I

MULTIVARIATE NORMALITY
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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, 0. C. MAY 1976

I
 I
1. REPORT YO. 2. GOVERNMENT ACCESSION NO.
0333970
NASA TN D-8226

7..AUTHORIS)
II 8 . PERFORMING
Mi70
ORGANIZATION REPOR r ,
H p o l d L. Crutcher* and Lee W. Falls 110.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
I WORK UNIT, NO.

George C. Marshall Space Flight Center

Marshall Space Flight Center, Alabama 358 12 i 11. CONTRACT OR GRANT NO,

13. T Y P E O F REPORi' & PERIOD COVEREI

12. SPONSORING AGENCY N A M E AND ADDRESS

National Aeronautics and Space Administration I Technical Note

Washington, D.C. 20546 SPONSORING AGENCY CODE

I
15. SUPPLEMENTARY NOTES
Prepared by Space Sciences Laboratory, Science and Engineering.
*U.S. Department of Commerce, National Oceanic and Atmospheric Administration, Environmental Data
Service National Climatic Center, Asheville, North Carolina 28801.
---
16. ABSTRACT
Sets of experimentally determined or routinely observed data provide information about the past,
present and, hopefully, future sets of similarly produced data.

An infinite set of statistical models exists which may be used t o describe the data sets. Some are
better than others. The normal distribution is one model. If it serves at all, it serves well. If a data set, or a
transformation of the set, representative of a larger population can be described by the normal
distribution, then valid statistical inferences can be drawn.

There are several tests which may be applied t o a data set to determine whether the univariate
normal model adequately describes the set. The chi-square test based on Pearson's work in the late
nineteenth and early twentieth centuries isoften used. Like all tests, it has some weaknesses which are
discussed in elementary texts.

This report provides extension of the chi-square test t o the multivariate normal model. Tables and
graphs permit easier application of the test in the higher dimensions. Several examples, using recorded data,
illustrate the procedures. Tests of maximum absolute differences, mean sum of squares of residuals, runs
and changes of sign are included in these tests. Dimensions one through five with selected sample sizes 11
to 101 are used t o illustrate the statistical tests developed in this report.

7.' KEY WORDS 18, D I S T R I B U T I O N S T A T E M E N T

Multivariate Normal Distribution
Statistical Analysis
Goodness of Fit Tests
UNCLASSIFIED - UNLIMITED
Multivariate Chi-square Test CAT. 65
Kolmogorov-Smirnov Test

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I I I I I 1I111

ACKNOWLEDGMENTS

Acknowledgment is made to the following employees of the National Climatic

Center of the Environmental Data Service, National Oceanic and Atmospheric
Administration, for their help: Mr. R. D. Crane (now deceased) and Mr. Bob Ford for
drafting, and Mrs. M. Larabee for typing the manuscript. Acknowledgment is made to Joe
Scollard, Duane Brubaker, and Raymond Smith of the National Aeronautics and Space
Administration, Marshall Space Flight Center, Computation Laboratory for their
assistance in computer programming.

Acknowledgment is made t o Professor E. S. Pearson and to the Biometrika

Trustees for permission to duplicate and abridge portions of the x2 tables (Tables 7 and
8) presented in Biometrika Tables for Statisticians, Volume I, 1956.

Acknowledgment is made t o Professor A. Hald and to John Wiley and Sons, Inc.,
for permission to abridge a portion of the x2 table, Table V, in Statistical Tables and
Formulas, 1952.

Acknowledgment is made t o Professor William G. Cochran and to The Annals of

Mathematical Statistics for permission t o quote from Professor Cochran’s paper, “The x2
Test of Goodness of Fit,” published in Volume 23, 1952.

Acknowledgment is made t o Univac Division, Sperry Rand Corporation, for

permission to use their electronic data computer programs available in their Univac
1106/1108 system, Stat-Pack, Program Abstracts, UP-4041 and Univac Large Scale
System, Math-Pack, Program Abstracts, UP-405 1.
 TABLE OF CONTENTS

Page

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1

I1. THE MULTIVARIATE NORMAL DISTRIBUTION . . . . . . . . . . 3

A. The Univariate (One-Dimensional) Distribution . . . . . . . . . . 6

B. The Bivariate Distribution . . . . . . . . . . . . . . . . . . 9
C. The Marginal and Conditional Distributions . . . . . . . . . . . 11
D. The Trivariate Distribution . . . . . . . . . . . . . . . . . . . 13 I

I11. THE x2 DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 16

iV . TECHNIQUES OF THE TESTS . . . . . . . . . . . . . . . . . . 18

.A . The Computation and Evaluation of Xz . . . . . . . . . . . . 19

B. Comparison of Computed and Ordered Empirical Chi-Squared
Values with the Cumulative xz Distribution . . . . . . . . . . . . 23
C. Random Number Generators . . . . . . . . . . . . . . . . . 25
D. Selection and Processing of Samples with Zero Means and Zero
Covariances . . . . . . . . . . . . . . . . . . . . . . . . 26
E. Selection and Processing of Sample Data (General) . . . . . . . . 35
F. Comparison of Ordered Random Chi-square Values with the
Theoretical Chi-square Values . . . . . . . . . . . . . . . . 45
G. Selected Tables of Randomly Generated Chi-square Values . . . . . 47
H. Selected Tables of Randomly Generated Chi-square Values Including
Tests for Symmetry . . . . . . . . . . . . . . . . . . . . . 48

V. APPLICATIONS TO REAL DATA . . . . . . . . . . . . . . . . 56

A. Temperature and Winds . . . . . . . . . . . . . . . . . . . 56

B. Wind - Surface . . . . . . . . . . . . . . . . . . . . . . 66
C. Wind -UpperAir . . . . . . . . . . . . . . . . . . . . . 66
D. Hurricane Motions . . . . . . . . . . . . . . . . . . . . . 72

VI . SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . 82

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

APPENDIX: PLOTTING DIAGRAMS WITH CONFIDENCE BAND (CENTRAL

0.96 PROBABILITY) FOR MULTIVARIATE NORMAL
DISTRIBUTIONS FOR VARIOUS DIMENSION AND
SAMPLE SIZE . . . . . . . . . . . . . . . . . . . . . 233

iii
 LIST OF ILLUSTRATIONS

Figure Title Page

1. Schematics of univariate normal distribution curves . . . . . . . . . 7

2. Normal distribution curve areas for selected intervals of the standard

deviation. . . . .
. . . . . . . . . . . . . . . . . . . . . 8

3. Bivariate normal frequency distributions . . . . . . . . . . . . . . 10

4. Bivariate distributions under a bivariate frequency surface . . . . . . . 12

5. Trivariate distribution: variances equal and covariances zero . . . . , . 15

6. Plot of x2 versus [ 1-p(x2)J prepared from data given in Table 1 . . . . . 24

7. 1000 data points generated by the Univac Stat-Pack random number

generator. . . . . . . . . . . . . . . . . . . . . . . . . . 27
2
8. Two-way plot of pairs of uniform random numbers generated by the
pseudorandom number generator of the Univac Stat-Pack . . . . . . . 30

9. Two-way plot of bivariate normal distribution (NID(0,l)) generated

by the Univac Stat-Pack random number generator . . . . . . . . . . 31

10. Three-dimensional sample size 7 median values of the 2nd, 50th, and
98th percentile values of 10 100 random chi-square values of the
multivariate normal distribution with zero mean, variance one and
covariance zero . . . . . . . . . . . . . . . . . . . . . . . 33

11. Three-dimensional sample size 3 1 median values of the 2nd, 50th and
98th percentile values of 10 100 random chi-square values of the
multivariate normal distribution with zero mean, variance one and
covariancezero .
. . . . . . . . . . . . . . . . . . . . . . 34

12. Examples of “Good Fit” of randomly generated normal variates for

differing dimensions (v) and sample sizes (n) . . . . . . . . . . . . 46

13. Examples of “Bad Fit” of randomly generated normal variates for

differing dimensions (v) and sample sizes (n) . . . . . . . . . . . . 49

14. Examples of “Good Fit” of randomly generated normal variates for

differing dimensions (v) and sample sizes (n) with the central 0.96
confidenceband . . . . . . . . . . . . . . . . . . . . . . . 50

iv
 LIST OF ILLUSTRATIONS (Continued)

Figure Title Page

15. Examples of “Bad Fit” of randomly generated normal variates for

differing dimensions ( v ) and sample sizes (n) with the central 0.96
confidence band . . . . . . . . . . . . . . . . . . . . . . . 51

16. Plot of data from Table 5 . . . . . . . . . . . . . . . . . . . 52

17. Plot of data from Table 5 . . . . . . . . . . . . . . . . . . . 53

18. Plot of data from Table 5 . . . . . . . . . . . . . . . . . . . 54

19. Zonal deviations of the wind (m/s) during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5 , 196 1,
0929-0934 EST . . . . . . . . . . . . . . . . . . . . . . . 57

20. Meridional deviations of the wind (m/s) during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5 , 1961, 0929-
0934 EST. . . . . . . . . . . . . . . . . . . . . . . . . . 58

2 1. Vertical deviations of the wind (m/s) during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5, 196 1,
0929-0934EST . . . . . . . . . . . . . . . . . . . . . . . 59

22. Temperature deviations in degrees Celsius during the Round Hill

Turbulence Measurements Program at Round Hill, MA, October 5 ,
1961,0929-0934 EST . . . . . . . . . . . . . . . . . . . . . 60

23. Four-dimensional chi-square ordered distribution of 300 consecutive

data points of wind component deviations (m/s) u’, V‘ and w’, and
temperature deviations in degrees Celsius during the Round Hill
Turbulence Measurements Program at Round Hill, MA, October 5 ,
1961,0929-0934 EST . . . . . . . . . . . . . . . . . . . . . 62

24. Four-dimensional chi-square ordered distribution of 300 consecutive

data points of wind component deviations (m/s), u’, v’ and w‘,and
temperature deviations in degrees Celsius during the Round Hill
Turbulence Measurements Program at Round Hill, MA, October 5,
196 1,0929-0934 EST with central Q.96 confidence band . . . . . . . 63

25. Four-dimensional chi-square ordered distribution of 100 consecutive

data points of wind component deviations (m/s), u‘, V’ and w‘, and
temperature deviations in degrees Celsius during the Round Hill
Turbulence Measurements Program at Round Hill, MA, October 5,
1961,0840-0842 EST . . . . . . . . . . . . . . . . . . . . . 64

V
 LIST OF ILLUSTRATIONS (Continued)

Figure Title Page

26. Four-dimensional chi-square ordered distribution of 100 consecutive

data points of wind component deviations (m/s), u‘, v’ and w’, and
temperature deviations in degrees Celsius during the Round Hill
Turbulence Measurements Program at Round Hill, MA, October 5,
1961,0840-0842 EST with the central 0.96 confidence bands . . . . . 65
-
27. Three-dimensional chi-square ordered distribution of 300 consecutive
data points of wind components, u, v, and w (m/s) in gust research
measured 4.7 meters above the desert at Palmdale, California using
a Vector Vane, March 2 1, 1969, 1330 PST . . . . . . . . . . . . . 67

28. Three-dimensional chi-square ordered distribution of 300 consecutive

data points of wind components, u, v, and w (m/s) in gust research
measured 4.7 meters above the desert ,at Palmdale, California using
a Vector Vane, March 3,1969, 1330 PST . . . . . . . . . . . . . 68

