28°90 | 28°35 | 6°7 ee ee eee ee SS | | 31-15 }30-61.| 99-94 | 6-8 a ee alee ee i ie | | fg3-02 | 38-41 | 3i-72'|| a9 ea si = Astras) |'34716)| 23-59 |, 220 ABD SD DB] DS _S.SY D. G DSVr Se Sy Gr Gr Gy Sates bs bs Bp b> te Ete te bp So Ce Co Co Oe Co Co Co Cs Co 09 % 8 @ % HW WWW SSHRWBAAUK SBN SSWHWWAAK BBN SF GHOBWAAGK SBWN IS BSHBAKK WBWNSBHBIAKHK WWHS 70 Tables for Statisticians and Biometricians TABLE XXXVIII To find Probable Error of B:. Values of VNXg,. Bi 0:00 | 0°05 | 0:10 | 0°15 | 0°20 | 0:25 | 0°30 | 0:35 | 0-40 | 0°45 | 0:50 | 0:55 | 0°60 | 0°65 | 0-70 | O75 Reon Genie aed a Poa esl ee | 1-41 | 1°60} 1:74] 1°93] 2:11} 2°28 329! 346) 365) 3:86!) 4:05 1:57 | 1°76] 1°89} 2°00} 2°10; 2-20 3:24) 344) 364) 3:84) 404 1°75 | 1:94] 2°07] 2°16] 2°20} 2-28 3°23} 3:43) 363) 3:83) 403 1°95] 2°16) 2°28} 2°35} 2°42) 2°49 3°27| 3°45] 3°64] 3:84} 4-03 2°18] 2°39) 2°53) 2°60) 2°72) 2°82 3°35] 350] 3°68] 3:86) 4:03 2°46] 2°68) 2°83) 2°97; 3:09} 3:19 3°53) 3°63) 3°75| 388) 403 2°78} 3°03) 3:24) 3°38) 3°52) 3°60 3°78| 383} 3°87] 3:95] 4:07 3°17| 3°48) 3°71| 3°87| 3°98) 4:03 4:06} 406] 406) 407) 415 3°64] 4:02] 4:26] 4°42) 4°52] 4°58 4:39] 434] 432) 431) 4:34 4:22) 4°65] 4:94] 5°11] 5°20} 5°22 480] 470} 463] 460] 4°61 4°90} 5°48) 5°76) 5°89) 5°95} 5:93 5°30] 5°20] 512) 5:05] 5:00 5°75 | 6:41] 6°72] 688] 6:90] 6°82 5°92} 5°79] 569} 560) 5:53 6°77| 7°55] 7:90] 8°00) 7:97] 7:83 6-70} 653] 639] 626) 614 8:00} 8°83} 9:22] 9°30] 9:22] 9-02 7:60} 738] 7:20} 7:02] 6:84 9°37 | 10°28 | 10°68 | 10°76 | 10-67 | 10°46 8°73} 844) 818] 7:92) 7°66 10°85 | 11°75 | 12°31 | 12°52 | 12°46 | 12°25 10:03} 9°63} 926] 890} 854 12°67 | 13°74 | 14°40 | 14°78 | 14°53 | 14:21 11°60} 11°06} 1054} 10°02} 9:55 14°78 | 15°98 | 16°78 | 17°09 | 16°93 | 16°53 13°49] 12°74] 12°02} 11°36] 10°80 17°50 | 18°83 | 19°83 | 20°03 | 19°78 | 19°36 15°30] 14°42] 13:60) 12°88] 12-27 20°80 | 22°50 | 23°68 | 23°81 | 23°34 | 22°67 17°54] 16°50} 15°54] 14°77] 14:06 24-74 | 26°83 | 28-47 | 28°05 | 27-24 | 26-29 | 25:25] 24-18] 23:03] 22:02] 21-01) 20°01} 19°04) 18-12] 17°23) 16°36 = — | 35:00 | 34°17 | 32°88 | 31°36] 29°77] 28:13] 26°60] 25:12] 23°82] 22°64) 21°54) 20°53] 19°56) 18°62 — — | 43:3 | 41°4 | 39°2 | 387-2 | 35:2 | 33:2 | 31:2 | 29-2 | 27-4 | 260 | 247 | 23:4 | 223 | 21°3 = — |55°3 |51°6 | 48-0 | 44:6 | 41-2 | 38-6 | 36:2 | 33:8 | 31°8 | 30:1 | 285 | 26:9 | 256 | 24:3 = — |72°7 | 66:0 | 59° 54°] | 49-5 | 45°7 | 42-1 | 39-2 | 36:8 | 348 | 32°9 | 31-0 | 292 | 27-7 — — |96°5 | 82-7 | 72-7 | 65:3 | 59:8 | 54:7 | 50°8 | 47-2 | 440 | 41:0 | 385 | 362 | 341 | 32-0 — | 75:0 | 68:0 | 622 | 56:9 | 52:2 | 482 | 45:1 | 42:2 | 396 | 372 — |101°3 | 87-2°| 76°8 | 68:3 | 62°0 | 56:9 | 52°7 | 49°71 | 45:9 | 428 — -- —_— — |140°0 |115-2 82:6 | 72°7 | 66:1 | 60:9 | 56°7 | 529 | 49°3 — — — — — |2045 |150°8 102°5 | 89:1 | 802 | 724 | 666 | 614 | 56:7 — _— — |103°6 | 94:4 | 85:9 | 78:0 <= — — /|130-4 |116-4 |1040 | 91:0 = — | — |175°2 |1448 |1244 |109°6 —_— — {2244 |178:0 |151°0 |132°6 == — | — |340°8 |246:0 |195°3 |163-2 96-2 122°3 325°7 |206-0 |154°2 |126°8 |110°1 | 96-9 | 866 | 781 | 712 | 656 0°80 4:24 4:23 4:22 4°20 419 418 420 4:26 4°40 463 4:98 5°47 6:03 6°67 741 8:22 9:14 10°34 11°77 13°42 15°58 17°72 20°2 23:1 26°3 301 34:7 40:0 4671 52-4 60°6 71:2 83:0 98°38 118-4 141-4 0°85 4:43 4-41 4:39 4°36 4°35 4°34 4°35 4°38 4°50 4°67 497 5°42 5°92 651 717 792 8:80 9°94 11:29 12°85 14:84 16°85 192 22°0 25°0 284 32°5 374 43:1 48°8 561 65°1 764 89°6 105°2 124°0 By. 0:90 | 0°95 | 1°00 | 1:05 | 1°10 | 1:15 | 1:20 | 1:25 | 1°30 | 1°85 4-62) 4:81] 5°00) 5:19} 5:38] 5°56] 5°75] 5:94] 612] 6:30 2-0 459| 4:77| 496] 5:15] 5:34] 5:53] 5:72| 5:90] 6:08] 6:27 Dap 4:56| 4°74] 493] 5:12] 5°31] 5:50] 5°69] 5:87] 605] 6:24 2:2 4:53| 4°71] 490] 5:08] 527] 5:46] 5°65) 5:84] 602] 6-21 23 451| 4:69] 4°87) 5:05] 5:23) 5-42] 5°61] 5:80] 599] 618 a4 4°50| 4:67] 485] 5:03] 5:21] 5:39| 558] 5:77] 5°96] 6-15 2-5 4°50| 4:67] 4:84] 5:01] 5:20] 5:36] 554] 5°72] 5°91] 611 26 4:52) 468] 4:84) 5:01] 5:18] 5:34) 5°51] 5°67] 5°85] 6:05 27 460| 4:72] 486] 5:03] 5°19] 5:34] 5:49] 5-65] 5°83] 6-02 28 4°73| 4:82] 493] 5:05] 5-20] 5:35] 5:50] 5-66] 5:82] 6:00 29 4:99] 5:03} 5-10] 5:18] 5:28] 5:39) 5:52] 5-66] 5:82] 5:98 3-0 5:38| 5:34] 5°36!) 5:37) 5:41] 5°48] 5:58] 5:70] 5°83] 5:97 31 5:83] 5°75| 5°67) 562] 560] 5°62] 5°68} 5:78] 5:90) 6:03 32 6:35| 6:22] 609] 6:00] 5:90} 5:94] 5:92} 5:95] 601) 6-12 33 6:95| 6°77| 661] 6-48] 629] 626] 624] 622] 6:22] 6:26 3h 7-64| 738] 717] 699] 684] 672] 663] 657] 654] 6:53 35 8°51} 8-23) 7°98! 7:70] 753] 740] 722] 7-09] 699] 6-98 36 9°58| 9:25] 8:96] 8-66] 836] 814] 7:90] 7°75] 761] 7-51 37 10°82| 10:37] 9:98) 9:62] 9:31] 9:03] 8-73] 851] 829] 811 3°8 12°31| 11:79] 11:30} 10°86] 10-41] 10°02} 9°64] 9:34] 9:03] 8-77 39 14°10) 13-42] 12:79} 12:20] 11°64] 11:13] 10°65] 10:21] 9°83] 9°51 40 16:01] 15°21] 14:44] 13°70] 13-00] 12°34] 11°73] 11-17| 10°67] 10:24 4 183 | 17:3 | 16-4 | 15:5 | 14-7 | 140 | 13:3 | 126 | 12:0 | 115 42 20°99 | 198 | 18:7 | 176 | 16°7 | 15°8 | 15°0 | 14:2 | 135 | 12°8 43 23°8 | 22°5 | 21°3 | 20:71 | 190 | 180 | 1771 | 161 | 15:3 | 14°6 Lh 26°8 | 25°3 | 23:9 | 29°6 | 21:4 | 20°3 | 19°3-| 18:3 | 17:3 | 16-4 Le 30° | 288 | 27:3 | 25°6 | 24:2 | 22°9°| 21-7 | 20°6 | 19°5 | 18-4 46 35:0 | 32°8 | 30°9 | 29:2 | 27-6 | 2671 | 24:7 | 23°3 | 22:0 | 20:9 47 40°3 | 37:7 | 35:3 | 33:2 | 31-4 | 29°7 | 28:0 | 26-4 | 25°0 | 23-6 48 46°8 | 43:1 | 40:2 | 37:8 | 35°6 | 33°6 | 31°6 | 29°8 | 281 | 266 v7) 52°3 | 48:8 | 45°5 | 42:6 | 40°0 | 376 | 35:4 | 33-4 | 31°5 | 29°8 5:0 60°6 | 56° | 52°6 | 49:1 | 45°8 | 43-1 | 405 | 38-0 | 35°6 | 33°6 51 70°5 | 65:4 | 60°6 | 56:3 | 52°5 | 49:3 | 46:3 | 43-4 | 40-4 | 38:0 52 81:9 | 75°6 | 70°2 | 65:0 | 60-2 | 562 | 52°5 | 489 | 45:5 | 42°6 53 96:0 | 87:6 | 80-4 | 74:0 | 68°3 | 63:4 | 58:8 | 54°7 | 51°0 | 47:7 54 111°2 | 99°6 | 91:2 | 84:0 | 77-4 | 71-2 | 65:7 | 61°2 | 56°9 | 52:9 55 131°2 |117-4 |105:2 | 96:0 | 87:3 | 79:3 | 72°8 | 67°8 | 63:2 | 58°6 56 1600 |142-4 |126:4 |113-4 |102-°2 | 93-0 | 84-4 | 77:3 | 711 | 65°6 57, 199°2 |175°8 |154:8 |134-2 |119°6 |107°0 | 97:2 | 88-4 | 806 | 74:4 5°8 2660 |221°6 |192°8 |163°6 |142°8 |128°0 |114°6 |104-:0 | 946 | 86-0 59 13781 |284-0 |231°5 |198°2 |171°6 |151°5 1136-2 |123-8 |112°8 |103-4 6:0 ies = = — |206°3 |186°3 |167°5 |150°0 |134:2 |121°5 61 _ = _ — |264 |232 |205 {280 |160 141 6-2 = = o- — |350 |297 |251 |216 |188 /|164 63 = == —_ — |510 |876 |308 |263 |225 |196 64 = = — — |889 [524 |387 {313 |264 |229 65 a —_ = = —_ = = aes = = 66 = = = = = _ = = = 6-7 as = = = = = 5 = = = 6°8 — = ate = = = = = = 6-9 = = = = a = = ae — = 7-0 Probable Errors of Frequency Constants TABLE XXXVIII.—(continued). Values of VN S,,. 71 Bs ~I i) TABLE XXXIX. To find the correlation in errors of B, and B;. Values of Rg,p. By eae : : 7 0-00 | 0:05 | 0-10 | 6-15 | 0°20 | 0-25 | 0:30 | 0°35 | 0-40 | 0-45 | 0-50 2:0 | 0-00 | 570 | -706 | -770 | -823 | -863 | -894| -917 | -935 | -949 | -960 2-1 | 0-00 | -557 | -685 | -755 | -798 | -838 | -870 | -895 | -914 | -936 | -948 2-2 | 0-00 | -551 | 672 | -728 | -771 | -814 | -847 | -874 | -896 | -919 | -934 2:3 | 0-00 | -550| -663 | -719 | -765 | -799 | -829 | -859 | -880 | -900 | -919 2-4| 0-00 | 551 | -660| -712 | -752 | -787 | -814 | -843 | -867 | -886 | -905 2-5 | 0°00 | -554 | -659 | -706 | -745 | -776 | 805 | -834 | -858 | -878 | -893 26 | 0°00 | -557 | “662 | -706 | -742 | -773 | -799 | -825 | -851 | -871 | -883 2-7 | 0-00 | -557 | ‘668 | -710 | -744| -773 | -800 | -825 | -846 | -863 | -876 2-8 | 0-00 | -556 | -674 | -716 | -750 | -779 | -803 | -826 | -842 | -858 | -871 2-9 | 0-00 | -550 | -680 | -724 | -760 | -787 | 810} -830 | -844 | -857 | -868 3:0 | 0-00 | -542 | -684| -738 | -774| -796 | -816 | -835 | -847 | -857 | -867 3-1 | 0-00 | -534 | 687 | -744 | -781 | -808 | -825 | -840 | -850 | -858 | -867 3-2 | 0°00 | -524 | -688 | -746 | -786 | -811 | -830 | -842 | 852 | -860 | -868 3°3| 0-00 | -512 | -688 | -747 | -788 | -814 | -832 | -845 | -855 | -863 | -870 3-4 | 0-00 | -501 | -686 | 748 | 790] -816 | -833 | -848 | -858 | -865 | -872 3-5 | 0°00 | -490 | -681 | -747 | -790 | 815 | -833 | -849 | -860 | -867 | -873 3°6 | 0-00 | -477 | -676 | -745 | -788 | -813 | -832 | -850 | -860 | -867 | -874 3-7 | 0-00 | -462 | -670 | -741 | -784| -810 | -831 | -848 | -859 | -867 | -874 3°8 | 0-00 | -450 | -662 | -736 | -779 | -803 | -828 | -845 | -858 | -866 | -874 3-9 | 0°00 | -438 | -654 | -720 | -770 | -796 | 822 | -841 | -856.| -866 | -875 4:0 | 0-00 | -422 | -645 | +713 | -760 | -788 | -816 | -837 | -853 | -865 | -873 4:1; — | — | -630| -702 | -748 | -780 | -807 | -830 | -849 | -862 | -871 42, — | — | 608 | -682 | -733 | -770 | -793 | -822 | -842 | -857 | -867 43; — | — | 580} -658 | -712| -753 | -784 | -811 | -832 | -848 | -860 4-4| — | — | 540 | -628 | -688 | -732 | -770 | -796 | -819 | -837 | -851 45 | — | — | -481| 590 | -657 | -709 | -749 | -780 | -804 | -824 | -841 4e| — | — | — | — | — | — 1-716] -754} -784 | -808 | -s28 47| — | — | — | — | — | — | 674] -723| -759 | -788 | -812 78 | arses zen vel v50 #9| — | — | — | — | — | — | 5321 -620| -680 | -738 | -z66 50| — | — |-— | — | — | — | 362) -534| -628 | 687 | -731 51 on = — — —— — a —s — — — 52 — — —4 —— — — mon 5 — — —t 5°3 —= — — — — — — — = — — 5-4| — | SS StS | = 65, —|—}|—|]—|]—]—]| = —|—-—|— Ta) | a er eee |e a7 —= — — — —e — — ——, — — — PS S| S|) ee nn FA | ae eet Se 6°0 = —— — — — —— ——— — — — — S22 a ee ee 6°2 — al — — ==, — — — — — 6°3 —— — — —— — — — oe —— 64/—}—|/—}]—};—-—]/—-—]—-|]— = 6°5 — a= — — —s — — — — 6°6 Aare === — — = = — — — —s — 6°7 = a — —_ — — — —— di — — 6°8 —_ — — a — —— — —d — — 6°9 — a — — — — — —s — — 70 — — Tables for Statisticians and Biometricians | | a Probable Errors of Frequency Constants 73 TABLE XXXIX.—(continued), Values of Raye, A, 0°80 | 0°85 | 0°90 | 0°95 | 1°00 | 1°05 | 1-10 | 1-15 | 1-20 | 1-25 | 1:80 | 1:35 | 1-40 | 1-45 | 1-50 ‘993 | -995 | -997 | -999 | 1-000 | 1-000 | 1-000 | 1-000 | 1-000 | 1-000 | 1-000 | 1-000 | 1-000 | 1-000 | 1-000 | 2-0 ‘989 | 991 | 994] -996| -998| -998| -999| -999| *999| 1-000 | 1-000 | 1-000 | 1-000 | 1-000 | 1-000 | 2-1 -983 | -986 | -989| -992] -995| -996| -997] -998| -998| -999| 1-000 | 1-000] 1-000 | 1-000 | 1-000 | 2-2 976 | 980 | -984| 988] -992] -993| -994] -995| -997] -998| -999] -999| -999/ 1-000| 1-000 | 2:3 ‘968 | 973 | -978| 983] -987| -989| -991| -993| -995] -996| -998] -998| -999| -999| 1-000 | 2:4 ‘958 | 965 | -972| -977| -982| -985| -988| -990| 992] 994] -996| -997] -998| -999| 1-000 | 2-5 947 | -956 | -964| 970] -976| “980] -984] -986| *988] -991| -993] -995| -997| -998] -999 | 2-6 937 | 947 | 957 | 963] -968| -973| -977| -980| ‘983] -986] -989| -992| -995| -997]| -998 | 2-7 ‘928 | -939 | 949 | -955| -960| 965] -970] 974] ‘978| -981| 985] -989| -992] -994] -996| 2-8 ‘921 | -932 | -942 | 947] -952| -957| -963] 968] ‘972| -976] -980] -984] -988] -990| -992| 2-9 ‘915 | 923 | -931| 937] 943] °948| -954] -960| ‘966| -971| 975] -979] -983] -986| -988 | 3-0 909 | -915 | -922| -929| -936| -942| -947] -953| °959] -965| -970| 974] -978] -981| -984| 3-2 ‘907 | -912 | 918 | 924] 930] °936| -941| -946| *952] -958| -963| -968] -973| -977| -980/ 3-2 ‘906 | -909| 914] 919] -925| 930] -935| ‘940| ‘946] 951] -956] -961] -966| -971| 975] 3-3 905 | 908 | -912| 916] -920] -925] -930] -935| *940] 945] -950] *954] -958| -964] 974] 3-4 "904 | 907] 910 | -914| -918| *922/ -926| -931| °936| -940| 944] 948] +952] 958] 965 | 3-5 ‘904 | 907 | 910] 914] -918| 921] -924] -928| ‘932] -935|/ -938| -942] -946] -952] -959| 3-6 ‘905 | -907 | 910 | -914| -917| -920] -923| -927| °930] -933/ -935| -937| -940] -946] -953| 3-7 ‘905 | -908 | -911| 914] 917] °920| -922] -925| ‘928] -930| -932| 934] -936] -941| -948| 3-8 -906 | -909 | -911| 914] 917] *919| -921] -924| -927] 929] -931| -933] -935] -939| -944/ 3-9 ‘906 | 909 | -912| -914| -917| ‘919] -921| -923| ‘926| -928] -930] -932/ -934|/ -936| -940| 4-0 ‘905 | -908 | -911| -914] -917| *919| -921| -923| °925] 927] -930| 931] -932| -933] -934| 4-7 905 | 907 | 910] 913] -916| -919| -921| -923| “924| -926| -929] -929/ -930| -930] -929| 4-2 -903 | -906 | 910] 913] -916| ‘918| -920| -922| *924] -926| -929| 928] -928| -927| -g24/ 4-3 -900 | -904| -908| 912] -916| -918| -920| -922| -923] -926| -928| 927] -927| +925] -922| 4-4 897 | 902 | 906] -910| -915| ‘918| -920| -922| “923) -926| -928| -927| -926| -923] -920| 4-5 *893 | -898 | 903 | 908] -913| ‘916| -919| -920] “922] -925| -927| -926| -925/ -923| -920| 4-6 “887 | -894| -900| 905] -910| “913| 917) -919| “921| -924| -926| 925| -925| -923] -922| 4-7 ‘881 | -890 | 896 | -901| -906] “910] -914] 917] “920] -923| -925| 926] -926| 925] -925 | 4-8 874 | 884 | 890] -895| -901| “907| -911| 915] 919] -922| -925| -926| -927| 927) -928| 4-9 863 | 875 | -883| 889] -896| -903| -908| -913| “918| -922| -925| -927| -928) -930] -932| 5-0 "851 | -864| 875 | -882| -890| “898| -905] -911| °917| -922] -925| -928] -931| -933! -936 | 5-2 837 | 852 | 866 | -875| -884| °892| -901| -909] 916] -921| 924] -928| -933| -937| -941 | 5-2 820 | -839 | -853| -865| -876| °885| -895| -904| 913] -918| -923] 929| -935} -940] -945 | 5-8 ‘798 | -818 | 837 | -853| -867| °877| -888| -898| ‘908]| -915| “921! “928] -935| 941] -947| 5-4 764 | 792 | 817| -837| -854| °867] -880| -890] 900] -910| -918| -925| 933) -940| -947| 5-5 — | — | -789]-815] -835] °852] -s6és| -880] 890] -904] 911] 917] 926] -935] -944/ 5-6 — | — |-750]-786] -811| °835| -854| -s69] -880| -892| -901| -909| 917] -927| -938 | 5-7 — | — |-701|-748| -783] 811] -835| -852/ -866| 879] -890| 897] -905| -915] -998| 5-8 — | — | 640] -700| -748| °781| -810| -828| °846] -861| -875| 883] -892| 901] -913| 5-9 — | — |-544!-639| +703] °746| -778| -so2| -825] -842| -857| -867| -879| -886|] -893| 6-0 —|—|—|—]| — | — | -741] -ve9] -796] -820) -837| -852| -866| -872| 873 | 6-2 =}—|—|—]| — | — | 691} -7a7| -ve2| -792| -815| 836) -852| -858| -856| 6-2 ee — | -628| -678| -724| -761] °790] -818| -838| -845| -842| 6-8 = — | -526| -606| -675| -724| -763] -793] -818| -831| -834] 6-4 =|/=—|—|—| — | — | 354] -526] -619| -680| -v26| -761| -791| 814] -831| 6-5 Se a a es MMe me a er ea ee ee |) | eal reo sa37 | 67 a en ean ea eal | ee | | ero e797) sas e8 a eae err ee eS S| ee | Se || | e001) aeas:|” “B57 | 6:9 a ce |e a da Soa ea (peo eee 468 | 602 eae | B, 10 Ba Ge hhhhhhhhs & ive) S tom SUSUSY ~2 o> 7 Na) © %» 93 S9 nS Sasa isa a) Ga ND Tis Co %*% 7a SSE GS 09 ~ 74 Tables for Statisticians and Biometricians TABLE XL. Zo find the Probable Error of the distance from Mean to Mode. Values of Mass or at By. (ee 0:05 | 0°10 | 0°15 | 0:20 | 0:25 | 0°30 | 0°35 | 0-40 0-45 | 0°50 | 0°55 | 0°60 | 0°65 | 0°70 | 0-75 | 0-80 354; — | —| —] —}| —| — | —] —] = 3703 | 2-44 2:10 | 1°80] 1°58 | 1-42 215/436] —|—|/—]|—] — |—]|—j;—]}]—]—]| — | 3810] 2°53] 2-16 1:87 |2°75/9:°65) — | — | —}| — |—}| —|—}] —j — — | soLsi7 1-64] 1:86| 3:00] — | — | —| — | —| —|]— |] —/} —|] — | — i} 1:46 |1-58|2°07; — | — | —| — |—]}] —}] —}] — |} — | | — |} 1:35 | 1-46 | 1°67 | 2°08 | 2°87] 4-04] 5:21; — | — | — | — pS p= = 1:28 | 1-37 | 1°58 | 1°98 | 2°60 | 3-42) 4-43/6-72} — | — —-|— —-|- 1-25 | 1:30 | 1-50 | 1°83 | 2°34 | 2-98) 3°75 | 5:06] 7-48; — | — | — ee 1-23 | 1-28 | 1-43 | 1°71 | 2-11 | 2-60] 3-17 | 4:12|5-28/ 7-45} — | — |} — | — | — | — 1-22 | 1-27 | 1-38 | 1°60 | 1°90 | 2-27] 2-69 | 3:20] 3°84] 4:78) 665; — | — | — | — | — 1-23 | 1:26 | 1°34] 1°51 | 1°73 | 1-98] 92-29 | 2°63 | 3°06 | 3°58 | 4-28 | 5-18) 6-43] 8-24) 10:89) — 1-25 | 1-27 | 1°32 | 1°44] 1-58] 1:76} 2-00 | 2:23 | 2°54 | 2-94] 3:42 | 3-93] 4:52] 5:50] 6°76 | 8°66 1:27 | 1-28 | 1°30 | 1°38 | 1-48) 1°60] 1-75 | 1:92 | 2°13 | 2-37 | 2°72| 3-12) 3:54] 4:21] 5-07 | 6-22 1-29 | 1-29 | 1-28 | 1°32 | 1°39 | 1:47] 1:55 | 1-68 | 1-83 | 2-03 | 2°27 | 2°57] 2:90) 3°39} 4:04] 4°66 1-30 | 1-29 | 1-28 | 1-29) 1°31 | 1-37| 1:45 | 1:54 | 1°63 | 1°79 | 2°00 | 2°24| 2-51) 2°88} 3°36 | 3°86 1:31 | 1-30 | 1-29] 1:27 | 1-25 | 1-30] 1-37 | 1:45 | 1°54] 1:66 | 1°83} 2°03] 2:26] 2°55] 2°89) 3-18 1-32 | 1-31 | 1-30 | 1:26 | 1-22 | 1-26] 1-32] 1:40] 1°50} 1°61] 1°74/ 1°89! 2°08) 2°31] 2°56 | 2°86 1°31 | 1-31 | 1-31 | 1-26 | 1-22 | 1-25 | 1-30] 1:37 | 1-46] 1:57 | 1°69] 1°82, 1:97] 2-14] 2°34) 2°62 1-30 | 1-31 | 1-32 | 1°28 | 1-25 | 1-27] 1-32 | 1:38] 1-46] 1-55 | 1°65/1-76| 1°88} 2-03] 2°20) 2-43 1:29 | 1-33 | 1-35 | 1°33 | 1°30] 1-32] 1-36 | 1:41 | 1-48 | 1:56 | 1°64] 1°73| 1°84] 1-96] 2°11 | 2-27 1-27 | 1°37 | 1-40 | 1°39} 1:39 | 1-40] 1-42] 1-46 | 1:51 | 1°58] 1°65] 1-73; 1°83} 1:94) 2:06 | 2-19 — | — | 1-47} 1:48) 1-50] 1-51) 1-53 | 1-55 | 1°57 | 1-61 | 1°66} 1°75; 1°85) 1-94] 2-03] 2-18 — | — | 1:58} 1°62} 1-64/1-65| 1-65 | 1-65 | 1°65 | 1-66) 1°70} 1°77) 1°85] 1:94] 2-03 | 2°12 — | 1°75 | 1°77) 1-78} 1-79] 1°78 | 1°76 | 1°75 | 1°75 | 1°78] 1°83} 1:90] 1:97] 2°05 | 2-13 — | 1-98 | 1:97 | 1:95 | 1-94 1-93 | 1-90 | 1:88 | 1-89] 1°92] 1°96 | 2°01) 2-07] 2-13} 2:20 — | — | 2-27 | 2°20 | 2-15] 9-11) 2-10 | 2-09 | 2-09 | 2-10 | 2°12 | 2°15; 2-18] 2:22) 2-26 | 2°32 —|}—|— | — | — | — | 2744 | 2-40 | 2°38 | 2°36 | 2°34) 2°36) 2°38) 2:40) 242 | 2-45 — | — | — |} — | — | — | 2°93 | 2°85 | 2-78 | 2-71 | 2°67 | 2°65 | 2°64] 2°63} 2°62 | 2°61 = — | — | — | — | 38°74] 3:52] 3:33 | 3-16] 3°07] 3-00} 2°95) 2°90] 2°85 | 2°81 —}/—|]—/—-| — | 5:44 | 4-64 | 4:16 | 3°87 | 3°63 | 3°45] 3°32| 3:24] 3°11] 3-05 - | — | — | — | — | — | 10-66 | 6°83 | 5°53 | 4:84 | 4:37 | 4°04) 3°79] 3°62) 3°47 | 3°37 =|) hay ay ; — | —|—!|—|—] — | 4:46] 4:21] 3-99 | 3-85 —}/—/}—j;—-);—]—}] — | —| =|] — | — |] — | 5°38} 5:05] 4°78 | 4-47 | | i ee a | — | 684) 6:19] 5°66 | 5:27 —;/-—|}—-—/]— —|;— |—]—|—!| — | — | 924] 7-96] 7-00 | 6-24 =f] )—peSehHleits | —]}= — | —_ 14:81/ 10°89) 8°87 | 7°64 — ee ee — — — — — _ = | => | — = | — | | —— — oe —_— —_— — | —_ — _ | —_ —_— a | — — —_— — — = —— — — — — — — — -~}—;/—};-;—-;—-}/—-}]-;-}—-;-]|J-|}-|-]-/- sh = | ey | eee ee rsa Pe ery he — ae — | = — | — — — = — — — — a 2 bay <= — =a —= =< = = | | | \ No sy H WTB WS TOF D DH Gk 09 29 29 09 0 wy RON TH © ON WB Hh G9 69 09 LO RO LO RO RO LO LY 19 NY LY BD RDO OF He OH OF KF PEE EH WOR DDOWNADOUUKRRrOW hae See 0°85 | 0:90 1:20] 1-13 1°64} 1-49 2:22.| 1:97 3'10| 2-63 = || Se 6:80| — 5°38 | 6:55 421] 4:95 3:°74| 4:34 3:34| 3°80 3:00) 3°35 2:74| 3:00 2°55 | 2°77 2°42) 2-60 2°35 | 2°50 2°35| 9-48 2°39 | 2°50 2°45 | 9°53 2°52| 92-60 2°64] 2°69 2:81] 9-82 3°02] 3°03 3'28 | 3:26 3°64| 3:55 4:04| 3-90 4:60] 4:35 5°33 | 4:98 6-21) 5:74 — | 6°69 = |) Galil — | 10°18 — | 13°53 — | 19°95 0°95 1:07 1°38 1°80 2°46 3°2 3°20 6°00 5°12 4°32 3°74 3°32 3°02 2°79 2°67 2°62 2°61 2°63 2°69 2°77 2°86 3°04 3°25 3°51 3°81 4°18 4-71 5°36 6°27 7°48 9°11 11°44 14°26 Probable Errors of Frequency Constants 1:00 1°02 1°29 1°66 2°15 2°81 3°94 TABLE XL—(continued). 1:05 | 1°10 “O77 IPA) de 154] 1 1:98} 1 2°56) 2° 3°40| 3 4 DD~AD TPB ww Wwe WROD WwWwWA IR OATO BTHROROWSABRNHODBDHMHMDDDONDOTAIHKHY DH ANOHAANDIDNOWOTWSHOSISCENDID So S wo _ a eae tag Wie Values of “* Sy Tee a) hm hb Oo TO OT Go bb OF Noo RONHFWORNTAWOONDREHAOBDAeW Worse onsrwone — D> OT OT BR Go 00 69 G9 CO 08 Co Oo Co 00 CO RR Or OD CO + DBONTWOTDRWWNNWWODHE OHH Or 7°89 NOTWNRFANODONTRE OONWHNAOLY FA TT lathe _ mDAOowwrh DH Hn Oi vO WEADSDANWGSIDARWMOEA SDAIN RHAONNENRODODEHW 3°50 DD > Er He sh v9 £9 09 09 09 BROOM WoORDDAH on Owowst WwW OCTON De Lt Ll lM! ul La Ul OWRK KH WoOoaIrorknade WHWWWNWWWWA AR OOOO ADADHAAARDATISONHNAH HDDS WEOAAAH OOIAATP HWW WW WO RR ROTA TO ESE WASTE OAIAAADAICOHWAHHO-10N ' CHOUWHSCORONADUNDWE ROH RAANNAW 75 145 | 1°50 “64 “60 *82 “78 1:02 ‘96 1:24} 1°16 151) 1°39 1°80] 1°64 2°10; 1°91 2°46) 2719 2°87 | 2°49 3°31 | 2°80 3°79 | 3:12 4:29 i ian Ped Hes COUR ICSAC COL CONF Hm tev Hees CUCM STO DOADODRARADHE KORE DUIBAAADSONATHDS-1ES WONDTAOWTNWEHAWSOTNONAOTNODNARHD DpReeHeeH i) Io OT Oe HH 2 9 2 2 CO HH TAF WAOCOCOATCKNNODOVUDG SAW TKSEN WASH AIGSSITTSSWD RR ORESOHID whee eH MB D_D DD Sd. DDD. GOV SSG Sy Oy Oy Sy By By HE Hy Hy Ay Ay A AH Ge Ge Og Cg Co Oe Oe Og Co Oy 1 1H WW WWW Wig © SCSWHBSAAUESBNSGSGHVWANKSGBNSSHVIAKE SBN SSHVIAAUKSGBBNSSGHRBSAHKHSBHNS 10—2 Tables for Statisticians and Biometricians 76 To find the Probable Error of the Skewness sk. TABLE XLI. Values of VN see B; DOAN RFR ARR AAR AAR NA AANA AA 2 3 3 5 11 wy [> eo) DMAMDMDODNOOANANHANAOAMOONR ERE EeAONAS 7 > Para lh | SHGHORBGHINASCSCSSAAHMHOEOSOHOMTANE | | | | | jt tii tiiiti So mia DwODHOAAATNAAA A AAG AA A GI 0 09 69 SH 10 Or S Dar ACH WDDAARDHONAHADOAMAIAOM-MODAGCOr S evox jl i fl | | | Stst@OREeerooneqmee rics movomoer Ii til ti tid lw yd Ss rq 0 19 DOADMMANANANANANTA KH BAH AAATAAAAM MH Hid00O 0 at ANE DWDMOPrMDDAATHAAMOATDNOATAYrWDWOADOSr - 3 eS ie GAUSS tel COG eran ote oto eel ma caer oi oC oA tel | S ao | CO HOD CO GU GU GU 4 tt tt rt ot GI GY GIG GY 09 09 HID CO I~ © | | | | | = Gr) MOR MOBHOMDHAHTODORHHR DOL DAHTHO al Ra} = | 1A hd | SPB fe SOS SO GP Be Be oy Oe Seine OP) GP a ey ee ee A A a S QA SHED OD GY GU GU II GY 09 09 HD 6 OD | iD fo} OOOMrMHAr-Ore DAME ATNMNDTOnPHA fe SWI Ree seer ope pacmocmc ve Ti TI A en a MOANA AAR RAR RRA AAAA HH S Lan SCOOP HDDHFOHMOOMOOMm M19 ODS s SMO Wee 2 eer ® pp poem eocommroe alli del the ee a | S (50) HMOANR ARR RR AAA RATAN GAIM OOH S| LLL LLL (883288 S88SSSSSSRSS8RR55 |p pit Tilt iti iii liiel S HMOANR AR RAR TARA ARR RA ANNANMOHM TH S|] Ll Ll) (S8SSS8SSSSSRSSSSRSseeerS yi ii tl tit li tii di S HMONANRA AAR RRA AR AAA AAANH HO w» DDODAMrMDMDHADMOAANAMAHADOODIDON OH OS oS) lfau lia i DOE ADSHAHAMAMOHOAMOSHSOHHoOOSD | | || /{ {1 i{1i1ilililill S HONANR AA RR ARR ARR RRR AANANHMWO } DMOMOhHONDOMADMDOPDWDOOANMHOCHrDADOAN s Sek te S Saooseananagaganassean | | | (|i itil irri ttt ti itt ti tttd Sree ae wee ay jerelahenenuties nem iew ser leln ue) enter werner err ley Te) R | [|| | (SBSSSSRRARARARABIBESS LTT ILI I tlt l ii tii tiitit 3S HANG HH SRA AAR AAA eRe SU eeebeeccen nese voces TELM MME Cet S AAS eee ae eee eee eA Seto (2S Pecne ees eee e tos Oa ii (a Te tna nee a S ee. i=) SCHDAAMDAHOOCHN|A MD HG OO OD OO 19 19 © S| Pee oveeiss niger ine beet ee ay hil lala ee tT S DANA A RRR RRA RR RNA ANA AN AA 19 SMDHDHHWAOVWVACHAAMNOEOAGAATS sees ese aigienig aoe MEE ele ele lat aren S Ba CS a I I 2 Hor HtHOMDMOMAADNOErAOANAROAaL 3 a Q OASHAHSIAGwO Ye HE wn eaAdwH = CORDON DONMS SSSR OSSEGSESSS 0°85 | 0°90 | 0°95 1°39 1°80 2°29 3°04 4°29 rs a D OX Ot 9 G2 O9 BO BO bo bo DORDANW SATA BSHAMH ARG coho hoe AISDAYW RaADae NPNNWwWWwWWwR Oo Sec ecseace lel ObnNw-I1 0 ao WOISGNRONeSANSTTIMDAAKAH BO OD ~1 > CUB iB ©9 Co Co bo BO BS tO WO bo SCWHONTOCONMDONMDWAWNOS Probable Errors of Frequency Constants 1°00 2°66 SSSataSseasnvsogn = BO © ATO Cr OTB IB Go Oo Co 09 bO BO DADWOTAWAAHE — tal | TABLE XLI—(continued). Values of VN Xx. 114 1°47 1°84 2°30 2°94 3°84 5°72 77 Ay 1°20 | 1:25 | 1°30 | 1°85 | 1-40 | 1°45 | 1°50 1:06 | 1-02 “99 95 “91 87 “83 1:37 | 0:32'|| 1:26) 1-20)) 1-15} L110) 1-05 1-72] 1°66] 1°59} 152) 1°45) 1°38) 1°31 2°12) 2:03) 1:94] 1:85) 1-76| 1°67) 1°58 2°63 | 2°49} 2°36] 2°23] 2°10] 1°98] 1:86 3°29] 3:06] 2°85} 2°66) 2°49] 2°32] 2°15 4:39 | 3:94] 3°54] 3°20] 2°93] 2°67] 2°44 6:25] 5°20] 4:46] 3°88] 3°42) 3°08] 2°74 a= 7°05| 5°68] 4°78] 4°03} 3°54] 3:05 = _— 7'45| 6:00] 4°85] 4°11) 3°37 = —_ = 7°80} 6°05) 4°77] 3°70 — _ — _ 5°57} 4:10 oe — —_ —_ _ _ 4°58 9°42) — — — — as — 7°08 | 9:02 — = = — 562} 6°89) 876) — = a = 4°77| 5°50] 6:42] 7:40] 8°57] 10:12) — 4:15] 4°61] 5°14| 5°84] 6°80] 844/ 11-00 3°67 | 4°03] 4:44] 5°04] 5°94] 7°12] 8°67 3°37] 3°65] 4°01] 4°55] 5°28} 6°18] 7-21 3:23] 3°48] 3°78| 4:22) 4°78] 5:44] 6-26 3:20] 3°40] 3°65| 3:97] 4°40) 4°91) 5°56 3'16| 3°34] 3°55| 3°79) 4°16] 455] 5°10 3°14] 3:29) 3°47] 3°67) 3°96] 4:28] 4°72 3°16] 3°28] 3°41] 3°58] 3°80] 4°06] 4°42 3:21] 3°30] 3°40| 3°53] 3°68] 3°88) 4°16 3°31| 3°36] 3°43] 3°51| 3°60] 3°73] 3:96 3:43] 3°44] 3°46] 3°51] 3°57) 3°65] 3°78 3°57 | 3°55] 3°53] 3°53] 3°55] 3°60) 3°67 3°76| 3°68] 3°63] 3°60] 3°58} 3°59} 3°62 4:00} 3°89} 3°80| 37 3°68 | 3°65) 3°63 4:30} 4°17| 4:06] 3°95] 3°87] 3°80) 3°75 4:77| 4:60] 4:44] 4°30] 4:19} 4°10) 4°01 5:34] 5°10] 4°90} 4°73] 4:59] 4°47] 4:38 5:99| 5:68| 5°44] 5°24] 5:06} 4°91] 4-78 6:78| 6°40] 6:10] 5°84] 5°62} 5°44] 5:28 7:90| 7:39] 6°96] 6°61] 6°33] 6°10} 5:89 9°31| 8°56] 8°03} 7°53] 7:15] 6°82] 6°54 11:19} 10:13] 9:30] 8°64] 8°08] 7°63) 7:24 13°69 | 12:21] 11°01] 10°02] 9°19} 8°55} 8-01 17°09 | 14°56 | 12°84 | 11°45 | 10°45 | 9°60] 8:92 21:30 | 17°44 | 15°07 | 13°20 | 11-90 | 10°83 | 10-07 _ —_— — — | 14:2 | 12°9 | 12-4 — — — — |1772 |15°6 | 14:7 — —_— — — |21°8 |19°6 | 18:3 —_ — —- — | 29:4 | 26:0 | 24:1 —_ —_ —_ 43°0 | 387°8 | 34:9 BSHVWAAK WOKS GB OHWA KM WH — > VIAADRBRAABAABAAAA AAA AAA THESE HH HAA Ce Co Go Ce Co Co Co Co Oo Cs WWW WWW swe WBN SGSHRGAUKSBBKNSE % ~ S : . SGMWVBAGCK SBN SGHIBAAK 78 Tables for Statisticians and Biometricians TABLE XLII. To give values of B;, Bs, Bs and By in terms TABLE XLII (a). Values of Bs. Bs 2:0 25 30 35 4°0 as) 5:0 a5 6°0 6:5 70 0:0 0 0 0 0 0 0-1 | 048493 | 0°68971 | 094286 | 1-25688 | 1°64906 |} 2°14375 0-2 | 0-91585 | 1-32958 | 1°78182 | 2°37049 | 3:°10000}] 4:01176 0°3 | 1-29873 | 1°85270 | 2°53043 | 3°36094 | 4:°38305] 5°65000| 7:2368 0-4 | 1°63902 | 2-34286 | 3-20000 | 4:24478 | 5:52277| 7:09474| 9-0462 0:5 | 1-94118 | 278126 | 3°80000 | 5:03582 | 6:53846| 8-37500 | 10-6364 ,| 0°6 | 2°20909 | 317350 | 433846 | 5°80178 | 7-44706 | _9-51429 | 12-0414 | 15-1585 "| O-7 | 244615 | 3°52441 | 4:82922 | 6-38287 | 8-26202 | 10:53182 | 13-2885 | 16-6624 0°8 | 2°65532 | 3°83820 | 5°25714 | 6°95698 | 899462 | 11:44375 | 14-4000 | 17-9932 0:9 | 2°83917 | 412064 | 5:64828 | 7-47488 | 9-65454 | 12-26250 | 15-3940 | 19-1758 | 23-7791 1°0 | 300000 | 4-36842 | 600000 | 7°94121 | 10:24999 | 13:00000 | 16:2857 | 20-2308 | 25-0000 1‘1 | 313980 | 4°59081 | 6°31618 | 8:36246 | 10°78796 | 18°66538 | 17-0877 | 21°1750 | 26°0857 | 32:0328 1-2 | 326038 | 4°78812 | 660000 | 8°74286 | 11°27443 | 14:26667 | 17°8105 | 22-0225 | 27:0546 | 33-1082 1°3 | 3°36330 | 4°96250 | 6°85454 | 9:08619 | 11°71461 | 14°81071 | 18-4633 | 22-7851 | 27°9217 | 34-0635 1-4 | 3°45000 | 5°11589 | 7-08235 | 9°39582 | 12°11304 | 15°303845 | 19:0536 | 23-4727 | 28-7000 | 34:9164 | 42-3613 1°5 | 3°52174 | 5-25000 | 7-28571 | 9°67501 | 12°47368 | 15°75000 | 19°5864 | 24-0937 | 29-4000 | 35-6786 | 43-1538 | ! | TABLE XLII (0). Values of Bs. B, 2:0 25 80 35 40 WS 5:0 55 6:0 65 70 0:0 | 5:00000 | 8:92856 | 15-0000 | 23-7288 | 31-0000 O:1 | 5°27356 | 9°41054 | 15°7973 | 25-7430 | 41°7660 | 69°3682 0°2 | 544361 | 9°75086 | 16-2648 | 26°4018 | 42:5000 | 69°4796 0°3 | 5°53293 | 9°86224 | 16°4907 | 26-6520 | 42°5613 | 68°5776 | 114°4732 0-4 | 5°55998 | 9:91072 | 16°5385 | 26°6144 | 42°1807 | 67:0888 | 109-4534 0:5 | 5°53802 | 9:87751 | 16-4545 | 26-3742 | 41-5076 | 65-2679 | 104-4652 0:6 | 5-47791 | 9°81734 | 16-2732 | 26-1077 | 40-6453 | 63-2707 | 99-6442 | 162-125 Bs) 0-7 | 5-38824 | 9-63895 | 16-0200 | 25-5026 | 39-6622 | 61-1946 | 95-0595 | 151-253 0:8 | 5:27513 | 9°45991 | 15-7148 | 24-9478 | 38-6061 | 59°1016 | 90-7143 | 141-707 0°9 | 5°14437 | 9°25645 | 15-3706 | 24-3462 | 37-5099 | 57-0279 | 86-6331 | 133-240 | 210-995 1-0 | 5-00000 | 902746 | 15-0000 | 23-7495 | 36-3971 | 55-0000 | 82-8022 | 125-664 | 195-C00 1:1 | 4:84537 | 8°78075 | 14-6111 | 23-0744 | 35-2835 | 53-0316 | 79-2091 | 118-839 | 181-299 | 286-374 1:2 | 4°68319 | 8°53522 | 14-2105 | 22-4107 | 34:1811 | 51°1309 | 75-8392 | 112-653 | 169-394 | 261-436 1:3 | 4°51562 | 8-27700 | 13-8032 | 21-7535 | 33-0971 | 49-3447 | 72-6772 | 107-016 | 158-930 | 240-845 1-4, | 4°34440 | 8-01454 | 13-3931 | 21-1002 | 32-0367 | 47-5471 | 69-7076 | 101-850 | 149°643 | 223-304 | 343-147 1°5 | 4:17097 | 7°75000 | 12-9832 | 20:4546 | 31:0037 | 45:90388 |} 66°9117| 97°112 | 141°333 | 208°129 | 313-704 Probable Errors of Frequency Constants 79 of B, and B, on the assumption that the Frequency falls into one or other of Pearson's Types. TABLE XLII (c). Values of Bs. Bs l 2:0 25 30 35 40 45 5:0 55 6°0 6°5 v0 0-0 0 0 10) 0 0 0-1 | 1:99086 | 4:39480 | 9°3207 | 19°9714 | 45-9387 | 128°529 0:2 | 359438 | 8:03374 | 16°5960 | 34°7825 | 76-6000 | 193-361 0°3 | 4°86677 | 10°68765 | 22-2196 | 45°7142 | 97-2263 | 228-104 | 668:284 0-4 | 5°85929 | 12°85477 | 26°5187 | 53°7090 | 111°0237 | 246-506 | 650°398 0°5 | 6°51704 | 14°51540 | 29-7545 | 59-4655 | 120°0543 | 255295 | 614°633 0°6 | 717383 | 15°7892 | 32°1362 | 63°9266 | 125-6629 | 258-147 | 581-205 | 1618°635 B,| 0-7 | 7-56616 | 16-6546 | 33-8306 | 66-2045 | 128°8283 | 257-225 | 550-107 | 1368-373 0°8 | 7°81963 | 17-2668 | 34:9714 | 67-8804 | 130-2010 | 253-872 | 521-257 | 1196°612 0°9 | 795777 | 17°6667 | 85-6658 | 68-7533 | 130°2587 | 248-937 | 495°375 | 1068°877 | 2769°42 | 1°0 | 800000 | 17°8291 | 36:0000 | 69:0644 | 129°3434 | 243-000 | 469°637 | 968°318 | 2280-00 1‘1 | 7°96281 | 17°8472 | 36:0487 | 68°7730 | 127°7158 | 236°441 | 446°547 | 886-541 | 1945-69 | 5313°80 1-2 | 7°86015 | 17°7503 | 85°8535 | 68°1357 | 125-5684 | 229-524 | 425-062 | 818-040 | 1700-98 | 4135-56 1°38 | 7°70375 | 17°5396 | 35°4754 | 67-2181 | 123-0362 | 222°562 | 405°663 | 759°486 | 1512-94 | 3388-18 1-4, | 750358 | 17°2423 | 34-9467 | 66-0678 | 120°5142 | 214°828 | 386°347 | 708-620 | 1863-20 | 2870-08 | 7265-31 15 | 7-26808 | 16°8768 | 34°2983 | 64°7210 | 117°2460 | 207-227 | 368°843 | 663°926 | 1240-65 | 2488-62 | 5719-68 TABLE XLII (d). Values of Bs. Bs | 20 2S 30 BS 40 45 5°0 55 60 65 7-0 0°0 | 14:0000 | 39°0649 | 105°000 | 290°678 | 868°015 O'1 | 16°4616 | 45°7741 | 124-835 | 855°508 | 1243°832 | 10228°33 0:2 | 17°7296 | 50°2472 | 132°998 | 369°894 | 1190-700 | 6204-69 0°83 | 18°1764 | 51°0927 | 134215 | 361-909 | 1089-739 | 4485-38 | 107697°95 0-4 | 18°0667 | 50°2458 | 131°337 | 344°886 | 977°506 | 3471°87 | 25413718 | 0-5 17°5474 | 48°7896 | 126-107 | 323°447 | 877°884| 2792°19| 13737°63 pes 16°8560 | 46°8558 | 119°601 | 3803°252 | 784:431 | 2303°07 9048°43 | 119230°33 0-7 | 15°9787 | 44°3106 | 112-492 | 277°658 | 701°500| 1934-79 6534°78 | 40994°77 0°8 | 15:0148 | 41°7081 | 105-200 | 255°716 | 628°450| 1648°52 5045°80 | 22660°09 0-9 | 14-0113 | 39:0906 | 97:984 | 235°072 | 564-277] 1420°51 4024°45 | 1483690 | 137288°7 1:0 | 13:0000 | 36°4119 | 91-000 | 216°137 | 507°894 |} 1235°50 3286°65 | 10612°25| 57584°9 1*1 | 12-0030 | 33°7916 | 84°339 | 198°263 | 456°575 | 1083-04 2741°39 8135°91 | 33078°5 | 797653-2 1-2 | 11°0354 | 31°3418 | 78°047 | 181°987 | 414°455 955°78 2322°13 6314°06 | 21891°8 | 155693°9 1°3 | 10°1070 | 28°9775 | 72°146 | 167°142 | 375°834 84897 199405 5108°55 | 15690°2 | 75009°7 1*4| 9°2240 | 26°7355 | 66°637 | 153°582 | 342°057 755°79 172618 4219°50 | 11846°9| 44891-9 | 565740 1°5| 83899 | 24°6268 | 61°512 | 141°477 | 310-976 676°32 1508-92 3544°82 9281°2 | 30280°3 | 180793 | 80 Tables for Statisticians and Biometricians TABLE XLIII. Probable Error of Criterion . Values of VN S,, for values of Bi, Be B; 05 10 15 “20 25 30 35 BO RATE WBN SS DBASE SBN SSMWVAAK SBN SG GHVBAACKBBHS Aaa De TL OY DT GL Gs Be Se SS Hs ty ty ty ty ty Oe Ge Og Oo Oo Oo Co Co Co Co % WW HW WWW WIS DAAAAD A HAW WWH RAH SOa™ E 1:04 1°30 1°35 1°34 1°33 1°30 1:27 1°83 1:97 1:98 1°93 1°84 1-74 164 351 3°71 3°42 3°00 2°66 2°42 2°22, 18°8 9°89 6°94 5°30 4°30 3°62 3°10 — |62°0 20°5 8°47 6°36 5°22 4°42 7°82 |91°7 — 49°7 20°2 12°2 8°48 2°99 | 11-7 70°2 —- 142 32°4 15°4 1°82 4°80 | 13°8 55°5 —_— 344 182 1°43 2°89 6°43 | 15°8 50°6 380 -- aleily/ 2°18 4:08 8:00 | 17-2 46°5 215 1°04 is) 3°08 5°18 9°36 | 24°6 44°3 ‘979 | 1°54 2°54 4:09 6°33 | 14°3 26°2 920} 1:41 2°20 3°32 4°91 8°84 | 1371 869 | 1°35 2-00 2°83 4:08 5:98 9°08 — 1°30 1°94 2°60 3°61 5-08 7°25 = 1°33 1:94 2°58 3°43 4°54 6-09 — 1-44 2°01 2°63 3°37 4:20 5°41 — 1°58 2°15 2°74 3°39 4:20 5°18 — 1°81 2°32 2°94 3°59 4°36 5°29 — — — _ —_ 4:90 5°63 —_— — _ — — 5°87 6°46 = _ — _ — 7°43 761 — — — — — 10°1 9°45 — — — —_ — 15'1 11°3 Probable Errors of Frequency Constants TABLE XLIII—(continued), Probable Error of Criterion ky. 81 Values of VN%,, for values of Bi, Ba By ‘75 | -80 | ‘85 | -90 | -95 | 1:00 | 1-05 | 1-10 | 1-15 | 1-20 | 1-25 | 1-80 | 1-85 | 1-40 | 1-45 | 1-50 949] 1:00 | 1:06] 1°12] 1°18] 1:25) 1°31] 1:39] 1:46] 1:56] 1-64] 1-75] 1°86] 1:97] 2-11] 2-25 -937| -992| 1:05} 1°10] 1:16] 1:22] 1:28] 1°35] 1:42] 1:49] 1:58] 1-67] 1:77] 1:88] 2-00| 2-12 ‘936| -987| 1:04] 1:09] 1°14] 119] 1:25] 1:31] 1°37] 1-43] 1°51] 1:59] 1-69] 1°79] 1:89] 1-99 ‘929| -982| 1:04] 1:08] 1712] 1:17] 1:22] 1:27] 1:32] 1-38] 1:45] 1-53] 1:62] 1-70] 1:79] 1:87 950] -990| 1:03] 1:07] 1°11] 1°15| 1°19] 1:24] 1:29] 1:34] 1-40] 1-47] 1:55] 1-62] 1:69| 1-77 972] -998| 1:03] 1:07] 1:10] 1:14] 1:18] 1:22] 1:27] 1:32] 1:37] 1-43] 1:49] 1:°55| 1-61] 1-68 1:01 | 1:03 | 1:06] 1:09] 1-12] 1°15) 1°18} 1:21] 1-26] 1:31] 1:36] 1:41] 1:46] 1:51] 1-57] 1-63 1:07 | 1:08 | 1:09] 1-11] 1:14] 1:17] 1:19} 1-21] 1-25] 1°30] 1:35] 1:39] 1-43] 1-48] 1-53] 1:59 117 | 115 | 1:16] 1:17] 1:18] 1:19] 1:21] 1:23] 1:27] 1:31] 1:35] 1-39] 1-43] 1-47| 1°52] 1-58 1:33 | 128 | 1-26] 1-26] 1-25] 1:25] 1:25] 1:27] 1:99] 1:32] 1:35] 1-39] 1-43] 1-47] 1:52] 1:57 153 | 1-47 | 1:42] 1°38] 1:36] 1:34] 1:34} 1:34] 1-35] 1:36] 1-38] 1-40] 1:43] 1-46] 1-51| 1-56 185 | 1:72 | 1°62} 1°56] 1:52] 1-49] 1°46] 1:44] 1:43] 1:43] 1:43] 1:44] 1:45] 1-47] 1:61] 1:57 234 | 2-13 | 1:96] 1°83] 1-75] 1:68] 1°62] 1°57] 1°54] 1°52] 1°51| 1:50] 1:50] 1:51] 1:54] 1:58 3:03 | 269 | 2-44] 2:23] 2:06] 1:94] 1-85] 1°77] 1°72] 1-67] 1°63] 1-60] 1°59] 1:58] 1°59] 1-61 4:08 | 350 | 3-06] 2:74] 2-49] 2:29] 2:14] 2-02] 1-92] 1°84] 1-78] 1°73] 1:70] 1-68] 1:68] 1-69 572 | 4°75 | 4:12] 3:56] 3°15| 2°83] 2°59] 2:38] 2-22] 2-09] 1-99] 1-91] 1:85] 1:81] 1-79] 1-7$ 8:85 | 690 | 5-71] 4°79] 4:15| 3:68] 3:22] 2:90] 2-67] 2-49] 2-31] 2:20] 2:09] 2-00] 1-96] 1-92 142 |10-4 | 8-22] 6°69] 5-70] 4:81] 4:17] 3-67] 3:31] 3:05] 2-81] 2-62] 2-45] 2-29] 2-19] 2-10 249 |181 |13°8 |1071 | 8-16] 6-71] 5°76] 4:92] 4-28] 3:84] 3-48] 3-16] 2-91] 2-68] 2:50] 2°36 655 |365 |21-7 |15°8 {12-2 |10°0 | 817] 6°75] 5:75] 5:00] 4:43] 3-93] 3:52] 3:18] 2-91] 2-72 242 (866 | 485 | 30-0 | 20-9 | 15-4 |12:0 | 9-61] 7:86] 6:60] 5:67] 4:90] 4:34] 3-88] 3-47] 3-19 — |374 |127 |62°9 | 38:3 | 26-0 |18-3 |14-4 ]11°5 | 9:15] 7:53] 6-45] 5-66] 4:98] 4-38] 3-80 — | — } — |200 |91:0 |51°6 | 32-7 | 93-4 |17-4 |18:4 |10°6 | 8-70] 7-50] 6:48] 5°52] 4-61 144 |478 — | — | 314 |112 | 70:0 | 42:9 | 29-1 | 21-3 {16:1 | 19:7 | 10:3 | 8-55) 7-24] 6:25 628 |135 — } — | — |580 |192 | 93:9 |55°8 |37-1 |265 |19-8 | 15-4 |12-1 | 9-74] 8-96 42:8 1683 |119 |280 | — | — | — | 286 |196 | 72:8 |46-4 | 32-2 | 23-9 |18°3 | 14-6 | 11°8 323 |44-7 |62°7 |99°6 |240 |742 | — | — | — (181 |91-0 | 58-9 | 41-0 | 30-0 | 22-3 | 16:9 962 33 | 4671 |68:0 |105 |240 |532 | — | — | — |260 |198 | 76-7 | 50-4 | 36:0 | 266 216 |271 |35-°0 | 47°3 |66:8 }104 |182 |413 | — | — | — |403 |172 |99-3 | 63-0 | 44:8 189 |226 |26°9 |33-7 | 44:0 |61°5 |84-2 ]115 | 337 = |=] | =] P= |e |g soe Ts |20'%% 24:6 |30-1 ||37:8 | 48-9 |66:0 |97-0 1157 |286 | — | — | — | — | — |172 172 |200 |23:0 |27-1 | 32:5 | 40-6 |51°5 |69°2 |99°8 |147 |253 | 559 = fia | = | = 173 |193 |21°7 |24:8 |29°5 |35-4 |43-1 154-7 |70°6 | 94-6 |138 | 216 oh) Fee ee 180 |19°7 |21:4 |23°9 |27-1 |31°5 |37-:0 | 44:2 |54°5 |69-°3 |93-2 |132 |205 | 380 | 196 |205 |21°6 |23°3 |95°6 | 28-6 |32°8 |38-0 | 44:8 |54:0 |68-6 |94:0 |130 |185 | — | — 223 |220 |92:5 |23°6 |25-2 |297-5 |30°5 |34:3 |39-4 | 45-4 154-7 |67°3 |86:5 ]116 |169 |275 — | — | — |24:5 | 95-8 |27-6 |29°8 |32°2 |35°6 1396 |44:8 | 51-4 |63-2 |83-0 |118 | 168 — | — | — |26:8 | 27-8 | 28-8 |30°0 |31°5 | 33:8 | 36-5 |39°8 | 44-4 |53-1 |67-0 | 85-2 |116 — | — | — |30-7 | 30:8 |31-0 |31°3 | 32-0 | 33-4 | 35-3 |38-2 | 49°2 | 48-4 |56-6 | 69:9 | 87-9 — | — | — |38-4 | 36-0 |34-4 |33-°5 |33-1 | 34-1 |36°0 |38-7 | 42-2 |47-:0 |53-1 | 61-9 | 74:8 — | — | — {50-4 | 42:5 |38°3 |36°8 | 36-4 | 36-5 | 37-9 | 40-0 | 43-0 | 46-6 | 51:3 | 57-8 | 66-2 a ny ee [pe | r-Om 4101 | 41-6) /(42°7 | 4406) || 47-2. 1150-3) 55-0) | 615 re ne en ele 0 (48-8 147-0) 11 46-1) | 46:0) ||46-7, 148-08 |/50°0 | 53:5) |/58-5 as ee ee enn aeseii54-Ol i 5icG) 49°91) 40-1 49-40 50-0) |'52-5, 5678 ee een 79-40 66-6) 158760 54-28 51-8 | 51-22 1/5078" 52-5) 55-5 es ee le = 988 84-0 66:9) (59-9) /55°9 53-9) (53-1 |/53°6 | 55-4 || 5 Bee ee ee : | Sr. aro) | eer =|=]/—])=]{]] =] fj y=] = = |j— | =] =] | = Ge |Gise jibe we | a A ie ee ree ee ee =|) BRS GaAs |. Gass — |) Se PS a ae ea re em ey ome Wh i oy aa le cela a | ee | eae | ea | ee aes a Se |) Se te Fg4' |96:0: 82:0 B. 11 AHAB BWBWHS ROU S te Ar~w~AatX~C S FIPIWAPSPPSAAAAAAATAAILS SHE EHH HLHH GH SHH HSH OHVPBHYY SSMBAUNK WRENS SHWBAAKR SBN SSHBWAGK WH: Bs Tables for Statisticians and Biometricians 82 TABLE XLIV. To find probable Frequency Type. Values of 1:77V NS, for given values of Bi, Bs (Semi-Minor Awis of Probability Ellipse). © HIDOMORDOAAVHAHOMORMADHOOD AHH SHOP A SHAH OO 09 hi * COD OCOC OCR RBA AA AAA SAGA GA GI 60 0 90 HH Hid 1 OI ODT = IDO OR SO AAG His Ol DO =O EDAD 0 OO NCO O txt | | | DODD CFF FAA BPR SRA AAA AA AA 0 09 0 HHId 1D OM OD BOMDASCSOAABHSOMASCABPOrANGOANOAM VOM OG agi | | | | [es Ml oes ol DODD OCF BAR AAA RA AAA AA GOD 09 09 HH HID OOO Or ODADAAABAHOMODAANHODOMOAPHNOAAAP orto | | | | RS Cal EN la ate Gp SC Cela IG SIC Cr Sills vate so ee a WT | | 6 | -s5| -6 65 6 OOM SSI GIGI GI OD 69 09 60 H HID 19 CO I~ CO SDROOHAAHDA HOM DSOADOEOPORMmNOOSAHA™ | 1) | 1 | | | | SS an Suey ee Sun aaa ae =a | | | | | | DOHODAANHDOHHOMAOAP GOH OPOOHOMD PANO | | COON FAA AAA SARA AAA A GI GIOD 09 09 Hom Hid 19 cI 1 | FF ODROSAAPDH POM DAANDHODAWOAHOONAOM | | | | | | | oy DPPOOAAPGHSPFGOMAROAD OH SOHOHHOGD | | | | | | | al DADOSOMAAGHVHAHOOAAAHOAANOONIN) | | | | COCR FAA AAA PRA AA AAAI A GOD 09 00 H Hd ‘9 Pee) CD CN SUC EGS Ge) pal Ree) Celta Ta GO ne) EOP rae oP Py COCO KF FARA RP RRR AP ANAAN AO OO HHH I | | | SISO SISOS OL OlO 1 Oa rata ae > | SHRM SPI OGK HASH RMD HH OGOHR HASH RM SH OSH HASARHD SHOGOHOWASARHDB AH OH VRS a. 83 Probable Errors of Frequency Types TABLE XLIV—(continued). Values of 177 VN 3, for gwen values of B,, B2 (Semi-Minor Axis a co} See eee eee ee EEE EEE EEELELEEe ert. RRR RRR Sr ebanel ies estes eriee Ss Be Ess Gol ak ae oy En oe hee Rea EIR NSIS DSRS tea cs ese Cer Darras eS SB Be raed eco ta ETS ETIONTES GH CLGs a COR ACSW ig Se 12 Ay 556 S65 SOOM MAAMATAGOSAAHODSSROOAA 11 | 1:15 E |: | E 1 5 | of Probability Ellipse). | 1:0 | 1-0 95 E LE DION OHPROAAHHOOSOAONMGHANGHAGPHOONS FPO ean COD COO SF RR RR MRM AAANAAANAMMMHHWINOSrOOM 2 a 11— 84 Tables for Statisticians and Biometricians TABLE XLV. To find probable Frequency Type. Values of 1°77 VN >, for values of B:, Bz, (Semt-Major Axis of Probability Ellipse). B, 0 | -05| -2 |-15| -2 | -25| «8 | 35] 4 | -45| 5 | -55| -6 | 65) -7 | -75 2-0 |1-6| 1:9 | 2-2 | 2:5 | 2:8] 3:1} 3-4] 3-7 | 4:1 | 4:4 | 4-7 | 5-0] 5:4] 5-7 | 6-1] 6-4 9-1 |1°7| 2:1 | 2-4 | 2-7 | 2°9| 3:2|3°5 | 3-8] 4-1) 4:4] 4:6| 4:9] 5-3 | 5-6| 6-0] 6-3 9-9 |1-9| 9:31 9-6 |2-9| 3-1] 3-4] 3-7 | 3-9| 4:1 | 4:4 | 4:6 | 4-9 | 5-2 | 5-5 | 5-9] 6-2 9-3 | 9-1| 2:5 | 2:8 |3:1|3°3| 3:5 | 3S | 4:0 | 4:2] 4:4] 4-6 | 4-9 | 5-2] 5-5 | 5°8| 6-1 2-4 | 2-4| 28/31 /3:3|35|3-7|3-9[ 4:1] 4:3] 4:5 | 4-7] 49] 5-1 | 5-4] 5-7 | 6-0 2-F |9-8| 3313-5 |3-7|3°8| 4:0 | 4:11 4:3 | 4:5] 4:7 | 4:8 | 5-0 | 5-2 | 5-4 | 5-7 | 5-9 2-6 | 33] 3:8| 4-0 |4-2| 4:3] 4:5 | 4-6 | 4:7 | 4:8 | 4:9 | 5-0 | 5-1] 5°3 | 5-5 | 5-7 | 6-0 2-7 |3-8| 4:41 4:61 4-7| 49] 5-0] 5:1] 5-2|5:3| 5:4 | 5-4 | 5-41.55 | 5-6 | 5-8 | 6-1 9-8 | 4-4] 5-0| 5:3 |5:4|5°6| 5-7] 5-8] 5:9 | 5-9| 5-9 | 58/58] 58 | 5916-0] 6-2 2-9 |\51| 5-7| 6-0 | 6-2 | 63| 6-4 | 6-5 | 6G | 6:5 | 6-5 | 6-4 | 6-3 | 6-3 | 6-3 | 6-4 | 65 3-0 |5-8| 6-5 | 6-9 | 7-1 | 7:2| 7:3 | 7-3] 7-3 | 7-2 | 7-1 | 7-0 | 6-9 | 68 | 6°8 | 6-8 | 6-8 3-1 |67| 7°51 8-0 18-3 | 8-4| 8-4 | 8-4 | 8-3 | 8-2| 8-0 | 7-9] 7-8] 7-7 | 7-6 | 7-5 | 7-4 3-2 |7-°8| 8-7 | 9:2 19-6 | 9-8] 9°8 | 9-7 | 9°5 | 9:4 | 9:2 | 9-0 | 8-8 | 8-6 | 8-4 | 8:3 | 8-1 3-319-2| 10| 11]11] 11] 11 | 11| 11] 11] 11] 10] 10]9°6| 9-4] 9-2] 8-9 3-/\11| 11 | 12/12] 13] 13] 13| 12] 12] 12] 11] 11] 11] 11) 10 35113|13| 14/14] 15] 15 | 14] 14] 14] 14] 13] 12] 12 12| 11 36\|15|16| 17\17|17| 17 | 16] 16] 16| 16] 15] 14] 14] 14; 13 3-718] 19 | 20/20 | 20| 20} 19] 19| 18| 18] 17] 17] 16 16| 15 3-8 | 91| 23 | 24| 24 | 24] 93 | 92] 22] 21] 21] 20] 19] 18| 18! 17 3-9 | 25 | 27| 28] 298 | 28] 97} 26| 25| 24) 24] 23| 22) 21] 20} 19 40 | 29| 32 | 34| 33 | 32] 31 | 30] 29] 28| 27] 26] 24) 23] 22) 21 41 | —| —| 43] 41 | 39| 37 | 36] 34] 32] 31| 30] 28) 26] 25) 24 42 | —| —| 55| 51 | 48| 45 | 42] 40] 38] 36] 34] 32| 30| 29) 27 P| 4-3 | —| —| 69| 62 | 57] 53 | 50] 48] 45| 43] 40) 37) 35] 33) 31 44 | —| —| 87| 76 | 69| 64| 60| 56] 53] 50] 46| 43] 40] 38] 36 45 | — 113] 98 | 86] 78 | 71| 66| 62| 57] 53| 49] 46] 43] 41 Gul | re | he 279) eS: eval Me ts | aces rs) a 7 | —| —| —|—|-——| —|110] 99] 90] 80] 72] 67] 62] 58] 55 48 | —| —|—|—|—]| —]140|124]110] 97] 86] 79] 73] 68] 64 9 | —| —| — | — | —| — | 200] 160) 135119] 105] 95) 86) 79] 74 5-0 | |) 2) SS) SS pe aaa ipa) 53) 130) |iui5)| 102) |or Fi ae) a i he a no to P| eh 134 | 120 ie || cae eg | | ea |e a | a a es Pe Gl (nea Te RE 9 Se Ss ee 55 |) 1) == —— | | a 0 297 | 230 Feros | pie | 1 SE SS ee ee ee Pe | Fe Po | | a |e | Ns pe fee |) ee a Se ee Ss FR a SS PS SSS | St SS a TL |e | (| ees S| Pee ee Nh a ee ee oy ee ee || ee | Se Pe Se PS eae | SS 7355 Flips pte Wes NY oe |e Pe Pe a) ee ae |) |W Se Sy SY SS Sd a ss —— (RON cel |) ee ee ee SS Se SY eS (59a fan | pe = SS eS | Sp SS Fi pa ean Uae (esi | mea ra | Fa en ee I ae Nie) Se My | VS Se See | SS SS | 67 Noone oe NSN aa iat || AS | GW, | a 6733 | ae ee RE a | | Pa Np ee | | Sh eee Mee | Be FEE (aaa pt (EN Ns ee ee Se eee |S 70 ae ww | | | | | | ODAAADARBABAARAAGD KRISH SHAR www wp ar~T08 11 136 167 [oat c WO MAWARAAAWAAAATA] eee tee See Re Eater ere ac aeet aerad aed rae tay ate ONO HOW NOSWOTARAAATHON 30 ot & w wmaocn IDS aawo 90 104 123 147 Probable Errors of Frequency Types Ro) ODATNWABHAADAHATNT-I1 DORNHOMK FOWRFDADDDDOMrNHKD bo Oo COTIRADTE &wWWwPH bd to oOnwmnmMmwoaoamw-10 0 Ob 6 a CO OAT ~T AT SI DT TT I I DOWDENSOSOHEWRABDS 11 TABLE XLV—(continued). Values of L'77VN S, for values of B,, By (Semi-Major Axis of Probability Ellipse). ™ S CO 0 COTATI TTT TATA AIO AHHWODAWYNNHNWKARHDOHMW 10 my S ou © OMIT ~T~T 1-1 HD OOH KOWOADAKRAADIDODOM wo H-~T 10 A 1:1 |1:15| 1:2 |1:25| 1°3 \1:35) 1-4 11-45) 1°5 DT FOES | P99) | LOE SL Uy |e | S992) OG LON Lh ea La Le 2 8G |9°0|9°4}9°8} 10} 10} 11) 11] 12 8°5 |8°8|9°2|96)9'9} 10} 11) 11] 12 8°3 | 8-7) 9:0}9°4)9°8; 10} 10) 11] 11 8-2 |8°5|8:9)92)96) 10) 10) 11} li 8-0 | 8°3| 8'7|9:0|9:'4)9°9) 10) 11} 11 79 |8°2|8:5|)88|9:2)9-'7| 10) 10} 11 7-7 | 8:0 | 8°3 | 8°7| 9:1} 9:5) 9:8} 10} 10 76 | 7°9|82|86|89|9-3]9°6] 10) 10 7'7 | 7°9 | 8:2 | 8:5 | 8°8 | 9:1} 9-4) 9°7) 10 7'8 | 8:0 | 8:2 | 8°5 | 8°7| 9:0} 9°3| 9°6| 99 8°1 | 8:2 | 8:3 | 8°5 | 8°7 | 8:°9 | 9:2 | 95 | 98 8°3 | 8:4] 8:5 | 8°7| 8:9 | 9:1) 9°3 | 9°5 | 9°7 8°8 | 8°8 | 8°9 | 9:0] 9:1 | 9:2} 9:4) 9:5 | 97 9:4 | 9°3 | 9°3 | 9°3 | 9°3 | 9:°4] 9°5|9°6| 98 10| 10] 9:9 | 9:9 | 9:8 | 9°7| 9-7 | 9:8 | 9:9 LS SLE SO). LOLOL} 10) sro 12) 12) 12 11} 11) 10} 10) 10) 11 14) 13) 13) 12) 12) Wey 2a) LL) at 15| 14] 14] 18] 13] 12] 12) 12) 12 17| 16] 15] 15} 14] 14] 13} 13] 18 18| 18) 17} 16) 15] 15} 14) 14) 14 20} 20] 19] 18] 17] 16) 15) 15] 15 BR CPM CAC Choy aie yaa) ala all) 3h 26] 25] 24] 23] 22) 21) 20) 19) 18 29| 28] 27| 25| 24) 23) 22) 21) 20 83| 32] 30) 28} 27) 26) 24) 23) 22 38| 36] 34| 32] 31] 29) 27} 26) 25 42| 40] 38] 36] 35] 33] 31} 29) 28 49| 46] 43| 41] 39] 37] 35] 33) 32 55| 52| 49| 46] 43] 41] 39] 37] 36 63| 58} 55) 52} 49| 46) 43} 41) 40 71| 66] 62! 58] 55] 51] 48) 46) 44 80| 75] 70] 65] 61) 57) 54) 51) 49 92| 85} 79| 74] 69] 64] 60) 57) 54 105; 96] 89| 83] 78| 73] 68| 64) 60 120} 110}102| 95] 88} 82] 76) 72) 67 136 | 126|116|108]|100] 93] 87) 81} 75 168 150] 136|125]|115}107] 99} 92] 85 200; 178 | 161 | 147 | 134] 123] 113) 104) 98 264 | 215} 190 | 171] 157 | 144) 130 | 120) 112 345 | 268 | 230 , 207 | 184| 167 | 150 | 138 | 127 480 | 364 | 294 | 250 | 215 | 194] 174] 159 | 144 680 | 477 | 370 | 299 | 252 | 224 | 201 | 181 | 165 1047 | G80 | 456 | 368 | 312 | 268 | 237 | 212} 191 — | — | — | — | — | — | 280| 248 | 223 |e —ilisoe)|2au 206 — | —]}] — | — | — | — | 412 | 362 | 320 — | — | — |] — | — | — | 525 | 446} 390 — | —} — | — } — | — | 809) 584) 491 85 86 Tables for Statisticians and Biometricians TABLE XLVI. Angle between Major-Awis and Axis of B, (Probability Ellipse) measured in degrees. To find probable Frequency T'ype. B, 0 |-05| 2 | -15| -2| +25) -8 | -35| 4 | 45). -5 | 55] 6 | 65) -7 | -75 2-0 | 0 | 12 | 23 | 28 | 31 | 33 | 35 | 86 | 37 | 38 | 39 | 40 | 41 | 41 | 42 | 42 2-1 | 0 | 11 | 21 | 25 | 28| 30 | 32 | 34 | 35 | 37 | 38 | 39 | 40 | 40 | 41 | 41 2-2 | 0 | 10 | 19 | 93 | 26 | 28 | 30 | 32 | 33 | 35 | 36 | 38 | 39 | 39 | 40 | 40 2:3 | 0 | 10 | 18 | 22 | 25 | 27 | 28 | 30 | 32 | 34 | 35 | 37 | 38 | 38 | 39 | 39 24| 0 | 9|17| 20 | 23 | 25 | 26 | 29 | 31 | 33 | 34 | 35 | 36 | 37 | 38 | 38 a5 | 01] 8|15/ 18] 21 | 23] 25 | 27) 29 | 31 | 38 | 34 | 35 | 36 | 37 | 37 26|0 | 7] 14/17] 20| 22 | 24 | 26 | 28 | 30} 31 | 33 | 34 | 35 | 35 | 36 27 | 0 | 7/131) 16] 19! 21 | 23 | 25 | 26 | 28 | 29 | 31 | 32 | 33 | 34 | 35 2-3 | 0 | 6 | 12) 15117] 19 | 21 | 23 | 25 | 27 | 28 | 30 | 31 | 32 | 33 | 33 2-9 | 01 6] 11! 14] 16| 18 | 20 | 22 | 23 | 25 | 26 | 28 | 29 | 30 | 31 | 22 30 | 0 | 5|10/13| 15) 17 | 19 | 21 | 22 | 24 | 25 | 27 | 98 | 29 | 80] 31 31|0 | 5] 9} 12| 141 16 | 18 | 20 | 21 | 23 | 24 | 26 | 27 | 28 | 29 | 30 32|0 | 5| 9|12|14| 16 | 17] 19 | 20 | 21 | 22 | 24 | 25 | 96 | 27 | 28 33|0 | 4} 8!11|13| 15 | 16] 18} 19 | 20] 21 | 22 | 24 | 95 | 26 | a7 34 | 0 | 4] 8j 10] 12] 14] 15/17 | 18] 19 | 20 | 21 | 22 | 93 | 24 | 25 35|\0| 3] 7) 9111/13 | 14] 15 | 17} 18 | 19 | 20 | 21 | 22 | 93 | 24 36 )0 | 3] 6] 8] 10| 12] 13] 14 | 15 | 16 | 18 | 19 | 20 | 21 | 22 | 23 37/0] 3] 5| 7] 9] 11] 12] 13] 14] 15 | 16 | 17] 18 | 19 | 20} 21 383/013) 5] 7] 9] 10] 11] 12] 13| 14] 15 | 16 | 17 | 18 | 19 | 20 39/0] 2] 4] 6| 8] 9/10] 11| 12] 18] 14] 15 | 16 | 17] 18 | 19 40) 0| 2) 4|-6] 7| 8] 9| 10] 11 | 12| 13} 14 | 16 | i6 | 17 | 18 Bet — V8 5a 6.) Sali ee) | 91 || toi al Fa) aa ib tea a Hy ee ey OGY SG) |G Iai |) ios |) St] Te as. |) De ye | 22 S28) 45 a1 252 16: P76) 9) 10) |/=10)\1) al ont S31) es tees Pee ORME ES Sa) re Se) Se Sy oy Wp ai | TN ey) he: pe eee Sa ese ON PL Che el) Gl) Gal) eV) hes) Oe) ie | | ie. FG || = ae =] Ail Bi Bl wl 7h Sil 9] tO | ad || a FS es a Gal Gee || RO ie 57 ee (Ree | an ee (ee ee pe ee ith au) ae) eile il, 10 Og eee fo ce ees SE SA Sie ah tanh Gal) ate TA Ba) D ep) |) = | | SS =| it] 3] Bi] BS) 4 al Bi el vi é Beit || = = |=) Sef") 64 6] 6] 7 || S|) = = All &i Bll % yas) ea |e ees | elf | 5 ie es ees |) By) All Bi Sh = = SE) ee eee) a ee ea Aalie Gxt || ee ie |) as | NSS eS pS te Se RH BB] 2 5:6) lees | ee ea = = | 57a | SS eae SS ee = = Ee a seme |e 11 NY ae a es a ee ee | a) ee | ee ee Teisomn| (eee | me | Ree ee ee | ees) es (ee BOs | aA ee SS a a ee eee ee | | ee |e G0 NN Se Se eS SS eS BB |e Ha aS a a | ee |e | ee ee Gee ee NE EEE fl SS ga ee ee | ee ieee (eee |e ee | eee | (Sel || |) |) es | Ss ef a fe SS | Sf SS eS SH | SY] = GBP eS eS Ie ia | ee i | ee Gis Wt ih a a | SN ee | em fee ce (ee CB Pay ee Ne | eft I Wee NN SP ee ee ee BB a ee lh a eer en || ee ee BES ee | Ce He A ee ee ee ee ee is es Ht) ey a eee eee | a et ee Ne | ee Probable Errors of Frequency Types TABLE XLVI—(continued). Angle between Major-Awis and Axis of B, (Probability Ellipse) measured in degrees. S He oe ee rFwmowe rs fo) 39 38 wo ce an bo Go we Co We we aornwneo bo bo SH OD 1 bo bo bt me bw 20 = No) 16 ee an oi bow ie or — ra | | | rwcmami®DEd 20 By tow Ora oO © 35 33 32 38 36 35 30 BNweEAADA 100 VIODABABAMAAAAANAAAAAAN AH HHH HEE & Co bo Oe Ge Co GH HH So WW WWHWWUWUWBWY SHDMBWAKKUBWKH SS HWAUHEK BBN SBVHWAAK SPH SGVHVNAKAE SW DTOBHNVAAE wWMWHS Bs ‘Lelans tri cone ans and B ecu Tables for Statist 88 Type of Frequency. wen the probability of a g mining Diagram XLVIT detei oT 4 = +14 25 1:2 43 TA 15 HETEROTYPIC Probable Occurrences in Second Small Samples 89 TABLE XLVIII. Percentage Frequency of Successes in a Second Sample after drawing “p” Successes in a First Sample “n”. “mM Successes n=6 m=5 Tes MRS R38 Nl ll SE 3 s ll fer) Je" S? NAAM SSNS AMHwSWBKD ALSSKS AMKwSWLS MTs ls MRS n=8)| m= 6, DAL Ss *kh DS Il ~10 Ra} NAS CS MWRS ” p=0 583333 26-5151 10-6060 3°5354 "8838 1263 53°8462 26°9231 12°2378 4°8951 1°6317 “4079 0582 61°5385 25°6410 9°3240 2°7972 6216 ‘0777 5771429 26°3736 109890 3°9960 11988 2664 0333 53°3333 26 °6667 12°3077 5°1282 18648 5594 1243 0155 64°2857 24°7253 8°2418 2°2478 *4495 0499 60-0000 25°7148 9°8901 3°2967 “8991 “1798 0200 56°2500 26°2500 11-2500 4°3269 174423 3934 0787 “0087 jo=il 31°8182 31°8182 21°2121 10°6060 3°7879 “7576 26°9231 29-3706 22-0280 13°0536 6°1189 2°0979 ‘4079 35°8974 32°6340 19°5804 8°7024 2°7195 4662 30°7692 30°7692 209790 1171888 4°6620 1'3986 2331 26 °6667 28°7179 21°5385 13°0536 6°5268 2°6107 “7615 1243 395604 32°9670 17-9820 771928 19980 2997 34°2857 31°6484 19°7802 9°5904 3°5964 9590 1399 30°0000 30-0000 20°7692 11°5385 5°2448 1°8881 "4895 0699 p=2 15-9091 26°5151 26°5151 189394 9°4697 2°6515 12°2378 22°0280 24°4755 20°3962 13°1119 6°1189 1°6317 19°5804 293706 26°1072 16°3170 6°9930 1°6317 15°3846 25°1748 25°1748 18°6480 10°4895 4°1958 "9324 12°3077 21°5385 23°4965 195804 13°0536 6°8531 2°6107 D594 23-0769 31°4685 25°1748 13°9860 5°2448 1°0489 18°4615 27-6923 25°1748 16°7832 8°3916 2°9370 5594 15-0000 24-2308 24-2308 18°3566 11-0140 5°1399 1°7132 “3147 p=3 70707 17°6768 25°2525 25°2525 17-6768 70707 48951 13°0536 20°3963 23°3100 20°3963 13°0536 4°8951 9°7902 21°7560 27°1950 23°3100 13°5975 4°3512 6°9930 16°7832 23°3100 23°3100 17-4825 9°3240 2°7972 5°1282 130536 19°5804 21°7560 19-0365 13°0536 6°5268 18648 12°5874 25°1748 27°9720 20°9790 10°4895 2°7972 9°2308 2071398 25°1748 22°3776 14°6853 6°7133 16783 6°9231 1671538 22-0280 22,0280 17-1329 10°2797 4°4056 10489 p=4 62937 17°4825 26°2238 26°2238 17°4825 6°2937 4°1958 12°5874 20°9790 24°4755 20°9790 12°5874 4°1958 2°8846 9°1783 16°5210 21-4161 21°4161 16°5210 9°1783 2°8846 p=s 90 Tables for Statisticians and Biometricians TABLE XLVIII—(continued). Percentage Frequency of Successes in a Second Sample “m” after drawing “p” Successes in a First Sample “n”. Successes p=0 p=l p=2 p=3 p=4 p=5 n=8) 0 52°9412 26°4706 12°3529 52941 2°0362 m=8{ of 26°4706 28°2353 21°1765 13°0317 67873 2 12°3529 21°1765 228054 19°0045 12°9576 8 52941 13°0317 19°0045 20°7322 18°1407 4 2°0362 6°7873 12°9576 18-1407 20°1563 5 “6787 2°9617 7°2563 12-9000 18-1407 6 "1851 1°0366 3°2250 7°2563 12°9576 vf ‘0370 2633 1:0366 2°9617 6°7873 8 “0041 ‘0370 *1851 “6787 2°0362 n=9 N 0 66°6667 42°8571 26°3736 15°3846 83916 m=5§ 1 23°8095 32°9670 32°9670 27:9720 20°9790 2 7°3260 16°4835 23°9760 27°9720 27:9720 3 1°8315 5:9940 11-9880 18°6480 24°4755 4 *33380 1°4985 3°9960 8°1585 13°9860 o 0333 "1998 “6993 1°8648 471958 n=9| O 62°5000 37°5000 21°4286 11°5385 57692 m=6{ 1 25°0000 32°1429 29°67038 230769 15°7343 2 8:9286 18°5439 94°7253 26-2238 23°6014 3 2°7472 8:2418 14°9850 20°9790 24°4755 4 “6868 2°8097 6°7433 12°2378 18°3566 5 +1249 6743 2°0979 4°8951 9°4405 6 0125 0874 3496 1:0489 2°6224 n=9| 0 58°8235 33°0882 17°6471 8°8235 4:0724 m=7{ 1 25°7353 30°8824 26°4706 19°0045 11°8778 2 10°2941 19°8529 24°4344 23°7557 19°4364 3 3°6765 1071810 16°9683 2175961 22°6759 4 171312 4:2421 9:2554. 15°1172 20°1563 ) "2828 1°3883 3°8873 8:0625 13°6055 6 "0514 3239 1°1518 30234 6°4788 7 “0051 “0411 "1851 *6170 1:6968 n=9| 0 55°5555 29-4118 14°7059 6°8627 2°9412 m=8{ 1 26°14388 29°4118 23°5294 15°6863 9:0498 2 11°4379 20°5882 23°5294 2171161 15°8371 3 4°5752 11°7647 18:0995 21°1161 20°1563 4 1°6340 56561 11°3122 16°7969 20°1563 5 "5027 2°2624 5°7589 10°7500 16°1250 6 °1257 “7199 2°3036 5°3750 10°'0782 7 *0229 1645 "6582 1°9197 4°5249 8 “0023 *0206 "1028 eyihh «issih n=9| 0 52°6316 26°3158 12°3839 54180 2°1672 m=9\ 1 26°3158 27-8638 20°8978 13-0031 6°9659 2 12°3839 20°8978 22°2910 18°5759 12°8602 3 5*4180 13-0081 18°5759 20°0047 1775042 4 2°1672 6°9659 12°8602 17°5042 19°0955 5 “7740 32151 7°5018 12°7303 17°1859 6 *2381 1°2503 3°6372 76382 12°7303 fi "0595 "3897 1°4029 36372 775018 8 ‘0108 ‘0877 *3897 1°2503 3°2150 9 ‘0011 “0108 "0595 ‘2381 “7740 n=5) 0 54°5454 27°2727 12°1212 4°5454 m=d) 1 27°2727 30°3080 22°7273 12°9870 2 ih i) 22°7273 25°9740 21°6450 3 4°5454 12°9870 21°6450 25°9740 4 12987 5°4112 12°9870 22°7273 & "2165 1:2987 4°5454 12°1212 Probable Occurrences in Second Small Samples Successes n=10 m= 5 n=10 m=10 WNWAAK ONS AY CsMnd Let Ss me WOVAAK SONGS DSEBWVANKSDHRS AY ®SWONHS p=0 68°7500 22°9167 6°5476 1°5110 2518 0229 52°3809 26°1905 12°4060 5°5138 2°2704 8514 2838 0811 0187 0031 0003 76°1905 19°0476 4:0100 6683 0786 “0049 61°5384 24°6154 9-2308 3°2107 1:0216 2919 0730 ‘0153 “0025 “0003 “0000 51°6129 25°8065 12°4583 5*7842 2°5707 10876 *4351 1631 0567 0181 0052 0013 0003 0001 “0000 “0000 TABLE XLVIII—(continued), p=1 45°8333 32°7381 15°1099 5:0366 1°1447 1373 26°1905 27°5689 20°6767 12°9736 70949 3°4056 1°4190 *4990 1403 0284 0031 57°1429 30°0752 100251 2°3588 *3685 0295 36°9231 30°7692 18-0602 8°7565 3°6485 1°3135 4032 1024 0203 0028 0002 25 °8065 26°6963 20°0222 12°8538 7°4156 3°9155 1°9033 *8512 “3482 *1289 0426 0122 0029 0006 0001 “0000 p=2 29°4643 33°9972 226648 10°3022 3°0907 “4807 12°4060 20°6767 21°8930 18°2441 12°7709 76625 3°9295 1°6841 D741 1403 ‘0187 42°1053 35°0877 16°5119 5°1599 1°0320 1032 21°5385 28-0936 229857 14°5941 76619 3°3874 1°2546 3795 0889 0145 0012 12°4583 20°0222 20°7638 17°3032 12°4583 79941 4°6342 2°4375 11607 4965 1882 0618 ‘0169 0037 0006 “0000 p=3 18°1318 30°2198 27°4725 16°4835 6°4103 1°2820 5°5138 12°9736 18°2441 19-4604 17:0278 125744 7°8590 4°0826 16841 4990 0811 30°4094 35°7757 22°3598 8:9439 2°2360 2752 12°1739 22°1344 23°7154 189723 12°2322 6°5238 2°8781 1:0279 "2827 0538 0054 5°7842 12°8538 17°3032 179953 15°7459 120490 8°2152 5°0297 2°7663 1°3589 5889 2204 0689 ‘0170 0029 0003 p=4 10°5769 24°0385 28°8461 224359 11-2179 2°8846 2°2704 70949 12°7709 17:0278 18°3377 16°5039 12°5030 7°8590 3°9295 1°4190 2838 21°4654 33°5397 26°8318 13°4159 4°1280 6192 6°6403 15°8103 21°3439 20°9694 16°3095 10°3613 5°3965 2°2614 ‘7269 1615 0189 2°5707 7°4156 1274583 15°7459 16°4305 14°7874 11°7361 8°2991 5°2415 2°9443 1°4548 “6200 2204 ‘0618 ‘0122 0013 p=5 5°7692 17°3077 26°9230 26°9230 17°3077 5°7692 8514 3°4056 76625 12°5744 16°5039 180043 16°5039 12°5744 76625 3°4056 8514 14°7575 29°5149 29-5149 18-1631 6°8111 1°2384 3°4783 1074348 172997 20°5034 18°9958 14°2469 8°7064 4°2644 1°5991 “4146 0565 1:0876 3°9155 79941 12°0490 14°7874 15°4916 14°2006 11°5313 8°3282 5°3344 3°0006 1°4548 *5889 “1882 0426 0052 9°8383 24-5958 30°2717 22°7038 10°3199 2°2704 17391 6°4073 12°8146 180913 19°7873 17°4128 12°4377 71073 3°1094 9423 1508 4351 1:9033 4°6342 8°2152 11°7361 14°2006 14:9480 13°8803 11°4309 8°3350 5°3344 2°9443 1°3589 4965 “1290 0181 6°3246 19-4604 29°1906 26°5369 14°5953 3°8921 8238 3°6613 8°7226 14°5376 186566 19°1897 159914 10°6609 5°4516 1/9383 3661 1631 *8512 2°4375 5°0297 8°2991 11°5313 13°8803 14-6968 13°7783 11-4309 8°3282 5°2415 2°7664 1'1607 “3482 ‘0567 12—2 91 92 Tables for Statisticians and Biometricians TABLE XLVIII—(continued). Percentage Frequency of Successes in a Second Sample “m” after drawing “p” Successes in a First Sample “n”. Successes p=0 p=l1 p=2 p=3 p=4 p=5 n=20| O 80-7692 64°6154 5171588 . 4070334 309349 23°5695 m= sf 16°1538 26°9231 33°3612 36°3940 36°8273 35°3542 2°6923 7°0234 1271313 17°3305 22-0964 26°0505 3512 1:2770 2°8884 5°1992 8-1408 11°5780 0319 “1520 “4333 ‘9577 1°8090 3°0647 0015 “0091 “0319 0851 1915 *3831 67°7419 45°1613 29-5884 19°0211 11°9763 7°3700 22°5806 31°1457 31°7019 28°1795 23-0313 17°6880 7:0078 15°0167 2171346 24°3861 24°8738 23°2155 2°0022 5°9325 10°8382 15°6071 19°3463 21°5333 5191 1-9965 4°5521 79661 11°7760 15°4158 3 | bo ) HA WP®HS Ny woh 1198 5750 1°5932 3°3250 5°7809 8°8091 6 0240 +1398 4618 11335 2°2940 4:0375 7 “0040 0278 “1079 *3084 “7210 1°4571 8 “0005 *0043 0193 0636 1707 “3946 9 0000 “0004 “0024 “0089 “0274 0722 10 0000 “0000 “0002 0006 *0023 “0068 n=20| 0 58°3333 33°3333 18°6275 1071604 5°3977 2°7859 m=15{ 1 25-0000 29°4118 25°4011 19-0508 13°0590 8°3578 2 10°2941 18-7166 22,2259 21°5090 18°2826 1471218 3 4°0553 1071381 155343 186411 19°1232 17-4841 4 1°5207 4°9056 9°3206 13°4987 16°3913 17°4841 5 5396 271584 49495 8°4849 12-0203 14°7942 6 1799 “8683 2°3569 4°7138 7°7053 10°8491 7 0558 3190 1:0101 2°3310 4°3590 6°9744 & “0159 “1063 3885 1:0256 271795 3°9421 9 0041 0318 1330 3989 “9581 + 1°9511 10 *0010 “0084 “0399 1353 *3658 8362 11 “0002 0019 “0102 0391 1188 3041 12 *0000 “0004 0022 “0093 0317 0907 13 “0000 0001 “0004 0017 0065 “0209 14 0000 0000 “0000 “0002 “0009 0033 15 “0000 “0000 0000 “0000 0001 “0003 n=20 51°2195 25-6098 12°4765 5°9099 2°7154 1:2068 m= of 25°6098 26-2664 19°6998 12-7783 775427 4°1377 0 1 2 12°4765 19-6998 20°2323 16°8602 12-2839 8:0929 8 5-9099 12-7783 16°8602 17°3419 15°1742 11°7715 4 2°7154 7°5427 12-2839 15°1742 15-6340 14:0706 5 12068 4:1377 8:0929 11°7715 140706 14°5245 6 “5172 2°1297 4:9048 8-2768 11°3473 13°3141 7 “0839 1°0326 2°7589 5°3399 8°3213 11-0186 8 9 *0315 4719 1°4462 3°1817 5°5954 8°3131 “0112 2030 “7070 1°7554 3°4638 5°7473 10 0037 “0819 3218 “8965 1:9757 3°6474 11 0012 0308 1358 “4226 1°0362 2°1221 12 0003 0107 "0528 “1829 “4974 1:1274 13 0001 0034 “0188 0720 2168 5429 14 0000 “0010 “0060 "0255 0848 2345 15 —_ 0003 ‘0017 “0080 +0293 0893 16 = “0000 0004 “0022 0087 0293 17 — —_ “0001 “0005 0022 “0080 18 = _ “0000 “0001 “0004 0017 19 = — —_— “0000 0001 “0003 20 = == = = 0000 “0000 Probable Occurrences in Second Small Samples Successes n= m= 25 5 > Tits Ce *®@ my WOMANS SCSWMRD DOANRANS SMHS p=0 83:8710 13°9785 1°9281 ‘2065 0153 -0006 72°2222 20°6349 5°4622 1°3242 2897 0561 0093 0013 0001 “0000 “0000 63°4146 23°7805 8-5366 2°9204 9472 "2894 0827 0218 0053 ‘0012 “0002 “0000 “0000 *0000 56°5217 25-1208 10°8476 45409 1°8380 “7172 +2690 “0965 0330 ‘0107 “0033 “0009 0002 ‘0001 “0000 TABLE XLVIII—(continued). p=1 69:8925 24-1008 51645 ‘7651 ‘0736 “0035 51°5873 30°3455 12°4141 4°1380 11680 “2803 "0564 “0092 “0011 0001 “0000 39°6341 30°4878 16°8485 7°8930 3°2888 12403 “4256 1326 0373 “0094 0020 0004 “0001 “0000 “0000 31°4010 28°5463 18°9202 10°8116 5°6036 2°6897 12069 5082 2009 0744 0257 “0082 “0024 “0006 “0002 “0000 p=2 57°8421 30°9868 9°1813 1°7656 2119 0123 36°4146 33°1041 186211 8-0091 2°8032 8119 1933 0368 0053 0005 0000 24-3902 28 °8832 21°8575 13°1550 6°7654 3°0643 1°2381 ‘4477 "1444 0412 “0102 “0022 “0004 0001 “0000 “0000 17°1278 23°8993 21°6231 15°8218 10-0864 5°7932 3°0491 14833 “6695 “2806 "1089 0390 “0128 ‘0038 *0010 “0002 “0001 p=3 47°5131 35°1949 13°5365 3°2488 4738 0329 25°3798 31°7248 23°0261 12-2806 5°1875 1°7786 “4940 "1086 *0179 “0020 ‘0001 14°7625 23°9392 23°2742 17°2894 10°6788 56953 2°6697 1°1072 “4060 *1307 0364 “0086 “0016 “0002 “0000 “0000 9°1614 1774502 20°2168 181951 13°8796 9°3504 5°6861 3°1589 1°61383 “7592 +3290 1308 “0475 “0156 0046 “0012 “0002 0000 p=4 38°7144 37°2254 17°8682 5°2115 9064 0741 17°4486 28°1430 25°3287 16°3035 8°1518 3°2607 1:0451 "2628 0493 “0062 “0004 8°7777 182869 21°9443 19°5777 14°2383 8°8100 4°7366 2°2329 "92.40 3337 “1038 0272 “0058 “0009 0001 “0000 4°7988 11°7044 16°6788 17:9618 16°0711 12°5094 8°6871 5°4604 3°1317 1°6449 “7916 3482 1393 “0502 “0162 “0045 ‘0011 “0002 “0000 p=5 31°2693 375232 21°8885 76134 1°5573 "1483 11°8200 23°6401 25°6780 195642 11°4125 5°2673 1:9313 5518 ‘1170 0165 0012 6°1203 13°1666 18°9753 19°9337 16°8191 11:9361 72943 3°8807 1°8017 "7266 *2515 0732 0173 *0031 0004 “0000 2°4579 73738 12°5733 15°8820 16°4186 14°5943 11°4669 8-0943 51816 3°0226 16088 *7800 3429 1357 “0477 ‘0147 “0039 “0008 “0001 “0000 93 94 Tables for Statisticians and Biometricians TABLE XLVIII—(continued). Percentage Frequency of Successes in a Second Sample “m” after drawing “p” Successes in a First Sample “n”, Successes p=0 p— p=2 p=3 p=4 p=sd n=25| O 50°9804 25°4902 12°4850 5°9824 2°8003 1°2784 m=25{ 1 25°4902 26-0104 19°5078, =12°7285 76094 4°2613 2 12-4850 195078 19°9229 166024 12-1751 8°1352 3 5°9824 12°7985 166024 169713 14:8499 11°6037 4 -2°8008-«=—Ss76094. «121751 = s«14:8499 ~=—-:15°1953.