29. Two-dimensional chi-square ordered distribution of 124 data points of

Cape Kennedy, FL, upper wind zonal and meridional components u
and v (m/s) at 8 km during the month of January; January 1, 1956 to
December 31, 1967. . . . . . . . . . . . . . . . . . . . . . 69

30. Two-dimensional chi-square ordered distribution of 124 data points of

Cape Kennedy, FL, upper wind components u and v (m/s) at 16 km
during the month of January; January 1, 1956 to December 3 1, 1967 . . 70

31. Two-dimensional chi-square ordered distribution of 124 data points

of Cape Kennedy, FL, upper wind zonal and meridional component
shears (m/s) between altitudes 12 km and 8 km, for the month of
January . . . . . . . . . . . . . . . . . . . . . . . . . . 71

32. Chi-square (vector deviation square) values of September-October 24

hour hurricane movements versus the empirical ( I-p(x2)) value of
occurrence (n=l 1) . . . . . . . . . . . . . . . . . . . . . . 76

33. Chi-square (vector deviation square) values of September-October 24

hour hurricane movements versus the empirical ( 1-p(x2)) value of
occurrence (n=21) . . . . . . . . . . . . . . . . . . . . . . 77

34. Chi-square (vector deviation square) values of September-October 24

hour hurricane movements versus the empirical (1-p(x2 )) value of
occurrence (n=3 1) . . . . . . . . . . . . . . . . . . . . . . 78

vi

._

-. ....... I
I

LIST OF ILLUSTRATIONS (Concluded)

Figure Title Page

35. Chi-square (vector deviation square) values of September-October 24

hour hurricane movements versus the empirical ( l-p(x2)) value of
occurrence (n=l 1) . . . . . . . . . . . . . . . . . . . . . . 79

36. Chi-square (vector deviation square) values of September-October 24

hour hurricane movements versus the empirical ( 1-p(x2)) value of
occurrence (n=2 1 ) . . . . . . . . . . . . . . . . . . . . . . 80

37. Chi-square (vector deviation square) values of September-October 24

hour hurricane movements versus the empirical ( 1-p(x2)) value of
occurrence (n=3 1) . . . . . . . . . . . . . . . . . . . . . . 81

vii
 LIST OF TABLES

Table Title Page

1. Fractiles (Quantiles) of the x2 Distribution . ... . . . . . .... 83

2. Medians of Selected Percentiles of 10 100 Random Chi-square Values

of the Multivariate Normal Distribution with Zero Means and
Covariances (NID (0,l ,O)) and Their Probability Plotting Positions
p = (1 - ((i - 0.5)ln)) for 3 Dimenisons with Sample Sizes 7 and 3 1 . . . . 84

3. Values of “c” Derived from Formula c = 1 - (v/4) - (4 In n/n) where v

is the Number of Dimensions, where n is the Number of Data, and
In is the Natural Logarithm . . . . . . . . . . . . . . .
. . . . 87

4. Probability Plotting Positions Based on Equation 1 - p = 1 - (i - c)/

(n - 2c + 1) where c = 1 - (v/4) - (4/n) (In n). v is Dimension
andivariesfrom 1 t o n . . . . . . . . . . . . . . . . . . . . 88

5. Median Values of Selected Percentile Values of 10 100 Random Chi-square

Values of the Multivariate Normal Distribution and Their Probability
Plotting Positions for Dimension ( v ) with a Sample Size (n) . . . . . . 96

6. Selected Percentile VaIues of Selected Statistics for Testing Normality . . 158

7. Maximum Absolute Difference Arrayed by Dimensions, ... . .. . 183

8. Maximum Absolute Difference Arrayed by Sample Size . . . . . . . 188

9. Mean Sum Squares Residuals Arrayed by Dimensions . . . . . . . . 193

10. Mean Sum Squares Residuals Arrayed by Sample Size . . . . . . . . 198

11. Runs Above or Below Line of Chi-square Versus Probability (1-p)

Arrayed by Dimensions . . . . . . . . . . . . . . . . . . . . 203

12. Runs Above or Below Line of Chi-square Versus Probability (1-p)

Arrayed by Sample Size . . . . . . . . . . . . . . . . . . . . 209

13. Runs Above or Below Median Chi-square Arrayed by Dimensions . . . 209

14. Runs Above or Below Median Chi-square Arrayed by Sample Size . . . 212

15. Number of n-tant Runs to be Expected for the x or x2 Vector from

Multivariate Normal (Wishart) Distributions . . , . . . . . . . . . 21 5

viii
 . .. .

LIST OF TABLES (Concluded)

Table Title Page

16. Tests of Bivariate Normality of 24 Hour Hurricane Movements for

Selected 5 Degree Latitude-LongitudeQuadrangles. . . . . . . . . . 219

17. Tests of Bivariate Normality of 24 Hour Hurricane Movements for

Selected 5 Degree Latitude-LongitudeQuadrangles, . .. . . .. . . . 222

ix
 LIST OF SYMBOLS

Symbol Definition

a above, greater than

b below, less than

C constant, summation of CY and p

d.f. degrees of freedom

f degrees of freedom

i subscripts, superscript, also indicates inverse as superscript

j subscripts, superscript, also indicates inverse as superscript

k number, degrees of freedom

1 subscript

In natural logarithm

1% common logarithm

m mean

n number of data, n-tant

P probability, number of fitted parameters

Pdf probability density function I .

r sample correlation

S sample standard deviation (Std. Dev.)

S2 sample variance

U longitudinal component of wind

V lateral component of wind

X
 LIST OF SYMBOLS (Continued)

Symbol Definition

W vertical component of wind

X variate, deviation from mean

Y variate, deviation from mean

C constant, correlation matrix

D determinant of correlation matrix

E expectation

I.D. identification number

K constant

K-S Kolmogorov-Smirnov test

MAD maximum absolute difference

MSSR mean sum squares of residuals

NID normally and independently distributed

Q a chi-square variate

R(u) random uniform variate

RAL runs above the line

RBL runs below the line

RAM runs above the median

RBM runs below the median

T2 Ho tellings “TZ”

X measured value of variate

XZ sum, square of X, square of deviation from expected divided by

the expected

xi

‘Il I1 I I I
 LIST OF SYMBOLS (Continued)

Symbol Definition

Y measured value of a variate

Y2 sum, square of Y

rejection level probability, parameter, constant

P power of the test, probability, parameter, constant

r gamma function

X chi or CHI

X2 chi-square

X20 computed chi-square from quadratic

x power (exponent)

population mean

V nu, dimensions, number of variates, degrees of freedom

P population correlation

population standard deviation

population variance

summation

variable

overbar, average

[ I matrix
, transpose, also deviation from mean

determinant of matrix

factorial

xii
 LIST OF SYMBOLS (Concluded)

Symbol Definition

00 infinity

d- square root

f,- algebraic signs, quadrant signs

degrees of k, v, f, d.f.
freedom

...
Xlll
 MULTIVARIATE NORMALITY

1. INTRODUCTION

One or more theoretical functions may be used to describe a set of experimental

or routinely observed data. It is necessary that, wherever possible, the selected function
satisfies the physical bounds or restraints of the data. Some sets of data contain
information conceming only one variable, while others contain information about two or
more simultaneously observed variables. The data may be homogeneous or heterogeneous,
i.e., a single type or a mixed type.

Edgeworth (1 9 16) and Geary (1 347) provide interesting discussions on the

applicability of the normal distribution. Edgeworth represents an earlier stage of
statistical development than does Geary. Edgeworth considers the normal law of error to
be a first approximation to the actual shape of a distribution. Corrections or
modifications to this form a second approximation. Presumably, as more empirical
evidence is gathered, further approximations might be made. Perhaps this is the way the
procedure really works. Models are used until they are found to be deficient; then a new
model is proposed. Geary indicates this in a rather interesting chapter on the fluctuations
in the attitudes of statisticians concerning the question of the occurrence of the normal
frequency distribution. This is still a relevant point. Up to approximately the end of the
last century, a main current in the thinking was a favorable inclination toward the
hypothesis of universal normality; Le., in a sense, everything was distributed normally and
departures from normality were only the result of sampling.

Geary (1947) indicates that near the first of the present century, with the
development of the theory of moments, concepts changed from that of universal
normality to that of universal nonnormality. This feeling was engendered by the use of
moments and the Pearsonian curve system. In this system the normal distribution
function is only one of the many functions. Another shift in thinking occurred
approximately 25 years later when Fisher showed that, when universal normality could
be assumed, inferences of wide practical usefulness could be drawn from samples of any
size. Geary states, “Normality is a myth; there never was, and never will be, a normal
distribution.” As Geary further states, “This is an over statement from the practical point
of view, but it represents a safer initial attitude than any in fashion during the past two
decades.” This was prior to 1947. This led to Geary’s hope that he had created a “prima
facie” case for the importance of testing for normality. It is this testing for normality in
the multivariate sense that is the basis for this report. The reader is referred to Fisher
(1924, 1950).

Tests are usually made without regard to alternatives, according to Cochran

(1954) who suggests that some alternatives should be considered. Cochran’s 1952 paper
also is excellent reading. Tests made without regard to alternatives are made without regard
to the Type I and Type I1 errors. A Type I error is the rejection of a null hypothesis
when it is true, whereas a Type I1 error is the nonrejection of the null hypothesis when it
is false. The probability of the Type I error is equal to the probability level selected for
testing, usually designated as a. The probability of the Type I1 error, p, is equal to one
minus the power of the test, usually written as (1 - p). The more powerful the test, the
less is the Type I1 error. Cochran (1954) indicates that most uses of the chi-square (x2)
test may be strengthened by: (1) use of small expectations in computing xz , (2) use of a
single degree of freedom or a group of degrees of freedom from the total x2, and (3) use
of alternative tests.

Where multivariate multimodal distributions are treated, techniques to separate

the data into unimodal homogeneous groups or clusters may be used prior to detailed
study. Among the several clustering techniques available, discriminant function, principal
component, or factor analysis may be used to effect the separation.

In this report only the multivariate single mode data set, which may be described
by the normal law of errors, is considered. This is restrictive but with the knowledge that
the population from which the data set has been selected may be considered to be
multivariate normal, valid inferences can be drawn and probabilistic statements can be
made.

There is a need for tests to determine whether a data set is multivariate normal.
As Andrews, Gnanadesikan, and Warner (1973) indicate and discuss, there are procedures
to help meet this need. These are: (1) likelihood ratio tests associated with
transformations to enhance joint normality; (2) goodness-of-fit tests such as the chi-square
(x2) and Kolmogorov-Smirnov (K-S), tests based on local densities, nearest distance tests;
and (3) informal graphical methods associated with radii and angles representation of the
data. The informal graphical methods are of particular interest in the last paper and this
paper also deals with graphical testing procedures.