~—S«13-6757 5 1:2784 42613 81352 11°6037 13°6757 14-0093 6 5682 22598 5:0451 82883 11°1185 12°8418 7 2453 11411 29344 54870 82990 ~—-10°7251 8 ‘1027 “5502 16103 33951 5°7456 82555 9 0416 2535 > *8365 19732 3°7128 5-9003 10 ‘0162 “1115 “4118 1-0801 22477 39335 il ‘0061 “0468 ‘1921 ‘5573 12771 2°4521 12 0022 ‘0187 “0848 2709 ‘6811 1°4304 18 “0007 ‘0070 0353 1238 “3406 ‘7802 Us “0002 0025 0138 0531 "1592 ‘3971 15 ee “0008 0051 “0212 0693 ‘1879 16 a ‘0003 ‘0017 “0079 0280 “0822 17 = “0001 ‘0005 “0027 “0104 -0330 18 = = -0002 ‘0008 “0035 “0120 19 = = “0000 “0002 “0010 ‘0039 20 = = = ‘0001 0003 ‘0011 21 = = = = ‘0001 0008 22 sy = = = = ‘0001 23 as = Be a = 24 = = i = = = 25 = eo = =: os — n=50| 0 91:0714 82°7922 75°1263 68-0389 61:4967 55-4676 m= + 1 82792 15°3319 21°2621 2671688 3071454 33-2805 2 ‘6133 17357-32711 51311 72349 -9°5087 3 0347 ‘1335 3207 ‘6157 1°0335 1°5848 4 0013 ‘0065 0192 “0440 “0861 “1517 5 “0000 -0001 ‘0005 “0014 ‘0033 “0066 n=50| 0 83°6065 69°6721 57-8633 47°8869 39:4857 32-4346 0) 1 13°9344 23-6177 -«29°9293 33-6048 «= 35-2551 35-3833 2 21256 54972 9-4513 13:5019 17-3070 20-6402 3 2932 1:0287 22503 39278 59827 8°3080 4h ‘0360 ‘1607 4296 ‘8910 15803 2°5164 5 0039 “0210 -0668 1614 +3282 5921 6 “0003 “0023 “0084 0233 0536 "1085 7 “0000 ‘0002 “0008 0026 “0067 ‘0152 8 “0000 “0000 ‘0001 “0002 “0006 ‘0015 9 ‘0000 “0000 “0000 “0000 “0000 “0001 10 ‘0000 -0000 -0000 “0000 “0000 “0000 n=50| 0 77:2727 59°4406 45°5092 34°6737 262849 19-8214 fat 1 17°8322 27-8628 32°5066 33°5552 32°3175 29-7321 2 39008 9-2876 146804 19:2530 22°6222 24:6927 3 8049-25965 52143 83429 11°6306 ~—«:14"7589 4 1558 “6385 15643-29695 = 48127 6-9910 6 ‘0281 “1405 “4083 ‘9011 16718 —-2°7465 6 0047 0278 0939 ‘2371 -4976 ‘9155 7 ‘0007 -0049 ‘0191 “0544 1279 ‘2616 8 ‘0001 0008 0034 ‘0109 0284 0642 9 = ‘0001 “0005 “0019 0054 0132 10 = = ‘0001 “0003 “0009 0024 11 = = _ = ‘0001 “0003 12 = = = — = “0000 18, 14, 15 = = = = = = Probable Occurrences in Second Small Sanples Successes n=50| 0 m= 20 i DSS SO WAH AS Co Ww 10 p=0 71-8310 20°5231 56513 1°4959 "3796 "0920 0212 “0046 ‘0010 “0002 “0000 67°1053 22°3684 7°2546 2°2857 “6984 2066 0590 ‘0163 “0043 “OO11 0003 “0001 “0000 “0000 50°4950 25°2475 12°4963 6°1206 2°9657 174210 “6731 3151 *1457 "0665 0800 0133 0058 “0025 “OO11 “0004 “0002 0001 TABLE XLVIII—(continued). p=1 51°3078 29°7437 12-4661 4:4655 1:4377 “4247 1161 0295 0070 0015 :0003 0001 44°7368 30°2276 14:9068 673492 2°4592 *8853 2994 “0956 “0289 “0083 “0022 “0006 ‘0001 “0000 “0000 25°2475 25°5026 19°1269 12°6198 77231 4°4875 2°5063 1°3552 “7126 3654 “1831 “0898 “0431 0203 “0093 “0042 0019 0008 “0003 “0001 p=2 36°4360 32°1494 18°2340 8°2882 3°2515 1°1380 3613 "1049 ‘0279 “0068 *0015 “0003 “0001 29°6230 3074346 20-2898 10°9546 51643 2°2004 "8629 3146 1073 0343 0103 “0029 ‘0008 -0002 “0000 “0000 12°4963 19°1269 19°3241 16°1034 11-9504 8°1873 5°2821 3°2480 1°9185 1°0942 “6049 *3250 “1699 ‘0866 0431 “0209 0099 “0046 0021 “0009 “0004 “0002 *OGO1 p=3 25°7195 30°7099 22°1018 12°2410 56902 2°3122 “8391 “2751 °0820 "0222 *0055 “0012 “0002 “0000 19°4782 27°0530 22°8617 15°0234 8°3826 4°1420 1°8546 “7627 -2904 "1029 -0340 "0105 “0030 “0008 “0002 “0000 “0000 6°1206 12°6198 1671034 16°2729 14°2388 11°2686 8°2677 5°7108 3°7517 2°3606 1°4298 8367 “4743 2610 1396 0726 “0368 "0182 “0088 “0041 “0019 0008 “0004 “0002 “0001 “0000 p=4 18-0421 273365 239720 15°7316 84901 39438 1°6163 5926 "1959 0585 0158 “0039 “0008 “0002 “0000 12°7149 22°3854 2370250 17°9083 11°5877 6°5376 3°3018 1°5167 “6398 “2494 “0901 0302 “0094 "0027 0007 “0002 “0000 “0000 2°9657 77231 11°9504 14°2388 14°3919 12°9527 10°6753 8°2014 5°9437 4:0975 2°7034 1°7147 1°0490 "6205 *BD57 “1978 “1068 ‘0561 "0286 "0142 “0069 "0032 ‘0015 “0007 -0003 “0001 “0000 “0000 p=5 12-5748 232150 24-1218 18-3785 11°3384 59480 2°7262 1-1089 “4039 "1323 "0390 0103 “0024 ‘0005 0001 “0000 8°2378 17°6525 2174900 1973831 14°3204 9°1130 5°1406 2°6162 1°2147 “5181 "2038 0741 “0249 *OO7T7 “0022 “0006 “0001 “0000 “0000 1°4210 4°4875 8°1873 11°2686 12°9527 13°0951 12°0038 10°1734 8:0780 60662 4°3381 2°9694 1°9531 1°2381 "7582 4493 "2580 "1437 ‘0777 “0408 “0208 0103 0050 0023 0010 “0005 “0002 “0001 “0000 95 96 Successes n=100} O 90°9910 m= 10f{ 1 SWAND Kits Ce % Lod SLANANY BOND mw Successes n=100 n= S st Tits Co to >W OVA At WONROS TABLE XLVIII—(continued). p=0 8:2719 “6830 0506 “0033 “0002 “0000 p=10 33°5855 36°9441 20°1513 771284 1°8005 “3376 0474 0049 “0003 “0000 p=0 95°2830 4°53738 1745 0051 “0001 “0000 87:°0690 11°3568 1°3947 1604 *0172 0017 “0002 “0000 3°9119 2°2212 “3388 "0492 0068 “0009 “0001 “0000 Tables for Statisticians and Biometricians p=1 82-7191 15°1778 1°8972 "1891 ‘0156 ‘0011 ‘0001 “0000 p=15 19°6056 33°0200 26°8727 138698 5°0127 1°3220 2571 0363 0036 0002 “0000 p=1 90°7457 8°7256 5083 -0199 :0005 0000 75°7121 19-9243 3°7027 “5730 0774 “0093 “0010 “0001 0000 69°5592 23°3813 5°6472 1°1584 "2122 "0354 “0054 “0008 ‘0001 “0000 ” p=2 p=3 p=4 p=d p=6 75°1302 68°1737 61°8023 55°9719 50°6412 20°8695 25°4855 2971520 31°9839 34-0855 3°5107 5°4097 7°4962 9°6874 11°9134 4416 8243 1°3455 2°0065 2°8031 "0442 0971 "1829 “3098 “4857 0036 0090 “0194 "0368 0641 “0002 0007 “0016 “0034 “0065 "0000 “0000 “0001 “0002 0005 = — “0000 “0000 0000 p=20 p=%5 p=30 p=3d p=40 11-0992 6:0712 3°1945 16083 “7697 25°8982 18°5708 12°3788 7°7198 4°5082 28°8081 26°8613 22°5639 17°3696 12°3485 20°0784 24°1644 25°4567 24°1112 20°8229 9°6930 14:°9554 19°6711 22°8554 23-9308 3°3813 66468 10°8709 15°4516 19°5797 “8619 271464 4°3483 75418 11°5470 1583 -4968 1°2424 2°6232 4°8456 0200 0788 "2425 6221 1°3845 “0015 0077 "0292 0908 2432 “0001 0004 0017 “0062 0199 p=2 p=3 p=4 p=5 p=6 p=7 86°3830 82°1896 78:1607 74:2914 70°5768 67°0123 12°5800 1671156 19°3467 22:2874 24°9514 27°3520 ‘9867 1°5956 2°3216 3°1517 4:°0737 5:0756 0488 0957 1642 2573 +3780 5287 “0015 “0034 “0067 0119 “0197 0306 “0000 “0001 0001 “0003 0004 “0008 65°7500 57:0221 49°3852 42°7116 36°8873 31-8110 26°1836 30°5476 33°3684 34°9458 35°5336 35°3456 65459 9°6321 12°7407 15°7096 18°4248 20-8110 12777 2:2767 3°5456 5:0426 6°7156 8°5076 *2091 4386 *7879 + =1'2724 1:°9006 2°6738 0295 0715 "1458 “2641 “4381 “6787 0037 “0100 0229 “0461 “0842 "1428 0004 0012 0031 “0068 0137 0252 “0000 0001 0004 0009 “0019 0037 — “0000 0000 0001 “0002 0005 _— — _ 0000 “0000 “0001 — — — _ _— “0000 57°8686 48:0604 39°8449 32:9751 27°2403 22-4613 2974247 32°8618 34°3491 34°4088 33°4530 31°8036 9°5567 13°4563 17°0252 20°0718 22°4994 24°2787 24716 4:2124 6:2724 8:5261 10°8479 13°1236 5480 1:0993 1°8873 2°9118 4°1535 5°5775 1077 "2490 4853 8394 1°3291 1°9649 0191 “0500 1093 "2099 3658 5913 “0031 0090 0219 "0462 “0881 *1547 “0004 “0015 0039 “0090 0187 0356 “0001 0002 0006 0016 0035 “0072 “0000 “0000 “0001 0002 “0006 0013 _ -- “0000 “0000 *0001 0002 — —- -— — 0000 “0000 Ta Percentage Frequency of Successes in a Second Sample “m” after drawing “p” Successes in a First Sample » ns. 45°7719 41°3280 35°5510 = 36-4659 141158 16°2472 3°7269 4°7658 “7174 1:0109 "1044 1609 “0115 0194 “0010 0017 ‘0001 0001 “0000 “0000 p=45 = p=50 *3473 1463 2°4580 1°2434 8°1229 4°9313 16°5036 12°0167 22°8255 19°9224 22°4513 23°4799 15°9030 19°9224 80093 =: 120167 2°7446 4°9313 5778 1°2434 0567 "1463 p=8 p=9 63°5933 60°3153 29°5020 31°4142 61463 7°2749 “7117 9287 *0454 “0649 0013 “0020 27°3928 235527 34-5611 33°3293 22°8233 24-4415 10°3611 12-2207 35865 4°6273 9959 1:3973 "2278 © °3459 0435-0711 0070 = --0122 ‘0009 “0017 “0001 “0002 0000 0000 18°4859 15°1848 29°7094 27°3600 25°4270 + 25°9920 152562 17°1690 771382 8°7832 2°7495 3°6775 “8994 1°3010 "2545 “3965 0630 “1053 0137 0245 0026 “0050 0005 “0009 “0001 0001 “0000 0000 p=9 37°2763 36°9072 - 18°2690 5°9052 1°3708 2374 “0309 “0030 “0002 “0000 p=10 57°1739 33°1007 84512 11813 “0899 0030 20°2198 31°7739 256636 14-0361 5°7796 18884 “5036 1112 0204 0031 0004 “0000 12°4488 24°8976 26°0397 18°8065 10°4578 4°7356 1°8040 “5898 “1675 0416 0091 “0017 0003 0000 Probable Occurrences in Second Small Samples TABLE Successes p=0 p=1 p=2 p=3 nm=100) 0 8071587 6471270 5171981 40°7920 m= 25} 21 16°0317 25°8576 31°2184 33°43861 2 31029 7°5681 12:2826 16°5799 3 *5802.-1°9024 9 338912) »=— 63556 4 1046 -4323 1:0701 2:0562 5 ‘0182 0908 2644 5855 6 0030 0178 0597 1501 1. “0005 0033 70125 0351 8 “0001 0005 “0024 “0075 Y) “0000 “0001 0004 0015 10 — 0000 0001 0003 il — — “0000 0001 12 — — — “0000 13 — _— _— — 1h — — _— —_— 15—25 — — — _— n=100) 0 66°8874 44:5916 29°6280 19°6185 m= “A 1 22°2958 29°9273 30°0284 26°6919 2 7°3322 14:8625 20°0189 22°3956 8 2°3780 6°4708 10°9693 14°8274 4 ‘7603 276038 5°3333 84691 6 “2396 9912 2°3852 4°3589 6 0743 3614 1:0008 2°0720 th 0227 1271 3987 9237 8 0068 0433 1520 “3901 9 0020 0143 0557 1572 10 “0006 "0046 0197 0607 11 0002 0014 0068 0226 12 = 0004 0022 0081 13 — “0001 “0007 0028 14 — — 0002 0009 15 — — 0001 0003 16 — — — ‘0001 Le _— — = = 18 — —_ — — 9 _- — -- — 20 — — — 21 — = — 22 _ — — _— 23 — — — 24-50 — —_ _ B. XLVIlI—(continued). p=4t 32°4330 33°5052 20°1031 9-0661 3°3806 1:0922 3138 “0815 0193 0042 “0008 “0001 “0000 12°9456 22°1671 22°4728 17°4789 11°4896 6°6996 3°5636 1°7600 8167 3590 "1504 0603 0232 ‘0086 0031 ‘0011 0004 “0001 p=5 25°7320 32°1649 22°7047 11°8013 4°9929 1:8077 O764 1647 "0426 “0100 “0022 “0004 “0001 “0000 85121 17°6113 20°9746 18°7745 13°9817 9°1228 5°3759 2°9173 1°4771 “7044 *3185 1373 0566 0224 0085 0031 ‘OO11 “0004 0001 fetes Thee p=6 20°3711 29-9575 24°3723 14°3734 6°8150 2°7378 *9606 *3000 "0844 “0215 “0050 “0011 “0002 “0000 5°5769 13°5550 18°5789 18°8406 15°7005 11°3492 773484 4°3512 2°3900 1:2301 “5978 "2758 1213 *0510 0206 0080 0030 ‘0011 “0004 ‘0001 p= 160915 27°2737 25°1757 16°6391 8°7536 3°8700 174841 5035 1531 0421 0105 0024 0005 “0001 0000 3°6405 10°1832 15°8126 17°9434 16°5656 13°1572 9°2958 5:9710 3°5398 1:9578 1:0184 “5012 2345 "1046 0447 0183 0072 0027 “0010 0003 “0001 p=8 12°6823 24°3890 25°2300 18°5020 10°7117 5°1757 2°1565 *7910 "2589 0763 0203 “0049 0011 0002 “0000 2°3676 75029 13°0370 16°3894 16°6252 14°4085 11°0430 76559 4°8771 2°8874 1°6022 8386 “4161 "1965 ‘0886 0382 "0158 0063 “0024 “0009 0003 “0001 97 p=9 9°9724 21-4922 24-6693 19-9086 12-5970 66134 2-9790 1:1761 +4127 -1299 ‘0369 ‘0095 0022 0005 ‘0001 0000 1°5339 54395 10-4710 14°4635 16°0095 15:0512 12°4505 9°2753 6°3248 39946 PBT aye 1°3088 “6871 "3425 *1627 ‘0738 *0320 0133 -0053 “0020 0008 “0003 “0001 13 p=10 7°8232 18-7076 23°6306 20°8423 14°3291 8°1327 3°9431 1°6693 6260 “2099 0634 0173 0043 “0009 “0002 “0000 “9900 3°8892 8°2261 12°3988 14°8876 15°1066 13°4281 10°7081 7°7895 5°2324 3°2752 1°9239 1-0664 “6601 “2797 1332 “0606 “0264 ‘0110 0044 ‘0017 “0006 0002 0001 98 1—250 = 2 SOMVRH Newton =~ ms Ba Be Sit Gs % WHA SOOND rss} ine Ss re RVs log |n 000 0000 301 0300 “778 1513 1°380 2112 2°079 1812 2°857 3325 3°702 4305 4°605 5205 5°559 7630 6559 7630 7601 1557 8°680 3370 9-794 2803 10°940 4084 127116 4996 13°320 6196 14551 0685 15°806 3410 17-085 0946 18°386 1246 19-708 3439 21-050 7666 22°412 4944 23°792 7057 25°190 6457 26°605 6190 28°036 9828 29°484 1408 30°946 5388 32°423 6601 33°915 0218 35°420 1717 36°938 6857 38°470 1646 40°014 2326 41°570 5351 43138 7369 44-718 5205 46°309 5851 47-911 6451 49°524 4289 51147 6782 52°781 1467 54°424 5993 56-077 8119 57°740 5697 59°412 6676 61-093 9088 62°784 1049 64:483 0749 Tables for Statisticians and Biometricians TABLE XLIX. Logarithms of Factorials. log|n from n=1 to n=1000. log |n 667190 6450 67906 6484 69°630 9243 71°363 3180 73°103 6807 74:851 8687 76°607 7436 78°371 1716 807142 0236 81-920 1748 83°705 5047 85497 8964 87:297 2369 89°103 4169 90916 3303 92°735 8742 94°561 9490 96:°394 4579 98:233 3070 100°078 4050 101929 6634 103°786 9959 105°650 3187 107°519 5505 109°394 6117 111°275 4253 113°161 9160 115054 0106 116-951 6377 118°854 7277 120°763 2127 122°677 0266 124-596 1047 126°520 3840 128°449 8029 130°384 3013 132°323 8206 134'268 3033 136°217 6933 138171 9358 140:130 9772 142-094 7650 144-063 2480 146°036 3758 148°014 0994 149°996 3707 151983 1424 153'974 3685 155970 0037 157°970 0037 log |n 159°974 3250 161°982 9252 163-995 7624 166°012 7958 168°033 9851 170:059 2909 172°088 6747 174122 0985 176°159 5250 178200 9176 180°246 2406 182°295 4586 184°348 5371 186°405 4419 188°466 1398 190°530 5978 192598 7836 194-670 6656 196°746 2126 198°825 3938 200°908 1792 202°994 5390 205-084 4442 207177 8658 209:274 7759 211°375 1464 213°478 9501 215°586 1601 217°696 7498 219°810 6932 221°927 9645 224°048 5384 226°172 3900 228299 4948 230°429 8286 232°563 3675 234°700 0881 236°839 9672 238°982 9820 241°129 1100 243°278 3291 245°430 6174 247°585 9535 249°744 3160 251°905 6840 254070 0368 256°237 3542 258°407 6159 260°580 8022 262°756 8934 log|n 264°935 8704 267°117 7139 269°302 4054 271°489 9261 273°680 2578 275'873 3824 278-069 2820 280°267 9391 282°469 3363 284°673 4562 286°880 2821 289°089 7971 291-301 9847 293°516 8286 295°734 3125 | 297°954 4206 300°177 1371 302°402 4464 304°630 3331 306°860 7820 309°093 7781 311°329 3066 313°567 3527 315°807 9019 318050 9400 320°296 4526 322°544 4259 324-794 8459 327047 6989 329°302 9714 331560 6500 333°820 7214 336°083 1725 338°347 9903 340°615 1620 342°884 6750 345°156 5166 347°430 6744 349°707 1362 351'985 8898 354:266 9232 356°550 2244 358°835 7817 3617123 5835 363°413 6181 365°705 8742 368:000 3404 370°297 0056 372°595 8586 374896 8886 log |n 377°200 0847 379°505 4361 381°812 9321 384122 5623 386°434 3161 -388°748 1834 391:064 1537 393°382 2170 395°702 3633 398024 5826 400°348 8651 402-675 2009 405°003 5805 407°333 9943 409°666 4328 412°000 8865 414:337 3463 416°675 8027 419°016 2469 421°358 6695 423-703 0618 426:049 4148 428°397 7197 430°747 9677 433°100 1502 435°454 2586 437°810 2845 440°168 2193 442°528 0548 444:889 7827 447°253 3946 449°618 8826 451°986 2385 454°355 4544 456°726 5223 459°099 4343 461°474 1826 463°850 7596 466°229 1575 468°609 3687 470°991 3857 473°375 2011 475°760 8074 478148 1972 480°537 3633 482°928 2984 485°320 9954 487°715 4470 490°111 6464 492-509 5864 log |x 494-909 2601 497-310 6607 499°713 7812 5027118 6149 504°525 1551 506°933 3950 509°343 3282 511-754 9479 514-168 2476 516°583 2210 518'999 8615 521°418 1628 523838 1185 526-259 7225 528°682 9683 531°107 8500 533°534 3612 535962 4960 538392 2483 540°823 6121 543°256 5814 545°691 1503 548°127 3129 550°565 0635 553004 3962 555°445 3052 557°887 7850 560°331 8298 562°777 4340 565°224 5920 567-673 2984 570123 5475 572°575 3339 575028 6523 577°483 4971 579°939 8631 582°397 7450 584:857 1375 587°318 0354 589°780 4334 592°244 3264 594-709 7092 597176 5768 599°644 9242 602°114 7462 604586 0379 607°058 7943 609°533 0106 612-008 6818 614-485 8036 Logarithms of Factorials TABLE XLIX—(continued). log |n 616:964 3695 619-444 3765 621-925 8191 624°408 6927 626°892 9925 629°378 7140 631°865 8523 634°354 4031 636°844 3615 639°335 7232 641°828 4836 644°322 6382 646°818 1825 649°315 1122 651813 4227 654313 1098 656°814 1691 659°316 5962 661°820 3869 664°325 5369 666°832 0419 669°339 8978 671°849 1003 674°359 6453 676°871 5287 679°384 7463 681°899 2940 684-415 1679 686°932 3638 689°450 8777 691°970 7057 694-491 8438 697°014 2880 699°538 0345 702063 0793 704°589 4186 707°117 0485 709°645 9652 712176 1649 714-707 6438 717:240 3982 719-774 4243 722°309 7184 724846 2768 727°384 0959 729°923 1720 732°463 5015 735°005 0807 737547 9062 740°091 9742 n log |n 742637 2813 745°183 8240 747°731 5987 750°280 6020 752°830 8303 755°382 2803 757°934 9485 760°488 8316 763°043 9260 765°600 2285 768°157 7357 770°716 4443 773°276 3509 775°837 4523 778°399 7452 780°963 2262 783°527 8923 786°093 7401 788°660 7665 791°228 9682 793°798 3421 796°368 8851 798°940 5939 801°513 4655 804087 4968 806°662 6846 809'239 0260 811°816 5178 814'395 1570 816°974 9406 819°555 8655 8227137 9289 824-721 1277 827°305 4589 829°890 9196 | 832°477 5069 835-065 2179 837°654 0496 840243 9992 842°835 0638 845-427 2406 848-020 5267 850°614 9192 853°210 4154 855°807 0125 858-404 7077 861-003 4982 863°603 3813 866°204 3542 868°806 4142 99 251—500 n log |n n log |n 401 | 871-409 5586 451 | 1002:893 0675 402 | 874-013 7846 452 | 1005:548 2059 403 | 876°619 0896 458 | 1008-204 3041 404) 879:225 4710 454 | 1010-861 3600 405 | 881-832 9260 455 | 1013519 3714 406 | 884-441 4521 456 | 1016-178 3362 407 | 887-051 0465 457 | 1018°838 2524 408 | 889°661 7066 458 | 1021-499 1179 409 | 892-273 4300 459 | 1024-160 9306 410 | 894-886 2138 460 | 1026°823 6884 411 | 897:°500 0556 461 | 1029-487 3893 412 | 900-114 9528 462 | 1032-152 0313 418 | 902-730 9029 463 | 1034:817 6123 414 | 905°347 9032 464 | 1037-484 1303 415 | 907:965 9513 465 | 1040°151 5832 416 | 910-585 0447 466 | 1042-819 9692 417 | 913-205 1807 467 | 1045-489 2860 418 | 915°826 3570 468 | 1048159 5319 419 | 918°448 5710 469 | 1050-830 7047 420 | 921:071 8203 470 | 1058-502 8026 421 | 923-696 1024 471 | 1056°175 8235 422 | 926°321 4149 472 | 1058-849 7655 428 | 928°947 7552 473 | 1061-524 6266 424 | 931-575 1211 474 | 1064-200 4050 425 | 934-203 5100 475 | 1066'877 0986 426 | 936-832 9196 476 | 1069-554 7056 427 | 939°463 3475 477 | 1072233 2239 428 | 942-094 7913 478 | 1074912 6518 429 | 944-727 2486 479 | 1077592 9873 430 | 947°360 7170 480 | 1080-274 2286 481 | 949-995 1943 481 | 1082-956 3737 2 | 952-630 6780 482 | 1085°639 4207 433 | 955:267 1659 483 | 1088:323 3678 484 | 957-904 6557 484 | 1091-008 2132 485 | 960°543 1449 485 | 1093-693 9549 436 | 9637182 6314 486 | 1096-380 5912 437 | 965-823 1128 487 | 1099-068 1202 438 | 968-464 5869 488 | 1101-756 5400 439 | 971107 0515 489 | 1104-445 8488 440 | 973°750 5041 490 | 11077136 0449 441 | 976°394 9427 491 | 1109°827 1264 442.\ 979:040 3650 492 | 1112519 0915 443 \ 981°686 7687 498 | 1115-211 9384 444 | 984:334 1517 494 | 1117-905 6654 445 | 986:982 5117 495 | 1120°600 2706 446 | 989°631 8466 496 | 1123-295 7523 447 | 992-282 1541 497 | 1125-992 1086 448 | 994:933 4321 498 | 1128*689 3380 449 | 997-585 6784 499 | 1181-387 4385 450 | 1000:238 8910 500 | 1184-086 4085 100 Tables for Statisticians and Biometricians 501—750 Table of log |n from n=1 to n=1000. log |r n log |r n log |r n log |r n log |r 1136-786 2463 551 | 1272°848 0029 601 | 1410°881 1614 651 | 1550°721 4519 701 | 1692-229 8994 502 | 1139-486 9500 552 | 1275°589 9419 602 | 1413°660 7579 | 652 | 1553°535 6995 702 | 1695076 2365 1142°188 5180 558 | 1278°332 6671 603 | 1416°441 0752 1556°350 6126 703 | 1697-923 1918 1144°890 9485 554 | 1281°076 1768 604 | 1419-222 1122 654 | 1559°166 1904 704 | 1700°770 7644 1147 ‘594 2399 555 | 1283°820 4698 605 | 1422-003 8676 655 | 1561°982 4317 705 | 1703°618 9536 S % 1150°298 3904 556 | 1286°565 5446 606 | 1424-786 3402 656 | 1564°799 3355 706 | 1706-467 7583 1153°003 3984 557 | 1289°311 3998 607 | 1427-569 5289 657 | 1567°616 9009 707 | 1709°317 1777 1155°709 2621 558 | 1292-058 0340 608 | 1430°353 4324 658 | 1570-435 1268 708 | 1712-167 2109 1158-415 9798 559 | 1294°805 4458 609 | 1433°138 0497 659 | 1573-254 0122 709 | 1715-017 8572 1161°123 5500 560 | 1297°553 6338 610 | 1435-923 3796 660 | 1576°073 5561 710 | 1717°869 1155 1163°831 9709 561 | 1300°302 5967 611 | 1438-709 4208 661 | 1578-893 7576 711 | 1720°720 9851 1166°541 2409 562 | 1303-052 3330 612 | 1441-496 1722 662 | 1581-714 6156 712 | 1723-578 4651 1169°251 3583 563 | 1305°802 8414 613 | 1444-283 6327 663 | 15847536 1291 713 | 1726°426 5546 1171°962 3214 564 | 1808°554 1205 614 | 1447°071 8011 664 | 1587°358 2972 714 | 1729°280 2529 1174°674 1286 565 | 1311°306 1690 615 | 1449-860 6762 665 | 1590°181 1188 715 | 1782134 5589 1177°386 7783 566 | 1314°058 9854 616 | 1452-650 2569 666 | 1593-004 5931 716 | 1784-989 4719 1180°100 2688 567 | 1316°812 5684 617 | 1455-440 5420 667 | 1595-828 7189 717 | 1737°844 9911 1182°814 5986 568 | 1319°566 9168 618 | 1458-231 5305 668 | 1598-653 4954 718 | 1740-701 1155 1185°529 7660 569 | 1322°322 0290 619 | 1461°023 2212 669 | 1601-478 9215 719 | 1743°557 8444 1188-245 7693 570 | 1825-077 90389 620 | 1463°815 6129 670 | 1604°304 9963 720 | 1746°415 1769 1190-962 6070 571 | 1327°834 5400 621 | 1466-608 7045 671 | 1607°131 7188 72 1193°680 2775 572 | 1330°591 9360 622 | 1469-402 4948 672 | 1609-959 0881 7 1196°398 7792 578 | 1833-350 0907 623 | 1472°196 9829 673 | 1612°787 1031 te: 1199°118 1105 574 | 1836°109 0026 624 | 1474°992 1675 674 | 1615°615 7630 7. 1201°838 2698 575 | 1838°868 6704 625 | 1477-788 0475 675 | 1618°445 0668 72 1749°273 1122 1752°131 6494 1754°990 7877 1757-850 5262 1760°710 8642 1204°559 2556 576 | 1341°629 0929 626 | 1480°584 6218 676 | 1621275 0135 726 | 1763-571 8009 1207°281 0662 577 | 1844°390 2687 627 | 1483°381 8894 677 | 1624-105 6022 727 | 17667433 3353 1210°003 7001 578 | 1847°152 1965 j28 | 1486°179 8490 678 | 1626°936 8319 28 | 1769-295 4667 1212°727 1558 579 | 1849°914 8751 629 | 1488°978 4997 679 | 1629-768 7016 729 | 17727158 1942 1215°451 4316 580 | 1852°678 3031 630 | 1491°777 8402 680 | 1632-601 2106 730 | 1775°021 5170 %W 8 Tie Co wy a 1218°176 5262 581 | 1855°442 4792 631 | 1494°577 8696 681 | 1635°434 3577 731 | 1777°885 4344 1220°902 4378 582 | 1358°207 4022 682 | 1497°378 5866 682 | 1638°268 1420 732 | 1780°749 9455 1223°629 1650 583 | 1360°973 0708 633 | 1500°179 9904 683 | 1641°102 5627 733 | 1783°615 0495 1226-356 7063 584 | 1863°739 4836 634 | 1502-982 0796 684 | 1643-937 6189 734 | 1786°480 7455 1229°085 0600 585 | 1366-506 6395 635 | 1505-784 8533 685 | 1646-773 3094 735 | 1789-347 0329 1231°814 2248 586 | 1869-274 5371 636 | 1508-588 3105 686 | 1649°609 6335 736 | 1792-213 9107 1234544 1991 587 | 1372°043 1752 637 | 1511°392 4499 687 | 16527446 5903 787 | 1795-081 3782 1237-274 9814 588 | 1374°812 5525 638 | 1514°197 2706 688 | 1655°284 1787 738 | 1797-949 4345 1240°006 5702 589 | 1377°582 6678 639 | 1517:002 7714 689 | 1658-122 3979 739 | 1800-818 0790 1242°738 9639 590 | 1380°353 5198 640 | 1519°808 9514 690 | 1660°961 2470 740 | 1803°687 3107 1245°472 1612 591 | 1883°125 1073 641 | 1522-615 8094 691 | 1663°800 7251 741 | 1806°557 1289 1248°206 1605 592 | 1385°897 4290 G42 | 1525°423 3445 692 | 1666°640 8312 742 | 1809°427 5328 1250°940 9603 | 593 | 1388°670 4837 O43 | 1528°231 5554 69S | 1669°481 5644 743 | 1812°298 5216 1253°676 5592 594 | 1391°444 2702 644 | 1531°040 4413 694 | 1672°322 9239 744 | 1815°170 0946 1256°412 9557 595 | 13894°218 7871 645 | 1533°850 0010 695 | 1675°164 9087 745 | 1818°042 2508 1259°150 1483 596 | 1396°994 0334 646 | 1536°660 2335 696 | 1678°007 5179 746 | 1820°914 9897 1261°888 1357 597 | 1399°770 0077 647 | 1539°471 1378 697 | 1680-850 7507 747 | 1823-788 3103 1264°626 9162 598 | 1402°546 7089 648 | 1542°282 7128 698 | 1683°694 6061 748 | 1826°662 2119 1267-366 4886 599 | 1405°324 1357 649 | 1545:094 9575 699 | 1686°539 0833 749 | 1829°536 6937 1270°106 8513 600 | 1408°102 2870 650 | 1547-907 8709 700 | 1689°384 1813 750 | 1832°411 7549 log [mr 1835°287 3949 1838°163 6127 1841-040 4077 1843°917 7790 1846°795 7260 1849-674 2478 1852°553 3437 1855°433 0129 1858°313 2546 1861-194 0682 1864-075 4529 1866-957 4079 1869°839 9324 1872°723 0258 1875°606 6872 1878°490 9160 1881°375 7113 1884-261 0726 1887°146 9989 1890°033 4896 1892-920 5440 1895°808 1613 1898696 3408 1901°585 0817 1904°474 3835 1907°364 2452 1910°254 6662 1913°145 6458 1916-037 1832 1918°929 2778 1921-821 9289 1924°715 1356 1927°608 8974 1930°503 2135 1933398 0831 1936-293 5057 1939-189 4804 1942-086 0066 1944-983 0836 1947-880 7107 1950°778 8872 1953°677 6124 1956°576 8856 1959-476 7061 1962°377 0732 1965°277 9863 1968°179 4446 1971°081 4475 1975:983 9943 1976 °887 0842 Logarithms of Factorials TABLE XLIX—(continued). log | 1979-790 7168 1982°694 8911 1985°599 6067 1988°504 8627 1991-410 6586 1994-316 9936 1997-223 8672 2000°131 2785 2003°039 2271 2005°947 7121 2008°856 7329 2011-766 2890 2014°676 3795 2017°587 0039 2020°498 1615 2023409 8517 2026-322 0737 2029-234 8270 2032°148 1109 2035°061 9248 2037°976 2679 2040°891 1398 2043°806 5396 2046°722 4668 2049°638 9208 2052°555 9008 2055°473 4063 2058°391 4367 2061°309 9912 2064°229 0693 2067°148 6703 2070°068 7936 2072°989 4386 2075°910 6047 2078°832 2912 2081°754 4974 2084°677 2229 2087 °600 4669 2090°524 2289 2093°448 5082 2096°373 3042 2099°298 6162 2102224 4438 2105°150 7863 2108°077 6430 2111-005 0133 2113°932 8967 2116-861 2926 2119-790 2003 101 751—1000 n log [rm 2125°649 5488 2128°579 9884 2131°510 9374 2134°442 3953 2137°374 3614 2140°306 8352 2143°239 8160 2146-173 3033 2149-107 2964 2152°041 7949 2154976 7980 2157°912 3053 2160°848 3161 2163°784 8298 2166°721 8459 2169°659 3638 2172°597 3829 2175°535 9027 2178°474 9224 2181°414 4417 2184°354 4598 2187°294 9763 2190°235 9906 2193°177 5020 2196°119 5101 2199°062 0142 2202°005 0138 2204°948 5083 2207 °892 4971 2210°836 9798 2213°781 9557 2216°727 4243 2219°673 3850 2222°619 8373 2225°566 7805 2228°514 2143 2231°462 1379 2234-410 5509 2237°359 4526 2240°308 8426 2243°258 7203 2246°209 0852 2249°159 9366 2252°111 2742 2255063 0972 2258°015 4052 2260°968 1976 2263°921 4740 2266°875 2337 2269°829 4762 log |n 2272°784 2010 2275°739 4075 2278°695 0953 2281°651 2637 2284°607 9123 2287 °565 0405 2290°522 6478 | 2293 °480 7336 2296°439 2975 2299°398 3389 2302°357 8573 2305°317 8521 2308°278 3229 2311°239 2691 2314-200 6902 23177162 5856 2320°124 9550 2323°087 7977 2326-051 1132 2329°014 9010 2331'979 1606 2334°943 8915 2337-909 0932 2340°874 7652 2343°840 9069 2346°807 5179 2349°774 5977 2352°742 1456 2355°710 1614 2358678 6443 2361°647 5940 2364°617 0099 2367-586 8915 2370°557 2384 2373°528 0500 2376°499 3259 2379°471 0655 2382°443 2683 2385°415 9339 2388°389 0618 2391°362 6514 2394°336 7023 2397°311 2140 2400-286 1860 2403°261 6178 2406-237 5089 2409°213 8589 2412°190 6672 2415°167 9334 2418°145 ca n 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 IT4 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 998 994 995 996 997 998 999 1000 log |n 2421-123 8376 2424-102 4745 2427-081 5674 2430-061 1158 2433°041 1192 2436°021 5771 2439-002 4890 2441-983 8545 2444-965 6731 2447-947 9443 2450°930 6677 2453-913 8428 2456°897 4691 2459-881 5461 2462°S66 0734 2465°851 0506 2468°836 4770 2471°822 3524 2474°808 6762 2477°795 4479 2480°782 6671 2483°770 3334 2486°758 4462 2489°747 0052 2492°736 0098 2495°725 4596 2498°715 3542 2501°705 6930 2504°696 4757 2507°687 7018 2510°679 3708 2513°671 4823 2516°664 0358 2519°657 0309 2522°650 4672 2525°644 3441 2528°638 6612 2531-633 4182 2534°628 6145 2537°624 2497 2540620 3233 2543°616 8350 2546613 7842 2549°611 1706 2552°608 9937 2555°607 2530 2558°605 9482 2561°605 0787 2564°604 6442 2567 604 6442 102 Tables for Statisticians and Biometricians TABLE L. Table of Fourth-Moments of Subgroup-Frequencies. Ordinate 2—11. Frequency 1—50. z=3)| z=4 z=5 | 2=6 oi xz=8 z—9 | 2=11 3 8 I be 3 16 81 256 625 | 1296 2401 4096 6561 14641 32 | 162 512 1250 | 2592 4802 8192 | 13122 | 29282 48 | 243 768 | 1875 | 3888 7203 | 12288 | 19683 | 43923 1024 | 2500 | 5184 9604 | 16384 | 26244 | 58564 80 | 405 | 1280] 3125] 6480] 12005 | 20480 | 32805 | 73205 96 | 486 | 1536 | 3750 | 7776 | 14406 | 24576 | 39366] 87846 112 | 567 1792 | 4375 | 9072 | 16807 | 28672 | 45927 | 102487 128 | 648 | 2048} 5000 | 10368 | 19208} 32768 | 52488 | 117128 144 | 729 | 2304] 5625 | 11664 | 21609] 36864 | 59049 | 131769 10 | 160) 810| 2560 | 6250 | 12960 | 24010 | 40960 | 65610 | 146410] 10 11 176 | 891] 2816 | 6875 | 14256 | 26411 | 45056 | 72171 | 161051 | 11 12 | 192| 972} 3072) 7500 | 15552 |. 