Mauchly (1940a) proposes a sphericity test which is often used. In the

two-dimensional (1 940b) (bivariate) sense he proposes an ellipticity test. Crutcher (1 957)
applies the latter test to upper wind distributions. Votaw (1 948) discusses the testing of
component symmetry in a multivariate normal distribution. Hald (1952a) uses the square
of the radii for the bivariate case. Kac, Kiefer, and Wolfowitz (1 955), Weiss (1 958), and
Anderson (1 966) discuss tests of normality based on density and distance methods. Healy
(1968) and Kessel and Fukunaga (1 972) have suggested procedures based on the squared
radii. Andrews, Gnanadesikan and Warner (1 973) describe graphical procedures based on
radii and angles. Considerable work is currently under way on this problem. More recent
tests and articles dealing with these problems may be familiar t o the reader but are not
apparent to the authors. An apology is tendered to any author whose work has
inadvertently been missed.

Examination of the marginal distributions, including all the univariate and the
bivariate combinations within the multivariate set, provides considerable insight.
Sometimes this is quite tedious.

2
 In this report the Monte Carlo or random sampling technique provides the
sampling used for the multivariate normal distributions. In the multivariate sense, each
vector has a magnitude and a direction. In the sense of scaled residuals, these are
magnitudes and directions from the centroid. In the multivariate normal and independent
case in large samples, the direction and magnitudes are not correlated and the sample
swarm is spherical in shape, hence, Mauchly’s sphericity test. The squares of the
magnitudes are distributed approximately as chi-square with degrees of freedom equal to
the number of variables and exactly as a constant multiple of a beta distributed variable,
Gnanadesikan and Kettenring (1 972). A chi-square probability plot of the squared radii in
the bivariate case versus the uniform probability plot of the angles should, for the null
hypothesis, produce a reasonably linear plot, Hald ( 1952a,b) and Andrews, Gnanadesikan
and Warner (1973); as shown by this group, a plot of the squares of the radii versus the
angles for the null hypothesis produces a random scatter on the unit square. Thus,
visually, the investigator can make the decision as to whether the data set is multivariate
normal.

Anderson (1 958) shows that sampling from a multivariate normal distribution

developed from one-dimensional normally and independently distributed variables; i.e.,
NID (0, u 2 ) will produce a multivariate Wishart distribution (1928, 1948) with NID ( p ,
u2 ) marginals, Smith and Hocking (1 972). This includes the univariate marginals. That is,
sampling from a known multivariate normal set with a zero mean and a dispersion matrix
u2 will produce sets which have means different from zero with the same dispersion
matrix. As Kendall and Stuart (1968) indicate, this may be regarded as a generalization
of the x2 distribution in the multivariate sense.

The Wishart distribution and its ramifications are so complicated and difficult that
it has not found much use. However, for small samples, in sampling from the Wishart
central multivariate normal distribution, the sampling follows the Wishart distribution.
The sample means may not be zero and the sampling distributions d o not follow the x2
for large samples.

Tests for multivariate normality also may be made from the viewpoint of the
roots of the dispersion matrices. Though these were examined in the preparation of this
report, they are not used and are not discussed further. The reader may refer t o Kcndall
and Stuart (1 968) and the references they provide.

II. T H E MULTIVARIATE NORMAL DISTRIBUTION

The normal distribution law in one dimension is due t o De Moivre (1733).

Usually, one associates it with the names of Gauss and Laplace. More than a century
after De Moivre, Bravais (1 846) developed and published his study of normal frequency
distributions in two and more variables. Contributions to the study of this problem were
made by Maxwell ( 1859), Bertrand ( 1888a,b), Czuber ( 1891 ), Pearson ( 1900), Kluyver

3
(1906), Student (1908, 19251, Strutt (1919), and Hotelling (1951). The distribution of
vector magnitudes in two and three dimensions, when the vector means and correlations
are zero and the variances are equal, often are referred to as Rayleigh and Maxwellian,
respectively, although sometimes these are interchanged. Where the means are not zero,
they are referred to as the generalized noncentral Rayleigh and Maxwellian distributions.
Sometimes only the projection of the multivariate distribution onto one axis is referred
to as Rayleigh or Maxwellian.

The probability density function (pdf) of the multivariate normal distribution

may be written as , ~

as indicated by Pearson (1900) for v correlated random variates, ( x l , x 2 , . . . ,xJ. The

main feature of the v-dimensional normal distribution is that all of its properties are
deducible or determined from the information contained in the means and the
covariances. Cochran ( 1952) and Elderton and Johnson (1 969) provide excellent
discussions of Pearson's work.

Denoting E as the expectation, 2 as a summation, and n as the number of data,

the following may be defined for use here, where the subscripts i, j range from 1 through
U.

where Xi and si2 are the sample estimates of the mean and variance.

Pearson (1900) showed that the quadratic form, Q of equation ( l ) , which is a

positive definite matrix, is distributed as x2 (chi-square) with v degrees of freedom.

4
 where the brackets [ ] indicate a matrix, [x‘] is the transpose of [XI where [XI = [ x l ,
x2, x g , . . . ,xv], and [ d ] and [p”] are the inverses of the covariance and correlation
matrices [aij] and [pij] , respectively. In equation (3c), x is a standardized variable. .

C is a constant for a distribution and may be written as

or

where I[oij] 1 is the determinant of the covariance matrix. In equation (4b) standardized
variables are used. If standardized variables are not used, then equation (4b) must be
divided by the product of the appropriate standard deviations. An element of the inverse
matrix oij is equal to the determinant of the cofactor (minor) of the element of the
matrix, u..,
1J
divided by the determinant of the covariance matrix. The determinant of the
cofactor of aij is the determinant of order (v-1) obtained by striking the ith row and jth
column of the covariance matrix multiplied by (-l)i+j, as shown by Anderson and
Bancroft (1 952) and Bendat and Piersol (1 966) among many others.

All marginal and conditional distributions of a multivariate normal distribution are

normal. The converse is not true as indicated by many writers. Cramer (1946) proves this
in a theorem. As Feller (1966) and Kowalski (1970) indicate, many an investigator
assumes multivariate normality if the marginal distributions are normal. If any tests are
rejected, the rejected distributions are transformed to normal and the assumption of
multivariate normality then is made. Transformations are discussed by Box and Cox
(1964) where they use the power from (X + Transformations of the total
distribution in terms of the dispersion matrix also are discussed by Joshi (1970) and
Andrews, Gnanadesikan and Warner (1973). In the succeeding sections of this report, a

11111111111111l111l111llIl Il I1I1 I1
 geometric illustration of this feature will be shown for the two-dimensional case. A few
standard statistical texts provide analogous illustrations. Extension to higher dimensions
hopefully will not be difficult for the reader.

A test of multivariate normality is tantamount to a test of normality of the total

distribution as well as all its marginal and conditional distributions.

The special cases of the uni-, bi-, and tri-variate cases now are discussed in greater
detail for better understanding.

A. The Univariate (One-Dimensional) Distribution

From equation (1) the pdf of the univariate normal distribution may be written

where

where ux 1 is estimated by its sample estimate sX1 .

Chi-square, x2, in this case may be written as

wherep, is estimated by its sample estimate X I . Chi then can be visualized as simply
1
the standardized variable ((X, - p x l ) / u x ). This is a scaled residual.
1

Figure 1a shows a univariatc normal distribution curve overlying a histogram. The

area of the histogram and the area bounded by the norinal curve and the variate axis can
be made equal and set to one. This figure will be re1'crred to later with respect to the
hachured areas. Figure l b also shows ;I normal distribution curvt: of which the total area
has been sectioned into twciity equiprobability ;ircas. The symmetrical portions are either
similarly hachured or blank. Thew also will be disciisscd later.

6
 .

L-L-I
-20
"F-
a. Overlaid on a histogram.

Observed, 0 Expected, E N=Number of class intervals

-4 -3 -2 -1 +1 +2 +3 +4

b. Twenty cquiprobability areas, each equal to 0.05 (similar area

markings show syinrnctric areas).

Figure 1 . Schematics of univariate nornial distribution curves.

I
 Figures 2a, b, c, and d also represent a normal distribution in one dimension.
These illustrate how some common inferential statements can be made. The captions and
indicated probability areas are believed t o be self-explanatory. The distribution is
determined completely by the mean and standard deviation. The latter is the square root
of the variance.

B L A C K A R E A ( p l u s white s t r i p
0.5
BLACK
AREA EQUALS 1.0000 * 0.51 a r e a ) EQUALS 0 . 6 8 2 6
h ?

Standard Deviation S t a n d a r d Deviation

BLACK BLACK
AREA EQUALS 0.2719 AREA EQUALS 0.0428

-03 -3 -2 -1 0 +1 +2 +3t+m -w -3 -2 -1 0 +1 +2 +3t+m

Standard Deviation S t a n d a r d Deviation

Figure 2. Normal distribution curve areas for selected

intervals of the standard deviation.

8
 B. The Bivariate Distribution

The pdf of the bivariate or two-dimensional normal distribution may be written as

In terms of the sample correlation statistics the estimate of C may be written as

with the x 2 estimate as

2
Here (1 - rx ) is the determinant of the correlation matrix, I[rl ] I,'and r, the sample
1 2
estimate, replaces p , the population correlation.

Figures 3a, b, and c show a bivariate normal frequency distribution where

conditional univariate normal distributions are indicated by the shaded univariate normal
distribution planes (a), (b), and (a and b), respectively. Each distribution plane is
perpendicular to c r parallel to the variate axes and perpendicular to the bivariate plane.
The units are in terms of standard deviations.

Figure 3d is another representation of the bivariate normal frequency (volume)

under the bivariate normal frequency surface. The representation is more in a polar
coordinate form rather than a Cartesian form. Here, the vector radii or x centered on the
vector mean sweep out circular (elliptic) cylinders bounded by the bivariate frequency
surface and the bivariate plane. The projection of the cylinders onto the bivariate plane
is visaalized as circles or ellipses. Certain volumes, interpretable as probabilities, are
contained within each cylinder. The volumes swept out are dependent on the magnitudes
of the radii. For example, a vector radius dependent upon the two variate standard
deviations will sweep out the central 0.40 probability core, while if the vector radius is
equal to one vector standard deviation (the square root of the S L I of ~ the squares of the
individual variate standard deviation), ( a x , + u x2 ')"2 will sweep out the central 0.63
probability circular (elliptic) cylinder. If the distribution is circular, the vector standard
deviation multiplied by [In (l/(l-p))l ' / z provides the vector radius of the p probability
ellipse. If the distribution is elliptical, the cigenvector-eigenvalue matrices provide the

11111111l1l111111ll lIl l l l l I1 I 1 l l l I
 a

Selected areas indicate conditional univariate normal distributions

(Modified from Hald, 1952).

Volume under a bivariate normal surface (The elliptic cylinders

cut out specified probabilities with specified vector radii centered
at the centroid at the tcrminus of V . ) (Crutcher, 1960)

Figure 3. Bivariate normal frequency distributions.

10
 -. ... .. . .~~ ~ ~ . .. . .. . .. .. - ., ,

orientation of and variances along the major and minor axes. When the square roots of
the eigenvalues are multiplied by [ 2(ln (l/(l-p)))l, the lengths of the semimajor and
semiminor axes for the p probability ellipse are obtained, Crutcher (1957, 1962).
Bertrand (1888a,b) provides limited tables for these. Brooks and Carruthers (1953) apply
these concepts to the specialized field of upper air wind distributions. Tables of x2 with
2 degrees of freedom (Table 1) can be used to determine the magnitude of the vector
radius (in terms of the vector standard deviation) to core out central elliptical (circular)
cylinders containing specified portions of the volume representing specified probabilities.
Groenewoud, Hoaglin, Vitalis, and Crutcher (1967) provide extensive and detailed tables
in two volumes for general use and some applications to problems in' geophysics in the
third volume. These permit elliptical (circular) coring of specified probability cores or the
probabilities of specified elliptical (circular) cylinders.