28812 | 49152 | 78732 | 175692 | 12 13 | 208 | 1053 | 3328 | 8125 | 16848 | 31213 | 53248 | 85293 | 190333 | 13 14 | 224] 1134] 3584 | 8750 | 18144 | 33614 | 57344) 91854 | 204974 | 14 15 | 240 | 1215} 3840} 93875 | 19440} 36015 | 61440 | 98415 | 219615 | 15 16 | 256 | 1296 | 4096 | 10000 | 20736 | 38416 | 65536 | 104976 | 234256 | 16 | y | 272 | 1877] 4852 | 10625 | 22032 | 40817 | 69632 | 111537 | 248897 | 17 18 | 288 | 1458 | 4608 | 11250 | 23328 | 43218 | 73728 | 118098 | 263538 | 18 19 | 304] 1589 | 4864 | 11875 | 24624 | 45619 | 77824 | 124659 | 278179 | 19 20 | 320 | 1620 | 5120 | 12500 | 25920 | 48020 | 81920 | 131220 | 292820 | 20 21 | 336 | 1701 5376 | 138125 | 27216 | 50421 | 86016 | 1387781 | 307461 | 21 22 | 352 | 1782 | 5632 | 13750 | 28512 | 52822 | 90112 | 144342 | 322102 | 22 23 | 368 | 1863 | 5888 | 143875 | 29808 | 55223 | 94208 | 150903 | 336743 | 23 24 | 384 | 1944 | 6144 | 15000 | 31104 | 57624 | 98304 | 157464 | 351384 | 24 25 | 400 | 2025 | 6400 | 15625 | 32400 | 60025 | 102400 | 164025 | 366025 | 25 26 | 416 | 2106 | 6656 | 16250 | 33696 | 62426 | 106496 | 170586 | 380666 | 26 27 | 432 | 2187 | 6912 | 16875 | 34992 | 64827 | 110592 | 177147 | 395307 | 27 28 | 448 | 2268 | 7168 | 17500 | 36288 | 67228 | 114688 | 183708 | 409948 | 28 29 | 464 | 2349 | 7424 | 18125 | 37584 | 69629 | 118784 | 190269 | 424589 | 29 3 480 | 2430 | 7680 | 18750 | 38880 | 72030 | 122880 | 196830 | 439230 | 30 31 | 496 | 2511 | 7986 | 19875 | 40176 | 74431 | 126976 | 203391 | 453871 | 31 32 | 512 | 2592 | 8192 | 20000 | 41472 | 76832 | 131072 | 209952 | 468512 | 32 83 | 528 | 2673 | 8448 | 20625 | 42768 | 79233 | 135168 | 216513 | 483153 | 33 34 | 544 | 2754 | 8704 | 21250 | 44064 | 81634 | 139264 | 223074 | 497794 | 34 85 | 560 | 2835 | 8960 | 21875 | 45360 | 84035 | 143360 | 229635 | 512435 | 35 36 | 576 | 2916 | 9216 | 22500 | 46656 | 86436 | 147456 | 236196 | 527076 | 36 37 =| 592 | 2997 | 9472 | 23125 | 47952 | 88837 | 151552 | 242757 | 541717 | 37 88 | 608 | 3078 | 9728 | 28750 | 49248 | 91238 | 155648 | 249318 | 556358 | 38 39 | 624 | 3159 | 9984 | 24875 | 50544 | 93639 | 159744 | 255879 | 570999 | 39 40 | 640 | 3240 | 10240 | 25000 | 51840 | 96040 | 163840 | 262440 | 585640 | 40 41 | 656 | 3321 | 10496 | 25625 | 53136 | 98441 | 167936 | 269001 | 600281 | 41 42 | 672 | 3402 | 10752 | 26250 | 54432 | 100842 | 172032 | 275562 | 614922 | 42 3 | 688 | 3483 | 11008 | 26875 | 55728 | 108243 | 176128 | 282123 | 629563 | 43 44 | 704 | 3564 | 11264 | 27500 | 57024 | 105644 | 180224 | 288684 | 644204 | 44 45 | 720 | 3645 | 11520 | 28125 | 58320 | 108045 | 184320 | 295245 | 658845 | 45 46 | 736 | 3726 | 11776 | 28750 | 59616 | 110446 | 188416 | 301806 | 673486 | 46 4v | 752 | 3807 | 12032 | 29375 | 60912 | 112847 | 192512 | 308367 | 688127 | 47 48 | 768 | 3888 | 12288 | 30000 | 62208 | 115248 | 196608 | 314928 | 702768 | 48 49 | 784 | 3969 | 12544 | 30625 | 63504 | 117649 | 200704 | 321489 | 717409 | 49 50 | 800 | 4050 | 12800 | 31250 | 64800 | 120050 | 204800 | 328050 | 732050 | 5U 1 & WA N+ CoM fon} req co bo re © Co ZV MD Tits Co TMH 3 DW SOND HA Ss TMH 41472 62208 82944 103680 124416 145152 165888 186624 207360 228096 248832 269568 290304 311040 331776 352512 373248 393984 414720 435456 456192 476928 497664 518400 539136 559872 580608 601344 622080 642816 663552 684288 705024 725760 746496 767232 787968 808704 829440 850176 870912 891648 912384 933120 953856 974592 995328 1016064 1036800 Verification of the Fourth Moment TABLE L—(continued), Ordinate 12—19. 114244 142805 171366 199927 228488 257049 285610 314171 342732 371293 399854 428415 456976 485537 514098 542659 571220 599781 628342 656903 685464 714025 742586 771147 799708 828269 856830 885391 913952 942513 971074 999635 1028196 1056757 1085318 1113879 1142440 1171001 1199562 1228123 1256684 1285245 1313806 1342367 1370928 1399489 1428050 38416 76832 115248 153664 192080 230496 268912 307328 345744 384160 422576 460992 499408 537824 576240 614656 653072 691488 729904 768320 806736 845152 883568 921984 960400 998816 1037232 1075648 1114064 1152480 1190896 1229312 1267728 1306144 1344560 1382976 1421392 1459808 1498224 1536640 1575056 1613472 1651888 1690304 1728720 1767136 1805552 1843968 1882384 1920800 50625 101250 151875 202500 253125 303750 354375 405000 455625 506250 556875 607500 658125 708750 759375 810000 860625 911250 961875 1012500 1063125 1113750 1164375 1215000 1265625 1316250 1366875 1417500 1468125 1518750 1569375 1620000 1670625 1721250 1771875 1822500 1873125 1923750 1974375 2025000 2075625 2126250 2176875 2227500 2278125 2328750 2379375 2430000 2480625 2531250 Frequency 1—50. 65536 131072 196608 262144 327680 393216 458752 524288 589824 655360 720896 786432 851968 917504 983040 1048576 1114112 1179648 1245184 1310720 1376256 1441792 1507328 1572864 1638400 1703936 1769472 1835008 1900544 1966080 2031616 2097152 2162688 2228224 2293760 2359296 2424832 2490368 2555904 2621440 2686976 2752512 2818048 2883584 2949120 3014656 3080192 3145728 3211264 3276800 83521 167042 250563 334084 417605 501126 584647 668168 751689 835210 918731 1002252 1085773 1169294 1252815 1336336 1419857 1503378 1586899 1670420 1753941 1837462 1920983 2004504 2088025 2171546 2255067 2338588 2422109 2505630 2589151 2672672 2756193 2839714 2923235 3006756 3090277 3172798 3257319 3340840 3424361 3507882 3591403 3674924 3758445 3841966 3925487 4009008 4092529 4176050 104976 209952 314928 419904 524880 629856 734832 839808 944784 1049760 1154736 1259712 1364688 1469664 1574640 1679616 1784592 1889568 1994544 2099520 2204496 2309472 2414448 2519424 2624400 2729376 2834352 2939328 3044304 3149280 3254256 3359232 3464208 3569184 3674160 3779136 3884112 3989088 4094064 4199040 4304016 4408992 4513968 4618944 4723920 4828896 4933872 5038848 5143824 5248800 130321 260642 390963 521284 651605 781926 912247 1042568 1172889 1303210 1433531 1563852 1694173 1824494 1954815 2085136 2215457 2345778 2476099 2606420 2736741 2867062 2997383 3127704 3258025 3388346 3518667 3648988 3779309 3909630 4039951 4170272 4300593 4430914 4561235 4691556 4821877 4952198 5082519 5212840 5343161 5473482 5603803 5734124 5864445 5994766 6125087 6255408 6385729 6516050 103 3 SB CNA NS sy 104 Tables for Statisticians and Biometricians TABLE L—(continued). Ordinate 2—11. Frequency 51—100. n | a=2 | e=3 | w=4 z=5 z=6 c=7 «z=8 z=9 z=11 n 51 | 816 | 4131 | 13056 | 31875 | 66096 | 122451 | 208896 | 334611 746691 | 51 52 882 | 4212 | 13312 | 32500 | 67392 | 124852 | 212992 | 341172 | 761332 | 52 53 848 | 4293 | 138568 | 33125 | 68688 | 127253 | 217088 | 347733 | 775973 | 58 54 | 864 | 4374 | 13824 | 33750 | 69984 | 129654 | 221184 | 354294 | 790614 | 54 55 | 880 | 4455 | 14080 | 34375 | 71280 | 182055 | 225280 | 860855 | 805255 | 55 56 | 896 | 4536 | 14336 | 35000 | 72576 | 184456 | 229376 | 367416 | 819896 | 56 5y 912 | 4617 | 14592 | 35625 | 73872 | 136857 | 233472 | 373977 | 834537 | 57 58 928 | 4698 | 14848 | 36250 | 75168 | 139258 | 2387568 | 380538 | 849178 | 458 59 | 944 | 4779 | 15104 | 36875 | 76464 | 141659 | 241664 | 387099 | 863819 | 59 60 | 960 | 4860 | 15360 | 37500 | 77760 | 144060 | 245760 | 393660 | 878460 | 60 61 | 976 | 4941 | 15616 | 38125 | 79056 | 146461 | 249856 | 400221 893101 | 61 992 | 5022 | 15872 | 38750 | 80352 | 148862 | 253952 | 406782 | 907742 | 62 63 | 1008 | 5103 | 16128 | 39375 | 81648 | 151263 | 258048 | 413343 | 922383 | 63 64 | 1024 | 5184 | 16384 | 40000 | 82944 | 153664 | 262144 | 419904 | 937024 | 64 65 | 1040 | 5265 | 16640 | 40625 | 84240 | 156065 | 266240 | 426465 | 951665 | 65 66 | 1056 | 5346 | 16896 | 41250 | 85536 | 158466 | 270336 | 433026 | 966306 | 66 67 | 1072 | 5427 | 17152 | 41875 | 86832 | 160867 | 274432 | 489587 | 980947 | 67 68 | 1088 | 5508 | 17408 | 42500 | 88128 | 163268 | 278528 | 446148 | 995588 | 68 69 | 1104 | 5589 | 17664 | 48125 | 89424 | 165669 | 282624 | 452709 | 1010229 | 69 70 | 1120 | 5670 | 17920 | 43750 | 90720 | 168070 | 286720 | 459270 | 1024870 | 70 71 | 1136 | 5751 | 18176 | 44375 | 92016 | 170471 | 290816 | 465831 | 1039511 | 71 2} 1152 | 5832 | 18432 | 45000 | 93312 | 172872 | 294912 | 472392 | 1054152 2 73 | 1168 | 5913 | 18688 | 45625 | 94608 | 175273 | 299008 | 478953 | 1068793 | 73 74 | 1184 | 5994 | 18944 | 46250 | 95904 | 177674 | 303104 | 485514 | 1083434 | 74 75 | 1200 | 6075 | 19200 | 46875 | 97200 | 180075 | 307200 | 492075 | 1098075 | 75 76 | 1216 | 6156 | 19456 | 47500 | 98496 | 182476 | 311296 | 498636 | 1112716 | 76 77 | 1232 | 6237 | 19712 | 48125 | 99792 | 184877 | 315392 | 505197 | 1127357 | 77 78 | 1248 | 6318 | 19968 | 48750 | 101088 | 187278 | 319488 | 511758 | 1141998 | 78 79 | 1264 | 6399 | 20224 | 49375 | 102384 | 189679 | 323584 | 518319 | 1156639 | 79 80 | 1280 | 6480 | 20480 | 50000 | 103680 | 192080 | 327680 | 524880 | 1171280 | 80 81 | 1296 | 6561 | 20736 | 50625 | 104976 | 194481 | 331776 | 531441 | 1185921 | 81 82 | 1312 | 6642 | 20992 | 51250 | 106272 | 196882 | 335872 | 538002 | 1200562 | 82 83 | 1328 | 6723 | 21248 | 51875 | 107568 | 199283 | 339968 | 544563 | 1215203 | 83 84 1844 | 6804 | 21504 | 52500 | 108864 | 201684 | 344064 | 551124 | 1229844 | 84 85 1360 | 6885 | 21760 | 53125 | 110160 | 204085 | 348160 | 557685 | 1244485 | 85 86 1376 | 6966 | 22016 | 538750 | 111456 | 206486 | 352256 | 564246 | 1259126 | 86 87 1392 | 7047 | 22272 | 54375 | 112752 | 208887 | 356352 | 570807 | 1273767 | 87 88 1408 | 7128 | 22528 | 55000 | 114048 | 211288 | 360448 | 577368 | 1288408 | 38 89 1424 | 7209 | 22784 | 55625 | 115344 | 213689 | 364544 | 583929 | 1308049 | 89 90 1440 | 7290 | 23040 | 56250 | 116640 | 216090 | 368640 | 590490 | 1317690 | 90 91 | 1456 | 7371 | 23296 | 56875 | 117936 | 218491 | 372736 | 597051 | 1332331 | 91 92 | 1472 | 7452 | 23552 | 57500 | 119232 | 220892 | 376832 | 603612 | 1846972 2 93 1488 | 7533 | 23808 | 58125 | 120528 | 223293 | 380928 | 610173 | 1361613 | 93 94 | 1504 | 7614 | 24064 | 58750 | 121824 | 225694 | 385024 | 616734 | 1376254 | 94 95 | 1520 | 7695 | 24320 | 59375 | 123120 | 228095 | 389120 | 623295 | 1390895 | 95 96 | 1536 | 7776 | 24576 | 60000 | 124416 | 230496 | 393216 | 629856 | 1405536 | 96 97 | 1552 | 7857 | 24832 | 60625 | 125712 | 232897 | 397312 | 636417 | 1420177 | 97 98 | 1568 7938 | 25088 | 61250 | 127008 | 235298 | 401408 642978 | 1434818 | 98 99 | 1584 | 8019 | 25344 | 61875 | 128304 | 237699 | 405504 | 649539 | 1449459 | 99 100 | 1600 8100 | 25600 | 62500 | 129600 | 240100 | 409600 | 656100 | 1464100 | 100 Verijication of the Fourth Moment 105 TABLE L—(continued). Ordinate 12—19. Frequency 51—100, n x=12 “2=13 ac=14 a=15 x=16 %=17 z=18 *%=19 n 51 | 1057536 | 1456611 | 1959216 | 2581875 | 3342336 | 4259571 5353776 | 6646371 | 51 52 | 1078272 | 1485172 | 1997632 | 2632500 | 3407872 | 43843092 5458752 6776692 | 52 58 | 1099008 | 1513733 | 2036048 | 2683125 | 3473408 | 4426613 5563728 6907013 | 53 54 | 1119744 | 1542294 | 2074464 | 2733750 | 3538944 | 4510134 5668704 7037334 | 54 55 | 1140480 | 1570855 | 2112880 | 2784375 | 3604480 | 4593655 5773680 7167655 55 56 | 1161216 | 1599416 | 2151296 | 2835000 | 3670016 | 4677176 5878656 7297976 | 56 57 | 1181952 | 1627977 | 2189712 | 2885625 | 3735552 | 4760697 | 5983632 | 7428297 | 57 58 | 1202688 | 1656538 | 2228128 | 2936250 | 3801088 | 4844218 6088608 7558618 | 58 59 | 1223424 | 1685099 | 2266544 | 2986875 | 3866624 | 4927739 6193584 7688939 | 59 60 | 1244160 | 1713660 | 2304960 | 3037500 | 3932160 | 5011260 6298560 7819260 | 60 61 | 1264896 | 1742221 | 2343376 | 3088125 | 3997696 | 5094781 6403536 7949581 61 62 | 1285632 | 1770782 | 2381792 | 3138750 | 4063232 | 5178302 6508512 8079902 62 63 | 1806368 | 1799343 | 2420208 | 3189375 | 4128768 | 5261823 6613488 8210223 63 64 | 1327104 | 1827904 | 2458624 | 3240000 | 4194804 | 5345344 | 6718464 | 8340544 | 64 65 | 1347840 | 1856465 | 2497040 | 3290625 | 4259840 | 5428865 6823440 8470865 65 66 | 13868576 | 1885026 | 2535456 | 3841250 | 4325376 | 5512386 6928416 8601186 66 67 | 1389312 | 1913587 | 2573872 | 3391875 | 4390912 | 5595907 7033392 8731507 67 68 | 1410048 | 1942148 | 2612288 | 3442500 | 4456448 | 5679428 | 7138368 | 8861828] 68 69 | 1430784 | 1970709 | 2650704 | 3493125 | 4521984 | 5762949 7243344 8992149 69 70 | 1451520 | 1999270 | 2689120 | 3543750 | 4587520 | 5846470 7348320 9122470 70 71 | 1472256 | 2027831 | 2727536 | 3594375 | 4653056 | 5929991 7453296 9252791 71 72 | 1492992 | 2056392 | 2765952 | 3645000 | 4718592 | 6013512 7558272 9383112 72 78 | 1513728 | 2084953 | 2804368 | 3695625 | 4784128 | 6097033 | 7663248 | 9513433 | 73 74 | 1534464 | 2113514 | 2842784 | 3746250 | 4849664 | 6180554 | 7768224 | 9643754] 74 75 | 1555200 | 2142075 | 2881200 | 3796875 | 4915200 | 6264075 7873200 9774075 | 75 76 | 1575936 | 2170636 | 2919616 | 3847500 | 4980736 | 6347596 7978176 9904596 76 77 | 1596672 | 2199197 | 2958032 | 3898125 | 5046272 | 6431117 | 8083152 | 10034717 | 77 78 | 1617408 | 2227758 | 2996448 | 3948750 | 5111808 | 6514638 | 8188128 | 10165038 | 78 79 | 1688144 | 2256319 | 3084864 | 3999375 | 5177344 | 6598159 | 8293104 | 10295359 | 79 80 | 1658880 | 2284880 | 3073280 | 4050000 | 5242880 | 6681680 | 8398080 | 10425680 | 80 81 | 1679616 | 2313441 | 3111696 | 4100625 | 5808416 | 6765201 8503056 | 10556001 | 81 82 | 1700352 | 2342002 | 3150112 | 4151250 | 5373952 | 6848722 | 8608032 | 10686322 | 82 83 | 1721088 | 2370563 | 3188528 | 4201875 | 5439488 | 6932243 | 8713008 | 10816643 | 83 84 | 1741824 | 2399124 | 3226944 | 4252500 | 5505024 | 7015764 | 8817984 | 10946964 | 84 85 | 1762560 | 2427685 | 3265360 | 4303125 | 5570560 | 7099285 8922960 | 11077285 | 85 86 | 1783296 | 2456246 | 3303776 | 4353750 | 5636096 | 7182806 | 9027936 | 11207606 | 86 87 | 1804032 | 2484807 | 3842192 | 4404375 | 5701632 | 7266327 | 91382912 | 11337927 | 87 88 | 1824768 | 2513368 | 3380608 | 4455000 | 5767168 | 7349848 9237888 | 11468248 | 88 89 | 1845504 | 2541929 | 3419024 | 4505625 | 5832704 | 7433369 9342864 | 11598569 | 89 90 | 1866240 | 2570490 | 3457440 | 4556250 | 5898240 | 7516890 9447840 | 11728890 | 90 91 | 1886976 | 2599051 | 3495856 | 4606875 | 5963776 | 7600411 9552816 | 11859211 | 91 92 | 1907712 | 2627612 | 3534272 | 4657500 | 6029312 | 7683932 | 9657792 | 11989532 | 92 98 | 1928448 | 2656173 | 3572688 | 4708125 | 6094848 | 7767453 | 9762768 | 12119853 | 93 94 | 1949184 | 2684734 | 3611104 | 4758750 | 6160384 | 7850974 | 9867744 | 12250174 gh 95 | 1969920 | 2713295 | 3649520 | 4809375 | 6225920 | 7934495 9972720 | 12380495 95 96 | 1990656 | 2741856 | 3687936 | 4860000 | 6291456 | 8018016 | 10077696 | 12510816 96 97 | 2011392 | 2770417 | 3726352 | 4910625 | 6356992 | 8101537 | 10182672 | 12641137 97 98 | 2032128 | 2798978 | 3764768 | 4961250 | 6422528 | 8185058 | 10287648 | 12771458 | 98 99 | 2052864 | 2827539 | 3803184 | 5011875 | 6488064 | 8268579 | 10392624 | 12901779 | 99 100 | 2073600 | 2856100 | 3841600 5062500 | 6553600 | 8352100 | 10497600 | 13032100 | 100 : i B. 14 106 Tables jor Statisticians and Biometricians TABLE L—(continued). Ordinate 2—7. Frequency 101—150. 101 1616 8181 25856 63125 130896 242501 101 102 1632 8262 26112 63750 132192 244902 102 108 1648 8343 26368 64375 133488 247303 103 104 1664 8424 26624 65000 134784 249704 104 105 1680 8505 26880 65625 136080 252105 105 106 1696 8586 27136 66250 137376 254506 106 107 1712 8667 27392 66875 138672 256907 107 108 1728 8748 27648 67500 139968 259308 108 109 1744 8829 27904 68125 141264 261709 109 110 1760 8910 28160 68750 142560 264110 110 111 1776 8991 28416 69375 143856 266511 111 112 1792 9072 28672 70000 145152 268912 112 118 1808 9153 28928 70625 146448 271313 113 114 1824 9234 29184 71250 147744 273714 114 115 1840 9315 29440 71875 149040 276115 115 116 1856 9396 29696 72500 150336 278516 116 117 1872 9477 29952 73125 151632 280917 117 118 1888 9558 30208 73750 152928 283318 118 119 1904 9639 30464 74375 154224 285719 119 120 1920 9720 30720 75000 155520 288120 120 121 1936 9801 30976 75625 156816 290521 121 122 1952 9882 31232 76250 158112 292922 122 123 1968 9963 31488 76875 159408 295323 123 124 1984 10044 31744 77500 160704 297724 124 125 2000 10125 32000 78125 162000 300125 125 126 2016 10206 32256 78750 163296 302526 126 127 2032 10287 32512 79375 164592 304927 127 128 2048 10368 32768 80000 165888 307328 128 129 2064 10449 33024 80625 167184 309729 129 13 2080 10530 33280 ~81250 168480 312130 130 151 2096 10611 33536 81875 169776 314531 131 152 2112 10692 33792 82500 171072 316932 132 133 2128 10773 34048 83125 172368 319333 133 184 2144 10854 34304 83750 173364 321734 184 155 2160 10935 34360 84375 174960 324135 185 136 2176 11016 34816 85000 176256 326536 136 137 2192 11097 35072 85625 177552 328937 187 158 2208 11178 35328 86250 178848 331338 138 189 2224 11259 35584 86875 180144 333739 139 140 2240 11340 35840 87500 181440 336140 140 141 2256 11421 36096 88125 182736 338541 141 142 2272 11502 36352 88750 184032 340942 142 143 2288 11583 36608 89375 185328 343343 143 144 2304 11664 36864 90000 186624 345744 144 145 2320 11745 37120 90625 187920 348145 145 146 2336 11826 37376 91250 189216 350546 146 147 2352 11907 37632 91875 190512 352947 147 148 2368 11988 37888 92500 191808 355348 148 149 | 2384 12069 | 38144 93125 193104 357749 149 150 | 2400 12150 38400 93750 | 194400 360150 150 Verification of the Fourth Moment TABLE L—(continued). Ordinate 8—14. Frequency 101—150. 107 n Zo 2=9 | z=11 | Z—12 2=13 101 413696 662661 1478741 2094336 2884661 3880016 101 102 417792 669222 1493382 2115072 2913222 3918432 102 103 421888 675783 1508023 2135808 2941783 3956848 103 104 425984 682344 1522664 2156544 2970344 3995264 104 105 430080 688905 1537305 2177280 2998905 4033680 105 106 434176 695466 1551946 2198016 3027466 4072096 106 107 438272 702027 1566587 2218752 3056027 4110512 107 108 442368 708588 1581228 2239488 3084588 4148928 108 109 446464 715149 1595869 2260224 3113149 4187344 109 110 450560 721710 1610510 2280960 3141710 4225760 110 111 454656 728271 1625151 2301696 3170271 4264176 111 112 458752 734832 1639792 2322432 3198832 4302592 112 113 462848 741393 1654433 2343168 3227393 4341008 113 11h 466944 747954 1669074 2363904 3255954 4379424 114 115 471040 754515 1683715 2384640 3284515 4417840 115 116 475136 761076 1698356 2405376 3313076 4456256 116 117 479232 767637 1712997 2426112 3341637 4494672 Late) 118 483328 774198 1727638 2446848 3370198 4533088 118 119 487424 780759 1742279 2467584 3398759 4571504 119 120 491520 787320 1756920 2488320 3427320 4609920 120 121 495616 793881 1771561 2509056 3455881 4648336 121 122 |: 499712 800442 1786202 2529792 3484442 4686752 122 128 503808 807003 1800843 2550528 3513003 4725168 123 124 507904 813564 1815484 2571264 3541564 4763584 124 125 512000 820125 1830125 2592000 3570125 4802000 125 126 516096 826686 1844766 2612736 3598686 4840416 126 127 520192 833247 1859407 2633472 3627247 4878832 127 128 524288 839808 1874048 2654208 3655808 4917248 128 129 528384 846369 1888689 2674944 3684369 4955664 129 130 532480 852930 1903330 2695680 3712930 4994080 130 181 536576 859491 1917971 2716416 3741491 5032496 IS. 182 540672 866052 1932612 2737152 3770052 5070912 1382 183 544768 72613 1947253 2757888 3798613 5109328 183 134 548864 879174 1961894 2778624 3827174 5147744 134 135 552960 885735 1976535 2799360 3855735 5186160 185 136 557056 892296 1991176 2820096 3884296 5224576 136 137 561152 898857 2005817 2840832 3912857 5262992 137 188 565248 905418 2020458 2861568 3941418 5301408 138 139 569344 911979 |° 2035099 2882304 3969979 5339824 139 140 573440 918540 2049740 2903040 3998540 5378240 140 141 577536 925101 2064381 2923776 4027101 5416656 141 142 581632 931662 2079022 2944512 4055662 5455072 142 143 585728 938223 2093663 2965248 4084223 5493488 143 144 589824 944784 2108304 2985984 4112784 5531904 144 145 593920 951345 2122945 3006720 4141345 5570320 145 146 598016 957906 2137586 3027456 4169906 5608736 146 147 602112 964467 2152227 3048192 4198467 5647152 147 148 606208 971028 2166868 3068928 4227028 5685568 148 149 610304 977589 2181509 3089664 4255589 5723984 149 150 614400 984150 2196150 3110400 4284150 5762400 150 108 Tables for Statisticians and Biometricians Ordinate 2—12. TABLE L—(continved). Frequency 151—200. 2416 2432 2448 2464 2480 2496 2512 2528 2544 2560 2576 2592 2608 2624 2680 2656 2672 2688 2704 2720 2736 2752 2768 2784 2800 2816 2832 2848 2864 2880 2896 2912 2928 2944 2960 2976 2992 3008 3024 3040 3056 3072 3088 3104 3120 3136 3152 3168 3184 3200 r=3 12231 12312 12393 12474 12555 12636 12717 12798 12879 12960 13041 13122 13203 13284 13365 13446 13527 13608 13689 13770 13851 13932 14013 14094 14175 14256 14337 14418 14499 14580 14661 14742 14823 14904 14985 15066 15147 15228 15309 15390 15471 15552 15633 15714 15795 15876 15957 16038 16119 16200 38656 38912 39168 39424 39680 39936 40192 40448 40704 40960 41216 41472 41728 41984 422.40 42.496 42752 43008 43264 43520 43776 44032 44288 44544 44800 45056 45312 45568 45824 46080 46336 46592 46848 47104 47360 47616 47872 48128 48384 48640 48896 49152 49408 49664 49920 50176 50432 50688 50944 51200 100000 100625 101250 101875 102500 103125 103750 104375 105000 105625 106250 106875 107500 108125 108750 109375 110000 110625 111250 111875 112500 113125 113750 114875 115000 115625 116250 116875 117500 118125 118750 119375 120000 120625 121250 121875 122500 123125 123750 128375 125000 195696 196992 198288 199584 200880 202176 203472 204768 206064 207360 208656 209952 211248 212544 213840 215136 216432 217728 219024 220320 221616 222912 224208 225504 226800 228096 229392 230688 231984 233280 234576 235872 237168 238464 239760 241056 242352 243648 244944 246240 247536 248832 250128 251424 252720 254016 255312 256608 257904 259200 362551 364952 367353 369754 372155 374556 376957 379358 381759 384160 386561 388962 391363 393764 396165 398566 400967 403368 405769 408170 410571 412972 415373 417774 420175 422576 424977 427378 429779 432180 434581 436982 439383 441784 444185 446586 448987 451388 453789 456190 458591 460992 463393 465794 468195 470596 472997 475398 477799 480200 618496 622592 626688 630784 634880 638976 643072 647168 651264 655360 659456 663552 667648 671744 675840 679936 684032 688128 692224 696320 700416 704512 708608 712704 716800 720896 724992 729088 733184 737280 741376 745472 749568 753664 757760 761856 765952 770048 778240 782336 786432 790528 794624 798720 802816 806912 811508 815104 819200 774144. x=9 990711 997272 1003833 1010394 1016955 1023516 1030077 1036638 1043199 1049760 1056821 1062882 1069443 1076004 1082565 1089126 1095687 1102248 1108809 1115370 1121931 1128492 11385053 1141614 1148175 1154736 1161297 1167858 1174419 1180980 1187541 1194102 1200663 1207224 1213785 1220346 1226907 1233468 1240029 1246590 1253151 1259712 1266273 1272834 12793895 1285956 1292517 1299078 1305639 1312200 ZA 2210791 2225432 2240073 2254714 2269355 2283996 2298637 2313278 2327919 2342560 2357201 2371842 2386483 2401124 2415765 2430406 2445047 2459688 2474329 2488970 2503611 2518252 2532893 2547534 2562175 2576816 2591457 2606098 2620739 2635380 2650021 2664662 2679303 2693944 2708585 2723226 2737867 2752508 2767149 2781790 2796431 2811072 2825713 2840354 2854995 2869636 2884277 2898918 2913559 2928200 3131136 3151872 3172608 3193344 3214080 3234816 3255552 3276288 3297024 3317760 3338496 3359232 3379968 3400704 3421440 3442176 3462912 3483648 3504384 3525120 3545856 3566592 3587328 3608064 3628800 3649536 3670272 3691008 3711744 3732480 3753216 3773952 3794688 3815424 3836160 3856896 3877632 3898368 3919104 3939840 3960576 3981312 4002048 4022784 4043520 4064256 4084992 4105728 4126464 4147200 r=12 n Verification of the Fourth Moment TABLE L—(continued), Ordinate 2—11, Frequency 201—250. 109 n Aa eo v=4 ei r=6 2=7 2=8 =—o c= 11 201 | 3216 | 16281 | 51456 | 125625 | 260496 | 482601 823296 | 1318761 | 2942841 202 | 3232 | 16362 | 51712 | 126250 | 261792 | 485002 827392 | 13825322 | 2957482 202 | 3248 | 16443 | 51968 | 126875 | 263088 | 487403 831488 | 1831883 | 2972123 204 | 3264 | 16524 | 52224 | 127500 | 264384 | 489804 835584 | 1838444 | 2986764 205 | 3280 | 16605 | 52480 | 128125 | 265680 | 492205 839680 | 1845005 | 3001405 206 | 3296 | 16686 | 52736 | 128750 | 266976 | 494606 843776 | 1851566 | 3016046 207 | 3312 | 16767 | 52992 | 129375 | 268272 | 497007 847872 | 1358127 | 3030687 208 | 3328 | 16848 | 538248 | 180000 | 269568 | 499408 851968 | 1864688 | 3045328 209 | 3844 | 16929 | 53504 | 130625 | 270864 | 501809 856064 | 1371249 | 3059969 210 | 3860 | 17010 | 58760 | 181250 | 272160 | 504210 860160 |} 1377810 | 3074610 211 | 3376 | 17091 | 54016 | 131875 | 273456 | 506611 864256 | 1884371 | 3089251 212 | 3392 | 17172 | 54272 | 132500 | 274752 | 509012 868352 | 13890932 | 3103892 213 | 3408 | 17253 | 54528 | 183125 | 276048 | 511413 872448 | 13897493 | 3118533 214 | 3424 | 17334 | 54784 | 183750 | 277344 | 513814 876544 | 1404054 | 3133174 215 | 3440 | 17415 | 55040 | 134375 | 278640 | 516215 880640 | 1410615 | 3147815 216 | 3456 | 17496 | 55296 | 135000 | 279936 | 518616 884736 | 1417176 | 3162456 217 | 3472 | 17577 | 55552 | 1385625 | 281232 | 521017 888832 | 1423737 | 3177097 218 | 3488 | 17658 | 55808 | 136250 | 282528 | 523418 892928 | 1480298 | 3191738 219 | 3504 | 17739 | 56064 | 136875 | 283824 | 525819 897024 | 1436859 | 3206379 220 | 3520 | 17820 | 56320 | 187500 | 285120 | 528290 901120 | 1443420 | 3221020 221 | 3536 | 17901 | 56576 | 138125 | 286416 | 530621 905216 | 1449981 | 3235661 222 | 3552 | 17982 | 56832 | 138750 | 287712 | 533022 909312 | 1456542 | 3250302 223 | 3568 | 18063 | 57088 | 139375 | 289008 | 535423 913408 | 1463103 | 3264943 224 | 3584 | 18144 | 57344 | 140000 | 290304 | 537824 917504 | 1469664 | 3279584 225 | 3600 | 18225 | 57600 | 140625 | 291600 | 540225 921600 | 1476225 | 3294225 226 | 3616 | 18306 | 57856 | 141250 | 292896 | 542626 925696 | 1482786 | 3308866 227 | 3632 | 18387 | 58112 | 141875 | 294192 | 545027 929792 | 1489347 | 3323507 228 | 3648 | 18468 | 58368 | 142500 | 295488 | 547428 933888 | 1495908 | 3338148 229 | 3664 | 18549 | 58624 | 143125 | 296784 | 549829 937984 | 1502469 | 3352789 230 | 3680 | 18630 | 58880 | 143750 | 298080 | 552230 942080 | 1509030 | 3367430 231 | 3696 | 18711 | 59136 | 144375 | 299376 | 554631 946176 | 1515591 | 3382071 282 | 3712 | 18792 | 59392 | 145000 | 300672 | 557032 950272 | 1522152 | 3396712 283 | 3728 | 18873 | 59648 | 145625 | 301968 | 559433 954368 | 1528713 | 3411353 284 | 3744 | 18954 | 59904 | 146250 | 303264 | 561834 958464 | 1535274 | 3425994 285 | 3760 | 19035 | 60160 | 146875 | 304560 | 564235 962560 | 1541835 | 3440635 236 | 3776 | 19116 | 60416 | 147500 | 305856 | 566636 966656 | 1548396 | 3455276 237 | 3792 | 19147 | 60672 | 148125 | 307152 | 569037 970752 | 1554957 | 3469917 288 | 3808 | 19278 | 60928 | 148750 | 308448 | 571438 974848 | 1561518 | 3484558 239 | 3824 | 19359 | 61184 | 149375 | 309744 | 573839 978944 | 1568079 | 3499199 240 | 3840 | 19440 | 61440 | 150000 | 311040 | 576240 983040 | 1574640 | 3513840 241 | 3856 | 19521 | 61696 | 150625 | 312336 | 578641 987136 | 1581201 | 3528481 242 | 3872 | 19602 | 61952 | 151250 | 3136382 | 581042 991232 | 1587762 | 3543122 248 | 3888 | 19683 | 62208 | 151875 | 314928 | 583443 995328 | 1594323 | 3557763 244 | 3904 | 19764 | 62464 | 152500 | 316224 | 585844 999424 | 1600884 | 3572404 2 3920 | 19845 | 62720 | 153125 | 317520 | 588245 | 1003520 | 1607445 | 3587045 246 | 3936 | 19926 | 62976 | 153750 | 318816 | 590646 | 1007616 | 1614006 | 3601686 247 | 3952 | 20007 | 63232 | 154375 | 320112 | 593047 | 1011712 | 1620567 | 3616327 248 | 3968 | 20088 | 63488 | 155000 | 321408 | 595448 | 1015808 | 1627128 | 3630968 249 | 3984 | 20169 | 63744 | 155625 | 322704 | 597849 | 1019904 | 1633689 | 3645609 250 | 4000 | 20250 | 64000 | 156250 | 324000 | 600250 | 1024000 | 1640250 | 3660250 110 Tables jor Statisticians and Biometricians Ordinate 2—9. TABLE L—(continued). Frequency 251—300. 