Extensions of these concepts to multivariate quality control can be made. Thomas

and Crigler (1 97.4) discuss tolerance limits for the multivariate radial error distribution.

Bates (1966) provides an excellent discussion of applications of the chi-square

goodness of fit for a bivariate distribdtion. The use of Bertrand's limited table and the
detailed table mentioned above or any other similar tables is dependent upon the
nonrejection of the null hypothesis concerning the bivariate normality of the data.

C. The Marginal and Conditional Distributions

Normality of the multivariate distributions implies normality of all marginal and

conditional distributions. The converse is not true, Cram& (1946), Joshi (1970),
Anderson (1 958), Freund (1 962), Kendall and Stuart (1 9681, Kowalski (1 973), etc.
Kowalski presents an excellent review of the situation.

Figure 4a is another representation of a bivariate normal distribution under the

bivariate normal frequency surface. This discussion embellishes Anderson's ( 1958)
presentation. It appears here as a limiting form of mean zero and variance one around the
centroid. Note the marginal distributions shown as a projection against the planes
perpendicular to the bivariate plane and as planes passing through the coordinate pair (0,
0) or origin of axes in the bivariate plane. Note also the conditional univariate normal
distribution perpendicular to the bivariate plane and passing through the centroid of the
distribution. Units are shown in terms of the standard deviations. Note the shaded
squares'in the bivariate plane located between plus and minus one standard deviation
along the variate axes. These will be referred to later.

Figures 4b, c, and d, in conjunction with Figure 4a, demonstrate this lack of
double implication. Figure 4a presents a bivariate normal distribution with the two
marginal or univariate distributions shown perpendicular to the bivariate plane, parallel
and perpendicular to the variate axes, and each one some distance from the centroid.
Figure 4a shows four shaded square areas in the bivariate plane, each between one and

11

' Il1 lIlllIl l1 1l 1 l l 1l1l 1l l l1 l1 Ill I I Ill I I Ill I I1 I

 L
h,

-7

Figure 4. Bivariate distributions under a bivariate frequency surface.

(a,b) Normal with marginal normals modified to: (c) Nonnormal
with one marginal normal and one marginal nonnormal
modified to: (d) Nonnormal with marginal normals.

i
two standard deviations, plus and minus, from the variate axes. These are repeated in
Figures 4b, cy and d. For reference these are numbered 1, 2, 3, and 4. In Figure 4b
volumes (equivalent to probabilities) are shown under the bivariate frequency surface and
above the shaded areas, 1, 2, 3, and 4.

In Figure 4c the volume labeled “1” is cored from the distribution by means of a
square corer or cookie cutter with sides equal to one standard deviation by a vertical
downward stroke above the area labeled “1.” The core is moved upwards and sideways
parallel to one variate axis and perpendicular to the other variate axis t o the left towards
“2” and is superposed above the bivariate frequency surface above “2.” The volumes
under the bivariate frequency surface above the bivariate plane areas “1” and “2” are
equal and symmetrically located by deliberate choice.

The bivariate frequency surface is discontinuous in the regions of “1” and “2.” A
hollow is evident at “ I ” and a spike is evident at “2” in Figure 4c. Note that the
marginal distribution on the left is not changed but that the marginal distribution on the
right shows a hump on its left and a dip on its right. These are located between one and
two standard deviations from the mean. It is pointed out that the vertical scales are not
the same for the bivariate as for the marginal distributions. The relative heights for
distributions are the constant C in the equations given previously. For the marginal or
univariate distributions these are ( 2 n f /’, for the bivariate distributions (2n)-’,and for
the trivariate distributions (2n)-3/’ . These are, respectively, 0.3989, 0.159 1, and 0.0635.
These represent the relative maxima of the various standardized frequency surfaces.

Figure 4d follows and is a modification of Figure 4c. Here, the volume over “3”
is cored, lifted out, moved to the right, and superposed over “4.”Please note that the
marginal distribution on the left remains the same, while the marginal distribution on the
right returns to its orginal form as the areas moved to the left and right, respectively, are
equal and at the same distance from the centroid. However, note that the bivariate
distribution is severely modified. It has two hollow spaces and two spikes which
obviously deform the surface. In fact, the conditional distributions between one and two
standard deviations from the centroid parallel to the left axis are discontinuous. It is
obvious that the illustration is simplified by moving equally shaped cores symmetrically,
yet this is sufficient for demonstration. Since a two-dimensional distribution is a marginal
of a three-dimensional, the analogy can be projected from any higher dimensions to lower
dimensions. Therefore, a simple test for multivariate normality resulting in nonrejection of
the null hypothesis automatically assures nonrejection of the null hypothesis for any lower
marginal or conditional distribution. This is the thrust of this report. Pearson’s (1900)
chi-square test is used.

D. The Trivariate Distribution

The pdf of the three-dimensional normal distribution can be written in covariance

form rather than the correlation form so as to show the similarity o r sameness with the
correlation form illustrated previously,

13
where

and

Symmetry of sij = di permits the use of the coefficient 2 as

Figure 5 schematically represents a trivariate normal distribution centered on the centroid

with mean zero and variances one with covariances zero.

Three schematic spheres or shells are shown. Certain probabilities will be

contained within spheres. A sphere (ellipsoid) centered on the centroid with a vector
radius equal in magnitude to the variate standard deviation will contain a central sphere
(ellipsoid) core representing 0.20 probability. An ellipsoid core swept or cut out of the
distribution and centered on the centroid with a vector radius of 1.0 vector standard
deviation will represent 0.608 probability. The vector standard deviation,

14
 ,/'

-. 1

Figure 5 . Trivariate distribution: variances equal and covariances zero.

is invariant and is equal to the square root of the trace of the covariance matrix; the trace is
the vector variance.

Other probability ellipsoids can be selected. The magnitude of the vector radius
required t o core specified probabilities around the centroid can be determined easily by
use of x2 tables with 3 degrees of freedom. No immediately available tables permit
ellipsoidal coring w i t h the trivariate frequency surface for off-center specified
probabilities or specified ellipsoids, as are available for the two-dimensional case. The
vector variance or trace of the covariance matrix is

%. + . . . +
2
ov2 = 0
% + xu

or after rotation and determination of the eigenvalue matrix

+ uv2 Y

where the letters a, b, . . . , v refer to variances along the principal axes, where the
variances are usually ordered as to magnitude. The mathematics and computer routines
are known to be available but cost prohibits suitable tabular preparation and publication.

Ill. THE x2 DISTRIBUTION

The following paragraphs follow Cochran’s ( 1952) cogent discussion of Pearson’s

(1900) paper on the x2 goodness of fit.

“In the standard applications of the test, the n observations in a random

sample from a population are classified into k mutually exclusive classes.
There is some theory or null hypothesis which gives the probability, pi,
that an observation falls into the ith class (i = 1, 2, 3 , . . . k). Sometimes
the pi are specified completely by theory as known numbers and
sometimes they are less completely specified as known functions of one or
more parameters, a l , a2 . . . whose actual values are unknown. The
quantiles mi = npi are called the expected numbers, where

16
 “The starting point in the theory is the joint frequency distribution of the
observed numbers xi falling in the respective classes. If the theory is
correct, these observed numbers follow aimultinomial distribution with the
pi as probabilities. The joint distribution of the xi is therefore specified by
the probabilities

“As a test criterion for the null hypothesis that the theory is correct, Karl
Pearson (1 900) proposed the quantity

x2 =
(xi2/mi) - n . 97

i= 1 -

If the probabilities pi remain fixed as n -+ 00, the limiting distribution of X2 when

the null hypothesis is true is the familiar x2 distribution,

where v is the number of degrees of freedom or the dimensions in x’. This particular
distribution also is known to be that followed by the quantity shown in equation (3) and
by the quantity

shown by many authors such as Cochran (1952) and Hald (1952a), and where the yi are
the standardized uncorrelated variates and are distributed normally and independently
with zero means and unit variances.

Now, x 2 , as a constant, is the equation of a generalized “ellipsoid” over which

the frequency of the deviations or errors is constant. Chi (x), as does x2, ranges from
zero to infinity. Chi is the vector radius or ray which sweeps out the ellipsoid in
v-dimensional space. I f the ellipsoid is viewed with respect to its principal axes rather
than its variate axes, it may be massaged, i.e., compressed or stretched along one axis at a

17

I
time until it forms a sphere. Effectively, this is accomplished by standardizing the
components along the axes by, dividing them by the respective standard deviations, i.e.,
scaling the residuals. The principal axes are located as eigen, latent, or characteristic
vectors. The respective eigenvalues, latent or characteristic roots, are the variances of the
transformed variates along the respective principal axes. The covariances and correlations
off the diagonal of the respective matrices are zero: Le., these are uncorrelated. Bartlett
(1934) discusses the vector representation of a sample. There the individual vector, of
course, would be a “x” as indicated previously.

Values of x2 through the first 15 degrees of freedom are given in Table 1. These
values are abridged from Pearson’s Tables (1900) with the permission of the publishers
and trustees of the estate and from A. Hald and S . A. Sinkbaek (1950) with the
permission of the authors and publishers. These same tabular values can be computed and
printed by the electronic computer and its peripheral equipment. Such computation is
done in the application of the chi-square tests in this paper. Many other texts serve as
references for the x2 distribution. Cram& (1 946) also offers a good discussion on the x2
distribution. Lancaster’s (1969) text on the x2 distribution is superlative.

IV. TECHNIQUES OF THE TESTS

The tests considered here are:

1. x2
2. x2 (Graph)
3. MSSR (Mean Sums of Squares of Residuals)

4. MAD (K-S) - Maximum Absolute Difference (Kolmogorov-Smirnov)

5. Signs Test

6. Runs above and below the line

7. Runs above and below the median.

Hald (1952a) suggests two procedures for the examination of an observed

two-dimensional distribution for normality. These are the first two mentioned.

1. The computation and evaluation of X 2 per equation (6) for the ellipsoidal or
circular shells.

2. The visual comparison of computed and ordered x2 values against the

appropriate theoretical cumulative x2 distribution (graphs).

18
These will be discussed separately and in some detail for the multivariate case which
includes the univariate case as a special case.

A. The Computation and Evaluation of X2

In the previous discussion it may not be apparent that the test depends upon the
application of “x2” in two forms. The quadratic form (x2) provides a form where each
observation as a x2 value of which x, the square root of x2, is the deviation of the
observational vector measured from the centroid. Thus, if there are 9 0 observations, there
are 90 x2 values or 90 vectors, each of which has direction and magnitude. Therefore, for
each data set there is a set of vectors with a x value representing the magnitude of each
vector. Each x value necessarily has been computed by means of the proposed model
using computed statistics, estimates of the distribution parameter(s). The test now
proceeds to check whether the vectors follow the proposed form.