21141 21222 21303 21384 21465 21546 21627 21708 21789 21870 21951 22032 22113 22194 22275 22356 22437 22518 22599 22680 22761 22842 22923 23004 23085 23166 23247 23328 23409 23490 23571 23652 23733 23814 23895 23976 24057 24138 24219 24300 65280 65536 65792 66048 66304 66560 66816 67072 67328 67584 67840 68096 68352 68608 68864 69120 69376 69632 69888 70144 70400 70656 70912 71168 71424 71680 71936 72192 72448 72704 72960 73216 73472 73728 73984 74240 74496 74752 75008 75264 75520 75776 76032 76288 76544 76800 156875 | 325296 | 602651 | 1028096 157500 | 326592 | 605052 | 1032192 158125 | 327888 | 607453 | 1036288 158750 | 329184 | 609854 | 1040384 159375 | 330480 | 612255 | 1044480 160000 | 331776 | 614656 | 1048576 160625 | 333072 | 617057 | 1052672 161250 | 334368 | 619458 | 1056768 161875 | 335664 | 621859 | 1060864 162500 | 3386960 | 624260 | 1064960 163125 | 338256 | 626661 | 1069056 163750 | 339552 | 629062 | 1073152 164375 | 340848 | 631463 | 1077248 165000 | 342144 | 633864 | 1081344 165625 | 343440 | 636265 | 1085440 166250 | 344736 | 638666 | 1089536 166875 | 346032 | 641067 | 1093632 167500 | 347328 | 643468 | 1097728 168125 | 348624 | 645869 | 1101824 168750 | 349920 | 648270 | 1105920 169375 } 3512¥6 | 650671 | 1110016 170000 | 352512 | 653072 | 1114112 170625 | 353808 | 655473 | 1118208 171250 | 355104 | 657874 | 1122304 171875 | 356400 | 660275 | 1126400 172500 | 357696 | 662676 | 1130496 173125 | 358992 | 665077 | 1134592 173750 | 360288 | 667478 | 1138688 174375 | 361584 | 669879 | 1142784 175000 | 362880 | 672280 | 1146880 175625 | 364176 | 674681 | 1150976 176250 | 365472 | 677082 | 1155072 176875 | 366768 | 679483 | 1159168 177500 | 368064 | 681884 | 1163264 178125 | 369360 | 684285 | 1167360 178750 | 370656 | 686686 | 1171456 179375 | 371952 | 689087 | 1175552 180000 | 373248 | 691488 | 1179648 180625 | 374544 | 693889 | 1183744 181250 | 375840 | 696290 | 1187840 181875 | 377136 | 698691 | 1191936 182500 | 378432 | 701092 | 1196032 183125 | 379728 | 703493 | 1200128 183750 | 381024 | 705894 | 1204224 184375 | 382320 | 708295 | 1208320 185000 | 383616 | 710696 | 1212416 185625 | 384912 | 713097 | 1216512 186250 | 386208 | 715498 | 1220608 186875 | 387504 | 717899 | 1224704 187500 | 388800 | 720300 | 1228800 z=9 1646811 1653372 1659933 1666494 1673055 1679616 1686177 1692738 1699299 1705860 1712421 1718982 1725543 1732104 1738665 1745226 1751787 1758348 1764909 1771470 1778031 1784592 1791153 1797714 1804275 1810836 1817397 1823958 1830519 1837080 1843641 1850202 1856763 1863324 1869885 1876446 1883007 1889568 1896129 1902690 1909251 1915812 1922373 1928934 1935495 1942056 1948617 1955178 1961739 1968300 Verification of the Fourth Moment 111 TABLE L—(continued). Ordinate 2—8. Frequency 301—350. n z—2 z2=3 a=4 2=5 z=6 z=7 z=8 n 24381 77056 188125 390096 722701 1232896 301 24462 77312 188750 391392 725102 1236992 302 24543 77568 189375 392688 727503 1241088 303 24624 77824 190000 393984 729904 1245184 304 24705 78080 190625 395280 732305 1249280 305 24786 78336 191250 396576 734706 1253376 306 24867 78592 191875 397872 737107 1257472 307 24948 78848 192500 399168 739508 1261568 308 25029 79104 193125 400464 741909 1265664 309 25110 79360 193750 401760 744310 1269760 310 25191 79616 194375 403056 746711 1273856 811 25272 79872 195000 404352 749112 1277952 312 25353 80128 195625 405648 751513 1282048 518 25434 80384 196250 406944 753914 1286144 S14 25515 80640 196875 408240 756315 1290240 815 25596 80896 197500 409536 758716 1294336 316 25677 81152 198125 410832 761117 1298432 317 25758 81408 198750 412128 763518 1302528 318 25839 81664 199375 413424 765919 1306624 319 25920 81920 200000 414720 768320 1310720 320 26001 82176 200625 416016 770721 1314816 321 26082 82432 201250 417312 773122 1318912 522 26163 82688 201875 418608 775523 1323008 823 26244 82944 202500 419904 777924 1327104 324 26325 83200 203125 421200 780325 1331200 825 26406 83456 203750 422496 782726 1335296 826 26487 83712 204375 423792 785127 1339392 327 26568 83968 205000 425088 787528 1343488 28 26649 84224 205625 426384 789929 1347584 529 26730 84480 206250 427680 792330 1351680 30 26811 84736 206875 428976 794731 1355776 331 26892 84992 207500 430272 797132 1359872 32 26973 85248 208125 431568 79953° 1363968 333 27054 85504 208750 432864 801934 1368064 SSL 27135 85760 209375 434160 804335 1372160 B35 27216 86016 210000 435456 806736 1376256 336 27297 86272 210625 436752 809137 1380352 B37 27378 86528 211250 438048 811538 1384448 SS, 27459 86784 211875 459344 813939 1388544 359 27540 87040 212500 440640 816340 1392640 540 27621 87296 213125 441936 818741 1396736 S41 27702 87552 213750 443232 821142 1400832 342 27783 87808 214375 444528 823543 1404928 543 27864 85064 215000 445824 825944 1409024 S44 27945 88320 215625 447120 828345 1413120 B45 28026 88576 216250 448416 830746 1417216 346 28107 88832 216875 449712 833147 1421312 S347 28188 89088 217500 451008 | 835548 1425408 348 28269 89344 218125 452304 837949 1429504 349 28350 89600 218750 453600 840350 1433600 350 112 Tables for Statisticians and Biometricians TABLE L—(continued). Ordinate 2—7. Frequency 351—400. n r=2 3) c—— a=5 351 5616 28431 89856 219375 352 56382 28512 90112 220000 8538 5648 28593 90368 220625 354 5664 28674 90624 221250 355 5680 28755 90880 221875 350 5696 28836 91136 222500 357 5712 28917 91392 223125 358 5728 28998 91648 223750 859 5744 29079 91904 224375 860 5760 29160 92160 225000 861 5776 29241 92416 225625 362 5792 29322 92672 226250 S68 5808 29403 92928 226875 864 5824 29484 93184 227500 865 5840 29565 93440 228125 866 5856 29646 ~ 93696 228750 367 5872 29727 93952 229375 568 5888 29808 94208 230000 869 5904 29889 94464 230625 3870 5920 29970 94720 231250 387 5936 30051 94976 231875 372 5952 30132 95232 232500 373 5968 30213 95488 233125 37. 5984 30294 95744 233750 B75 6000 30375 96000 234375 376 6016 30456 96256 235000 377 6032 30537 96512 235625 378 6048 30618 96768 236250 379 6064 30699 97024 236875 380 6080 30780 97280 237500 381 6096 30861 97536 238125 382 6112 30942 97792 238750 383 6128 31023 98048 239875 384 6144 31104 98304 240000 385 6160 31185 98560 240625 386 6176 31266 98816 241250 387 6192 31347 99072 241875 388 6208 31428 99328 242500 589 6224 31509 99584 243125 890 6240 31590 99840 243750 891 6256 31671 100096 244375 3592 6272 31752 100352 245000 3893 6288 31833 100608 245625 S94 6304 31914 100864 246250 395 6320 31995 101120 246875 3896 6336 32076 101376 247500 397 6352 32157 101632 248125 898 6368 32238 101888 248750 399 6384 32319 102144 249375 400 6400 32400 102400 250000 454896 456192 457488 458784 460080 461376 462672 463968 465264 466560 467856 469152 470448 471744 473040 474336 475632 476928 478224 479520 480816 482112 483408 484704 486000 487296 488592 489888 491184 492480 493776 495072 496368 497664 498960 500256 501552 502848 504144 505440 506736 508032 509328 510624 511920 513216 514512 515808 517104 518400 842751 845152 847553 849954 852355 854756 857157 859558 861959 864360 866761 869162 871563 873964 876365 878766 881167 883568 885969 888370 890771 893172 895573 897974 900375 902776 905177 907578 909979 912380 914781 917182 919583 921984 924385 926786 929187 931588 933989 936390 938791 941192 943593 945994 948395 950796 953197 955598 957999 960400 Sol 352 853 SL 855 356 S57 358 859 360 S61 S62 363 S64 365 366 S67 368 369 3870 3871 S72 373 STL 37, 3ST6 3T 378 79 380 381 S82 B83 584 885 386 387 588 589 890 8o1 892 S98 394 895 596 897 598 3899 400 ———_—_—) Poisson's Exponential Binomial Limit 113 TABLE LI. Tables of e™m?*/x!: General Term of Poisson's Exponential Expansion (“ Law of Small Numbers”). aa Bicefelo-e a cre enee || 8 | waramrcrane | 0-1 02 ™m 08 904837 090484 004524 “000151 “000004 Let 382871 "366158 201387 073842 020307 004467 000819 “000129 “000018 “000002 818731 163746 016375 001092 000055 “000002 ‘740818 "222245 033337 003334 000250 “000015 “000001 O-4 670320 268128 053626 “007150 000715 000057 “000004 12 “301194 361433 *216860 "086744 026023 “006246 “001249 “000214 *000032 “000004 “000001 8 21 22 13 *272532 354291 230289 099792 032432 008432 001827 000339 “000055 “000008 “000001 14 0-5 0-6 07 0-8 606531 303265 075816 012636 001580 000158 ‘000013 000001 15 5648812 329287 098786 019757 002964 “000356 000036 000003 16 496585 347610 "121663 “028388 004968 “000696 000081 000008 “000001 i) *449329 359463 143785 038343 007669 001227 “000164 “000019 000002 1°8 09 406570 365913 164661 049398 “011115 002001 000300 000039 “000004 1°9 10 367879 367879 "183940 061313 015328 003066 000511 ‘000073 “000009 “000001 2-0 SMNVA AK MRS 8 246597 345236 241665 112777 039472 011052 002579 “000516 “000090 “000014 000002 “223130 *384695 251021 "125510 047067 014120 003530 “000756 “000142 000024 “000004 .201897 .823034 - 258428 "137828 055131 “017642 004705 “001075 “000215 “000088 “900006 ‘000001 182684 310562 263978 149587 063575 021615 “006124 001487 “000316 “000060 “000010 “000002 165299 297538 "267784 “160671 072302 “026029 “007809 “002008 “000452 “000090 “000016 000003 25 26 27 2:8 “149569 284180 269971 170982 ‘081216 “030862 009773 002653 “000630 “000133 “000025 “000004 “000001 29 135335 270671 270671 180447 090224 ‘036089 012030 003437 000859 “000191 “000038 *000007 “000001 3-0 WONANK SMHS “122456 "257159 270016 “189012 099231 041677 ‘014587 004376 7001149 “000268 “000056 000011 “000002 B. 110803 243767 268144 196639 *108151 047587 017448 005484 “001508 “000369 “000081 “000016 “000003 000001 | “100259 *230595 "265185 “203308 116902 “053775 020614 006773 001947 “000498 000114 “000024 “000005 “000001 090718 217723 261268 “209014 125409 060196 *024078 “008255 002477 ‘000660 000158 “000035 “000007 ‘000001 “082085 205212 256516 213763 133602 ‘066801 027834 009941 003106 “000863 “000216 000049 “000010 ‘000002 074274 193111 251045 217572 "141422 073539 031867 “011836 003847 001111 000289 “000068 “000015 “000003 “000001 067206 "181455 "244964 220468 148816 080360 "036162 013948 004708 001412 “000381 “000094 “000021 “000004 “000001 060810 “170268 238375 "222484 "155739 “087214 “040700 016280 005698 001773 “000496 000126 “000029 “000006 “000001 “055023 159567 231373 "223660 162154 094049 045457 “018832 006827 “002200 000638 "000168 “000041 000009 “000002 049787 149361 "224042 "224042 “168031 100819 050409 021604 “008102 002701 “000810 “000221 000055 ‘000013 “000003 “000001 15 a a eas eomenl| s | WHDSDAVRANE WMHS mw S ili lat cS) aa Si So rs Tables for Statisticians and Biometricians a7 024724 091477 169233 208720 193066 "142869 “088102 046568 0215388 “008854 003276 001102 000340 “000097 “000026 “000006 000001 47 009095 042748 100457 157383 "184925 "173830 136167 091426 053713 028050 013184 005633 “002206 “000798 000268 “000084 “000025 “000007 000002 a ae] 003346 “019072 054355 103275 114 TABLE LI—(continued). | ™ a 31 32 33 Sh 35 36 0 | -045049 | 040762 | 036883 | -033373 | 030197 | -027324 1 | -1389653 | -130489 | -121714 | +113469 | -105691 | :098365 2 | +216461 | *208702 | -200829 | :192898 | -184959 | 177058 3 | 223677 | -222616 | *220912 | :218617 | :215785 | -212469 4, | 173350 | *178093 | *182252 | *185825 | °188812 | :191222 5 | (107477 | *113979 | *120286 | *126361 | 132169 | -137680 6 | °055530 | 060789 | "066158 | -071604 | 077098 | 082608 7 | 024592 | -027789 | -031189 | ‘034779 | 038549 | -042484 8 | 009529 | 011116 | ‘012865 | -014781 | -016865 | 019118 9 | 003282 | 003952 | -004717 | ‘005584 | 006559 | -007647 10 | 001018 | ‘001265 | -001557 | :001899 | :002296 | :002753 11 | ‘000287 | 000368 | -000467 | :000587 | 000730 | 000901 12 | -000074 | 000098 | °000128 | -000166 | :000213 | :000270 18 | 000018 | 000024 | -000033 | :000043 | 000057 | -000075 14 | -000004 | -000006 | 000008 | :000011 | :000014 | -000019 15 | °000001 | -000001 | :000002 | -000002 | -000003 | :000005 16 _— _ _ “000001 | -000001 | 000001 17 _ _ = _ _ _ z| 41 42 43 44 45 46 0 | 016573 | 014996 | 013569 | 012277 | 011109 | -010052 1 | (067948 | -062981 | -058345 | ‘054020 | 049990 | :046238 2 | -189293 | *132261 | -125441 | ‘118845 | -112479 | :106348 3 | °190368 | *185165 | -179799 | *174305 | °168718 | -163068 4, | °195127 | *194424 | -193284 | -191736 | *189808 | :187528 5 | -160004 | -163316 | :166224 | -168728 | °170827 | *172525 6 | *109336 | °114321 | -119127 | °123734 | °128120 | -132270 7 | 064040 | 068593 | -073178 | ‘077775 | ‘082363 | :086920 8 | -032820 | -036011 | :039333 | 042776 | 046329 | 049979 9 | 014951 | *016805 | :018798 | 020913 | 023165 | 025545 10 | -006130 | ‘007058 | -008081 | 009202 | -010424 | -011751 11 | -002285 | -002695 | -003159 | ‘003681 | -004264 | 004914 12 | :000781 |.:000943 | -001132 | 001350 | -001599 | -001884 13 | 000246 | -000305 | :000374 | 000457 | -000554 | -000667 14 | 000072 | -000091 } :000115 | 000144 | -000178 | :000219 15 | 000020 | 000026 | :000033 | -000042 | -000053 | :000067 16 | 000005 | ‘000007 | -000009 | ‘000012 | :000015 | -000019 17 | ‘000001 | -000902 | -000002 | -000008 | :000004 | :000005 18 — —_ “000001 | 000091 | -000001 | :000001 19) a ae = = =| a | 51 52 53 5h 55 5-6 0 | :006097 | ‘005517 | -004992 | ‘004517 | ‘004087 | -003698 1 | ‘031093 | ‘028686 | -026455 | ‘024390 | :022477 | -020708 2 | ‘079288 | 074584 | 070107 | ‘065852 | 061812 | ‘057982 3 | °134790 | -129279 | -193856 | °118533 | °113393 | -108234 | | a8 022371 085009 161517 204588 194359 147713 093551 “050785 024123 “010185 “003870 001337 “000423 "000124 000034 “000009 “000002 3g 020242 078943 153940 200122 195119 152193 098925 “055115 026869 011643 004541 ‘001610 “000523 000157 “000044 000011 000003 “000001 40 018316 073263 “146525 “195367 195367 156293 "104196 059540 029770 013231 "005292 001925 “000642 000197 “000056 “000015 “000004 “000001 Rh RRR NAAE WMH TOANAAK LOND | 48 49 50 58 003028 017560 050923 “098452 008230 “039503 094807 151691 “182029 "174748 139798 095862 057517 030676 014724 006425 “002570 “000949 000325 “000104 000031 “000009 000002 “000001 007447 ‘0836488 089396 146014 178867 175290 143153 “100207 061377 033416 016374 007294 002978 “001123 000393 000128 “000039 ‘000011 000003 000001 59 002739 016163 “047680 093771 ‘006738 033690 "084224 "140374 “175467 175467 "146223 "104445 065278 036266 018133 008242 “003434 001321 000472 “000157 “000049 000014 000004 “000001 60 002479 014873 044618 089235 me SODVWAMNH WINS | 8 | xz Sail 4 | -171857 5 | 175294 6 | °149000 7 | *108557 § | 069205 9 | -039216 10 | 020000 11 | :009273 12 | 003941 13 | 001546 14 | -000563 15 | 000191 16 | 000061 17 | 000018 18 | ‘000005 19 | -000001 20 — 21 — z 61 0 | 002243 1 | 013682 2 | 041729 3 | 084848 4, | 129393 5 | *157860 6 | 160491 7 | 139856 8 | 106640 9 | ‘072278 10 | 7044090 11 | (024450 12 | 012429 13 | 7005832 14 | 002541 15 | :001033 16 | -000394 17 | ‘000141 18 | 000048 19 | ‘000015 20 | 000005 21 | 000001 22 _— “168063 174785 “151480 "112528 073143 042261 021976 -010388 “004502 “001801 000669 "000232 000075 “000023 “000007 000002 6-2 "002029 012582 “039006 “080612 "124948 "154936 -160100 141803 *109897 075707 046938 026456 ‘013669 “006519 002887 “001193 000462 000169 000058 “000019 “000006 “690002 Poisson's Huponential Binomial Limit TABLE LiI—(continued). *164109 1738955 153660 116343 ‘077077 045390 024057 011591 “005119 002087 “000790 “000279 “000092 “000029 “000008 “000002 “000001 63 001836 “011569 036441 076527 “120530 151868 "159461 *143515 ‘113018 079113 “049841 "028545 “014986 007263 “003268 ‘001373 “000540 000200 “000070 000023 “000007 “000002 “000001 ah 55 56 “160020 172821 "155539 119987 “080991 "048595 026241 “012882 *005797 002408 “000929 “000334 “000113 “000036 000011 “000003 000001 64 “001662 010634 “034029 072595 "116151 "148674 *158585 144992 115994 082484 052790 030714 016381 “008064 003687 001573 000629 000237 “000084 000028 “000009 “000003 ‘000001 155819 “171401 157117 "123449 084871 “051866 028526 014263 006537 *002766 001087 “000398 000137 “000044 “000014 “000004 “000001 65 001508 009772 “031760 068814 111822 "145369 157483 "146234 118815 085811 °055777 032959 017853 “008926 004144 001796 000730 000279 “000101 “000034 “000011 000008 | “000001 151528 169711 158397 126717 088702 055192 030908 015735 007343 003163 001265 000472 000165 “000054 “000017 000005 “000001 66 001360 008978 "029629 065183 107553 141969 156166 "147243 “121475 “089082 058794 035276 “019402 “009850 004644 “002043 000843 *000327 “000120 000042 “000014 “000004 “000001 57 *147167 “167770 159382 “129782 092470 "058564 033382 017298 008216 003603 001467 000557 -000199 000067 “000021 “000006 “000002 67 aT oo 5:9 115 6°0 142755 165596 “160076 132635 096160 061970 035943 “018952 “009160 004087 001693 “000655 000237 “000081 000026 “000008 “000002 “000001 68 138312 “163208 160488 “135268 099760 065398 "038585 020696 010175 004618 001946 “000766 “000282 “000098 “000032 “000010 000003 “000001 6:9 133853 "160623 160623 137677 103258 068838 041303 022529 “011264 “005199 “002228 000891 000334 “000118 “000039 000012 “000004 “000001 Ut) 001231 008247 027628 061702 103351 “138490 154648 “148020 123967 092286 061832 037661 021028 010837 005186 002317 “000970 000382 "000142 “000050 000017 “000005 “000002 “001114 007574 025751 058368 099225 "134946 “152939 148569 126284 “095415 064882 040109 022728 011889 005774 002618 ‘001113 “000445 “000168 “000060 “000020 “000007 “000002 000001 001008 006954 023990 055178 095182 131351 151053 148895 "128422 098457 067935 “042614 “024503 013005 ‘006410 “002949 001272 000516 000198 000072 “000025 “000008 000003 “000001 “000912 006383 022341 052129 091226 127717 “149003 “149003 130377 "101405 070983 *045171 026350 014188 “007094 003311 “001448 000596 “000232 “000085 “000030 “000010 “000003 000001 15—2 SH COVA NS CsPMHSD WONVA AY WONDS 8 | 1S MRNA nts SMS 116 “087364 124057 “146800 “148897 132146 "104249 074017 “047774 028267 015438 007829 “003706 “001644 “000687 -000271 -000101 “000036 “000012 “000004 “000001 Tables for Statisticians and Biometricians *005375 019352 046444 “083598 “120382 "144458 "148586 °133727 "106982 077027 “050418 030251 016754 008616 004136 001861 “000788 “000315 “000119 “000043 000015 “000005 “000002 043799 079934 116703 "141989 "148074 135118 "109596 080005 053094 032299 018137 009457 004603 002100 000902 000366 000141 000051 “000018 000006 000002 “000001 8-1 000304 “002459 “009958 “026885 054443 088198 “119067 137778 "139500 125550 “101696 074885 “050547 “031495 018222 “009840 004981 002373 001068 “000455 “000184 &2 000275 002252 009234 025239 051740 084854 115967 “135848 “139244 “126866 "104031 077550 | 052993 033426 “019578 |} 83 “000249 002063 “008560 023683 049142 “081576 “112847 "133805 "138823 *128025 *106261 ‘080179 *055457 ‘035407 020991 ‘010703 | -011615 “005485 | ° 006025 002646 | ‘002942 001205 | “000520 | - ‘000213 | ° 001356 000593 a | TABLE LIi—(continued). 004523 016736 041282 076372 “113031 *139405 "147371 136318 112084 “082942 055797 “034408 “019586 “010353 005107 *002362 “001028 000423 “000165 000061 000021 000007 “000002 000001 EDA "000225 | “001889 0079383 022213 “046648 078368 “109716 "131659 138242 129026 "108382 “082764 057935 037435 “022461 012578 “006604 003263 001523 000673 000283 015555 “038889 072916 "109375 “136718 "146484 137329 “114440 “085830 058521 036575 021101 0113804 005652 “002649 “001169 000487 “000192 “000072 “000026 “000009 “000003 “000001 85 “000203 001729 007350 020826 044255 075233 106581 “129419 137508 “129869 “110388 “085300 060421 “039506 023986 “013592 007221 “003610 001705 “000763 “000324 014453 “036614 069567 “105742 “133940 “145421 “138150 116660 “088661 *061257 038796 022681 012312 006238 002963 001325 “000559 000224 000085 *000031 000011 000004 “000001 6 000184 001583 “006808 019517 041961 “072174 "103449 127094 “136626 130554 112277 087780 062909 041617 025565 “014657 ‘007878 003985 “001904 “000862 ‘000371 013424 "034455 “066326 “102142 “131082 “144191 138783 118737 091427 063999 041066 "024324 013378 ‘006867 ‘003305 001497 “000640 “000259 “000100 000037 ‘000013 “000004 “000001 &7 000167 “001449 006304 ‘018283 039765 “069192 “100328 "124693 135604 "131084 114043 “090197 “065393 043763 027196 ‘015773 ‘008577 004389 “002121 ‘000971 000423 78 “000410 “003196 012464 032407 “063193 “098581 128156 "142802 139232 “120668 “094121 “066740 043381 “026029 “014502 “007541 003676 001687 000731 “000300 *000117 “000043 “000015 “000005 “000002 “000001 000371 002929 011569 030465 ‘060169 095067 125171 141264 139499 “122449 096735 069473 045736 027794 015684 “008260 “004078 “001895 “000832 000346 ‘000137 “000051 ‘000018 000006 000002 “000001 000335 002684 ‘010735 028626 057252 “091604 "122138 “139587 139587 124077 099262 072190 *048127 029616 “016924 009026 004513 002124 “000944 000397 “000159 “000061 *000022 000008 “0000035 “000001 | | &°8 000151 001326 005836 “017120 037664 066289 "097224 "122224 "134446 “131459 “115684 092547 “067868 “045941 “028877 “016941 ‘009318 004823 002358 “001092 “000481 &9 000136 001214 “005402 “016025 035656 063467 094143 119696 “133161 131682 117197 094823 070327 “048147 “030608 ‘018161 “010102 005289 002615 001225 “000545 9°0 000123 ‘001111 004998 014994 033737 060727 091090 117116 131756 131756 “118580 097020 072765 050376 “032384 019431 “010930 ‘005786 002893 001370 000617 WONVA AS SMHS WDMNVAASYLCDOHRS Poisson's Hxponential Binomial Limit 117 TABLE LI—(continued). m xz az 81 8:2 83 Sh 85 86 8-7 8:8 8-9 9-0 21 | -000071 | -000083 | -000097 | 000113 | -000131 | 000152 | -000175 | -000201 | 000231 | -000264 | 27 22 | -000026 | 000031 | -000037 | 000043 | -000051 | 000059 | -000069 | -000081 | -000093 | ‘000108 | 22 23 | -000009 | -000011 | -000013 | -000016 | -000019 | 000022 | -000026 | -000031 | -000036 | -000042 | 23 24, | 000008 | -000004 | -000005 | 000006 | -000007 | 000008 | -000009 | -000011 | 000013 | -000016 | 24 25 | -000001 | -000001 | -000002 | -000002 | -000002 | -000003 | 000003 | -000004 | -000005 | -000006 | 25 ae = = — | -000001 | -000001 | -000001 | -000001 | «000001 | -000002 | 000002 | 26 Ei = = = = == = = — | 000001 | -000001 | 27 a | 91 9:2 9°3 94 95 96 97 9°8 99 | 100 | « 0 | -000112 | 000101 | -000091 | 000083 | -000075 | 000068 | “000061 | -000055 | -000050 | -000045 | 0 1 | -001016 | -000930 | -000850 | -000778 | 000711 | -000650 | 000594 | -000543 | -000497 | -000454| 7 2 | -004624 | -004276 | -003954 | -003655 | -003378 | ‘003121 | ‘002883 | -002663 | 002459 | 002270] ¢ 3 | 014025 | -013113 | 012256 | -011452 | -010696 | -009987 | 009322 | -008698 | ‘008114 | 007567 | 3 4 | 031906 | -030160 | -028496 | -026911 | -025403 | -023969 | 022606 | -021311 | -020082 | 018917| 4 5 | 058069 | -055494 | 053002 | -050593 | 048266 | -046020 | 043855 | -041770 | ‘039763 | 037833 | 5 6 | -088072 | -085091 | -082154 | -079262 | -076421 | -073632 | 070899 | -068224 | 065609 | -063055 | 6 7 | 114493 | -111834 | -109147 | -106438 | -103714 | -100981 | 098246 | -095514 | ‘092790 | 090079 |_ 7 8 | -130236 | -128609 | -126883 | 125065 | -123160 | -121178 | 119123 | -117004 | “114827 | -112599| 38 9 | °131683 | -131467 | *131113 | -130623 | -130003 | -129256 | “128388 | -127405 | -126310 | *125110| 9 10 | *119832 | -120950 | 121985 | -122786 | -123502 | -124086 | “124537 | -124857 | “125047 | -125110 | 10 11 | 099133 | 101158 | -103090 | -104926 | -106661 | -108293 | -109819 | -111236 | -112542 | “113736 | 12 12 | -075176 | -077555 | ‘079895 | -082192 | -084440 | -086634 | ‘088770 | 090843 | -092847 | -094780 | 12 13 | 052623 | -054885 | 057156 | -059431 | -061706 | 063976 | ‘066236 | 068481 | -070707 | -072908 | 13 14 | *034205 | -036067 | 037968 | -039904 | -041872 | 043869 | “045892 | -047937 | -050000 | 052077 | 14 15 | 020751 | -022121 | ‘023540 | -025006 | -026519 | -028076 | ‘029677 | 031319 | 033000 | -034718 | 15 16 | 011802 | -012720 | 013683 | ‘014691 | -015746 | -016846 | 017992 | -019183 | ‘020419 | 021699 | 16 17 | -006318 | -006884 | ‘007485 | -008123 | -008799 | 009513 | -010266 | 011058 | -011891 | ‘012764 | 17 18 | -003194 | -003518 | -003867 | -004242 | -004644 | -005074 | ‘005532 | 006021 | -006540 | 007091 | 18 19 | 001580 | -001704 | -001893 | -002099 | -002322 | -002563 | -002824 | -003105 | -003408 | -003732 | 19 20 | 000696 | -000784 | °000880 | -000986 | -001103 | -001230 | -001370 | -001522 | -001687 | -001866 | 20 21 | -000302 | -000343 | “000390 | -000442 | -000499 | -000563 | -000633 | -000710 | -000795 | -000889 | 21 22 | 000125 | 000144 | -000165 | -000189 | -000215 | -000245 | -000279 | -000316 | 000358 | 000404 | 22 23 | 000049 | -000057 | -000067 | 000077 | -000089 | -000102 | -000118 | -000135 | -000154 | -000176 | 23 24, | 000019 | -000022 | -000026 | -000030 | -000035 | -000041 | 000048 | -000055 | 000064 | -000073 | 24 25 | 000007 | -000008 | 000010 | -000011 | -000013 | 000016 | -000018 | 000022 | 000025 | -000029 | 25 26 | -000002 | -000003 | -000008 | -000004 | -000005 | -000006 | -000007 | -000008 | -000010 | -000011 | 26 27 | “000001 | -000001 | “000001 | -000001 | «000002 | 000002 | “000002 | -000008 | -000004 | -000004 | 27 o5\2 = _ — | -000001 | -000001 | -000001 | «000001 | -000001 | -000001 | 28 ee = = a = = = = — | 000001 | 29 meor | 12 | i0-8 | to, |? ioe) 106 | 107 |) t08 | t0-9 | 11-0 | 0 | -000041 | 000037 | “000034 | -000030 | 000028 | -000025 | 000023 | -000020 , 000018 | 000017 | 0 1 | -000415 | -000379 | -000346 | 000317 | «000289 | -000264 | 000241 | -000220 | -000201 | -000184| 1 2 | -002095 | -001934 | -001784 | -001646 | -001518 | -001400 | -001291 | -001190 | -001097 | -001010| 2 3 | 007054 | -006574 | -006125 | 005705 | -005313 | -004946 | 004603 | -004283 | -003984 | -003705 | 3 118 Tables for Statisticians and Biometricians TABLE LI—(continued) 10°8 10-4 10 ‘017811 035979 “060565 ‘087387 110326 “123810 125048 “114817 096637 075080 054165 ‘036471 023022 013678 “007675 “004080 “002060 “000991 000455 000200 “000084 “000034 “000013 “000005 “000002 ‘000001 “000015 “000165 “000931 003445 009559 021221 039259 062253 ‘086376 "106531 “118249 119324 *110375 "094243 074721 055294 038360 025047 015446 009023 016764 034199 058139 “084716 “108013 “122415 124863 115782 098415 077218 056259 038256 “024388 014633 “008292 004451 002270 “001103 ‘000511 000227 “000096 000039 “000015 “000006 “000002 “000001 11°2 000014 “000153 “000858 “003202 “008965 020082 “037487 059979 “083970 "104496 117036 119164 “111220 “095820 076656 057236 “040065 “026396 016424 009682 015773 032492 nor “055777 ‘082072 “105668 “120931 *124559 "116633 “100110 “079318 058355 “040071 “025795 “015629 “0089438 “004848 “002497 “001225 “000573 “000257 “000110 “000045 ‘000018 ‘000007 “000003 “000001 11°3 “000012 000140 000790 002976 008406 018997 035778 057755 081579 *102427 115743 118899 111964 097322 078553, 059177 “041793 027780 ‘017440 010372 014834 | 080855 053482 | 079458 “103296 119364 124139 ‘117368 101719 081375 “060450 041912 027243 016666 009629 “005271 “002741 001357 “000642 000290 000126 “000052 *000021 “000008 000003 “000001 Its “000011 000128 000727 “002764 007879 017963 034130 055584 079206 “100328 1143874 118533 112607 098747 “080409 ‘061110 043541 029198 018492 ‘011095 | | 013946 "029287 051252 ‘076878 *100902 ‘117720 "123606 | "117987 | *103239 083385 062539 043777 ‘028729 017744 ‘010351 -005720 ‘003003 “001502 ‘000717 -000327 000143 -000060 000024 000009 000004 -000001 11°5 000010 000116 “000670 “002568 “007382 “016979 "032544 053465 076856 098204 112935 118068 113149 “100093 “082219 063035 045306 030648 019581 011852 10°6 ‘013107 ‘027786 049089 074334 098493 “116003 122963 "118492 104667 0853844 “064618 045663 "030252 018863 “011108 006197 003285 001658 000799 000368 000163 “000069 “000028 000011 “000004 “000002 , 000001 11°6 “000009 “000106 ‘000617 002385 006915 016043 031017 051400 074529 096060 111480 117508 113591 101358 083982 064946 ‘047086 032129 020706 012641 10°7 012313 026350 046991 071830 096072 114219 "122215 “118882 “106003 087248 ‘066683 047567 “031810 “020022 “011902 006703 003586 001827 “000889 “000413 000184 000079 “000032 “000013 “000005 “000002 “000001 11°7 000008 000097 000568 “002214 006476 015153 029549 “049388 072231 093900 “109863 “116854 113933 102539 085694 066841 048877 033639 021865 013465 10°8 011564 024978 “044960 069367 093646 112375 “121365 "119159 107243 “089094 068730 049485 033403 021220 012732 007237 003908 002010 000987 000463 “000208 “000090 000037 ‘000015 000006 000002 “000001 11°8 000008 “000089 000522 “002055 006062 ‘014307 028137 047432 069962 091728 *108239 116110 114175 103636 087350 068716 ‘050678 035176 023060 "014322 "010856 | ‘023667 042995 066949 091218 "110475 *120418 *119323 “108386 ‘090877 070754 051415 035026 022458 -013600 ‘007802 004252 -002207 -001093 000518 000235 -000108 000043 000017 -000007 000003 000001 11°9 ‘000007 000081 000481 001907 "005674 013504 026782 “045530 067725 089548 “106562 “115281 114320 "104647 “088950 070567 052484 036739 024288 015212 11:0 “010189 “022415 041095 064577 ‘088794 “108526 119378 119378 "109430 092595 072753 053352 036680 023734 “014504 008397 004618 “002419 001210 “000578 “000265 000117 000049 “000020 “000008 000003 “000001 12°0 000006 000074 000442 ‘001770 005309 012741 025481 043682 065523 087364 104837 114368 114363 105570 “090489 072391 054293 038325 025550 016137 8 | WM WANA SMHS Poisson's Exponential Binomial Limit 119 TABLE LI—(continued). m & x ee eet Se) it |b tia | tare || 1) tae | “tye, | 120 20 | ‘005008 | -005422 | -005860 | -006324 | -006815 | 007332 | -007877 | 008450 | 009051 | 009682 20 21 | 002647 | -002892 | -003153 | -003433 | -003732 | -004050 | -004388 | 004748 | -005129 | -005533 | 21 22 | -001336 | -001472 | -001620 | -001779 | -001951 | 002136 | -002334 | 002547 | -002774 | -003018 | 22 23 | -000645 | -000717 | -000796 | -000882 | -000975 | 001077 | 001187 | 001307 | -001435 | -001575 | 23 24, | -000298 | -000335 | “000375 | -000419 | -000467 | 000521 | -000579 | 000642 | -000712 | -000787 | 24 25 | -000132 | -000150 | -000169 | -000191 | -000215 | 000242 | -000271 | -000303 | -000339 | -000378 | 25 26 | -000057 | -000065 | -000074 | -000084 | -000095 | -000108 | -000122 | 000138 | -000155 | -000174 | 26 27 | -000023 | 000027 | “000031 | -000035 | -000041 | 000046 | -000053 | 000060 | -000068 | -000078 | 27 28 | -000009 | -000011 | -000012 | -000014 | -000017 | -000019 | 000022 | -000025 | 000029 | -000033 | 28 29 | -000004 | -000004 | «000005 | -000006 | 000007 | -000008 | -000009 | -000010 | -000012 | ‘000014 | 29 30 | 000001 | -000002 | -000002 | -000002 | -000003 | «000003 | -000003 | -000004 | «000005 | -000005 | 30 81 | — | -000001 | -000001 | -000001 | -000001 | -000001 | -000001 | -000002 | -000002 | -000002 | 31 32; — = = = a = — | -000001 | -000001 | -000001 | 32 eee 7) | we | 12s | 7o% | yee | 126 | 12-7 | 128 | 1289 | 180 | 0 | :000006 | 000005 | “000005 | -000004 | -000004 | 000003 -000003 | -000003 | -000002 | 000002 | 0 1 | 000067 | 000061 | -000056 | -000051 | -000047 | 000042 | -000039 | -000035 | -000032 000029 | 1 2 | 000407 | -000374 | 000344 | -000317 | 000291 | 000268 | -000246 | -000226 | -000208 | 000191 | 2 8 | 001641 | 001522 | :001412 | 001309 | -001213 | 001124 | -001042 | 000965 | -000894 | 000828 | 3 4 | 004966 | -004643 | 004341 | -004057 | -003791 | -003541 | -003307 | -003088 | -002882 | -002690| 4 5 | 012017 | -011330 | 010679 | -010062 | -009477 | -008924 | -008400 | -007905 | -007436 | 006994 | 5 6 | -024233 | -023037 | ‘021892 | -020794 | -019744 | ‘018740 | 017781 | 016864 | 015988 | 015153] 6 7 | 041889 | 040151 | ‘038467 | 036836 | -035258 | 033733 | -032259 | -030837 | -029464 | 028141 | 7 8 | 063358 | -061230 | -059142 | 057095 | -055091 | 053129 | -051212 | -049339 | ‘047511 | 045730 | 8 9 | 085181 | -083000 | :080828 | -078665 | -076515 | 074381 | 072266 | -070171 | -068100 | 066054) 9 10 | 103069 | -101261 | -099418 | 097544 | -095644 | 093720 | 091777 | 089819 | 087849 | ‘085870 | 10 11 | -113376 | -112308 | *111168 | -109959 | -108686 | *L07352 | ‘105961 | -104516 | -103023 | *101483 | 11 12 | 114321 | -114180 | *113947 | -113624 | -113215 | -112720 | -112142 | -111484 | -110749 | *109940 | 12 13 | *106406 | -107153 | *107811 | -108380 | -108860 | *109251 | *109554 | -109769 | 109897 | “109940 | 13 14 | 091965 | -093376 | *094720 | -095994 | -097197 | :098326 | 099381 | -100360 | 101263 | “102087 | 14 15 | 074185 | -075946 | 077670 | -079355 | -080997 | ‘082594 | -084143 | -085641 | -087086 | ‘088475 | 15 16 | 956103 | -057909 | 059709 | -061500 | -063279 | 065043 | -066788 | 068513 | -070213 | ‘071886 | 16 17 | 039932 | -041558 | 043201 | -044859 | -046529 | 048208 | -049895 | 051586 | -053279 | 054972 | 17 18 | -026843 | -025167 | 029521 | -030903 | -032312 | :033746 | 035204 | -036683 | 038183 | ‘039702 | 18 19 | ‘017095 | 018036 | ‘019111 | :020168 | -021258 | -022379 | 023531 | -024743 | 025925 | -027164 | 19 20 | 010342 | 011033 | ‘011753 | 012504 | -013286 | 014099 | 014942 | -015816 | 016721 | -017657 | 20 21 | -005959 | -006409 | -006884 | -007383 | -007908 | -008459 | -009036 | 009640 | 010272 | -010930 | 21 22 | -003278 | -003554 | ‘003849 | 004162 | -004493 | 004845 | -005216 | 005609 | -006023 | -006459 | 22 23 | 001724 | 001885 | 7002058 | -002244 | -002442 | “002654 | -002880 | -003122 | -003378 | 003651 | 23 24 | 000869 | 000958 | 001055 | -001159 | -001272 | -001393 | -001524 | -001665 | -001816 | -001977 | 24 25 } -000421 | -000468 | ‘000519 | 000575 | -000636 | 000702 | -000774 | -000852 | -000937 | -001028 | 25 26 | «000196 | -000219 | ‘000246 | 000274 | -000306 | 000340 | -000378 | -000420 | 000165 | -000514 | 26 27 | -000088 | -000099 | -000112 | -000126 | -000142 | -000159 | -000178 | -000199 | -000222 | -000248 | 27 28 | -000038 | -000043 | -000049 | -000056 | -000063 | 000071 | -000081 | 000091 | -000102 | “000115 | 28 29 | -000016 | 000018 -000021 | -000024 | -000027 | 000031 | -000035 | -000040 | -000046 | “000052 | 29 30 | :000006 | -000007 | -000009 | 000010 | -000011 | 000013 | -000015 | 000017 | -000020 | -000022 | 30 81 | -000002 | -000003 | -000003 | 000004 | -000005 | -000005 | -000006 | -000007 | -000008 | 000009 | 81 32 | -000001 | -000001 | -000001 | -000002 | -000002 | -000002 | -000002 | -000003 | -000003 | -000004 | 32 el eS = — | 000001 | -000001 | 000001 | -000001 | -000001 | -000001 | -000002 | 33 CP Nee = a a2 ua ge ay = = 3 000001 | 1 OANA Kw &BNRD WWNVAASY CMHS 10 11 | 20 15"1 000002 | 000027 | 000175 “000766 002510 006575 014356 | 026867 043994 | 064036 083887 099901 109059 109898 ‘102833 089807 073530 056661 041237 028432 018623 011617 006917 003940 002151 “001127 “000568 “000275 “000129 “000058 “000025 “000011 “000004 000002 “000001 | “000001 | 000011 “000075 *060352 *0012389 “003494 ; 008212 “016541 “029153 045673 *064399 *082547 | Tables for Statisticians and Biometricians TABLE LiI—(continued). hn we) @® 000002 000024 00016] “000709 002341 “006180 013596 “025639 042304 062046 “081901 “098281 108109 109773 “103500 “091080 075141 058345 *042786 “029725 “019619 012332 007399 004246 “002336 001233 000626 000306 “000144 “000066 “000029 “000012 “000005 000002 000001 142 “000001 “000010 “000069 “000325 “001153 003275 “007752 015726 “027913 “044040 062537 “080730 000002 “000022 “000148 000657 “002183 005807 012872 024458 “040661 “060088 079916 “096626 “107094 “109566 "104087 "092291 076717 “060019 044348 031043 020644 013074 007904 “004571 002533 “001348 000689 “000340 “000161 000074 000033 000014 “000006 “000002 000001 143 “000001 “000009 “000063 “000300 ‘001073 003070 007316 014946 026715 042447 060700 078910 1S*4 “000002 000020 000136 “000608 002035 005455 012183 023322 039064 “058161 077936 “094940 106017 "109279 104595 093439 “078255 061683 045920 032385 021698 013846 008433 004913 002743 001470 000758 000376 000180 “000083 000037 "000016 “000007 “000003 000001 000001 “000008 000058 000277 “000999 002876 006902 “014199 025559 “040894 058887 077089 | ‘075270 m™ 13°5 13°6. | 18°7 | 13°8 13°9 14'0 000001 “000019 000125 000562 001897 005123 011526 022230 037512 056269 075963 093227 104880 108914 105024 094522 079753 063333 047500 033750 022781 014645 008987 005275 002967 001602 000832 000416 000201 000093 “000042 000018 000008 “000003 000001 145 *000001 “000007 000053 000256 “000929 002694 006510 013486 "024443 039380 057101 “000001 “000017 000115 000520 001768 “004810 “010902 021181 036007 054410 073998 091489 “103687 108473 105373 095539 “081208 “064966 049086 035135 023892 015473 009565 “005656 003205 001744 “000912 “000459 000223 000105 000047 000021 000009 “000004 “000001 “000001 14°6 000007 000049 000237 “000864 002523 006139 012804 023367 037907 055343 073456 “000001 “000015 000105 “000481 001648 004514 010308 020173 034547 052588 072046 089730 102441 107957 "105644 “096488 082618 “066580 050675 036539 “025030 016329 010168 006057 003457 “001895 “000998 “000507 000248 000117 “000053 “000024 “000010 “000004 “000002 “000001 147 000006 “000045 000219 “000803 002362 005787 012152 022330 “036472 053614 071648 000001 “000014 000097 000445 001535 004236 009743 019207 033132 050802 *070107 087953 “101146 “107370 105836 097369 083981 068173 052266 037962 026193 017213 010797 006478 003725 002056 001091 000558 000275 000131 “000060 “000027 “000012 “000005 000002 “000001 148 000006 “000041 “000202 000747 002211 “005454 011530 021331 035078 051915 069850 “000001 000013 “000089 “000411 “001429 003974 009206 018280 031762 049054 068185 086162 099804 106713 "105951 098181 "085295 069741 053856 039400 027383 018125 011452 006921 “004008 002229 001191 000613 000305 000146 “000068 “000030 “000013 “000006 “000002 000001 149 000005 000088 000186 “000694 002069 005138 010937 020370 033723 050247 068062 “000001 000012 “000081 000380 001331 003727 “008696 017392 030435 047344 “066282 084359 098418 105989 "105989 098923 "086558 071283 055442 “040852 028597 019064 012132 007385 004308 002412 001299 “000674 “000337 000163 000076 000034 000015 000006 “000003 000001 15°0 “000005 000034 000172 “000645 001936 004839 010370 019444 032407 048611 066287 WOHNVA AS MARS wa KRODOOHDVWANHRSBENG 141 096993 "105200 “105951 099594 “087768 072795 057023 042317 029834 020031 012838 007870 004624 002608 001414 000739 000372 *000181 000085 000039 000017 000007 000003 “000001 095530 "104349 105839 “100195 088923 074277 "058596 043793 031093 021025 013570 008378 004957 002816 001538 000809 000410 “000201 “000095 “000044 “000019 “000008 “000003 “000001 “000001 Powssow's Huponential Binomial Linit 094034 | 103437 105654 100723 “090021 075724 060158 045277 032373 022045 014329 008909 005308 “0030386 “001670 “000884 000452 000223 “000106 “000049 “000022 000009 “000004 000002 “000001 TABLE Li—(continued). 121 144 092507 102469 105396 “101181 091063 077135 061708 *046768 033673 023090 015114 009462 005677 003270 001811 000966 000497 000247 “000118 “000055 000025 000011 000005 “000002 “000001 145 090951 "101446 “105069 “101567 "092045 078509 063243 "048264 084992 024161 “015924 010039 006065 “003518 001962 001054 000546 000273 000132 “000062 “000028 “000012 “000005 “000002 “000001 146 089371 100371 104672 101881 092967 079842 064761 049763 036327 025256 016761 010640 006472 003780 002123 “001148 000598 000301 000147 “000069 “000032 “000014 “000006 “000002 “000001 “087769 099247 “104209 *102125 093827 7081133 "066259 051263 037678 026375 017623 011264 “006899 004057 "002294 “001249 ‘000656 “000332 ‘000163 000077 “000035 “000016 “000007 000003 ‘000001 148 086148 “098076 “103681 102298 094626 082380 067735 "052762 039044 027517 018511 011911 007345 004348 002475 001357 ‘000717 000366 “000181 “000086 “000040 “000018 “000008 “000003 “000001 “000001 9 084510 096862 “1038089 "102402 095361 083581 069187 *054257 040422 “028680 “019424 012584 007812 004656 002668 001473 000784 000403 “000200 000096 000045 “000020 “000009 “000004 “000002 “000001 15°0 082859 “095607 102436 102436 ‘096034 084736 ‘070613 055747 041810 029865 020362 013280 “008300 004980 002873 001596 “000855 “000442 000221 ‘000107 “000050 000023 “000010 000004 000002 | < “000001 16 122 Tables for Statisticians and Biometricians TABLE LI. Table of Poisson-Exponential for Cell Frequencies 1 to 30. Cell Frequencies 5 8 20 => op 18 fa | 17 Be ae a<_ 15 mn 14 BE! as Se |p ve wo | 11 on | 10 a aS 9 012 "050 = 8 034 123 O77 28 7 091 “302 623 | 1033 23 6 248 730 | 1:375 | 2°123 | 2-925 Re 5 674 | 1°735 | 2°964 | 4:238 | 5-496 | 6°708 = 4 ————| 1:832 | 4:043 | 6197 | 8:177 | 9-963 | 11:569 | 13-014 : 3 —_—_| 4:979 | 9-158 | 19-465 | 15°120 | 17-299 | 19-124 | 20-678 | 22-022 2 2 |__| 13-534 | 19-915 | 23-810 | 26-503 | 28°506 | 30:071 | 31:337 | 32:390 | 33-282 1 | 36-788 | 40-601 | 42-319 | 43-347 | 44-049 | 44-568 | 44-971 | 45-296 | 45-565 | 45-793 Actual] 36-788 | 27-067 | 22°404 | 19-537 | 17°547 | 16-062 | 14-900 | 13-959 | 13-176 | 12°511 26°424 | 32°332 | 35°277 | 37-116 | 38°404 | 39°370 | 407129 | 40°745 | z 1 2 8 4 Bot 16 7 8 9 10 22 21 19 = 1 41-259 | 41-696 3 3 2 | 8-030 | 14-288 | 18-474 | 21-487 | 23-782 | 25-602 | 27-091 | 28-338 | 29-401 | 30323 # 3 | 1:899 | 5-265 | 8-392 | 11-067 | 13-337 | 15-276 | 16-950 | 18-411 | 19-699 | 20-845 s 4 366 | 1:656 | 3°351 | 5°113 | 6-809 | 8-392 | 9-852 | 11-192 | 12-492 | 13-554 ey 5 -059 -453 | 1:191 | 2:136 | 3-183 | 4-262 | 5°335 | 6:380 | 7°385 | 8-346 = 6 -008 ‘110 “380 813 | 1:370 | 2-009 | 2°700 | 3-418 | 4146 | 4:875 = 7 001 024 “110 284 +545 883 | 1:281 | 1:726 | 2-203 | 2-705 S 8 -000 -005 029 “092 -202 363 “572 823 | 1110 | 1-428 8 9 a ‘001 -007 “027 ‘070 140 241 “372 “532 ‘719 > 10 = -000 “002 -008 “023 051 096 “159 242 346 Za 11 = a -000 “002 007 ‘018 036 “065 105 "160 a: | 12 = = = ‘001 -002 -006 013 “025 044 ‘071 BS | 18 as = = -000 001 002 “005 “009 017 030 ss = Us = = = = -000 ‘001 -002 ‘003 ‘007 013 Bell Gib = = = = oe 000 001 001 “002 “006 Ze | 16 =3 = = = = == “000 000 ‘001 “002 ae | 17 = = = = = = = = “000 ‘001 Ag 18 — = = = = = a at = ‘001 ° 19 = = — | — — — — = = 000 3 20 = = = = aa im = SO a es a a = = ey 7 = = fe 3 23 = = = aes = = = = me == 3 2 =a any ca ae Paul aA = = = = 4 25 = = = = = = = = = = g 26 = = = = = ax es = = a Pn (eZ — _ — — _ — ~ - ~ : ay 28 ae ae = = = ae = = =e = Per cent. occurrence of values differing by w or more in defect from Actual from Actual. Per cent. occurrence of values differing by # or more in excess 11 002 020 121 “492 1:510 3°752 7861 147319 23°198 34°051 45°989 11°938 42073 31°130 21°871 14'596 9-261 5593 3°219 1-769 929 “467 225 104 047 020 008 003 001 ‘001 “000 12 Table of Poisson's Exponential TABLE LII—(continued). Cell Frequencies 13 “000 003 022 105 374 1:073 2°589 5°403 9:976 16°581 25°168 35°317 46°311 10994 V4 15 16 17 —ooo _— “000 000 “000 001 001 “002 004 004 “009 018 021 040 067 086 138 "206 279 401 543 “763 1:000 1:260 1-800 27199 2°612 3°745 4°330 4912 6°985 7°740 8-467 11°846 | 12-699 | 13°502 18°475 | 19°312 | 20°087 26°761 | 27°451 | 28-084 36°322 | 36°753 | 37°146 46°565 | 46°674 | 46°774 10°244 9°922 9°629 43'191 | 43-404 | 43°597 33°588 | 34:066 | 34°503 25°114 | 25°765 | 26°367 18°053 | 18-776 | 19°451 12°478 | 137184 | 13°852 8:297 8°923 9°526 5°311 5°825 6°329 3°275 3°669 4:064 1-947 2°232 2°523 1117 1°312 1°516 619 “746 882 B31 “411 "497 ‘172 219 "272 086 114 144 042 057 074 020 028 036 009 014 ‘O17 004 006 008 “002 003 003 001 | 002 | 002 ‘600 | 001 | 001 a: 000 | 000 18 123 19 20 “000 *000 ‘001 “002 004 ‘007 ‘O15 026 052 ‘078 "151 209 387 *500 “886 1:081 1°832 27139 3°467 3901 6:056 6613 9-840 | 10°486 14:975 | 15°651 21°479 | 22-107 29°203 | 29-703 37°836 | 387142 46948 | 47:026 9°112 8-884 43°939 | 44-091 35°283 | 35°630 27°451 | 27°939 20°686 | 21°251 15:099 | 15°677 10°675 | 11°219 7313 7°789 4°856 5§°248 3°127 3°433 1°954 2°182 1°185 1'348 “699 *809 *400 "473 223 "269 121 149 “064 “O81 033 "042 ‘017 "022 “008 ‘O11 “004 “005 “002 “003 ‘001 ‘001 “000 ‘O01 — “000 124 Tables for Statisticians and Biometricians TABLE LIIl—(continued). Cell Frequencies x 24 25 a 22 = = 6 21 as Ea 8 20 “= 000 = 19 “000 ‘001 op 18 ‘001 “002 Ba 17 005 008 ge | 16 ‘015 | 022 sg | 15 043 “059 2 14 ‘109 142 SE] 73 952 | 314 ea | 12 ‘540 ‘647 #8 | 11 1-072 | 1-240 os | 10 1:983 | 2-229 =F 9 3-440 | 3:775 aA 8 5626 | 6-048 ie 7 8713 | 9:204 28 6 12:827 | 13-358 | 13-867 = 5 18-025 | 18-549 | 19-048 a 4 24:263 | 24-730 | 25-172 ° 3 31-391 | 31-753 | 32-094 & 2 39°168 | 39-387 | 39-593 1 47-283 | 47-340 | 47-392 Actual 8:115 7952 n 1 44-603 | 44°708 | 44:810 3 2 36°812 | 37:062 | 37:299 4 3 29°620 | 29-982 | 30°326 5 4 93-927 | 23-660 | 24:0 a 5 17°748 | 18-211 | 18-655 = 6 13:213 | 13-669 | 14-110 | 7 9°585 | 10-007 | 10-418 = 8 6-777 | 7°146 pS 9 4670 | 4-978 bh 10 3138 | 3-385 ee ee! 2°057 | 2°246 Gey | 7G 1:315 | 1:456 BS | 18 “821 ‘921 Be e uy “500 570 eo 15 298 “345 g8| 16 173 | -204 ac | 17 098 118 ae 18 ‘055 ‘067 © 19 -030 ‘037 3 20 016 | -020 5) 21 ‘009 ‘011 5 22 ‘005 006 3 23 003 ‘003 ° Y 002 “002 = 25 ‘001 ‘001 3 26 000 “000 : 5 a7 = = “000 “000 ‘000 -000 ‘001 Ay 28 | — = = == = —_— “000 tt ae ES © WH DOWMVANEK WH g ~ iS) oe a OS NNR S$Ses wigs SSSLSERI SSRSeses TABLE LITT. 017 4533 034 9066 052 3599 069 8132 087 2665 104 7198 “122 1730 139 6263 *157 0796 174 5329 191 9862 *209 4395 “226 8928 244 3461 261 7994 *279 2527 ‘296 7060 3141593 331 6126 349 0659 366 5191 "383 9724 401 4257 *418 8790 436 3323 453 7856 *471 2389 “488 6922 506 1455 523 5988 541 0521 558 5054 575 9587 693 4119 610 8652 628 3185 645 7718 663 2251 “680 6784 “698 1317 “715 5850 “733 0383 “750 4916 “767 9449 “785 3982 802 8515 820 3047 837 7580 855 2113 “872 6646 890 1179 907 5712 925 0245 942 4778 959 9311 977 3844 994 8377 1012 2910 1:029 7443 1-047 1976 Angles, Arcs and Decimals of Degrees Lenetus or Circunar Arcs Are 1:064 6508 1:082 1041 1099 5574 1:117 0107 1°134 4640 1151 9173 1-169 3706 1°186 8239 1:204 2772 1-221 7305 1:239 1838 1:256 6371 1:274 0904 1:291 5436 1:308 9969 1°326 4502 1:343 9035 1361 3568 1°378 8101 1396 2634 1°413 7167 1°431 1700 1:448 6233 1-466 0766 1°483 5299 1500 9832 1518 4364 1°535 8897 1553 34380 1570 7963 1588 2496 1:605 7029 1°623 1562 1640 6095 1:658 0628 1675 5161 1-692 9694 1°710 4227 1-727 8760 1°745 3293 1°762 7825 1°780 2358 1°797 6891 1815 1424 1°832 5957 1°850 0490 1°867 5023 1°884 9556 1:902 4089 1:919 8622 1:937 3155 1:954 7688 1:972 2221 1-989 6753 2007 1286 2°024 5819 2042 0352 2059 4885 2'076 9418 2°094 3951 Deg. 121 122 123 124 125 126 127 128 129 150 131 132 133 134 185 156 137 138 Are v Deg. Are 2111 8484 | Z| -01667 | -000 2909 2129 3017 | 2} -03333 | 0005818 2146 7550 | 3 | -05000 | -000 8727 21642083 | 4] -06667 | 001 1636 21816616 | 5! -08333 | -001 4544 21991149 | 6 | -10000 | -001 7453 2-216 5682 | 7 | -11667 | 002 0362 22340214 | 8 | :13333 | -002 3271 2251 4747 | 9 | -15000 | 002 6180 2:268 9280 | 10 | -16667 | -002 9089 2-286 3813 | 12 | -18333 | -003 1998 2303 8346 | 12 | -20000 | -003 4907 2:3212879 | 13 | -21667 | 003 7815 2:388 7412 | 14 | -23333 | 0040724 2°356 1945 | 15 | -25000 | 0043633 2373 6478 | 16 | -26667 | -004 6542 2:3911011 | 17 | -28333 | -004 9451 2°408 5544 | 18 | -30000 | -005 2360 2°426 0077 | 19 | -31667 | -005 5269 2:443 4610 | 20 | -33333 | -005 8178 2-460 9142 | 22 | -35000 | -006 1087 2-478 3675 | 22 | -36667 | 0063995 2°495 8208 | 23 | -38333 | -006 6904 25132741 | 24 | -40000 | :006 9813 2530 7274 | 25 | -41667 | -007 2722 2548 1807 | 26 | -43333 | 007 5631 2°565 6340 | 27 | -45000 | -007 8540 2583 0873 | 28 | -46667 | :008 1449 2-600 5406 | 29 | -48333 | 008 4358 2617 9939 | 30 | -50000 | :008 7266 2°635 4472 | 31 | -51667 | :0090175 2°652.9005 | 32 | 53333 | -009 3084 2'670 3538 | 83 | 55000 | -009 5993 2-687 8070 | 34 | -56667 | 009 8902 2-705 2603 | 35 | -58333 | 0101811 2°7227136 | 36 | -60000 | :0104720 2°740 1669 | 37 | -61667 | -010'7629 2°757 6202 | 38 | -63333 | -011 0538 2-775 0735 | 39 | -65000 | :011 3446 2°792 5268 | 40 | -66667 | -011 6355 2°809 9801 | 41 | -68333 | -011 9264 2°827 4334 | 42 | -70000 | 0122173 2°844 8867 | 48 | 71667 | -012 5082 2°862 3400 | 44 | -73333 | 0127991 2:879 7933 | 45 | -75000 | -013.0900 2°897 2466 | 46 | -76667 | 0133809 2:914 6999 | 47 | -78333 | 0136717 2-932 1531 | 48 | -80000 | 013 9626 2-949 6064 | 49 | -81667 | 0142535 2-967 0597 | 50 | -83333 | 0145444 2°9845130 | 51 | -85000 | 014 8353 3-001 9663 | 52 | -86667 | -015 1262 3019 4196 | 53 | -88333 | :015 4171 3-036 8729 | 54 | -90000 | ‘015 7080 3°054 3262 | 55 | -91667 | -015 9989 3:071 7795 | 56 | -93333 | -016 2897 3:089 2328 | 57 | -95000 | -016 5806 3106 6861 | 58 | -96667 | -016 8715 31241394 | 59 | -98333 | -017 1624 3°141 5927 | 60 | 1:00000 | -017 4533 Deg. 00028 00056 00083 00111 00139 00167 00194 00222 00250 00278 00306 00383 00361 | 00389 00417 00444 00472 00500 00528 00556 00583 | 00611 00639 00667 00694 00722 00750 00778 00806 00833 00861 00889 00917 00944 00972 01000 01028 01056 01083 ‘01111 01139 01167 01194 01222 01250 01278 01306 0:1333 01361 01389 01417 01444 01472 01500 01528 01556 01583 01611 01639 ‘01667 Are 000 0048 000 0097 000 0145 000 0194 “000 0242 000 0291 000 0339 000 0388 000 0436 000 0485 000 0533 000 0582 000 0630 000 0679 "000 0727 000 0776 000 0824 000 0873 000 0921 000 0970 000 1018 ‘000 1067 000 1115 000 1164 000 1212 000 1261 000 1309 000 1357 000 1406 000 1454 000 1503 000 1551 000 1600 000 1648 000 1697 000 1745 000 1794 000 1842 000 1891 000 1939 000 1988 000 2036 000 2085 000 2133 000 2182 “000 2230 000 2279 000 2327 000 2376 000 2424 000 2473 000 2521 000 2570 000 2618 000 2666 000 2715 000 2763 000 2812 000 2860 000 2909 126 Tables for Statisticians and Bivmetricians TABLE LIV. The G(r, v)-Integrals. a r=1 r=Z ¢ ——— es log F(r, v) A | a log F (r, v) log H (r, v) A a | 0 | 0-301 0300 Sil 0196 1199 | 0-196 1199 a 1 | -301 0609 oe | G19 | 196 2052 | -196 1391 ee es 2 | -301 1538 28 | eai | 196.4614 | -196 1966 eco | eee $ | -301 3087 ae 625 | 1968890 | -196 2924 1349 | 382 4 | so1sse2 J 2175 630 | 1974890 | -196 4964 Ty | eee 5 | -301 8067 636 | 198 2627 | -1065985 380 6 | 3021508 J 343 | 645 | 1992118 | 1968087 | 2102 | azo 7 | 3025594 | 285 | 654 | -2003385 | -197 0567 el eo 8 ‘303 0335 5 406 666 201 6452 "197 3424 3231 375 9 | 303 5741 cane 679 | -2031349 | -197 6655 cnet | aoe 10 | -304 1895 693 | -2048110 | -198 0260 370 11 | -304 8603 ee 710 | 2066774 | -198 4934 aoe | see 12 | -305 6091 vase | 728 | -2087382 | -198 8574 aon | 364 13 | -306 4307 sor6 | 749 | 2109985 | -199 3280 ie ae 1, | 307 3271 cost | 771 | 213.4631 | -199 8344 aren S| 208 15 | -308 3006 y 796 | -2161383 | -200 3764 352 16 | 3093538 | 799383 822 | 2190303 | -200 9537 pie 347 rv | 3104802 | 153°3 | 53 | -222 1462 | -201 5657 cree eee is holier) been 885 | -225 4936 | -202 2190 ry hee 19 | 3130189 | 430?) 919 | 2290807 | -202 8921 pee ee 20 | -314 4200 958 | -2329167 | -203 6054 327 21 | 3159169 | Teecg | 999 | 2370114 | 2043814 a 320 22 | 3175137 | 20°38 | 1045 | 2413755 | -205 1294 a 23 | 3192150 | ji058 | 1093 | -2460203 | -205 9387 oe, a0 24 | 3210356 | 48506 | 1146 | -2509584 | -206 7787 aro | 300 25 | -322 9507 1204 | -2562034 | -207 6487 291 26 | -324 9963 oreo 1266 | -261 7697 | -208 5478 one 284 ev | 3271685 | 33722 | 1334 | -267 6733 | -209 4753 eee. igi 28 | 3294740 | 330° | 1407 | 2739311 | -210 4302 ooo ip ae 29 | -331 9202 | S908 | 1486 | -2805618 | 2114118 | 4 Core | 256 30 | -334 5150 1573 | -2875852 | -212 4190 247 3 337 2672 ake 1667 | -2950232 | -2134509 | 4 ares 236 s2 | -3401860 | 225c8 | 1769 | 3028992 | -2145064 | joes | 226 33 | -343asis | 30298 | 1881 | 3119398 | 2155846 | [Osoe | “213 34 | 3465656 | S253° | 2002 | 3200695 | -2166842 | jY7eq | 203 35 | -3500496 | * 2134 | -329 4214 | -217 8041 191 36 | 353 7469 pickle 2279 | -339 3271 | 2189431 | } on 178 sy | -3576792 | 39202 | 9436 | 3498221 | 2201000 | j jy9, | 165 38 | 3618410 | 4icc, | 2609 | 3609451 | 2219734 | }ifa5 | 152 s9 | -3662708 | 47028 | 2799 | 3727382 | 2224621 | J oon | 138 40 | -370 9805 3007 | 3852475 | -293 6644 125 | g1 | -375 9908 | G03 | 3035 | -3985e32 | 2248791 | | Song | 109 42 | 3813246 | $8938 | 346 | -4126205 | 2261048 | J 550, | 94 4s | 3870070 | Bo82e | a7e4 | -4275995 | -297 3307 | 1 Sto 77 4 | 3930658 | GU°88 | 4069 | -4435266 | 2285824 | | San, | 62 45 \ 399 5316 -460 4745 | -229 8313 Tables of the G(r, v)-Integrals TABLE LIV—(continued). 127 r=3 r=4 log F(r, v) | log H(r,v) | A | A? | log F(r,v) | logH(r,v) | A A? | log F(r, v) | 0:124 9387 | 0-275 4537 0-071 1811 | 0-309 7418 | |. 0-028 0289 195 0847] -275 4674 ri 2731 -071 3902] -309 7523 ae 211] -028 3019 125 5230| -275 5084| 410/973] 072.0177] 309 7840 | 258/211] -029 1221 126 2545| 275 5767| 68 272] -073 0650] “309 8366 | 25" /210] -030 4908 127 2807] 275 6721 | ,99° | 271| -074 5342] 309 9103] 237/209] -032 4110 128 6039| 275 7918 | 1°?6 a70] -076 4285] “310 0049 208] -034 8865 1496 1154 -130 2270] -275 9444 | 1496 | 069] -o78 7517] -310 1204 207] -037 9224 -132 1533] -276 1209 | "88 | 267] 081 5088] “310 2565 | 136) | 206] 041 5250 134 3870] -276 3242 | 202 | 266] -os4 7055] -310 4132 1771 | 204] 045 7016 ‘136 9331 | -276 5540 Ba 263] -088 3486] “310 5904 | 3543 |202] -050 4609 139 7969| 276 8101 | °°" | 261] 092 4458] -310 7877 201] -055 8130 2822 2174 ; ‘142 9850] -277 0993 |258] -097 0060] -311 0051 198] -061 7690 146 5043) -277 4003 | 2082 | 256] -102 0399] -311 2493 ae 196] -068 3415 -150 3626 | -277 7338 | 2229 | 252] -107 5555] -311 4990 2740 | 193] 075 5446 “154 5688 | -278 0926 | 3285 | 249] -113 5680] 311 7751 | 54,,| 190] 083 3937 159 1322) -278 4762 | 2°87 | 945] -120 0895] 312 0701 187] -091 9061 164.0636] -278 8844 | 482/241] -127 1349] “312 3838 | 393’ | 184] -101 1002 -169 3743] -279 3167 | 4222 | 937] -134 7199] -312 7159 3501 | 180] *110 9967 175 0768] -279 7726 | 460 | 233] +142 8621] -313 0660 | 33-7 |176] 121 6176 181 1848] -280 2519 | 5/93 | 228] 151 5802] 313 4337 | 3244 |172| -132 9872 ‘187 7130] -280 7539 | 9229 223] -160 8948] -313 8186 168] -145 1317 5243 4018 : -194 6774 | -281 2782 | °243| 217] -170 8981 | -314 2204 164] -158 0795 a 0955} -281 9243 | 2489 | 919] -181 4042] -314 6385 ea, 160] -171 8614 209 9858] -282 3915 | 7073 | 206] -192 6491 | 315 0726 | 435! | 154] -186 5105 218 3638] -282 9794 | 89 | 200] -204 5907 | “315 5222421? |150] -202 0627 297 2664) -283 5873 | °°79| 194] -217 2596] 315 9866 144] -218 5568 236 7023] -284 2145 | 272) 188] -230 6885} -316 4655 | 4759 |138| -236 o346 246 7020] -284 8604 |645| 180] -244 9127] -316 9583 | °7°/ | 134) -254 5413 ‘957 2933) -285 5244 abe 173| 2599707] “317 4644 | 278° | 197| 274 1955 268 5060] -286 2057 | 65/3 | 165] -275 9034] “317 9833 | 505 | 121| -204 8399 280 3725 | +286 9035 158] -292'7555| 318 5143 116] °316 7413 7136 ‘ 5426 292 9278 | -287 6170 150] -310 5754] 319 0569 109] +339 8909 306 2096 | -288 3456 | 7288 | 141] -329 4149] -319 6104 | 273° | 102] 364 3553 320 2589 | -289 0883 | 7427 133] -349 3304] -320 1741| 2°37 | 96] -390 2059 “335 1201 | -289 8443 | 7760] 194] -370 3832] “320.7474 | 2755| 89] -417 5203 -350 8413| -290 6127 115| -392 6390] 321 3297 82| -446 3897 7800 5 =] .a97 99 5904 ‘476 8841 367 4747 | -291 3927 | 8°| 106] -416 1697] -321 9201 75| -476 884 385 0v70 | "202 1892 7905 “97 | -441 0529] “322 5181 | 68? | 68] -509 1932 403 7099] -292 9834 | 8002) 87] -467 3733] -323 1228 |6047| eo} -543 9072 493 4403] -293 7923/2000) 77| -495 2227 | “323 7335 | 613),| 53) -579 2529 444 3416 | -294 6089 67| 524-7011] -324 3495 45} 617 3872 8232 “s 6205 eid -466 4933) -295 4391 | 9282) 57] +555 9177] -324 9700 38| -657 7483 -489 9829 | -296 2610 | 8289| 46] -s88 9916} 325 5943 | 6209 | 30) -700 4872 514 9055 | -297 0945 | 833° 36] -624 0530] 326 2216 | 6572 | 92] -745 7688 541 3658] 297 9316 | a4, | 23) “G61 2446] “326 8510 | B20? | 12] -793 739 569 4783 | -298 7710 | 8394 "700 7225 | “327 4817 ‘844 6999 log H (r, v) 0°329 0589 329 0673 329 0930 329 1357 329 1956 329 2723 329 3661 *329 4765 329 6037 "829 7474 329 9075 330 0838 330 2761 330 4842 330 7079 330 9469 331 2010 331 4700 331 7532 332 0508 332 3621 332 6870 333 0250 333 3757 333 7387 334 1137 334 5001 334 8976 335 3057 335 7239 336 1516 336 5884 337 0339 337 4874 337 9485 338 4164 338 8908 *339 3709 339 8563 340 3463 340 8404 341 3378 341 8381 342 3405 342 8446 343 3495 Tables for Statisticians and Biometricians 128 r=6 ~ log F (r, v) | log H(r, v) A 0 | 1991 9999 | 0-341 4849] 1} -992 3379] “341 4921| 572 2) -993 3526] 341 5137| 226 S| -995 0459] 341 5496 | 3°9 4} -997 4213} -341 5999 | 503 5 | 0:000 4836] -341 6644 6] -004 2390] -341 7432 ig 7 | -008 6950] -341 8360 | ,928 8} -013 8607} -341 9498 | 1068 9] -019 7468} -342 0635 13 45 10 | -026 3653] -342 1980 11} -033 7300] -342 3461 vee 2] -041 8562] -342 5075 17 Ae 13] -050 7609] -342 6823 1878 14 | -060 4633} “342 8701 | 1878 15 | -070 9839] -343 0707 16] -082 3457] -343 2839 ae 17 | -0945734] -343 5095 ean7 18 | -107 6941] 343 7472 | 2877 19 | -121 7375] -343 9968 | 2496 20 | +136 7352] -344 2579 21 | 1527219] -344 5302 lee 22) -169 7350] “344 8135 | 2033 23 | -187 8149] -345 1074 | 2039 24 | -207 0053] 345 4115 | 3041 25 | -2273532| -345 7256 26 | -248 9095 | -346 o4ga | 3236 27 | 271 7291] 346 38i9 | 3928 28 | -295 8713] “346 7235 | 3415 29 | -321 3998] -347 0734 | 3408 30 | -348 3836] 347 4311 $1} 376 8974] -347 7965 a 92| -407 0214] -348 1689 | 3724 33 | -438 8428] -348 5480 aaES 54 | -472 4556] -348 9332 | 3552 35 | -507 9618] -349 3242 36 | 545 4718] -349 7204 pe 87 | 585 1052] -350 1213 | 4008 98 | -626 9922] -350 5265 | 4052 89 | °671 2739} 350 9354 | 4°90 4o\ -718 1040] -351 3477 | 4122 41 | “767 6502) -351 7627 | 4150 42} 8200951] -352 1799 | 4172 43) -875 6383) -352 5989 | 4100 44 | -934 4983] -353 o191 | 4202 46} 996.9145] -353 4399 | 42 TABLE LIV—(continued). log F (r, v) 1-961 0819 144] -961 4851 144] -962 6953 143] -964 7151 143] -967 5483 142] -971 2008 141} -975 6796 140} -980 9940 139} -987 1544 137} +994 1735 136 | 0-002 0658 134] -010 8465 133} -020 5347 131} -031 1503 128} -042 7157 127} -055 2551 123} -068 7956 121] -083 3665 119] -098 9997 115] +115 7298 112} +133 5946 109} +152 6347 106} -172 8941 103} +194 4206 99] :217 2653 95] 241 4839 92] -267 1362 88] -294 2868 84] -323 0053 78] -353 3670 76] -385 4529 71] -419 3506 66] -455 1549 62} -492 9680 57] +532 9005 52] +575 0721 47] -619 6127 43] 666 6629 38] -716 3753 33] “768 9159 28] -824 4653 22] -883 2199 17] 945 3944 12] 1-011 2229 7} 080 9618 154 8920 r=7 log H(r, v) 0°350 1576 350 1638 350 1824 350 2134 350 2567 350 3123 *350 3801 350 4601 350 5522 350 6562 350 7720 350 8995 351 0386 351 1891 351 3508 "351 5235 351 7071 *351 9012 352 1058 352 3206 "352 5452 352 7795 353 0232 353 2760 353 5375 353 8075 354 0857 354 3717 354 6652 354 9659 355d 2732 355 5871 *355 9070 *356 2324 *356 5632 356 8988 357 2388 357 5829 397 9306 358 2814 358 6350 358 9910 359 3488 359 7080 360 0682 360 4290 A2 124 124 123 123 122 122 121 120 118 117 116 114 112 19 14 10 6 log F(r, v) 1-934 0080 934 4765 935 8831 938 2304 941 5232 945 7679 “950 9729 ‘957 1486 964 3073 972 4634 981 6333 ‘991 8357 0:003 0914 015 4237 028 8583 043 4233 059 1498 076 0714 094 2250 113 6505 "134 3912 *156 4939 180 0093 "204 9923 231 5019 259 6019 289 3613 320 8543 354 1610 389 3678 426 5682 465 8626 ‘507 3601 551 1780 597 4436 646 2944 697 8795 “752 3605 809 9127 870 7267 935 0097 1:002 9876 074 9063 151 0352 ‘231 6680 317 1271 r=8 log H (r, v) 0'356 5570 356 5624 356 5788 356 6059 356 6440 356 6928 356 7524 "356 8226 356 9034 356 9947 357 0964 357 2083 357 3304 357 4625 357 6044 857 7560 357 9170 358 0874 358 2669 358 4553 “358 6524 .358 8579 359 0716 359 2933 359 5226 "359 7594 360 0033 360 2540 360 5112 360 7747 361 0441 361 3191 “361 5993 361 8844 362 1741 362 4681 362 7659 “363 0671 363 3715 363 6787 363 9883 364 2998 364 6130 B64 9274 365 2427 365 5584 77 146 Tables of the G& (r, v)-Integrals 129 TABLE LIV—(continued). r=9 r=10 r=11 log F'(r,v) | log H(r,v) | A A? | log F(r, v) | log H(r, v) A A2 | log F(r, v) | log H(r, v) A 0 | 1-909 9294] 0361 4744) 4, 1-888 2505] 0365 3717| 44 T'868 5367 | 0°368 6367| 4, 1} -910 4635] -361 4793| ,4?