Considerable discussion prevails in the literature as to how best to categorize the

data or establish class intervals. Following the arguments of Pearson (1900), suggesting at
least one vector or observation in each category, and the arguments of Cochran (1952)
and Mann and Wald (19421, use as few in each category as possible and use a relatively
large number of categories to ensure a power of the test of about one-half and thereby
reduce the Type I1 error. Cochran also argues for allowing the number in an interval to
dip as low as one. Most investigators use a less conservative approach and suggest 5 as a
minimum number allowable in a category while others, still less conservative, suggest 10.
Bates (1966), in providing a good discussion of the test for a bivariate distribution,
provides a survey on these approaches. The less conservative the approach is, the greater
is the Type I1 error.

The question arises as to whether the class intervals may be equal or unequal in
terms of variate scale or probability. Pearson (1900), Mann and Wald (1942), Cochran
(1952), and Hald (1952a) suggest the use of equiprobability class intervals. Vessereau
(1958), Kempthorne (1 967), and Roscoe and Byars (1 97 1) discuss these problems.
Figures l a and 1b, respectively, schematically illustrate equal intervals of scale and
probability. In future work the use of equiprobability intervals will be investigated, but in
this report equal class intervals of scale will be used except where “pooling” is necessary
to obtain at least five observations in each interval.

Mann and Wald recommend the following formula to determine the number of
categories to use for the univariate distribution. This formula is

19
points, and “c” is a constant. “c” is determined so that Cl/fi)l e
-
” the number of intervals based on “N”, the number of observations or data
where “ k ~ is

C
-x2/2
dx is equal to a

preselected critical region. For a rejection level of 0.05, c = 1.64; Le., it is the normal
variate value that satisfies the above integral.

Another formula is proposed by Sturges (1 926). It is

Dahiya and Gurland (1973) provide further insight into the rather tenuous problem. They
.state, “Even for this case there is no unique answer and the best choice of ‘k’ depends on
the alternative distribution.” Their results do provide a range of choices of ‘k’ for several
different alternatives in testing for normality. Their formula maximizes the power,
thereby decreasing the Type I1 error. According to Mann and Wald (1942), if the number
of intervals used is k = kN,the power of the test is equal to or greater than one-half for
all the alternatives. Therefore, the probability of a Type I1 error is equal to or less than
0.50. If k is not equal to kN, there is at least one alternative where the Type I1 error is
less than 0.50.

Some investigators suggest that the number of intervals can be one-half of kN

without seriously impairing the test. Hamdan (1963) implies that the use of a large
number of intervals (categories) as obtained by the formula of Mann and Wald perhaps is
not as good as a lesser number, for example 10 to 20.

Tate and Hyer (1973) indicate that the power of the chi-square probabilities is
weakened when the expectations are small. They also indicate that increasing k, the
number of class intervals, does not increase the power of the test. This argument parallels
and supports that of Mann and Wald and Hamdan.

Another formula, commonly used and incorporated in the computer program in

this study, is given by Mills (1955), Brooks and Carruthers (1953), and Panofsky and
Brier (1968). It is

For a given N, KN < kN.

20
 If the multinomial distribution or the multivariate distribution, both in terms of
hypercubes, is .used and the dimensions are high, the number of cells or hypercubes is
great; the probability for each hypercube, especially in terms of equal probability cubes,
must be low. This requires a large number of observations for resolution. Techniques to
compute the probability for the two-dimensional cases are adapted from the work of
several investigators. Tables t o do this are available in the Bureau of Standards
publication (1959) and from Owen (1 956, 1962). Crutcher (1962) provides some
examples of the use of the tables. Milton (1 970) provides techniques and procedures for
the multivariate case.

Spherical shells, circular annular rings or area bands, or even rectangular or square
shells or cubical shells are special cases of the ellipsoidal shell, also called the hypercube.
Figure l b shows the univariate distribution where the matched hachured areas are added.
Here, the normal distribution is divided into 10 shell areas where the matched hachured
area represents one shell representing a probability of 0.10. In the usual sense, as often
employed in making the x2 test, each matched hachured area would be separated into its
two parts, providing 20 smaller equiprobability sections, each of 0.05 probability; the
two matched hachured areas would be a shell in one dimension.

Cochran’s (1952) extensive experience provides us with ideas of what t o d o for

attribute and continuous data. For continuous data, such as will be used in examples in
this report, Cochran (1954), following Williams (1950) for the one-dimensional case,
proposes an expected value of 12 per shell for n = 200, 20 per shell for n = 400, and 30
per shell for n = 1000. At the tails, pool (if necessary) so that the minimum expectation
is five. Since the number of data may be small, the major operation will be to broaden
the shells, especially the outer shell, such that each shell has an expectation of at least
five.

For a given number of shells, the degrees of freedom available decrease with
increasing dimensions. The available degrees of freedom are k-p-1 where k is the number
of shells (categories), p is the number of parameters fitted, and the “1” represents the
normalizing of the volume (area) to equal that under or within the normal surface. The
number of parameters fitted is the number of means and covariances estimated. For the
one-, two-, and three-dimensional cases, there are 2, 5 , and 9 means and covariances,
respectively, providing k-3, k-6, and k-10 degrees of freedom. Thus, 10 equiprobability
shells for a one-dimensional case provide 7 degrees offreedom for testing. The general
formula for calculation of the degrees of freedom is k-((v+l)(v+2)/2).

In the case of two dimensions, the shells (or categories) may be elliptical cylinders
bounded on the bottom by the bivariate plane and on the top by the bivariate frequency
surface. Figure 3d ‘may be viewed for concept. It does show equiprobability concentric
cylindrical shells. In the case of three or higher dimensions, the shells (hyper-shells) are
concentric around the centroid. Again, for concept, examine Figure 5. It, too, does not
necessarily show equiprobability shells.

If 10, 20, or 25 equiprobability intervals are considered that allow for

equiprobability intervals of 0.10, 0.05, or 0.04, respectively, then all that is required is

21
 simply to compute the x or x2 values serving as limits of the equiprobability levels and
distribute the computed x or x2 values of the data set. However, it is not required that
the intervals be equal in either scale or probability. If graphical procedures and visual
counting are used, it is the fractional multiples of the x which are used as vector radii of
the ellipses or circles. Squares, cubes, rectangles, or hypercubical or rectangular shells
can be used. The shapes need not be specified, but the user must be able to compute the
expected and observed values within each category. Ordinarily, the circles (ellipses) or
squares are the easiest to use. However, with the computer, the appropriate x or xz
values and their inverse values for any number of sets or intervals may be computed and
used. The differences between the expected and observed number of x or xz values in an
interval are squared and this is divided by the expected. Here, x2 values are computed
and used, which bypasses the necessity to compute the x values and the necessary square
roots of the xz values used as boundaries of its equiprobability shells. As indicated,
graphical procedures require the x values. These results then are summed as indicated in
equation (10) and checked against ,the appropriate degrees of freedom in Table 1 at the
preselected probability 'level for testing. If a computer is used, this can be included in the
program. Here, a, the probability level selected for rejection, is 0.05. The theoretical
value of a for the X2 criterion is computed and printed.

Application of the usual x2 test to the X2 statistic is fraught with difficulties. A

small number of observations and a large number of dimensions (variates) decrease the
chance of each hypercube (sphere shell) containing observations as the number of
hypercubes increases. The expected frequency in a hypercube cannot be less than zero, so
for a starter the number of observations sliould at least equal the number of hypercubes.
The minimum count expected in each hypercube would be one. The resulting x2 test
would be very conservative.

With the degrees of freedom (d.f.) equal to (k-((v+l) ( v + 2 ) / 2 ) ) and with the
dimensions v known, k, the number of shells, may be determined. For any test the d.f.
must be one or greater. If k is set equal to 5 log,, n, for a degree of freedom equal to
one; n must be greater than 7, 25, 159, 1585, and 25, 119 for dimensions 1, 2 , 3 , 4, and
5, respectively. For higher degrees of freedom, the number of observations is greater. For
2 degrees of freedom the numbers required are, respectively, 10, 40, 252, 2512, and 39,
81 1. With X2 then computed and with the d.f. known, the significance level can be
computed. The investigator then can make the decision to reject or not to reject on the
basis of a previously selected significance level. For example, if a = 0.05, any probability
level less than 0.05 automatically calls for rejection. The program permits the option of
selecting the level of rejection, a,prior to computation so that the decision making
process stays as honest as possible. Both the computed significance level and the prior
selected level are printed.

Only the x2 distribution is utilized so there is no test of symmetry. These x2

tests are deficient in this respect. To this extent these tests may be considered necessary
but, perhaps, not sufficient. However, the use of the x2 test for goodness of fit is well
established. Its insufficiency in this regard does not negate its general worth as a goodness
of fit test. Later sections provide for other tests.

22
 B. Comparison of Empirical Chi-Squared Values Against the
Cumulative x2 Distribution

Pearson (1 900) provides theoretical cumulative distribution curves. Hald (1952b)

gives tabular presentation of both the x2 variate and its inverse form. The discussion now
follows that of Hald (1952a) for the two-dimensional form. Extension is made to the
multivariate form.

On semilogarithmic paper the theoretical x2 values may be plotted against the

theoretical probabilities provided by the x2 model. Manually, these values are obtained
most easily by inverse x2 tables such as Table 1. With electronic data computer program
modules and with appropriate peripheral equipment, the xz values and their respective
theoretical probabilities may be computed and graphed. Andrews, Gnanadesikan, and
Warner (1 973) show a graphing procedure for the x2 values.

The distribution function of x2 depends only on the degrees of freedom, f. The

probability that x2 belongs to the interval (x2, x2 + d(x2)) as shown by Hald (1952a) is

As indicated and shown by Huld ( 1 952a) for the two-dimensional distribution (v=2), the
theoretical line passes through the point ( 0 , l ) (chi-square versus logarithm of probability
with a slope of -0.217 for common logarithm and -0.5 for natural logarithm). For x2
equal to zero, 1 is absolute certainty or a probability of 1.00. Figure 6 shows the
theoretical lines for v = 1,2,3,. . . ,12 that respective v-dimensional distributions may be
expected to follow. Hald’s example (1952a) is an example of a two-dimensional
distribution. The ordered calculated x2 values plotted against the empirical probabilities
obtained from 1.00 - ((i-(c))/n), where c is set to 0.5, shown by Hald show a pattern
apparently randomly scattered about the straight line. This formulation will be discussed
in more detail in a later section.

If the empirical plot appears to be distributed more or less randomly about the
theoretical line, the investigator may reasonably infer that the data from which the
sample was obtained are distributed norinally. However, this satisfies only an intuitive
feeling based on experience. Although such decisions are valid, it is comforting if some
helpful numerical or graphical decision process can be evolved and developed. The
example that Hald employs appears to serve the decision to not reject the null
hypothesis. N o quantitative test is provided.

It is the purpose of this section to develop tables and graphs which may be used
to help make decisions as to the v-variate normality of distributions.

23
1l11l I ~ II I I 1111111111Il Ill

0 1 2 3 4
X2
5
- 6 7 8 9 10

Figure 6. Plot of x2 versus [ l-p(x2 )I prepared from data given in Table 1 (Note
straight line for f=2 with intercept (0,l) and slope -0.5; slope is -0.217 if
common logs are used; f is degrees of freedom for chi-square and v
for dimensions).