| 97 | -888 8502] “365 3761| ,3¢| 87 | -869 2023] 368 5407| 5) 2] -912 0669} -361 4938 | 345 97 | -s00 6508] -365 3892| }°1| 87 | -871 2002] 368.5527 15) S| 9147497] -361 5180| 34°) 97 | -893 6556] “365 4111 | 21?! 87 | -874 5345] 368 5725| 5° 4] 918 4961] -361 5520| 33?) 96} -897 8705] -365 4416 | 308) 87 | -879 2114] -368 6004 | 3/0 5 | -923 3346] -361 5955 96 | -903 3037] °365 4808 86 | -885 2402] 368 6361 6 | -929 2675] -361 6485 | 231/95] -909 9608] -365 5287) 479| 86 | -892 6324] 368.6796) 5) 7| 936 3067] -361 7111) $26| 94] -917.8700] 365.5851 264| 85 | -901 4027] 368 7310| >5t 8} 944 4661| 3617831) §79| 94] -927 0317] 365 6500| 829) 84] -911 5681} 368 7900| 228 9| 953 7619] -361 8644| $13) 92) -937 4602] 365 7233| 793| 83] -923 1487] -368 8568 | OC 10 | 9642128] -361 9550 92 | -949 2033] °365 8050 82] -936 1674] -368 9311 11} -975 8398] -362 0547 | ,997| 90 | -962 2575] -365 so49| 899) 82] -950 6505] 369 0129| 855 12 | -988 6667] -362 1635 | 1988! 89 | -976 6581] "365 9929 | PA'| 80} -966 6268] 369 1022| 52° 18 | 0-002 7197 | -362 2812 |154/| 88 | -992 4345] -366 0990 | 16) | 79 | -984 1288] 369 1987 | ,)0- 14] -018 0278] -362 4075 | 1385 | 86 | 0-009 6193] “366 2129 | 1522) 78 | 0-003 1923] “369 3024 | 173. 15 | -034 6230] -362 5426 84] -028 2479] “366 3346 76 | -023 8567) °369 4132 16 | -052 5403] -362 6860 | 14°9| 83 | -o48 3595] -366 4639 | 1203! 75 | -046 1651] “369 5308 | 191! 17 | 071 8179} -362 8378 | 1?55| 81 | -069 9966| -366 6007 | 1368) 73 | -070 1646] -369 6553 | 1377 18 | -092 4974] -362 9976 |199| 79 | -093 2058] -366 7447 | }41| 72 | -095 9063] -369 7864 | 3-6 19 | -114 6239] -363 1654] 16/8 77 | -118 0374] -366 8960 | 1257 | 69 | “123 4460] 369 9240 | 1100 20 | -138 2465] -363 3409 75 | 144.5461] 367 0541 | 1°8!| 6g | -152 8439] -370 0679 a1} -163 4180] -363 5239 |18°0| 73 | -172 7910] -367 2190 | 1849| 66 | -184 1654] “370 2179 | 122) 22 | -190 1960] -363 7142 |1903| 71 | -202 8360] 367 3904| 171*| 64 | -217 4809] -370 3739 | 19)g 23 | -218 6422] -363 9115 | 1°73| 63 | -234 7504] “367 5682| 1775! 61 | -252 8669] -370 5357 | G05 24 | 248 8237) -364 1157 |306"| 66 | -268 6087] “367 7522| 1880/59 | -290 4057] “370 7030 | 14° 25 | -280 8124] -364 3264 63 | -304 4913] -367 9420 57 | -330 1857] °370 8757 26 | -314 6864] -364 5435/2171) 61 | -342 4851] -368 1376 | 196) 55 | -a72 3033] -371 0536 | 1450 27 | 350 5295] -364 7666 |5971| 58 | -382 6838| 368 3386 |5°10| 52 | -416 8615| -371 2365 | 1375 28 | -388 4393] -364 9955 | 2200| 55 | -425 1882| -368 5448/2062 50 | -463 9718] “371 4241 | 155) 29 | -428 4925] 365 2300 |535>| 52 | -4701076| “368 7559/5122 | 47 | -513 7544| “371 6162 | 1964 30 | -470 8156] -365 4697 50 | -517 5593] -368 9718 44] -566 3390] 371 8125 5 9 200 81} 5155153] -365 7144 | 2447 47 | -567 6703] 369 1922 | 3593 | 49 | 621 se57] “372 0129 | 5015 52} 5627146] -365 9636 |3°93| 44 | -620 5776| “369 4167 | 35°)| 39 | -680 4854] “372 2171 | 5578 88] -612 5463| -266 2173 | 2°37) 40 | -676 4293] -369 6451 | 5551 | 36 | “742 3617] 372 4249 | 5915 84] -665 1541| -366 4750 |59""| 37 | -735 3855] -369 8772 | 2221 | 33 | 807 6710] 372 6369 | 314) 35 | -720 6932] -366 7365 34] -797 6194] °370 1126 31 | -876 6045 | °372 8500 36 | -779 3322| -367 0013 |2649) 31 | -s63 3188] -370 3510 | 2955 | 28 | -949 3690} 373 0669 | 3151 87} 841 2534 -367 2693 |569°| 98 | -932 6869] 370 5923 | 313 | 95 | 1-026 1888] “373 2862 | 5076 | 88 | -906 6549] -367 5400 |3707| 24 | 1-005 9444] 370 8360 | 243" | 29 | -107 3073} 373 5078 | 555 39} 9757519] 367 8131 |5/31| 21 | -083 3313] 371 0819/3479 19 | -192 9888] “373 7314 | 55°. 40 | 1-048 7784] -368 0884 18 | -165 1080] 371 3297 16 | -283 5209] 373 9567 | 7” 5 A 2 Lad 41} -125 9992| -368 3654 |2/7°) 14 | -251 5588] -371 5790 | 34°4| 13 | ‘379 2165] -374 1834 | 550) 42 | -207 6624] -368 6438 |2/54| 11 | -342 9931 | -371 8296/3296! 10 | -480 471 | 374 4113 | 500 43 | -294 1013] -368 9233 |279>| 7 | -439 7401] -372 0813 |3958| 6 | -587 4953| 374 6400 | 5553 44 | °385 6379| 369 2036 | 2893) 4] -542 1966] -372 3335 |322°| 4] -700 8587] 374 8693 | 5558 45 | -482 6360| -369 4842 “650 7407 | °372 5861 | 2°? 820 9540 | -375 0990 B. 17 130 Tables for Statisticians and Biometricians TABLE LIV—(continued). r=12 r=13 r=14 logF(r,v) | logH(r,v)| A | A? | log F(r, v) | logH(r,v)| A | A? | log F(r, v) | log H(r, v) 0 | 1:850 4619 ] 0371 1582] 3 1-833 7746 | 0373 3653| 94 1-818 2772 | 0:375 2489 1| -851 1933] -3711619| ,°0| 73 | 8345719] -373 3636] ,21| 68 | ‘819 1404] 375 2520 2} 853.3889] -3711729| j9| 73 | 836 9652] “373 3787| tgq| 67 | “821 7316] 375 2614 3] 857.0528] 371 1912| 4°2| 73 | -840 9592] 3733956] 3°?! 67 | 826 0558] 375 O71 4 | 862 1923) 371 a166| 395/72 | “$46 5615] 373 4192] 5o5| 67 | 892 1212] 375 2990 5]: 1] -371 2494 72 | -853 7829] -373 4494 67 | -839 9395] -375 3271 6 | -876 9402] -371 9893| 4°”| 71 | 862 6374] -373 4864] 29°| 66 | 8495258] -375 3614 7) 886 5774| -371 3364| £75| 71| -873 1421] -3735299| $3°) 66 | -860 8985] 375 4019 8} 897 7474| 371 3907| P42| 70 | 885.3174] -373 5800| Pea| 65 | 874.0798] “375 4484 9] -910 4722] -371 4519| ©3| 69 | -899 1873] -373 6365 65 | -889 0954] -375 5010 10 | 9247770) -371 5201| ®°2| 69 | -914 7789] -373 6995 | ©8°| 64 | -905 9746] -375 5595 11} -940 6901] -371 5952| 2°)! 68 | -932 1233] -373 7689 oe 63 | -9247510] -375 6239 12 | -9582436| -371 6771| S19| 67 | -951 2549] -373 8445 | 2°6) 62 | -945 a618] 375 6942 13 | 977 4727] -371 7656 | S86! 66 | -972 2124] -373 9263) S18! 61 | -968 1485] -375 7702 14 | -998 4167] 371 8608 65 | -995 0382] -374 0142 60 | -992 8571] -375 8518 15 |0:021 1187] -371 9624 | 2°8| g4 | 0-019 7792 | -374 1081 | 925] 59 | 0-019 6382] -375 9390 ( 1080 997 16 | -045 6258] -372 0703 | 1949| 62 | “046 4865] “374 2078] G91 | 58 | -048 5469] “376 0317 17 | -071 9896] -372 1845 |1242| 61 | -075 2161] -374 3132 | 19> 56 | -o79 6435] 376 1297 18 | -100 2660] 372 3048 | 1503| 59 | -106 0288] -374 4243 | 1141) 55 | -112 9939] -376 2329 19 | -130 5159] -372 4310 58 | +138 9908] -374 5409 53 | -148 6693] -376 3412 20 | 162 8054] -372 5630 | 1°2°| 56 | -1741737| -374 6628 | 1219| 52] -186 7470] -376 4544 21 | -197 2059] -372 7006 | 18%] 55 | -211 6551] -374 7899 | 1221] 51 | 227 a107| -376 5725 22| -233 7945| -372 8437 | 1431] 53 | 251 5187| -374 9201 | 1822! 49 | -270 4509] 376 6952 23 | 272 6547] -372 9921 |1384| 51 | -293 8552] -375 0591 |1870| 47 | “316 2653] -376 8225 24) -3138765| -373 1456 49 | -338 7623] -375 2008 46 | 364 8594] -376 9542 25 | -357 5571} -373 3040 | 1°84| 47 | -386 3455] +375 3472 | 1464| 43 | -416 3469] -377 0901 26 | -403 8013} -373 4672 | 1632) 45 | -436 7186] -375 4978 |1206| 49 | 470 8506} 377 2301 27 | -452.7221] -373 6349 | 1877 | 43 | -490 0042-375 6527 | 1742] 40 | 528 5029] 377 3739 28} 504 4412] -373 8069/1720! 41 | -546 3346] -375 8115 |1?82| a8 | 589 4465] 377 5215 29 | -559 0903] -373 9831 39 | -605 8596 | -375 9742 | 1627! 36 | -653 8352] -377 6726 30 | 6168111] -374 1631 | 18°!| 37 | -668 7120] -376 1405 | 19°3| 34] -721 8353] -377 8270 31 | -677 7567] -374 3469 1933 35 | -735 0790] -376 3102 | 169| 32 | -793 6258] 377 9846 52 | -742.0993| -374 5341 | 1873) 33 | -805 1331] -376 4831 | 1728) 30 | “869 4003] 378 1452 33 | 809 9965] -374 7246 | 190°) 30 | -879 0680] 376 6589 | 1/22 | 28 | 049 3679] 378 3085 34,1 -881 6625] -374 9182 28 | 957 0932| -376 8376 26 | 1-033 7545) °378 4745 35 | -957 2989] -375 1145 | 1963) 95 | 1-039 4354] -377 o1gs | 1812] 93 | -122 8047] -378 6428 36 | 1-037 1322] -375 3133 | 1°88 93 | -126 3402| -377 2024 ie 21 | -216 7831] -378 8138 a7 | 121 4073] -375 5144| 2017 | 21 | -218 0735] -377 3881 |18°7| 19 | -315 9768] -378 9857 38 | -210 3905] -375 7176 | 2032| 18 | 314 9240| 377 5757 |18%8| 17 | -420 6970] +379 1599 39 | 3043704] -375 9296 16 | -417 2053] -377 7649 14] +531 2818] -379 3356 40} 403 6615] -376 1291 | 2°8°| 13 | -525 2853] -377 9556 | 1997| 19 | -648 0989] -379 5127 41 | 508 6058] 376 3370/2078 | 11 | 639 4542] -378 1475 | 1958| 10 | “71 5487] “379 6909 2| -619 5764] -376 5458 |2°8°| 8 | -760 1977] 378 3403/1958 7 | -902 0674] 379 8699 43 | 736 9806] -376 7555 |2097| 5 | -8s7 9308] -378 5338 |1935| 5 | 2-040 1318] -380 0497 44) 861 2638] -376 9657 |2102| 3 | 2023 1367| -378 7279 | 1843 | 2] 186 2627) “380 2299 5 | 9929140] -377 1762/2 166 3448 | -378 9299 | 194 341 0309 | -380 4103 Tables of the & (r, v)-Integrals TABLE LIV—(continued). 131 log F (r, v) } log H (r, v) log F (r, v) | log H (r, v) log F (r, v) } log H (r, ») A 1'803 8114] 0:376 8754| 4, 1-790 2485 | 0:378 2941] 9 1-777 4825 | 0°379 5425 | og 8047405] -3768783| 28/59 | -791 2436] -378 29c9) 28| 55] -778 5436] -3795450/ 78 807 5297] 376 8871| ,$6| 58 | -794 2308] -378 3051) ,87| 55] -781 7289] -379 5528] 5° 812 1842] -376 9018] 349] 58] -799 2158] -378 3188| 127 55 | -787 0445] -379 5607 | 329 818 7130] 376 9222| 3°3| 58 | -806 2081] -378.3380/ 392| 54] -794 5004] -379 5838| 35) 827 1285] -376 9485 58 | 815 2211] -378 3626 541 -8041110] -379 6070| ~ 837 4469] -376 9805| 375] 57 | -826 2719] -378 3927 | 300| 54] -s15 so45] -379 6352 a 849 6881] -3770183| 3/5| 57 | -839 3819] -378 4281] 3°8| 53] -s20 8736] -379 6686 | 324 863 8758] 377 0617| 42° | 56 | -854.5764] -378 4688] 4P°| 53] -8460751] -379 7070| 35) 880 0376] -377 1108] 547] 56 | 871 8848] -378 5149| £0)| 52] -8645306] 379 7503| 783 898 2051] -377 1655 55 | -891 3410] -378 5661 52 | -885 2758] -379 7986 918 4141| -377 2256] G07) 54] -o12 9831] -378 6226 | $04) 51 | -908 3516] 379 8517 a 940 7047 | -377 2912| £05 | 54] -936 8541] -378 6841| 6)°| 50 | -933 8035] 379 9096] 950 965 1215| -37 3622| 770] 53] -963.0016| -378 7507| Soo| 49] -961 6821] 379 9723] G4 991 7138] -377 4384| 202) 52 | -901 4782] -378 9222| 723| 49 | -992 0436] -380 0396 | O15 0-020 5357| -377 5199 51 | 0-022 3418] -378 8985 48 | 0-024 9495] -380 1115 ‘051 6467| -377 6064] $9? -055 6559] -3789796| S11) 47 | -o60 4672] 380 1878 | 268 085 1115] -377 6979| gay 091 4895 | -3790654| S°9| 46 | -098 6704] 380 2686] 228 121 0005 | 377 7942 | 304 129 9181] -379 1558| 904] 45 | 129 6392] -380 3537] 59) “159 3904] -377 8953 | 1557 171 0234] 379 2506] 948) 43 | -183 4606} 380 4430] 535 -200 3641] -378 0011 214 8939] -379 3498 42 | -230 2288) -3805363| 9°? 2440113] -378 1113 | 1102 261 6258] -379 4532 | 1025| 41 | -280 0460] 380 6336 ees 290 4293] -378 2259 | Tigo 311 3226] 379 6606 | 1078) 40 | -333 0225 -380 7348 | 1)15 "339 7229-378 3448 | To90 364 0964] 379 6721 | 1114) 38 | -389 2774] “380 8397 | 1). ‘302 0008] 378 4677 | 1569 420 0681 379 7874 | 1199) 37 | 448 9393 380 9482 | 1955 447 3985 | -378 5946 ‘479 3681 | -379 9063 5 | -513 KR 506 0343] “378 7252 | 1304 542 1372] 380 02891225) 34 | -579 0504] “381 1756 pee “568 0547 | -378 8595 | cs -608 5269] “380 1548 |1222| 33 | 649 8104] 381 2941 | 175 633 6129} -378 9973 | 211 678 7007 | “380 2839 | 129? 31 | -724 6013] “381 4157 | 1515 702 8740] -379 1383 | T4149 ‘762 8356] 380 4162] j55> 29 | 803 6106 81 5402 | 157 776 0162) -379 2825 831 1213] -380 5514 | 1352 "853 2318] -379 42906 | 1475 913 7633 | -380 6893 | 1359) 96 | -975 1103] 381 7973 | 1503 934 7285 | 379 5795 | noe 1-000 9834] -380 8299 | 140° 94 | 1-068 0549] 381 9296 | 1316 1-020 7304] -379 7320 | 1770 093 0211 | -380 9729 |1430| 93 | -166 1294] -382 0642 | 1300 111 4801} -379 8869 190 1352] 381 1181 |1422/ 91 | -269 6092] 382 2009 | 130 207 2399 | -380 0440 | 1°71 -292 6060} -381 2654 |/473| 19 | -378 7922] -382 3395 : 1591 ‘ 1492 : one 1404 Bs aise Vicar GO ae Il eve cee eeu sose |2002) 16-1) gis seca |) sau enzo | 1420 1626 1525 615 58: 220 | 1435 527 5412) -380 5267 | 1°50 635 3206| -381 7180 |1525) 13 | -743 9935 | -382 7655 | 14‘ si] 3 8 ail 8] eae] RII Se) BL sete] ae ae 772.0144 | -380 8560 896 8681 | -382 05 025 1 5 ‘904 7199} -381 0223 oo 2-038 8307 | -382 1826 fae 8 | -173 7686] 383 2028 Tee 2-045 0157 | -381 1894 | 1728 -188 9051} -382 3393 |1957| 6 | 333 6229| -383 3503) 7755 193 4131] -381 3572 | 1678 347 6370] -382 4966 | 1573] 4 | -502 6906] “383 4984] 743) "350 4709 | -381 5254 | 1682 ‘515 6232) 382 65431277) 9] -681 6063} 383 6468 | 115° 516 8011] -381 6938 693 5170] -382 8122 871 0649 | °383 7953 17—2 42 Tables for Statisticians and Biometricians TABLE LIV—(continued), 132 r=18 ?° log F (r, v) | log H (7, v) O 4 1°765 42 0°380 6494 1} °766 5520] -380 6518 2 769 9355 J 380 6591 8 775 5818] -380 6713 4} ‘783 5015] ‘380 6884 5] ‘793 7099} +380 7103 6 | ‘806 2262] 380 7370 7 | °821 0746] -380 7685 8 |] °838 2835] -380 8048 9] ‘857 8863] -380 8457 10 | ‘879 9209] -380 8913 11 904 4307] *380 9415 12 | ‘931 4638] -380 9962 13 | ‘961 0741] ‘381 0554 14] ‘993 3209] -381 1190 15 | 0°028 2696 | -381 1869 16 | °065 9915] +381 2591 17 | ‘106 5648] -381 3354 18 | ‘150 0744] -381 4157 19 | *196 6125] -381 5001 20 | ‘246 2790] -381 5882 21 | ‘299 1823] -381 6802 22] °355 43891] °381 7757 3} 4151758] -381 8749 24} °478 5288] °381 9774 25 | °545 6451] -382 0832 26 | °616 6833] -382 1921 27 | °691 8145] -382 3041 28 | °771 2230] °382 4189 29 | °855 1078} °382 5365 80 | °943 6834] °382 6567 $1 | 1:037 1813] -382 7794 82] 1385 8513] °382 9043 33 | *239 9636] -383 0314 34} °349 8099] -383 1605 35 | *465 7060] -383 2915 36 | ‘587 9937] °383 4241 87 | °717 04385] °383 5583 88 | °853 2571] °383 6938 39 | -997 0711] ‘383 8305 40 1 2°148 9600} 383 9683 41 | °309 4403] +384 1069 42 | ‘479 0754] 384 2462 4% | °658 4799] °384 3860 44 | ‘848 3263] 384 5262 45 4 3°049 3507 | +384 6665 log F(r,v) | logH(r,v)| A 754 0014 | 0381 6376 23 *755 1945 J 381 6399 70 ‘758 7762} °381 6469 116 ‘764 7532 | °381 6584 162 ‘773 1369] “381 6746 208 ‘783 9431} 381 6954 “797 1925} °381 7207 299 812 9104} °381 7505 343 831 1269] °381 7849 388 851 8772} °381 8237 432 875 2015 | °381 8669 901 1455] -381 9144] 378 929 7603] -381 9663 | 229 961 1026] 382 0224] 26! 995 2351 | -382 0826 | 603 0-032 2269] “382 1470 072 1535] “382 2154] 755 115 0974] 382 2877 | x@y 161 1483] 382 3638 | 209 210 4036] *382 4437] 935 262 9690] “382 5273 318 9588] -3826144| oo, "378 4966] °382 7049 | ga, 441 7157 | -382 7988} 974 508 7603 | “382 8960 | 1599 ‘579 7858} “382 9962 654 9597] 383 0994 | 154) 734 4627 | “383 2055 | 199 818 4896] “383 3143 | 3574 907 2506] “383 4257 | 1439 1-000 9723 | -383 5396 099 8993 | °383 6558 1184 "204 2956} -383 7742 1204 314 4464] -383 8946 1223 430 6601 | “384 0170 1241 553 2702} °384 1411 682 6377 384 2667 | 15°) 819 1539] -384 3938 | 1271 963 2435) -384 5222 | 1284 2115 3673] -384 6518 | 1295 9] -276 0267] -384 7823 7 | 445 7673] -384 9136 1320 5 | “625 1841} °385 0455 1324 4] ‘814 9263] -385 1780 1328 2 13:015 7041} °385 3108 1330 228 2952 7 °385 4437 log F'(r, v) 1743 1485 “744 4077 *748 1876 “754 4955 “763 3431 ‘774 7474 “788 7299 805 3174 824 5417 "846 4398 871 0541 898 4326 928 6293 ‘961 7039 ‘997 7224 0036 7574 "078 8894 124 2043 172 7969 ‘224 7699 280 2346 *339 3115 402 1306 468 8327 539 5695 614 5047 693 8148 “777 6902 ‘866 3362 "959 9740 1:058 8424 163 1992 273 3224 389 5124 *512 0941 “641 4188 ‘777 8668 921 8503 2073 8165 234 2509 403 6815 582 6831 ‘771 8822 ‘971 9629 3183 6730 “407 8314 r=20 log H (7, v) 0°382 5253 | *382 5275 382 5341 *382 5451 *382 5605 382 5802 "382 6042 *382 6326 382 6652 382 7021 382 7432 "382 7883 382 8376 *382 8909 *382 9482 383 0093 383 0743 383 1430 *383 2153 383 2912 *383 3706 "383 4534 "383 5394 "383 6286 383 7209 383 8162 383 9142 *384 0150 384 1184 384 2243 *384 3325 *384 4429 384 5554 "384 6698 384 7860 384 9039 *385 0233 *385 1440 *385 2660 385 3891 *385 5130 "+385 6378 385 7632 385 8890 386 0151 *386 1414 ala SEEER naan = bo 09 oo oo ww. opovoor bo We to WO Go Go Go Oo bo DBOornww Hm OO) SIT bo bo bo bo bo Nwoga~s oul sell eel aeelll <>) OsT OO eee Nwagd CO bo Pp Tables of the G (7, v)-Integrals 133 TABLE LIV—(continued). r=91 r=22 r= 23 log F (7, v) | log H (r, v) log F (7, v) | log H (r, v) log F(r, v) | log H(r, v) | A a2 o | 1-732 8121 | 0-383 3271 1-722 9451 | 0384 0548 1-713 5069 [0384 7182| 1] 7341352] -383 3292 724 3364] 384 0568 714 9643} -3847202| }9| as 2| -738 1155) -383 3354 728 5130} -384 0628 719 3391 | -384 7259] 8%) 38 3] -744.7542) -383 3459 735 4826] -384 0728 “726 6397 | 384 7355| ,26/ 38 4| -754 0660] -383 3606 745 2584 | -384 0867 736 8798 | -384 7488 | 135) 38 5 | -766 0683] -383 3793 757 8590 | 384 1047 "750 0786 | -384 7660 38 6] -780 7641] -383 4022 773 3082] °384 1266 766 2613] -384 7869| 302) 37 7 | -798 2414] -383 4293 791 6354 | -384 1593 785 4585] -384 8116| 347) 37 8| -818 4736] -383 4604 ‘812 8757| -384 1820 807 7070] -384 8400 | 29%) 37 9| 8415196] °383 4955 837 0698] °384 2155 833 0493 | -384 8721 | 297) 36 10] -867 4241] -383 5346 864 2646] “384 2529 ‘861 5346] -384 9078 36 11} 396 2374] -383 5776 -894 5129] -384 2940 893 2180] 3849471 | 223! 36 12} -928 0162] -383 6245 927 8740] -384 3388 928 1617} °384 9899 | 429 | 35 13 | -962 8933] °383 6753 964 4139] 384 3872 966 4346| -385 0363] 404) 34 14 | 0-000 7283| -383 7299 0-004 2054] -384 4393 0-008 1129} -385 0861) £95) 34 15 | -041 8074] °383 7881 047 3287] °384 4949 053 2804] -385 1393 33 16 | -086 1444] -383 8500 093 8713 | -384 5540 “102 0289] 385 1958| 36°| 33 171 -133 8306] -383 9154 143 9291] -384 6164 “154 4586} -385 2556| 229 | 32 18 | -184.9653] -383 9843 197 6061 | “384 6822 -210 6783 | -385 3185 | 623) 31 19 | -239 6563] -384 0566 255 0156 | °384 7513 270 8064| -385 3845 | 660) 30 20 | -298 0208] -384 1322 316 2801 | “384 8235 334.9713 | -385 4536 29 21] -360 1851) -384 2111 381 5322| -384 8987 403 3116] -385.5256| 12) | 28 | 22 | -426 2861] -384 2930 450 9155 | -384 9770 475 9774 | -385 6004 | 778] 98 23 | -496 4716| -384 3780 524 5848 | -385 0581 553 1308] 385 6780| 275! 97 24 -570 9011] -384 4659 602 7073 | -385 1420 “634 9466] “385 7583 | £03| 26 25 | -649 7464) -384 5566 685 4632 | °385 2286 ‘721 6136] °385 8411 25 26 | -733 1933] -384 6500 773 0472 | -385 3178 ‘813 3351] -385 9264 on 24 27 | -921 4416] -384 7460 ‘865 6689 | -385 4094 910 3304] 386 0141| 877| 93 28 | -914 7070] -384 8445 963 5543) °385 5034 1-012 8362] -386 1040| 89°) 92 29 | 1-013 2292] -384 9453 1-066 9472 | -385 5996 121 1074] 386 1961 | 221| 20 30 | -117 2380] -385 0484 176 1108} °385 6980 235 4191] -386 2902 19 81 | -227 0250] +385 1535 291 3286] -385 7984 356 0681 | “386 3862 a 18 32 | -342 8756] -385 2607 ‘412 9072 | -385 9007 -483 3750] 386 4840| 978) 17 33 | -465 1055] -385 3696 541 1773] -386 0047 ‘617 6859 | “386 5836 | ,2?°| 16 34] 594.0558] -385 4803 676 4967 | “386 1104 759 3748 | -386 6846 | 1011) 34 35 | -730 0957] -385 5926 819 2523] -386 2175 908 8466 | -386 7871 13 86 | -873 6248] °385 7063 ‘969 8630] -386 3261 2-066 5394] +386 8910 ae 12 37 | 2-025 0761) -385 8213 2-198 7827 | -386 4359 -232 9279 | -386 9960 | 1050) 3) 38 | -184 9195] -385 9375 296 5039 | -386 5468 -408 5272] -387 1021 | 1961 jo 39 | 353 6651] 386 0547 | 473 5612 | “386 6586 593 8968 | -387 2091 | 1070) 40 | -531 8676| -386 1728 |: 660 5361 | -386 7713 789 6446 | -387 3169 | 1978 5 41 | -7201308] -386 2916 ‘858 0615] 386 8848 996 4325 | -387 4954 nae 5 421 9191130] -386 4110 3-066 8272 | -386 9987 3-214 9823] -387 5344 |1000) 4 | 43 | 3-129 5327] -386 5308 987 5865) -387 1131 "446 0817] 287 6438 | 1004) 3 44 °352 1756) 386 6510 521 1629] -387 2278 690 5919] -387 7535 | 1087/4 45 \ 587 9021) -386 7712 "768 4580 | -387 3426 949 4561 | 387 8633 | Tables for Statisticians and Biometricians TABLE LIV—(continued). 134 r=24 log F (r, v) | log H (7, v) 1:704 4618 | 0°385 3256 ‘705 9852 | -385 3275 ‘710 5584] -885 3330 ‘718 1899] -385 3421 ‘728 8942 | -385 3549 “742 6914] °385 3714 ‘759 6077 | °385 3915 ‘779 6750} -385 4151 802 9317] °385 4423 *829 4225] -385 4730 *859 1983] -385 5073 892 3170] -385 5450 928 8434] -385 5860 ‘968 8495] -385 6305 0:012 4148] +385 6782 059 6268 | -385 7292 110 5814] -385 7834 165 3831] +385 8406 224 1458] -385 9009 286 9928] -385 9642 354 0583} +386 0304 425 4870] -386 0994 501 4356} +386 1711 582 0734] +386 2455 667 5830] -386 3225 ‘758 1612] -386 4018 854 0205 | -386 4836 "955 3899] -386 5676 1-062 5163} -386 6538 175 6661] -386 7420 295 1263] -386 8322 421 2069] +386 9242 554 2426] +387 0180 “94 5945] -387 1134 842 6534] -387 2102 998 8417] -387 3085 2163 6169] +387 4080 337 4747 | °387 5086 520 9526 | -387 6103 ‘714 6347 | -387 7128 ‘919 1558] -387 8162 3°135 2068 | *387 9201 363 5411] °388 0246 *604 9809] -388 1295 860 4254] °388 2346 4130 8591] °388 3398 r=25 r=26 A | & | logF(r, v) | log H(r, v),| A log H(r,v) | A | a2 8 1-695 781 | 0385 8838| ,| _ | 1-687 4284] 0°386 3984| 44 15) 37 | 697 3569] 385 8855] 58/35 | -689 0840] 386 4001| 57 | 34 32 | 87] -702 1393] -385 8908] 83] 35] -694 0540] “386 4052] 35 | 34 123| 36] -710 1019] -385 8996] ,55| 35 | -702 3476] “386 4136] 478 | 34 18 | 36 | -721 2704] 385 9119] 122) 35 | +713 9805] -386 4254| J59 | 34 36 | -735 6660] -385 9277] 1°8| 35 | -728 9746] “386 4406 33 201) 36 | -753 3158] -385 9470| 193] 35 | -z47 581] -386 4591] 33 | 23 979| 28 | “774 2534] 385 9697 | 56) | 34 | °769 1658] “386 4810} 95] | 33 siz | 35 | -7985185] -385 9958 | 367] 34 | 794 4395] “386 5061 | Sey | 33 30, | 35 | -826 1578] “386 0253 | 325/34 | -823 2972] “386 5345 | 376 | 32 | 34] -857 2243] -386 0582 33 | -855 5847| -386 5661 32 377) 341 -s91 7784] 386.0943) 362] 33 | -s01 5743] 386 6009 | 375 | 31 ty | 34 | 929 8876] -386 1338) 397) 32 | -931 2665] 386 6388 | 44] | 31 477| 23 | ‘971 6270] 386 1764 | 4-.| 32 | 974 7393] “386 6798 | 440 | 30 51a | 32 | 0017 0795] -386 2223) 4°5| 31 | 0-022 0791} *386 7239 | zy | 30 32 | -066 3362] 386 2712 31 | -073 3806] 386 7710 29 Pty | 31 | 119 4971] -386 3232) 22°) 30 | -198 7480] “386 8210 0 | 29 O73 | 30 | 176 6711] -386 3782) 9°)| 29 | -188 2945] “386 8738| p57 | 28 633 | 22 | 237 9768] 386 4361 | Gog | 29 | “252 1435] “386 9295 | 5g4 | 27 662| 22 | "303 5430] “386 4969 | Go. | 28 | “320 4290] “386 9880 | g7; | 27 “| 28 | -373 5093] °386 5604 27 | 393 2963] 387 0491 26 oor | 27 | 448 0267 | -386 6267 | 66°) 96 | -470 9025 “387 1128 637 | 25 744| 20 | 527 2584] “386 6955 | 777) 25 | 553 4176] “387 1790 | gar | 24 veo | 26 | 611 3809} 386 7669 72°| 25 | -641 0250] “387 2477] 749 | 24 ro. | 22 | -7005843] 386 s408| 729) 24 | -733 9226] -387 3187] 733 | 23 24 | -795 0741] °3869170] ‘°~| 23 | 832 3242] 387 3920 22 oi6| 23 | -895 0715] -386 9955 | 78°| 29 | -936 4600] “387 4674 rs ae g62| 22 | 1000 8153] “387 0762] 65, | 21 | 1046 5783] 387 5450| 796 | 20 oe, | 21 | 112 5627] “387 1589| $47 | 90 | -162 9470] 387 6246 | gi, | 19 po5 | 19 | -230.5913] “387 2436) B7| 19 | 285 8547] “387 7060] gag | 18 18 | -355 2004] -387 3302 18 | -415 6129] -387 7893 17 oog| 17 | -486 7129] “387 4185 | 803) 17 | 552 5576] “387 8742 araie 238) 16 | 625 4776] -387 5085 | 99| 16 | -697 0516] 387 9608 ggo | 15 op4| 15 | -771 8709] -387 6001 | 226| 14 | -849 4867 | 388 0488 | go4 | 14 269] 14 | -926 3001] -387 6931 | 290| 13 | 2-010 2864] -388 1382 | go7 | 13 2/13 | 2-089 2053] 387 7874 12 | -179 9088] -388 2289 12 295) 11 | -261 0633] -387 8829 | 97°) 11 | 358 g499| 388 3208 oe 10081 10 | -442 3906] -387 9795 | 990) 10 | “547 6471} “388 4137] 35 | 9 017! 9 | -6337476| -3880771| 3/8) 9 | -7468833] 3885075) 947 | 8 1026) 8] -s35 7436) “388.1756 | 85| 7 | -957 1916] -388 6022 | 954 | 7 6 | 3-049 0373] 388 2748 6 | 3-179 2602] -388 6975 6 a0] 6 | -274 3517] -388.3746| ,393| 5 | -413 8384] -388 7935 ie 1045) 3 | -512 4708] 388 4749| 197] 4] -661 7426] -388 8900] o¢g | 4 1048! 3 | -764.9515| -388.5756| 194 2] 923 8647] -388 9868] yr | 2 1051) 1 | 4030 6307} -388 6765 | 190) | 1 | 4201 1786] 389.0838 | g75 | 1 312 6342 | -388 7775 494 7524 | -389 1810 Tables of the G (r, v)-Integrals TABLE LIV—(continued). r=27 r=28 ¢° log F(r,v) | log H(r,v) | A A? | log F(r, v) } log H(r, v) A o | 1-679 3877 | 0-386 8744 1-671 6341 | 0'387 3160| 4, 1] -681 1094] 386 8760| 1° | 33] -673 4219] -3873176| 40 2| -636 2778] -386 8809 4° | 33] -678 7887] 387 3223| 2, 3} -6949025] -386 se91| ,8! | 32] -687 7445] -387 3301 | 41, | gee) Ban es = eae 6 | -741 7096] -386 9329 178 | 32 | -736 s4s3] -387 9724| 575 7| -7643877] -386 9539 | 319 | 32 | -759 8968] -387 3927 | 333 8| -790 6698] -3s6 9781 | 342 | 31 | -787 1876] -387 4160 | 334 9 | -820 6063] -387 0055 | 273 | 31] -818.2728] -387 4424 | So 10) -854 2546| -387 0359 31 | -853 2121) °387 4717 11 | -891 6799] -387 0694 | 33° | 30] -802 ov] -387 5041 | 373 on ee 14 | 0-027 3887] -387 1879 | $24 | 29 | 0-032 9863] -387 6183 | 430 15 | -080 7353) -387 2332 28 | -088 3780] °387 6620 16 | -138 3092] -387 2814 | 452 | 28 | -148 1586] -387 7084| 4, Qieeae| 2 a| 8/5] See| wee| es 19 | -337 6258| -387 4422 | 263 | 26 | -355 1112] -387 8635 | 75° 20 | -413 3944] -387 5010 25 | -433 7811] °387 9203 21} -4940896] -387 5624 | 613 | 24] +517 5657| -387 9794 | 27? 22] -579 8882] -387 6261 | 88 | 24] -606 6479] -388 0409 231 -670 9806] -387 6923 | °81 | 93 | -701 2256) -388 1047 | ©38 24) -767 5727) -387 7607 | 04 | 22 | -801 5122| -388 1706 | Gay 25 | -869 8863] -387 8312 21 | -907 7380| -388 2387 26 | -978 1616] -387 9039 | 727 | 20 | 1-020 1512] -388 3088 | 795 27 | 1-092 6538 | +387 9786| £47 | 19] -139 0194] 388 3808 | /2° 28 | -213.6439| -388.0552| 75° | 1g] -264 6312] -388 4547 | 229 29 | -341 4310| -388 1337 ae 17 | -397 2979 | -388 5303 | 278 30 | -476 3386] -388 2138 16 | -537 3550] °388 6077 31} -618 7157] -388 2956 | 818 | 15 | -685 1648] 388 6865 te = | elec eas Rea edb 84 | 2-094 5869| -388 5499 | $92 | 12 | -179 1790] -388 9317 ae 35) -270 9268] -388 6372 11 | -362 2366] -389 0159 36] -456 9512] -388 7257 | $0? | 10 | -555 3447] -389 1012 os paces || sestoors (et ill era cia (seuiarce | 822 39 | 3-078 9562] -388 9967 | 912 | 7 | 3-201 0137] -389 3625 | 87? 