24
 Sets of vectors, in terms of components, are generated randomly from a random
number generator with presumably no correlation between components. It is not
important whether there is correlation. It is important that there be no autocorrelation.
One hundred sets of such vectors in subsets of 5, 7, 1 1, 15, 21, 3 1, '5 1, and 101 are
computed for Y = 1, 2, 3, 4, and 5 . Odd numbers in the samples permit easier selection
of the medians. Observed x2 values are computed for each vector and ordered and
plotted against their empirical probabilities for comparison against the v-dimensional
theoretical x2 and theoretical probability values.

C. Random Number Generators

No random number generator is truly random. Each is a pseudorandom generator.

Any reference in this paper to a random number generator implies a pseudorandom
generator. The idea is to produce a sequence of integers or numbers which, in spite of
being produced by a fixed procedure, will serve as random variables in computer
simulations.
-
As fixed procedures are used, the sequence of numbers produced by any random
number generator will repeat. The length of interval over which nonrepetition of any
pattern occurs is in a sense a measure of the goodfiess of the generator. Given an
initialization number, the sequence generated exists in some hyperplane, hypercube, or
hypersphere of space. Changing of the initialization numbers is, in a sense, a change of
the hyperspace. If a series of random number generators can be so perturbed, or
perturbed in other ways, then the final output may be expected to exist in a more truly
random fashion in hyperspace.

The random number generator used here by permission is the Univac-Stat-Pack in

the 1108 Computation Library of the Marshall Space Flight Center (MSFC). The
generator first generates a random number over the interval 0 to 1; i.e., R(U) is the
uniform variate (0.00 < R(U) G 1.0). From a sequence of R(U) numbers a random
normal number R(N) is generated as R(N) = [ZR(U) - (N/2)], where the summation runs
from 1 t o N. If N is chosen as 12, the distribution of the output has an expected mean
of zero and a variance of 1. Doubling of the summation sequence from 12 t o 24 simply
doubles the variance or increases the standard deviation by the square root of 2. It was
found during the course of this work. that this generator has the unhappy facility of
truncation; Le., ail insufficient number of large deviations is produced. However, after
much work with the rather extensive output, it was decided to not rerun the problem
with a new generator. The truncation problem, though unresolved, did not create an
untenable situation.

Following are a few brief comments on other random number generators. One is
selected which will be used if this work is to be extended.

25
 Box and Muller ( 1958), Hull and Dobell (1962), Knuth (1969), Marsaglia (1972),
and Marsaglia, Anathanarayanan, and Paul ( 1 972) are among the many who have worked
on the problems associated with random number generators. Marsaglia’s work is found to
be good. He indicates that congruential generators are not suitable for precision Monte
Carlo use. Some means should be used to perturb the generator so as to destroy its gross
lattice structure in v-dimensions. Through the years Marsaglia and his colleagues have
developed such a perturbed generator. Many universities now utilize this program. This is
known as “The McGill Random Number Package ‘Super Duper’.’’ It has been tested by
the senior author and by Mr. R. L. Joiner of the National Climatic Center (NCC) and will
better fulfill our future requirements.

Figures 7a, b, and c show processing output of a sequence of 1000 random

numbers generated by the Univac-Stat-Pack generator. This is a congruential random
number generator. Some perturbing was done by using different initializing numbers for
each subsample. It was modified to permit extension of the range from k6 to + I 2
standard deviations. Figure 7a shows the distribution of normal deviates versus their
percentage frequency. Figure 7b shows the lag correlation graph correlogram of data out
to 100. Figure 7c shows a spectrum analysis of the same set of data with confidence
bands. The spectrum is white; Le., no pasteling is evident.

Figure 8 further illustrates the output of the random number generator. Pairs of
uniform random numbers are plotted pairwise. Each dot represents a pair. No pattern is
evident to the authors. This does not imply that there is no pattern. Comparison of this
plot to a plot of the “Super Duper” output at the NCC indicated that the output of the
Univac-Stat-Pack package would serve our purposes. Obviously, to do otherwise would
require an extensive and expensive rerun of the problem. A further check using the
Box-Muller technique supported this decision.

Figure 9 further illustrates the output of the random number generator where the
paired values are normal. The distribution appears to be a zero mean cluster or swarm
with an expected high density near the center. There are 10,000 data points. For
purposes of this study, except for the slight truncation error, the output of the
Univac-Stat-Pack normal random number generator is considered to be sufficient.
Preference in future work will be given to the McGill University “Super Duper” program.

D. Selection and Processing of Samples with Zero

Means and Zero Covariances

The random number generator described previously produces a sample from a

univariate normal distribution. Pairs of these numbers produce a sample of a bivariate
normal distribution. Triplets, quadruplets, and quintuplets form trivariate, quadrivariate,
and pentavariate distributions. Combinations of still more will produce the respective
multivariate distribution. This report treats distributions only through the pentavariate.

26
 -3 -2 -1 0 1 2 3 4

STANDARD NORMAL VARIATE

a. Cumulative distribution of 1000 univariate normal data points.

Figure 7. 1000 data points generated by the Univac Stat-Pack random

number generator.

27
 I l l l l
80 90 100
LAG
b. Lag correlation coefficients for a sequence of 1000 normal data points

Figure 7. (Continued).

28
I

HARMONICS
c. Spectral density plot for 1000 normal data points.

Figure 7. (Concluded).

29
 0 0.1 02 0.3 0.4 0.5 0.6 0.7 0 -8 0.9 1.o

X 1 UNIFORM DISTRIBUTION

Figure 8. Two-way plot of pairs of uniform random numbers generated

by the pseudorandom number generator of the Univac Stat-
Pack (10 000 points).

Since the mean of the distribution in each case from which the sampling is made is
zero, the medians of the samples should follow the curve of chi-square values given in
Table 1 and illustrated in Figure 6. In actuality, only with large samples will this be an
exact condition. A calculated mean of zero, diagonalized unit (identity) covariance, or
correlation matrix will be that exception. It is important to discuss this feature before
proceeding to sample means and sample covariances in application. The assumption is

30
 10

z 8
0
I-
3
K
I- 6
Fi
a
E
0
z
I
...
x"

-12 -10 -8 4 . i 3

.. ..

X I NORMAL DISTRIBUTION

Figure 9. Two-way plot of bivariate normal distribution (NID(0, 1)) generated

by the Univac Stat-Pack random number generator (1 0 000
points; range of the components is 24 standard deviations).

made, for purposes of the following discussion, that any sample in v-dimensions obtained
by means of a random number generator will have a mean of zero and that the actual
correlation is zero; i.e., the distributions are central spherical. Confidence bounds are
.empirically determined by the computation of medians of percentile values of computed
chi-squares. The chi-squares are the squares of the vector radii. Various sample sizes of 5 ,
7, 9, 11, 15, 21, 25, 31, 51, and 101 were computed for dimensions 2, 3, 4, and 5 . Only
two of these tables are shown here. One hundred samples of each sample size were
determined, and selected percentile values were obtained. This procedure was repeated
100 times and the median value selected and printed.

31
 Table 2 provides the median values of selected percentiles of 10 100 random
chi-square values of the multivariate distribution with zero means and zero covariances
(NlD(0,1,0)) and their probability plotting positions, p = (l-((i-OS)/n)) for three
dimensions and a sample size of 7. In Table 2 and subsequent tables v is dimension and n
is sample size. The chi-square values are computed as x$ = Zx?, i = 1, 2, 3, where xi is a
random number from a random normal generator, with a mean of zero, a variance of 3,
and with covariances equal to zero. In other words, sample variances, sample covariances,
and sample means are not computed and used in this procedure.

The values shown in Table 2 were obtained as follows for a sample size of 7:

1. A sample of seven chi-square values was obtained.

2. The sample values were ordered from low to high.

3. This sample was set aside.

4. Steps 1, 2, and 3 were repeated to obtain 100 samples of size 7, each ordered
from low to high. A two-dimensional matrix of 7 columns and 100 rows then was
available.

5. The values in each column then were ordered from low to high so that
selected percentiles in each column would be selected. The second number in each
column would be the second percentile.

6. Steps 1 through 5 were repeated to obtain 101 similarly formed

two-dimensional 7 by 100 matrices.

7, A selected percentile value in the ordered column, for example the second
percentile value in the first ordered column, then was available for each layer of a stack
of 101 two-dimensional matrices. The number 101 and the odd numbered sample sizes 7,
11, and others were used to permit easy selection of median values. The stack of 101
second percentile values was ordered and the median values were selected. It is this
median value which appears in Table 2 and other similar tables. It is this value which is
selected and plotted as an example in Figure 10.

The various percentile values shown in Table 2 may be used to establish

confidence bands. For example, the 2nd and 98th percentile values establish the central
0.96 probability confidence band. A set of seven ordered chi-square values computed
from such a three-dimensional distribution as described previously, when viewed against
the probability plotting points, sbould fall within the limits. If this is true, then the null
hypothesis that the population from which the sample was taken is not different from
the three-dimensional multivariate normal distribution need not be rejected. The level of
significance is 1.OO - 0.96, or 0.04.

32
 1.oo

.80

.60

.40

N
- O
20

X
I

n
I .10
r

.08

.06

-04

.02

.o 1

-
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

x2
Figure 10. Three-dimensional sample 'size 7 median values of the 2rid, 50th,
and 98th percentile values of 10 100 random chi-square values of the
multivariate normal distribution with zero mean, variance one and
covariance zero (These provide the 96 percent central
confidence band).

Figure 10 presents the 0.02, 0.50, and 0.98 probability levels, giving the expected
central curve and the central 0.96 confidence band. Table 2 and Figure 11 illustrate the
same concept with the larger sample size of 31. Here, some symmetry is evident in the
flaring of the confidence band. A difference between the central median curves and the
corresponding curves in Table 1 or Figure 6 signifies the departure of the distribution
from the normal distribution even though the mean is specified as zero. Even more
marked will be the departure (the Wishart distribution) from the normal for small
samples.

33
 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4

x-2
Figure 1 1. Three-dimensional sample size 3 1 median values of the 2nd, 50th,
and 98th percentile values of 10 100 random chi-square values of the
multivariate normal distribution with zero mean, variance one and
covariance zero (These provide the 96 percent central
confidence band).

The problem faced by most researchers, including the authors, deals with small
samples and populations for which the means, variances, and covariances are unknown. It
was these problems and the lack of suitable techniques to adequately answer questions
dealing with these problems that led the authors to develop and prepare this report.
Hopefully, it will be useful to others.

34
__--
 E. Selection and Processing of Sample
Data (General)

Following are the final instructions to the electronic computer programmer after
viewing the results of several generators.

1. Select 100 samples each of 5, 7, 11, 15, 21, 25, 31, 51, and 101 sets of
random normal variables using a random noimal variate generator with zero mean,
variance 1, and covariance zero with a range of +12 s. It is important here to note the
difference in approach in subsection D and this subsection. Subsection D assumes and
uses no sample correlations. This subsection assumes and uses the sample correlations
though the correlations presumably are not different from zero, having been obtained
from uncorrelated and independent random numbers. However, in the real world of
sampling, there may or may not be correlation among the variates. In this study these
correlations are used even though a Type I1 error may be committed. That is, we may be
using a non-zero correlation even though correlations should be zero. Also, in the real
world of sampling, non-zero means may be obtained. Here,’ it is assumed that the data
will be distributed about the sample mean in the functional form of the basic underlying
distribution. Of course, Hotelling’s (1 93 1) T2 test could be used to test the difference of
the mean from zero. However, this test is not performed since it is believed that the
empirical tables developed here will assist in decision making. In this report we are testing
for multivariate normality. For small samples, the distribution is the multivariate Wishart.
The data sets selected from the random normal generator will be dimensional one
through five. Retain the random normal variables in the order in which they are selected.
For example, prepare sets for the two-dimensional distribution by taking sequential
. numbers, Le., (first, second), (third, fourth), (fifth, sixth), etc. There will be 100 samples
of 5 pairs, 100 samples of 7 pairs,. . . , 100 samples of 101 pairs. Assign each of the 100
samples an I.D. number ranging from 1 to 100.

Include in this part 1 options to select one variate, two variates (pairs), three
variates (triplets), four variates, and five variates. Five variates are the maximum for this
study. The first paragraph of part 1 describes the procedure for generating 100 samples
each of 5, 7, 11, 15, 21, 31, 5 1, and 101 pairs (two variates) of random normal variables.
For triplets (three variables), the random normal variates again are selected in the
sequential order that they were generated, i.e., (first, second, third), (fourth, fifth, sixth),
(seventh, eighth, ninth), etc. Follow this same procedure for four variates and five
variates. There are five options in part 1 which are:

Option 1: One-dimensional case (one variate) designated as X1 .

Option 2: Two-dimensional c a e (two variates or “pairsyy)designated as (X, ,X2 j.

Option 3: Three-dimensional case (three variates or “triplets”) designated as

(XlYX27 x 3 1.

35
 Option 4: Four-dimensional case (four variates) designated as (X, ,X2 ,X3 ,X4).

Option 5 : Five-dimensional case (five variates) designated as (X, ,X2,X3,X4 ,X5).

For example, follow the procedure below for the 100 samples of 11 pairs (option 2, two
dimensions, n = 11). Use this same procedure for the other options.

2. For each of the 100 samples of 11 pairs compute the following parameter
estimates where n = 11 :

Equations (1 6) through (1 9) provide estimates of the population parameters, pxl, p x 2 ,

2 2
ax2’ ax2’ and p x , x 2 .
Equation (1 9) gives the correlation coefficient rX1 x2 between the variables X1
and X2. The general equation for correlations is

36
 WhenQ = j , then rx or r .x. = 1; Le., in the following correlation matrix, all diagonal
Q Q % J
elements equal 1. (Note Q and j = 1, 2, 3, 4, 5 to include all options.) Five dimensions
are the maximum used in this paper. Using equation ( 2 0 ) , form the correlation matrix C
as follows:

Bivariate Case
- - - - -1 -
rll r121 r13 r14 r15
I
I
rZ1 r221 r23 r24 r25
______- J
C = r31 r32 r33 r34 r35

Note that this matrix is a symmetric matrix with rl = r2 , r, = rj 1 , etc. Consequently, ,

our notation will be confined to the upper right triangle above and including the diagonal
elements of the correlation matrix. Solve for the determinant of C; call this determinant
D.

Now, the general equation for xo2 is

xo2 = D
1
[ (
RXl,X1
x1
"
X
,
-x,
-
)' + R x 2 , x 2 (),XS
x, -x2 2

+ . * . Rx5,x5(
x5 -x5
ix;)
2

x, -x, x, - 2 2
+ 2RX1,X2( -S F x)i:(
)

+ 2 R x 1 ,x3(
x1 -x,
% --)( ---) x3 -x3
sx3 + * :Rx1,x5( ;xr)( i;)-x,
x, -x, x5
'

+ 2RX2,X,( -,>-x, (x3yg-)

xz -x3 - + ' -x,
2 R x 2 , x s ( xasx, --)(sx,-)
x, -x5

x4 -x4 x,
+ 2RX3,X4( <c)(
x3-x3 x4-x4
);s- + * 2 R x 4 , x 5 ( -sx,)(
-575
sy)]' (22)

37

I
 where RXeX. is the cofactor of rij (the elements of correlation matrix C ) , Whittaker and
1 J
Robinson (1 954).

3.a. The option chosen in part 1 of this subsection will determine the dimensions
to be used in the general equations. As before, we will use option 2 (two dimensions,
100 samples of 11 pairs) for illustration.

b. For n = 11, compute Xl, X 2 , sxl, s x 2 , r l 1 , r 1 2 , r Z 1 , and r 2 2 using

equations (161, (1 71, (18), and (20). Note that r x l ,x2 - - r l 2 , rx = -
r13, r x 1 , x 4 -
1, 3
-
- 1-2 3, r x 2 , x 4 -
- r24 , etc., in matrix (21 1.
rl4 , r x 2 , x 3

c. Form the correlation matrix C. For the two-dimensional case (bivariate) this
will include elements r l 1 , r l 2 , r2 , and r2 only. Solve for D, the determinant of matrix
C.

d. Introducing the 11 observations along with the parameters computed in 3 . b

and 3.c into equation (22), we obtain 11 values of x$ Retain the random order of the
xO2values for future use in the analysis of runs.
e. Order the 11 values of xo2obtained in part 3.d in order of increasing
magnitude. Compute the empirical probability of exceeding xo2corresponding to the n =
11 ordered x2values using

n-i-c+ 1 12-i-c
[1 -p(xo2)] = ___ + 3
n-2c+ 1 12 - 2c

fox i = 1, 2, 3, . . . , 11. “c” equals [ 1 - (v/4) - (4/n(ln n))] , where v is the dimensions;
i.e., if v is 2, the dimensions are 2. In is the natural logarithm.

f. Prepare Table A. This table will be the 1 1 ordered xo2values versus [ 1 -

p(x;)]. Note there will be 100 table A’s corresponding t o the ‘100 samples of size 11.
Print out only the first 11 samples of the 100 samples in all cases and options. (The
others must be computed and retained, however.)

g. Plot the ordered xo2values versus [ 1 - p(xo2)] on linear versus log,

coordinates. Plot the theoretical fit to these points using the Univac 1108 Electronic
Computer subroutine 12.2, CHI, x, Univac Stat-Pack. Use the appropriate degrees of
freedom (d.f.); i.e., for the two-dimensional case, d.f. = 2 and for the three-dimensional
case, d.f. = 3, etc. Plot only the first 1 1 samples of the 100 samples in all cases and
options.

38

I
 .. .

h. Repeat 3.b through 3.g for each of the 100 sets of l l pairs. Establish a
matrix 11 by 100; Le., prepare a table of 11 columns with 100 items in a column.

i. Order from low values to high values the 100 values of xo2available in each
column. The smallest x2value will be in the upper left diagonal corner of the array.

j. Repeat 3.a through 3.i for each of 101 similarly arranged arrays.

k. Arrange these arra.ys so that in essence a modular form, 1 1 X 100 X 101, is

obtained. In general form this would be an n X 100 X 101 module.

1. Order from low values to high values the 101 values of xo2available in each
vertical column. The smallest xo2value will be in one corner of the three-dimensional
array, while the largest value will be in the opposite end of the array diagonal.

m. Various statistics may be selected from these arrays. The first selection is the
median of the vertical columns, which is the 51st value. The medians are chosen without
interpolation for the 2nd, 5th, IOth, 50th, 90th, 95th, and 98th percentiles. These are
the respective positions in each of the 11 vertical columns of the 1 1 X 100 X 101 array
in the final diagonalized form. Print these values by percentile versus “i,” i = 1, 2 , , . . ,
11.

n. Repeat these procedures for each sample size and dimension selected, n = 5,
7, 11, 15, 21, 25, 31, 51, and 101 and v = 1, 2, 3, 4, and 5.

0. Maximum Absolute Difference (MAD). Compute the theoretical cumulative

probabilities- [ 1 - p(x2)] for the n = 1 1 ordered xo2values using the Univac 1108
subroutine 12.2, CHI. Add these values of [ 1 - p(xz)] to Table A. Compute the MAD
between E1 - p(xo2)] and [ 1 - p ( x z ) ] . Note there will be 100 values of MAD
corresponding to the 100 samples of size n - 1 1.

p. Compile Table B. This will be a table of the 100 values of MAD

corresponding to the 100 samples of size n = 1 1 . Order these 100 values of MAD in
order of increasing magnitude in the table. Give the corresponding sample number in the
table, i.e., sample number 1, 2, 3, . . . , 100.

q. Using the Univac 1108 subroutine 13.4, CHIN, for the n = 11 values of [ 1 -
p(xo2)] in Table A, calculate the corresponding values of x2 located on the theoretical
straight line. Add these values of x2 to Table A.

r. Compute the mean sum of squares of the residuals (MSSR) from the
theoretical line using
 1l 11llIl I I

This is ‘the.general equation for MSSR. For the two-dimensional case we are using for
illustration (n = 1 I), equation (24) will be

There will be 100 values of MSSR corresponding to the 100 samples of size n = 1 1, Add
these values of MSSR to Table B corresponding to their respective sample number.

4. Analysis of Runs.

a. The random order of XoZvaluesretained in part 3.d is used here for analysis of
runs. Also, the paired random xo2and corresponding x2 values from Table A are to be
retained for this analysis.

b. Runs Above and Below the Theoretical Line. (In the case of the
two-dimensional distribution the line is straight with a slope of -0.5 when plotted on
semilogarithm paper where the logarithm is the natural logarithm.) Examine the paired
random xo2 and x2 values described in part 4.a. If x$ < x 2 , this is an element below the
line and will be denoted as “b.” If x: > x 2 , this is an element above the line and will be
denoted as “a.”

We now have a series of a’s and b’s, the total number of elements being 11. (We
have 100 of these series of 1 1 elements corresponding to the 100 samples of size n = 1 1.)
An example might be:

aa b aa b aaa bb

1 2 3 4 5 6

A run is defined as a sequence of identical observations that are followed or

preceded by a different observation or no observation at all. In this example there are six
runs in the sequence of n = 1 1 observations. In Table B record the number of runs above
(a) the line and the’ number of runs below (b) the line for the 100 samples. For this
example there are three runs above the line and three runs below the line.

c. Runs Above and Below the Median. For each sample of n = 11 compute the
median of the x$ values in Table A as follows. Arrange the xoz values in order of
increasing magnitude. If n is odd, the median is the middle item. If n is even, the median
is the mean of the values of the two middle items. Examine the original random set of

40
 .............................................................. ..
I

x2 values described in part 3.d. Let “a” denote those values of x$ greater than the
median. Let “b” denote those values of x2less than the median. Again, we have a series
of a’s and b’s as in part 4.b.

In Table‘ B record the number of runs above (a) the median and the number of
runs below (b) the median for the PO0 samples of n = 11.

d. Runs of Quadrant Signs. Retain the random order of xozobservations

described in part 3.d. Assign to each Xo2value the corresponding quadrant signs. These are
the signs of (X, - x,)
and (X, - E,) in equation (22). For the two-dimensional case an
example would be (n = 1 1)’

+-}
+- 2

:I}
-+
3

--)
-- 4

(Note there will be three signs for the three-dimensional case, four signs for the
four-dimensional case, etc.)

Now, a run is defined as in part 4.b. In this example there are six runs in the
sequence of n = 11 observations.

All 100 samples of n = 11 groups of quadrant signs will be combined. Form a

frequency distribution of the number of runs of quadrant signs; Le., the random variable
is the number of runs per sample of n = 11. For a single sample of n = 11, there could
be at most 11 runs and at least 1 run. The total frequency for this distribution will be >,
100. Label this frequency distribution Table C.