40 | -309 7991| -389 0885 6] -440 6310] -389 4511 41} 553 6412] -389 1809 | 924 | 5] -693 7374] -389 5402 | 892 42) -811 3309} -3s9 2738 | 929 | 3] -961 2127] -389 6298 | $98 43 | 4-083 7948 | -389 3670 2 | 4-244 0178] -389 7197 44) "372.0436 | -389 4604 | $34 | 1 | -543 2028] -389 8098 | 978 45 \ -677 1878) -389 5540 859 9176} 389 9000 135 r=29 A? | log F(r, v) | log H(r, v) T-664 1478 | 0-387 7268 31] ‘666 0017] °387 7283 31] °671 5669} °387 7328 31] -680 8538} °387 7404 31} ‘693 8799} °387 7510 31] °710 6695] °387 7647 31} °731 2544} -387 7813 30 | °755 6734} °387 8008 30 | °783.9728] °387 8234 30 | ‘816 2068] ‘387 8488 30} °852 4372} °387 8772 29 | °892 7340} °387 9084 29 | °937 1757] °387 9424 28 | °985 8495] °387 9792 28 | 0°038 8519] -388 0187 27 | ‘096 2888} -388 0609 27 | °158 2763] °388 1057 26 | °224 9410] ‘388 1531 25 | *296 4208} °388 2031 25 | °372 8653] °388 2555 24 | °454 4367] °388 3103 23 | °541 3106] °388 3674 23 | °633 6767] °388 4268 22 | °731 7398] °388 4883 21 | °835 7212} °388 5520 20 | °945 8594] °388 6177 19 | 1062 4115] -388 6854 18 | °185 6549] °388 7550 18 | °315 8886] °388 8263 17] ‘453 4351] -388 8993 16 | °598 6420} °388 9740 15 | °751 8846} 389 0501 14] ‘913 5680] ‘389 1277 13 | 2°084 1297] 389 2067 12 | °264 0426] ‘389 2868 11] °453 8181} °389 3682 10 | °6540100] 389 4505 9] °865 2184] °389 5338 7 | 3°088 0940} -389 6179 7 | °323 3486] °389 7028 5] 5717357] °389 7883 4] ‘834 1065] °389 8744 3] 4°111 3678] -389 9608 2] °404 5148] ‘390 0476 1] ‘714 6355] 390 1346 5°042 9213] °390 2217 30 29 Oo~10 0 mS pw 136 Tables for Statisticians and Biometricians TABLE LIV—(continued). WWW Oits Go 1H © r=31 r=32 log F(r, v) | log H (r, v) log F (r, v) Plog H(r, v) | A | A? | log F(x, v) | log IZ (r, v) 1°656 9109 658 8309 664 5945 0-388 1099} -388 1113 oa 29 “388 1157 29 1649 9073 | 0°388 4679 1°643 1226 | 0-388 8034 "651 8935] °388 4694 28 | ‘645 1748] °388 8048 °657 8555 | °388 4736 * | 28 | °651 3354] °388 8089 “674 2126] °388 1231 a | 29 | ‘667 8048} 388 4808 99 28 | ‘661 6158] °388 8158 687 7031] 388 1333 5 | 29 | ‘681 7598] °388 4906 127 28 | °676 0352] °388 8254 705 0914] 388 1465| 182 | 99 | -699 7466] -388 5034 28 | ‘694 6208] °388 8378 726 4101] -388 1625 | 161 | 29 | -721 7993] -388.5189| 129 | 28 | -717 4073] -388 8528 ‘751 6995 | -388 1815 ae 29 | -747 9592] +388 5372| 511 | 28 | -744 4379] “388 8706 781 0077 | +388 2032 | 218 | 98 | -778 2762] 388 5583 | 545 | 27 | 775 7637] -388 8910 ‘814 3906] -388 2278| 346 | 28 | -8i2 8080] -388 5822 | 555 | 27 | “811 add 388 9140 ‘851 9121] -388 2552 28 | -851 6207] 388 6086 27 | -851 5483] -388 9397 "893 6447] -388 2854| 302 | 27 | -s94 7893 388 6378 a8 26 | -896 1530] -388 9680 “939 6698] -388 3183 | 32? | 27 | -942 3977] -388 6696] 344 | 26] 945 3449-388 9988 990 0775] -388 3538 | 396 | 26 | -994 5394] -388 7041 | 3-5 | 26 | -999 2205] -389 0322 0-044 9676] -388 3920 | 989 | 26 J0-051 3173] “388 7410 | 34 | 25 | 0-057 8864] -389 0680 ‘104 4498] -388 4328 25 | -1128450] -388 7805 | °? | 95 | -121 4595] -389 1062 “168 6443] -388 4762 | 433 | 95 | -179 2464] -388 8225 412 | 24] -190 0681] 389 1469 -237 6820] -388 5220| 428 | 94] -250 6573] 388 8668 ie7 | 23 | '263 8522] -389 1899 311 7056] -388 5703 | 453 | 24 | -327 2249] -388 9136 | 454 | 23 | 342.9638] “389 2351 “390 8701] -388 6209 aud 3] -409 1093] -388 9626] <7, | 22 | -427 5684] -389 2826 -475 3432] -388 6739 22 | -496 4842] -389 0138 22 | -517 8452] -389 3323 565 3065] -388 7291 | 222 | 99 | -589 5372] -389 0673 He 21 | -613.9879] 389 3840 660 9566] -3887865| °“4 | 21 | -688 4714] -389 1998 oog | 20] “716 2063] -389 4379 762 5053] -388 8460 | >9> | 20 | -793 5058] -389 1804| 25, | 20 | -824 7266] -389 4937 ‘870 1816] -388 9076 o 20 | -9048771] -389 2400| gy; | 19 | -939 7929 "389 5514 ‘984 2323] -388 9711 19 | 1-022 8405] -389 3015 18 | 1-061 6692] -389 6109 1-104 9235] 389 0366| 6°4 | 18 | -147 6710] -389 3648 633 | 18 | -190 6391 | 389 6723 232 5423] -389 1038 | 672 | 17 | -279 6653] -389 4299 Gav | 17 | °327 0091] “389 7353 ‘367 3981] 389 1727 ep 16 | -419 1433] -389 4966 | go4 | 16] “471 1094] -389 8000 509 8245] -389 2433 | 700 | 16 | +566 4498] “389 5649| Gog | 15 | -623 2962] 389 8661 ‘660 1814] +389 3155 15 | -721 9568] -389 6348 14 | -783 9534] -389 9338 | ‘8188570 | -389 3891 is 14 | -886 0657] -389 7060 ee 13 | -953 4956] -390 0028 986 2707 | -389 4642 | 790 | 13 | 2-059 2096] 389 7786 | 432 | 13 | 2-132 3701 | “390 0732 2-162 8749] -389 5405 | 783 | 19 | -241 8567] -389 8525 | 450 | 12 | -321 0601] “390 1447 349 1593] -389 6180 | 77> | 11 | 434.5127] “389 9275 | 42) | 11 | 520 0879 390 2174 545 6529] -389 6966 10 | -637 7246] -390 0035 10] +730 0183] -39 ‘752 9288] -389 7762 bed 9 | -852 0847] -390 0806 io 9 | -951 4628} -390 3657 971 6080] -389 8567 | 80 | 8 | 3078 2349] 390 1585 | 4°" | 8 | 3-185 0841] “390 4412 3202 3639} -389 9380| 819 | 7 | -316 8712] -390 2372) 441) 7 | -431 6009] "390 5174 “445 9277] 390 0201| 850 | 6 | 568 7494] -390 3166 | 455 | 6 ‘691 7938 '390 B44 -703 0947 | -390 1028 | 827 | 5] -834 6915] -390 3966 51 -966 5111] * ‘974 7302] -390 1859 | 832 | 4 | 4-115 5919] -390 4771 So, | 4 | 4-256 6765] “390 7498 4261 7776] -390 2695 | 896 | 3 | -412 4256] -390 5580| 815 | 3| "563 2967] -390 6282 “565 2668] -390 3534 | 829 | 2] -726 2571) -390 6392| g14 | 2] “887 4707] -390 9069 886 3234] 390 4375 | S41 | 1 [5-058 2499] -390 7206 | g15 | 1 | 5230 3999] -390 9857 5°226 1804 *S90iD217)| = ‘409 6782 *390 8021 i "593 3997 “BOL 0646 Tables of the G (r, v)-Integrals r=s30 log F'(r, v) | log H (r, v) A A? | log F (r, v) 1-636 5434] 0°389 1183] 5 1-630 1576 638 6617] -3891197| 13 | 27 | -632 3421 -645 0208] -389 1937| ¢? | 27 | -638 8996 -655 6323] -3891304| $7 | 26 | -649 8424 670 5163| -389 1397] ,3° | 26 | -665 1908 -689 7005 | -389 1516 26 | -684 9738 713 2210] -389 1662 toe 26 | -709 2283 741 1222] -389 1835 | $42 | 96 | -738 0001 773 4568 | -389 2033 | 39° | 26 | -771 3436 “810 2865 | -389 2256 | 226 | 25 | -809 3225 851 6818 | -389 2505 25 | -852 0090 897 7225] -389 2780 | 344 | 25 | -899 4858 948 4980] -389 3078 | 29? | 24 | -951 8449 0-004 1077 | -389 3402 | 343 | 24 | 0-009 1887 -064 6615 | -389 3749 | 347 | 24 | 071 6306 -130 2802] -389 4120| ®”! | 93 | -139 2949 -201 0960} -389 4514 ts 23 | -212 3180 277 2533 | -389 4931 | 434 | 22 | 290 8487 358 9091] -389 5370] 13) | 22 | -375 0487 “446 2340] -389 5830| {05 | 21 | -465 0938 539 4127] -389 6312 20 | -561 1746 638 6453] -389 6814 aes 20 | -663 4971 744 1480 | -389 7336 | 227 | 19 | -772 2842 856 1542] -389 7877 | Pay | 19 | 887 7765 974 9160 | -389 8437 | 2°) | 18 | 1-010 2336 1-100 7050} 389 9014 17 | -139 9357 233 8145} -389 9609 | (9? | 17 | 277 1848 "374 5602} -390 0220 | 647 | 16 | -422 3065 523 2830] -390 0847 | 615 | 15 | “575 6518 680 3501] -390 1489 | 622 | 14 | -737 5995 846 1578] -390 2145 13 | -908 5576 2-021 1335] -390 2815 ae 13 | 2-088 9669 205 7387 | °390 3497 | 68° | 19 | -279 3029 -400 4717 | -390 4190 | £93 | 11 J -480 0791 -605 8714] -390 4895 | 799 | 10 | -691 8509 822 5204] -390 5610 9 | -915 2186 3-051 0494] -390 6333 ie 8 | 3-150 8322 292 1420] -390 7065 | 732 | 7 | -399 3963 546 5396 | -390 7805| 199 | 6 | 661 6747 815 0471 | 390 8551| 12° | 6 | -938 4971 4-098 5399 | -390 9302 5 | 4-230 7654 397 9704 -391 0058 | 196 | 4] -539 4613 “714 3773| -391 0818| 469 | 3] -865 6549 5-048 3939] -391 1581| 763 | 9 | 5-210 5144 402 7596] -391 2345 | 76 | 1| 575 3166 777 8311 | -391 3111 -961 4600 B. 137 TABLE LIV—(continued). r=34 r=35 log H(r, v) A A? | log F(r, v) | log H (r, v) A A2 0°389 4146} 1. 1:623 9542 | 0°389 6937 13 389 4159| 13 | 96 | -626 2048] -389 6949] 35 | 25 389 4198| 2? | 26 | -632 9608| 389 6987| 3 | 25 389 4262| 6° | 26 | 644 2348] -389 7050] 33 | 25 "389 4353 | 28 | 26 | -660 0478] 389 7138| ,}5 | 25 389 4469 26 | -680 4295] -389 7251 25 389 4611 | 14? | 95 | -705 4180] -389 7889 | 745 | 25 "389 4778| 164 | 25 | -735 0604] “3897551 | jg- | 24 "389 4970 | 507 | 25 | -769 4129] -389 738 | 51 | 24 “389 5187 | 314 | 25 | -808 5407] “389 7948 | 535 | 24 389 5429 24 | -852 5188] -389 8183 24 5G 389 5695 | 266 | 94 | -901 4317] 389 s442| 32. | 23 389 5985 | 370 | 24] -955 3744] 389 8724| 3,5 | 23 "389 6299 337 23 | 0°014 4525 | “389 9029} 35. | 23 “389 6636 | 227 | 23 | 078 7824] 389 9356] 35, | 22 389 6996 23 | -148 4925) -389 9706 | °° | 22 2 389 7379 | 35? | 99 | -223 7229] -390 0078 eibel 389 7783 | fon | 21 | 304 6270| -390 0471] 474 | 21 389 8209 | {ie | 21 | 391 3713] “390 0884 | 434 | 20 -389 8656 | 4a" | 20 | -484 1369] 390 1319] 454 | 20 389 9123 20 | -583 1198] -390 1773 19 389 9611 a 19 | “688 5323] “390 9246 | 4 | 19 390 0117 | 227 | 19 | -800 6039] -390 2738 | 519 | 18 390 0643| 222 | 18 | -919 5823] -390 3248| 553 | 17 “390 1186 | P43 | 17 | 1-045 7350] “390 3776 | S45 | 17 -390 1746 17 | -179 3502] -390 4321 16 390 2324 | °7 | 16 | -320 7390] -390 4881 eer ite 390 2917 | 0°32 | 15 | -470 2366] -390 5458 | 54) | 15 “390 3525 | 609 | 14 | -628 2047] -390 6049 | Go; | 14 390 4148 | 65? | 14 | 795 0329] 390 6654 | gig | 13 -390 4785 13 | -971 1417] 390 7273 13 390 5435 | 850 | 19 |a-156 9847 390 7904| G15 | 12 -390 6097 | 682 | 11 | 353 0516] 390 8547 | g54 | 11 “390 6770 684 11 | 559 8711} 390 9201 | ge, | 10 -390 7454 | 684 | 10 | -778 0150] 390 9865 | g7, | 9 -390 8148 9 |3-008 1016] -391 0539 9 i -390 8850 oe 8 | -250 8000] -391 1222 a 8 390 9561| 411) 7] -506 8356] “391 1912] go7 | 7 391 0278 | 238 | 6 | -776 9950] 291 2609| ~53 | 6 391 1002 | #24) 5 | 4-062 1324] 391 3312| fo | 5 -391 1732 | “ 4] 3631764] -391 4021 4 go1 2466 | 734) 4] 681 1377] 391 4734 pre Aled 391 3203 | 733 | 3 | 5-017 1183] -3915450| 715 | 3 391 3943 | 149 | 9] -372 3207] -391 6169] fo) | 2 391 4686 | 742 | 1] -748 0596] “391 6890/ 455 | 1 391 5429 | 6-145 7749 | -391 7612 18 138 Tables for Statisticians and Biometricians TABLE LIV—(continwed). r=36 r=37 r=38 e Se ees | log F(r, v) | log H(r,v) | A | A? | log F(r,v) | log H(r,») | A | A} log F(r,v) flog H(r,v)| A | 2? —= —_—_—— — ——— = 0 | 1-617 9231 | 0389 9572] 45 1-612 0550 | 0390 2063] 45 1-606 3413 | 0.390 4421) 4, | 1| -620 2399] -389 9584| 22 | 24] -614 4378] -390 2074] 37 | 24] -608 7902] “390 4433| 3° | 23 2} 627 1943] -389 9620/ 3/ | 24] -621 5908] 3902110] 3° | 24] -616 1417] "390 4468| $2 2s 3| -638 7995] -389 9682 6! | 24] -633 5272] -300 2170] $9 | 24] -628 4093] “390 4526) By | 23 4} 6550771} -389 9767| ,8° | 24] -650 2694] -390 2253 | 57 | 24] -645 6161] “390 4607] 15, | 23 5 | -676 0575| 389 9877 24] 671 8485] -390 2360) 197 | 23} -667 7940] -390 4711 | 1° | 93 6} -701 7800] -390 0011 | 134 | 24 | -698 3051] 390 2490 130 | 93 | -694 9847 | -390 4837 ie 23 7 | -732 2932] -390 0168 | 198 | 24] -729 6890] -390 2643 | 197 | 23 | -727 2393] “390 4987 | 129 | 23 8 | -767 6546] -390.0350| 18? | 24] -766 0594] -390 2820| 344, | 23] -764 6187 | “3905159 144 | 22 9 | 807 9316] -390 0555 | 30° | 23 | -807 4855] -390 3020 | 599 | 23 | -So7 1940] “3905353 596 | 22 10 | -853 2010] -390 0783 23 | 854.0464] -390 3242 22 | 855.0464] 390 5569 22 11 | -903 5502] -390 1035 ao 23 | -905 8318] -390 3486 =o 22 | -908 2680] -390 5808 535 | 22 12| -959 0766] -390 1309 | 374 | 22} -962 9420] -390 3753 | 524 | 22 | 966 9620] “390 6067 | 54) | 21 13 | 0-019 s889| -390 1605 | 396 | 92 | 0-025 4886] -390 4041| 305 | 21 | 0-031 2499] 390 6348 355 | 21 14 | -086 1070] -390 1924 | 318 | 22] -093 5948] -390 4351 | 359 | 21 | 101 2374] “390 6650 | 355 | 20 15 | +157 8628] -390 2264 21 | -167 3965] -390 4682 211 -1770849| -390 6972 20 : 361 ie : 352 . ony ees 16 | -235 3007] -390 2625 | 35) | 21 | -247 0418] -390 5034] 3°2 | 20 | -258 9378] 390 7314) 365 | 20 17 | 318.5782] -390 3007 | 35° | 90 | -332 6929] -390 5405 | 3/° | 90 | -346 9624] 390 7676 | Say | 19 18] -407 8669] -390 3409 | 495 | 20 | -424 5260] 3905797 | {71 | 19 | -441 3400] “390 8057 | ‘oq | 19 19 | 503 3529| -390 3832 | 447 | 19 | 522 7325] -390 6208 | 434 | 19 | 542 2671| “390 8457| Jy, | 18 20 | -605 2380] -390 4273 19 | -6275199] -390 6637 18 | -649 9569] -390 8876 18 are wi [ 460 . 448 ars. a1} -713 7407] 390 4733 | 460 | 18 | -7391128] -390 7085 | 468 | 18 | -764 6400] 390 9312] 423 | 17 22 | -829 0968] -3905212| 472 | 18 | -857 7536] -390 7551 | fog | 17 | “886.5655 | “3909765 | 475 | 17 23 | -951 5615] -390 5708 | #76 | 17 | -983 7045] -390 8033| 493 | 16 } 1-016 0028} 391 0235 | 128 | 16 24 | 1-081 4096 | -390 6221 | 958 | 16 | 1-117 2483] -390 8532| 57? | 16 | -153 2423] -391 0721] 559 | 16 25 | -218.9381] -390 6750 16 | -258 6900} -390 9048 15 | -298 5974] -391 1223 | ©? | a5 | - | 26 | -364 4667 | -390 7296 aa 15 | -408 3585] -390 9578 | P20 | 15 | -452 4059] “391 1739 ory pie 27 | 518 3405] 390 7856| Pee | 14] -566 6085] -301 0123 | 25 | 14] 615 0321] “391 2270| P15 14 28 | -680 9313] -390 8431 | 24° | 14 | -733 8222] -391 0682 | 2°2 | 13 | -786 8688] “391 2815 | 2°? | 13 29 | -852 6402] °390 9019 13 | -910 4119} -391 1255 13 | -968 3394] °391 3372 | 22 | 12 30 | 2-033 8997 | -390 9621 | 9° | 19 | 2-096 8223] -391 1840| °°? | 12 | 2-159 9006 | -391 3942 | °/° | 12 s1| -225 1766] -391 0234| 614 | 12 | -298 5330] -301 2437 | P27 | 11 | -362 0454] 391 4523 | 20) | 11 32} -426 9744] 391 0859 | 65> | 11 | -501 0620] 301 3046 | 608 | 10 | 575 3055] “391 5115] Gg | 10 33 | -639 8374] -301 1495 | 635 | 10 | -719 9685] -391 3664| 629 | 9 | -s00 2556] -301 5718] g15 | 9 34 | 864.3536] -391 2141 9 | -950 8571] 391 4293 | ©25 | 9 | 3-037 5167] 391 6330 9 35 [3-101 1591] -391 2796 | © | 8 |3-194 3816] -391 4930 | ©87 | 8 | -287 7604] -391 6950 ©! | 8 36 | -350 9494] -391 3460 ce 8 | -451 2499] -301 5576 | 646 | 7 | -551 7138] 391 7579 te 37 | -614 4497] -391 4131 | 672 | 7 | -722 2990] -301 6229 | 6°3 | 6 | -830 1647] “391 8215| G15 | 6 38 | -892 4902] 391 4809 | 678 | 6 | 4-008 1507] 391 6888 | 622 | 6 | 4-123 9677] °391 8857| Gig | 6 39 | 4185 9427] -391 5492 | 684 | 5 | -309 9183] -391 7553| 66° | 5] -4340507] 391.9505] ges | 5 yo | -495 7624] -391 6181 4] -628 5140] -391 8224| © | 4] -761 4293] 392 0157 A 41 | 822 9894] -391 6874 | 893 | 3} -965 0066] -391 8898) 74] 3 | 5107 1807] 3920814 | Go% | 3 2 | 5-168 7569] 391 7571 | 627 | 3 [5320 5613] -391 9576| 6/8 | 3] -472 5925] -3921474| Geo | 2 3 "534 3023 *391 8270 | 701 2 “696 4499 "392 0256 682 9 “858 7544 392 2136 664 2 4 | 9209782] -391 s971| 703 | 1 ]6-094 0627| -302 0938 | 682 | 1 | 6-267 3043] -392 2800] gas | 1 5 16-330 2655] -391 9673 | 7 514 92221 -392 1621 | 88 699 7361] °392 3465 Tables of the G (r, v)-Integrals TABLE LIV—(continued). r=39 r=40 ?° log F'(r, v) | log H(r, v) A log F'(r, v) | log H (r, v) A 0 | 1-600 7740] 0-390 6658) 4, 1-595 3459 | 0°390 8782| 4, 1} 6032801} -390 6669| 4! 597 9271] -3908793| 55 2] -610 8391] -3906703| 34 -605 6756 | -390 8826| 22 3] -623 4380] 390 6760| 2° “618 6058] -390 8881 | 2 4] -641 1093] -390 6338 | 7 | 636 7416] 390 8958| 44 5 | -663 8860] -390 6940 660 1171 | -390 9057 6 | 691 8107] -390 7063) $74 688 760] 390 9177 | 130 7 | -724 9361] -390 7209 | 146 722 7721] "390 9319 | 145 8 | -763 3246] 390 7377 | 168 762 1697 | 390 9483 | 182 9 | 807 0490] -390 7566 | 3°? 807 0434] -390 9667 | 35? 10} -856 1930] -390 7777 857 47891 -390 9873 905 11 | -910 8510] -390 8009 | 32. ‘913 5732] -391 0099 a 12 | -971 1287] -390 8262 | 393 975 4348 | -391 0346 | 34 13 | 0-037 1440] 390 8535 | 374 0-043 1845] -391 0612 | 50" 14 | 109 0268] -390 8829 | 394 “116 9556] 391 0899 | 37 15 | 186 9202] -390 9143 -196 8949} -391 1205 | * 16 | -270 9806] -390 9477 | 338 283 1630] “391 1530 | 329 17 | 361 3789| -390 9829 | 3°? 375 9350] 391 1874 | 3a 18 | -458 3010] -391 0201 | 37) 475 4017 | “391 2236 | 305 19 | °561 9488] -391 0591 | 390 581 7701] -391 2616 | 35. 20 | -672 5410] -391 0998 695 2648] -391 3014, a1| 7903143] 391 1493| 420 816 1285] -391 3428 | 74 22 | -915 5247] -391 1865 | {°° 944 6237] -391 3859 | fi 23 | 1-048 4484} -391 2323 | 498 1-081 0339} °391 4305 | 444 24 | -189 3837 | -391 2796 | 428 225 6650] “391 4767 | 4° 25 | ‘338 6522} -391 3285 878 8470] ‘391 5243 | ~'‘ 26 | -496 6007] -391 3788 | 20? 640 9357] -391 5734 31 27 | -663 6033 | -301 4306 | Pa) 712 3146] 391 6238 | 3 28 | “840 0630] -391 4836 | 23) 893 3974] -391 6756 | 24 29 | 2-026 4145] -391 5379 | 24? 2-084 6300] -391 7285 | 279 30 | -223 1268] -391 5935 986 4933 | -391 7827 5 552 31 | -430 7056] -391 6501 Be -499 5062] -391 8379 | 775 32 | -649 6970] -391 7078 | 22 724 2290 | 391 8942 | 303 -880 6908] -391 7665 | 287 | 9 | 961 2666) “391 9514) 3/7 34 | 3-124 3244] -301 8261 | $96 | 8 | 3-211 2729] -392 0095 | 35) 35 | -381 2874] -391 8866 8 | -474 9551] -392 0685 36 | -652 3259] -391 9479 ae 7| -753 0789] -392 1982 | 227 37 | -938 2488] -392 0098 | 619 | 6 | 4-046 4738] 392 1886 | G74 38 | 4-239 9332] -302 0724| 626 | 5 | -a56 0307] 392 2496 | Grp 39 | 558.3315] -392 1355 | 622 | 5 | -682 7535] -392 3112 G55 40 | 8944793} -392 1991 °° | 4 | 5-027 6774] -392 3732 41 | 5-249 5035] -392 2631 S40 | 3] -391 9676] -392 4355 | B57 ye | 624 6328] -302 3274, O43 | 2] -776 8842] -392 4982 | 254 48 | 6-021 2079| -392 3919 | 649 | 2 | 6-183 8028} -392 5612 | 625 #4 | -440 6950) -392 4566 | 647 | 1] 6142272] -392 6242 | Go) 45 | 884 6991] 392 5213 7-069 8037 | -392 6874 139 r=4] A? | log F(r,v) J log H(r,v) | A | A? |} T-590 0501 | 0°391 0801] |, a2 | -592 6975] -391 0812] 3) | 21 22 | -600 6445] 391 0844] 22 | 21 22} -613 9059] -391 0898] 32 | 21 22} -632 5064] 3910973) 4° | 21 22 | -656 4807] -391 1069 | 21 | 22 | -685 8737] 391 1187 | on 21 21 | -720 7406] 391 1325| 720 | 21 a1} -761 1472] 391 1485| Too | 21 21 | -807 1702] 391 1665 | S50 | 20 21] -858 8973] -391 1865 20 20] -916 4280] -391 2086 | 321 | 20 20 | -979 8734] 391 2327 | 3a) | 20 20 | 0-049 3576] “391 2587 | 30 | 19 19} -125 O71} 391 2867 | S59 | 19 19 | -207 0023] -391 3165 789 | 19 19 | -295 4780} -391 3483 ou 18 18 | -390 6238 | -391 3818 322 | 18 18 | -492 6351} -391 4172 | 229 | 17 17} ‘601 7243] -391 4542 388 17 17] -718 1215} -391 4930 | 16 16 | -842.0755] 391 5334 | 495 | 16 | 16 | -973 8556] 391 5754] 435 | 16 15 | 1-113 7524] -391 6190 | 429 | 15 15 | -262 0794] 391 6640 {2s | 14 14] -419 1750} -391 7105 14 14 | 585 4038] “391 7584 | 472 | 13 13 | -761 1593] -391 8076 | 55. | 13 12 | -946 8652 -391 8581 | 777 | 12 12 | 2142 9789] -391 9097 | 234 | 11 11 | -349 99331 -391 9626 ul 10 | -568 4405 | -392 o164 | 223 | 10 10 | -798 8947 | 3920713 | 2-9 | 9 9} 3-041 9761] -392 1272 | 2a | 9 8 | -298 3551] -392 1939 | 997 | 8 8 | -568 7567] 392 2414 | 8 7 | -853 9658] -392 2997 | 283 | 7 6 14154 8329} -392 3586 595 io 5 | -472 2803] -392 4181 | B45 | 5 5 | 807 3096] -3924782| Foo | 5 415-161 0098] 392 5386 | 4 3 | -5345660] -392 5995 ae 3 2} -929 2701] -392 6607| Ry | 2 1 | 6-346 5322] -392 7221 | Pr | 1 o | -787 8038] 392 7836| Pie | 1 7-255 0429 | -392 8452 1s—2 140 Tables for Statisticians and Biometricians TABLE LIV—(continued). r=42 r=43 log F(r, v) | log H (r, v) |, log F (r, v) 1-584 8804 | 0391 2724] 1, 1-579 8310 | 0391 4556] 45 1°574 8962 587 5939 | -391 9734) 3° | 21 | -582 6106] -391 4566| 10° ‘577 7420 595 7394] -391 2766) 3) | 21 | -590 9546] -391 4597] 31 586 2845 609 3321] -3912818| °° | 21 | -604 8785| -391 4648) 3 -600 5397 “628 3972] -391 2801) 22 | 91 | 624 4083] -391 4720] 22 620 5341 652 9704] -391 2985 21 | -649 5803] -391 4812 646 3049 683 0975] -391 3100] }3° | 21 | 680 4416] -391 4924] 152 677 9004 “718 8352] 391 3235 | 122 | 20] -717 0501] -391 5056 | 132 715 3798 760 2510] 391 3391| 378 | 20 | -759 4750] 391 5208 | 175 758 8138 807 4232] -391 3567 | 148 | 20 | -807 7965] -391 5380) 17° -808 2846 -860 4419] 391 3763 20 | -862 1068] 391 5571 863 8866 919 4089] -391 3978 | 53° | 19 | 922 5103] -391 5781 | 240 925 7265 ‘984 4383] 391 4213] 537 | 19 | -989 1236] -391 6011 | 22° 993 9238 0-055 6569] 391 4467 | 575 | 19 | 0-062 0767| -391 6259 | 308 0-068 6113 133 2048] -391 4740] 5°3 | 19 | 141 5130] 301 6526 | 267 149 9361 217 2361] 391 5032 18 | -227 5903] -391 6810 238 0595 307 9195] 391 5341] 310 | 18 | -320 4814] 391 7113 an 333 1584 405 4390] 391 5669| 379 | 17 | -420 3748] -391 7433 | 320 435 4256 509 9950] -391 6014 309 | 17 | 527 4755) -391 7770 | 322 545 0710 ‘621 8050] -391 6376 | 2° | 16 | -642. 0063] -391 8123 3°? 662 3227 741 1047] -391 6754 16 | -764 2086] -391 8493 "787 4277 ‘868 1492] -391 7149] 39° | 16 | -s04 3438] -391 8878 | 35° 920 6532 1-003 2143] -391 7559 | 45° | 15 | 1-032 6937| -301 9279 | 49! 1-062 2883 146 5976] 391 7984| 475 | 15 | -179 5637) -391 9694 | 41° 212, 6450 -298 6206] “391 8424] 4°0 | 14 | -335 2826] -392 0124 | 430 372 0600 459 6298] -391 8878 13 | 500 2054] -392 0567 540 8966 ‘629 9989 | -391 9345 | 04 | 13 | 674 7149] -392 1024] 4° 719 5463 810 1308} 391 9826 | 459 | 12] -859 2234] -392 1493 | 46° 908 4315 2,000 4600] -392 0318 | £93 | 12 | 2-054 1759] -392 1974| 482 2-108 0073 201 4548 | -392 0823 | 78 | 11 | 2600519] -392 2467 | 493 318 7645 ‘413 6204] -392 1338 10 | -477 3687] -392 2970 541 2327 KRldz 637 5019} 392 1864] 726 | 10 | -706 6845] -392 3484 | 214 775 9829 873 6876] 392 2400] P28 | 9} -948 6018] -392 4007| 232 3-023 6317 3-122 8129] -392 9945 | P45 3-203 7711] -392 4540 | 23° 284 8450 "385 5647 | -392 3499 | P>4 -472 8957 | -392 5081 | 241 560 3426 662 6857 | 392 4060 "756 7363] 392 5629 850 9027 954 9802] -392 4629 | PO? 4-056 1162] -392 6185 | 28 4157 3682 4-263 3195] 392 5204 581 3 | 371 9277] -392 6747 567 5 | *480 6520 588 6485] 392 5785| Fo, | 5 | -705 1384] -392 7314] 77) *821 7444 931 9935 | -392 6371] Fo5 | 4 | 5-056 7991] -392 7886 | --- 5°181 7208 5294 4700} 392 6962 4] -428 0520] +392 8463] ?/ ‘561 7502 677 2993 | -392 7556| 224 | 3 | -820 1404] -392 9044) 200 963 1049 6-081 7839 | -392 8153| 504 | 2 | 6234 4197] -392 9627 | 7 o- 6°387 1719 509 3896 | 392 8752] Go, | 1] -672 3690] 393 0212 | 27 “835 4649 961 6887 | -392 9353 | go | 1 |7:135 6055] 393 0799 | 2o. 7°309 6389 7-440 4103] 392 9954| 7 ‘625 8999 | -393 1386 | °°‘ “811 5060 0°391 6305 391 6316 391 6345 391 6395 391 6465 391 6554 391 6664 B91 6793 391 6942 B91 7109 391 7296 "391 7502 “391 7726 391 7769 391 8229 391 8508 391 8803 391 9116 B91 9445 B91 9791 *392 0152 *392 0528 392 0920 392 1326 392 1746 “392 2179 392 2625 392 3084 392 3554 392 4035 *392 4528 392 5030 392 5541 “392 6062 392 6590 392 7126 “392 7669 392 8218 392 8773 *392 9332 +392 9896 393 0463 393 10383 *393 1605 393 2178 "393 2752 3S bo oo aa wocew i) “IDnDOO ee “1-1 a loner} — on ell ell ell edie Coto He He OF Bee ee OoOnrrb to me bow eo “10M OO Tables of the G (7, v)-Integrals TABLE LIV—(continued). peas aes ¢° log F'(r, v) | log H(r, v) A A? | log P(r, v) | log H(r, v) A A? y | 1570 0711] 0391 7975 | 45 1-565 3509 | 0391 9572| 1 1} 572 9830| 3017984] 3° | 20 | -568 3289] 391 9581/ 30 | 19 2} 581 7241 391 8014] 32 | 20| 577 2685] -391 9610) 30 | 19 3] 5963106] -301 8063] 42 | 19 | -592 1863] -391 9658] {9 | 19 4} 616 7696] -301 8131] 68 | 19] -613 1100] 391 9725] 8 | 19 6 | -643 1393] -391 8219 19 | -640 0785 | -391 9811 19 6 | -675 4689] 391 8326 a 19 | -673 1423] 391 9915 ao 19 7} -713 8192] -391 8452 | 138 | 19 | -712 3634] -392 0039 | 125 | 19 8 | -758 2623] -391 8598 | 4) | 19 | -757 8157] -392 o181| Té7 | 18 9 | -808 8824] -391 8762] 183 | 19 | -809 5852] -392 o342| 18) | 18 10 | -865 7761] -391 8944 18 | -8677706| 392 0520 18 11} -929 0524] -391 9145 | 3) | 18 | 932 4834] -392 0717] 397 | 18 12 | -998 8337] -391 9365 | 32° | 18 J 0-003 8487] -392 0932 | 343 | 18 13 | 0-075 2558] -391 9602 | 52 | 18 | -082 0054] -392 1164 | 332 | 17 14 | 158 4691] -391 9857 | 3° | 17 | -167 1071] 392 1413 | 363 | 17 15 | -248 6386 | -392 0129 17 | -259 3228] -392 1679 17 16 | 345 9453] -392 0418 | 3°? | 17 | -358 8372] -392 1962 | 383 | 16 17} -450 5863 -392 0724 | 298 | 16 | -465 8522] -392 2261 | 29? | 16 18 | -362 765} -392 1046 | 352 | 16 | 580.5873] -392 2576 | 24 | 16 19 | -682 7492] -392 1383 | 325 | 15 | -703 2808] -392 2906 | 33h | 15 20 | -810 7568] -392 1737 15 | -834 1911] -392 3252 15 a1] -947 0729] -392 2105 | 38° | 15 | 973.5979] -392 a612 | 3°? | 14 22 11-091 9931] -392 2488 | 35 | 14 J 1-121 8031] -392 3987 | 3/2 | 14 93| -245 8365] -392 2885 | 37 | 14 | -279 1333] -392 4375 | 358 | 13 24 | -408 9476] -392 3295 | 45 | 13 | -445 9406] 392 4776| 40° | 13 25 | +581 6979] -392 3719 13 | -622 6047| 392 5191 12 26 | -764 4880] -392 4155 | {20 | 12 | -so9 5352| -392 seis | 427 | 12 27 | 957 7499] -392 4603 | 465 | 12 | 2-007 1738] -392 6056 | 435, | 11 28 | 2-161 9490] -392 5063 | 3°” | 11 | -215 9964] -392 6506 | fey | 11 29 | -377 5876 | -392 5534 | 447 | 10] -436 5163] -392 6967 | 4°) | 10 30 | -605-2071| 392 6015 10 | -669 2873] -392 7437 10 31] -845 3918] -392 6506 | 525 | 9] -914 9064] -392 7918| 480 | 9 | LA £9 1 . Fon 29.17 . 3 ed peted ete ae ee et ae 34,1 -647 9002] -392 8032 | 247 | 7] -735 5636] -392 9410 | 202 | 7 ! : 524 513 35 | -945 1800] -392 8556 7 | 4-039 56314 -392 9993 6 36 | 4-258 7310] -392 9087 | ?37 | 6 | -360 1998] -393 o442| 25? | 6 37 | -589 4872] -392 9624 5 | -698 4283] -393 0967 | 222 | 5 38 | 938 4614] -393 0166 | 247 | 5 |.5-055 2844] -393 ve Bee Ob 39 | 5306 7534] -3030713| >a! | 4] -431 8025] -393 2033 | 22> | 4 4o | -695 5595) -393 1264 3 | -829 4749] -393 2572 3 41 | 6106 1805] -303 1818 | >? | 3 ]6-249 3623] -393 3115| 243 | 3 Fe ee oalecaee 2) GG Waeiisesel aeaiaoy (247 | 1 44, | -483 7836] -393 3496 | 25! | 1] 6580347] -393 4755 | 245 | 1 ro ese Ti 15) | 96 | 561 755 | 549 45) -997 2234) 393 4057 8-183 0474] -393 5304 141 rahi a. log F (r, v) | log H(r, ») A A? 1-560 7311 | 0392 1100] 5637753] 3921110] 92 | 19 *b72 9134] °392 1138 47 19 -588 1625] 3921185 | 4 | 19 -609 5508} 3921250] 88 | i9 637 1182) -392 1334 18 670 9162] 392 1437] 39° | 18 711 0082 | “392 1557 | q3y | 18 757 4696 | -392 1697 | q3> | 18 810 3885 | -392 1854| 197 | 18 869 8656] -392 2029 18 936 0149] “392 2221 | 39 | 17 0-008 9642 | -392 2431 | 350 | 17 088 8555] 392 2658 | 34) | 17 ‘175 8458 | -392 2902 | 56r | 17 270 1076) °392 3163 16 2 ‘371 8299] “392 3440 | 348 | 16 481 9188 | 392 3732 | 3y0 | 16 598 4987 | -392 4041 Sys | 15 723 9132] 392 4364 | 333 | 15 857 7263] °392 4702 14 1-000 9236] -392 5055 | 3°3 | 14 151 7141 | -392 5421 | 387 | 13 312 5311] -392 5801 | $59 | 13 “483 0345 | -392 6194 | 408 | 13 663 6124] -392 6600 re 854 6834] -392 7018 | Joo | 12 2-056 6988) 392 7447 | 440 | 11 270 1448 | 392 7887 | fay | 11 -495 5462 | -392 8338 | 4, | 10 733 4685 | -392 8799 9 9345222] -392 9269] 11) | 9 3-249 3662] “3929748 | 42° | 8 528 7119 | -393 0235 | fo, | 8 823 3284| -393 0729 | 20) | 7 4134 0476] -393 1231] °°? | 6 5 461 7700] -393 1740 | 208 | 6 807 4710] 393 2254 | 215 | 5 5172 2090] 393 2773 | 252 | 6 557 1330] 393 3297 | 255 | 4 963 4920] -393 3824 3 _. | 531 6392 6458] -393 4355 | P31) 3 846 0761] “393 4889 | 228 | 2 7-325 4006] 393 5424| 22° | 1 832 3876| -393 5961 | 22! | 0 8-368 9732 | -393 6498 142 Tables for Statisticians and Biometricians TABLE LIV—(continued). r=48 r=49 Pg = = log F(r, vy) }log H (vr, v) | A | A® f log F(r, ») log F(r, v) 0 | 1:556 2075 | 0°392 2565 9 | 1-551 7763 | 0:392 3969 9 1-547 4336 1} 3593178] -392.2574| 59 | 18] 5549527] -392 3978 9 | 18 | 550 6762 2] 368 6545] 3922601] 28 1s] -s64 4879] -392 4005| 3! | 18 | -560 4099 3 | 5849348] -392 2647 | 18] -580 3995] -392 4050| § | 18 | -576 6529 4 | 6060878] -392 2711) 65 | 18 | 6027172] 3924113] §° | 18] -599 4352 5 | -6342541| -392 2794 18 | -631 4824] -392 4193 18 | -628 7993 6 | -868 7863] -392 2894 ae 18 | -666 7488] -392 4291 ee 18 | -664 7999 7| -7o9 7492] 392 3012 | 128 | 18 | -708 5826] -392 4407 | 158 | 17 | -707 5046 8 | -757 2198] “392 3149 | 126 | 18 | -757 0624] -392 4541 | 133 | 17 | -756 9936 9} 811 2881] -3923302| 194 | 17 | -812 2801] -392 4692| 12s | 17 | °813 3607 zo | 8720569] -3923474| 17! | 17] -874 3406] -392 4859 17 | 876 7129 11 | -939 6427] -392 3662 ee 17 | -943 3629] -392 5044| 50° | 17 | -947 1718 72 | o-014 1760] “392 3868 | 398 | 17 Jo-019 4803] -302 5245 | 39) | 17 | 0-024 8733 13 | -095 8020] 392 4090 | 32° | 17 | -102 8409] -392 5463 339 | 16 | -109 9685 14} -184 6808] 392 4329 | 332 | 16 | -193 6083] -392 5697 | 334 | 16 | -202 6245 15 | -280 9888] -392 4584 | 16 | -291 9695] -392 5947 16 | -303 0249 16 | -3849189] -392 4855 a 16 | -398 1005] -392 6213 ae 15 | -411 3709 17 | -496 6818} -392 5142| 38% | 15 | -512 2374] 392 6403 | 38% | 15 | -597 8818 18 | -616 5066} -392 5444| 31? | 15 | -634 6070] -392 6789 | 306 | 14] 652 7964 19 | -744 6421 | -392 5760| 357 | 15 | -765 4636] -392 7099 | 33? | 14 | -786 3739 20 | -881 3579} -392 6092 14] -905 0822] -392 7424 14] -998 8954 21 | 1:026 9460] -392 6437 | 348 | 14 | 1-053 7610] -392 7762 | 338 | 13 | 1-080 6e4g 221 -181 7216] -392 6796 ee 13 | -211 s218} -392 8114) 39° | 13 | -242 0110 23 | 346 0254] -302 7168 | 372 | 13} -379 6125] -392 8478 | 36° | 12 | -413 2886 24 | 5202250] -392 7553 | 38° | 12 | -557 5084] -392 8855 | 377 | 12 | -504 8907 25 | -704 7169] -392 7950 12 | -745 9141] -392 9244 12] -787 2004 26 | -899 9284] -392 8359 pe 11 | 945 2662} -392 9645 | 475 | 11 | -990 6930 27 | 2-106 3205 | -392 8779 | 420 | 11 | 2-156 0351 | -393 0057 | 412 | 11 | 2905 8389 28 | 324.3901] -392 9210| 431 | 10] 37s 7283] 303 0479 | 422 | 10 | -433 1556 29 | -554 6729} -392 9652| 441 10} 613.8925] -393 0911] 43° | 10] -673 2014 30 | -797 7467] -393 0103 | 862 1178 | -393 1353 9 | -926 5783 | 21 | 3-054 2350] -393 0563 | 48° 9 | 3-124 0408] -393 1804| *! | 8 | 3-193 9359 32] 3248107] 393 1032 | 489 | 8] -400 3485] -303 2263| 492 | 8 | -475 9753 33 | -610 2007] -3931509| 477 | 8 | -691 7826] -393 2731 fe 7 | -773 4538 34} 911 1903} -393 1993| 48° | 7] -999 1454] -393 3205| 47* | 7 | 4-087 1898 35 | 4-228 6293] -393 2485 6 | 4-323 3042] -393 3686 6 | -418 0685 26 | +563 4374] 393 2982 oe 6 | -665 1980] -393 4174 oe 6 | 767 0481 | 37 | -916 6109] -393 3486 | 203 | 5 [5-025 8441] -393 4667 | 493 | 5 | 5135 1668 38 | 5-289 2309] -393 3994| 298 | 4] -406 3461] -393 5165 4| 523 5508 39 | 682 4708] -393 4507| 213 | 4] 807 9020] -393 5667 one 4] -933 4228 40 | 6-097 6055] -393 5024| 917 | 3 | 6-231 8144] -393 6174 3 | 6-366 1119 41 | 536 0267} 393.5543 220 | 2] -679 5011] -393 6683 | 202 | 3] -823 0651 2} -999 2449] -393 6066 | 227 | 2 7-152 5073] -393 7195 | 212 | 2 | 7-305 8593 3 | 7-488 9134] -303 6590| 224 1] -652 5197) 393 7708| 14 | 1] 816 2157 44 | 8006 8381] -393 7116| 226 | 1 |8-181 3822] -393 8293) 91? | 1 | 8-356 0160 35 | 554 9966 | 741 1137-393 8739 997 3206 393 7642 7=50 log H (r, v) 0°392 5316 "392 5325 *392 5352 392 5396 932 5457 392 5536 392 5633 392 5746 392 5877 392 6025 "392 6189 392 6370 “392 6568 392 6781 392 7010 “392 7255 392 7516 392 7791 392 8080 “392 8384 “392 8702 392 9034 “392 9378 392 9736 393 0105 393 0486 393 0879 393 1282 393 1696 393 2120 393 2553 “933 2995 393 3445 393 3903 393 4368 "393 4840 393 5318 *393 5801 393 6289 393 6781 "393 7277 393 7776 393 8278 393 8781 393 9286 393 9791 A2 el ed led T1000 aa) 1-7 OO bobo we Pe eR OF OF AAQAAD 1-1 - a) _ mm bo bo CO me He Or Or O~I~1 HC wooo Miscellaneous Constants in Frequent Use nd — ey — ee — fe ee — TABLE LV. Miscellaneous Constants. m 3141 5926 54 log 497 1499 log 2x -798 1799 log = 1-201 8201 - log = 1-600 9100 @ 2°718 2818 28 +367 8794 41 log e 434 2944 82 loge?? 036 1912 07 logloge ‘637 7799 16 centimetre = 393 70432 ins. inch = 2°539 9772 cm. square cm. = 155 00309 sq. ins. square inch= 67451 4842 sq. cms cubic cm. = ‘061 025386 cub. ins. cubic inch =16:386 623 cub. cm. kilogram = 2204 6212 Ibs. avoir. lb. avoir, = °453 59265 ke. radian =57'295 7795 degrees. degree = 017 4532 925 radians. CAMBRIDGE. PRINTED BY JOHN CLAY. M.A. AT THE UNIVERSITY PRESS, 143 a: , prey" gat s2 1p pe Sos neerene tate, os t ') ‘ Fi f Sigs enero TL AH 3 0112 019