41

. . . ...
I 1I IIllll1111ll ~ 11111l1l1111

Also, print out as Tqble D the first 11 sets of signs (of the total 100 samples) for
sample size n = 11. This will be an 11 X 11 table. (See the suggested printout format.)

5 . Cumulative Probabilities. Prepare Table' E as follows.

Using the data from Table B, order the following in order of increasing
magnitude :

a. MSSR

b. Number of runs above theoretical line

c. Number of runs below theoretical line

d. Number of runs above the median

e. Number of runs below the median.

6. Repeat parts 2 through 5 of subsection 1V.E for the 5, 7, 15, 21, 25, 31, 51,
and 101 paired samples (Option 2).

7. Repeat 'parts 2 through 5 of subsection 1V.E for:

a. Option 3 : three-dimensional case

b. Option 4: four-dimensional case

c. Option 5 : five-dimensional case.

8. Combine Tables A through E for Option 2 for all sample sizes n = 5, 7, 11,
15, 21, 25, 31, 51, and 101.

9 . Repeat part 8 for Options 3, 4,and 5.

10. Suggested Printout Format:

Table A: 2 dimensions n = 11

42
 Table B: 2 dimensions n= 11

_ . ~ ~ _-
Sample Runs Above Runs Below Runs Above Runs Below
No. Line Line Median Median
.~ ~
~
~ - - .. __-.-

Table C: Runs of quadrant signs

2 dimensions n= 11
c - .

Number Runs Fi
1
2
3
4
5
6
7
8
9
10
11

43
 Table D: 2 dimensions n = 11

++ - - +-
++ +- ( 1 1 columns) -+
+- -+ - -
+- -+ ++
- - ++ -+
- - ++ ++
-+ - - -+
+-. +- +-
++ -+ ++
+- - - ++
++ +- -+

Table E: 2 dimensions n = 1 1 Ordered Values

Runs Above Runs Below

Median Median

44
 F. Comparison of Ordered Random Chi-square Values
with the Theoretical Chi-square Values

The calculation and ordering of the chi-square (x2) values is the first step in the
visual examination of multivariate (v-variate) distributions for normality. The familiar x2
test is used though it offers no test for asymmetry. The graphs, either of the original data
in histogram form or in the cumulative percentage graphical techniques discussed here, do
offer qualitative assessment of asymmetry. If the null hypothesis is not rejected, then
symmetry is assumed in the assumption of normality. Other tests provided for in this
report do permit symmetry tests.

Equation (3) permits the calculation of x2 values by use: of the estimates of the
population parameters, i.e., the sample statistics. These empirical data are ordered and
processed as in subsection 1V.E. The difference between the data treatments in
subsections 1V.E and 1V.F is that in the former the means and covariances were zero.

There remains the difficult and somewhat controversial problem of the

determination of the plotting points for the empirical data. Blom (1958) discusses the
calculation of empirical probabilities for the univariate distribution. The general formula
is p = ((i - cx)/(n - a - /3 + 1)). Simplifying this wherea = /3 and setting a = /3 = c, p = ((i -
c)/(n - 2c + 1)). Letting c equal 0, p = i/(n + l ) , while letting c equal 0.5, p = (i - 0.5)/n.
Earlier workers use c equal to 0.25 or 0.33. Sarhan and Greenberg (1962) discuss this
formula. Gringorten (1 963) prefers to use c equal to 0.44.

Undoubtedly, the more general formula, Blom (1 958), would be generalized still
more when higher dimensions are used. Apparently, the more general formula would need
to consider the sample size n, the ith ordered positior,, and the dimension v.
Considerable work by the authors reveals no simple solution. The plotting graph paper
serves for comparison only of sample data against the random generator or Monte Carlo
results. Therefore, the simple formula p = (i - c)/(n - 2c + 1) will be used for all sample
sizes and all dimensions. From plots of the deviation of sample data from the theoretical
curves indicated in Figure 6, a formula to determine an appropriate value of c was
obtained. Tnis formula is c = 1 - (v/4) - (4/11) In n, where v is the dimension and In is the
natural logarithm.

Table 3 provides values of “c” derived from the formula c = 1 - (v/4) - (4/n) In n.
Table 4 provides probability plotting positions determined from the formula (1 - p) = 1 -
((i-c)/(n - 2c + 1)). These are shown for sample sizes 5, 7, 9, 11, 15, 21, 25, 31, 35, 41,
45, 51, and 101 for one through five dimensions.

Before proceeding further it is important to discuss the problem of small samples.

With large samples, 500 for example, the curves or plots obtained will approach the
theoretical in the sense portrayed in Figure 6 and figures of the type subsequent to and
similar to Figure 12. The confidence bands will widen and flare rapidly at the lower

45
 1.000

1.000

NO

:o.ioo-
X

- cc

0.01 o------.--~-L-
0 2 3 4 5 6 7 8 9 10 ii 12
\
i3 14 15x02
0oio-
0 i 2
-- 3 4 5
-

6
33-

7 8 9
L
IO 11 1 2 13
i
.
14 15X;
u = 4 N = i i
u = 4 N = 5 i
SAMPLE N 0 . 7
SAMPLE N 0 . 6

Figure 12. Examples of “Good Fit” of randomly generated normal variates for differing
dimensions (v) and sample sizes (n).
probability levels of (1-p). Thomas and Crigler (1974) discuss, in general, tolerance
limits in the multidimensional radial error distribution. The assumption is made that the
multivariate mean is zero and the multivariate dispersion is circular in form; Le., the
vector variance is the square root of the number of dimensions. This assumption is not
usable for small samples where sample estimates of the mean and variance are used. This
fact caused many difficulties in assessing the situation. These difficulties are inherent in
small sampling problems. The problems of Student (1925) are revisited, and the reader is
reminded that this is the problem of the multivariate Wishart distribution previously
discussed. The decision was made to develop the tables and graphs using the multivariate
normal form of distribution.

The crux of the problem is that the values of maximum chi-square (x2) versus (1
- p) converge to some value. Therefore, the confidence bands converge. This is not
apparent in the tables and graphs because the median values of the sample percentiles are
used. The convergence is more noticeable as the sample size gets smaller. For example,
see Figure 18. Briefly, the maximum possible chi-square (x2) is (n-1)2/n when the
unbiased estimate of the population variance in each dimension is used. If the biased
estimate of the population variance is used, the maximum chi-square (x2)is (n-1). Thus,
the computed maximum chi-square values to be obtained in a sample of 5 , no matter
what the dimensions may be, will be 3.2 and 4.0, respectively, for the case of the
unbiased and biased variances. Therefore, the confidence bands beginning at a probability
of 1 diverge from x2 equal zero and then converge toward the values given previously. If
computed maximum chi-square values are greater than the limiting values given, there is
an error in computation. If the error is small, it may be due t o numerical rounding
errors. Actual convergence is to a number less than the maximum given. It is a function
of both sample size and dimension.

As the number in the sample increases, the probability of reaching a maximum

chi-square diminishes and the confidence bands converge more slowly. As the dimensions
increase, the probability of reaching a maximum chi-square increases. As both the sample
size and dimensions increase, the possibility of increasing magnitude of chi-square
increases. With infinite n and infinite dimension, the chi-square limit is infinite.

G. Selected Tables of Randomly Generated

Chi-Square Values

Subsection 1V.D gives the procedures to generate random normal variates and
calculate, select, and order chi-square values. Subsection 1V.E discusses the comparison
problems.

Table 5 provides median values of selected percentiles of 10 100 random

chi-square values of the multivariate normal . distribution. Their probability plotting
positions are indicated. The order and the plotting probabilities are given.

47
 Figure 6 provides the background of theoretical curves against which ordered
chi-square values of a random sample may be plotted. Curves appear for dimensions one
through twelve or 12 degrees of freedom. Curves of only the first 5 degrees of freedom
are used here. Figures 12a through 12d show some selected values of what visually appear
to be good fits. These data are taken directly from the computer output t o microfilm
where 11 samples of each dimension one through five and sample sizes of 5, 7, 11, 15,
21 , 3 1, 5 1, and 101 were prepared. Visually appearing bad fits also are obtained. Figures
13a through 13d are examples of what some of the bad fits might look like in future
sampling. It must be remembered that when data are ordered, there is a correlation
involved in the sequential ordering. Therefore, the usual change in a curve from point to
point is relatively slow. Ordinarily, there are no abrupt changes or breaks.

Comparison of the good fits with the bad fits illustrates how well the ordered
random chi-square values shown as dots approach the expected theoretical line as
indicated by the solid .central curve. Figures 14a through 14d show Figures 12a through
12d with confidence bounds of 0.02 and 0.98 probability. Figures 15a through 15d show ,

Figure's 13a through 13d with the same confidence bounds. Figures 14a through 15d
indicate that the visual inspection was correct. This would be expected for these figures
are selected from those that were considered to be the best and the worst fits. The
confidence bounds are taken from tables prepared as designed under subsection 1V.E.

If a percent or more of the empirical data fall outside the confidence band, the
distribution can be said to be significantly different from the respective k-variate normal
distribution. (a is the rejection level.) The usual Type I or Type I1 error may be made.
This procedure is mechanistic, yet it does serve to provide objective guidelines for
decision.

Figures 16, 17, and 18 show several features of the tabulated results of this study.
These figures are for one dimension with a sample size of 5 , two dimensions with a
sample size of 51, and five dimensions with a sample size of 7, respectively. Figure 16
shows the squeezing in of the curves near the probability level of 0.50. This is
characteristic of many of the data ensembles. Figure 17 illustrates that with a sufficiently
large number of data, the confidence bands tend to flare as shown in Figures 10 and 11.
Figure 18 shows marked characteristics of the small samples. Remember that the
maximum possible chi-square is equal to (n-1)2/n or n-2+(l/n) for the unbiased sample
and n-1 for the biased sample. Therefore, the upper bound for the empirical data sample
must at times be less than that implied by the data curve plotted from the usual
chi-square tables.

H. Selected Tables of Randomly Generated Chi-square

Values Including Tests for Symmetry

In subsection 1V.E the instructions to the computer also resulted in tabular

preparation of:

48
-
N o I l ? i \ i I 1
X
o_ 0.100
I
r

0.01 0

- A, a -

N O
Y \
‘ ’

0.010 0.010
0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 1 5 X i 1,’
u = 4 N=21 u =4 N =51
SAMPLE N 0 . 3 SAMPLE NO.11

Figure 13. Examples of “Bad Fit” of randomly generated normal variates

for differing dimensions (v) and sample sizes (n).
 1.000

-
- 0
X
o_ 0.100

.-
I

---
I-

7
-
.

-I.--

--a-t- 2 2

o. o I o ---------2h-, 0.0I o-----c------L+-

0 I 2 3 4 5 6 7 8 9 1 0 1 l X o
2
0 1 2 3 4 5 6 7 8 9 l O l l X :
u : 2 N:I5 v - 2 11-21
SAMPLE NO. I S A M P L E NO. 1
1.000

-
-0
X
0.100
I
c

0.0I 0
I

; I
-c~t&~---r-+~-----4--
: ’\!

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 5 X i 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 1’5 X,’
Y r 4 I:I 1 1 2 4 Nr51
S!. ‘PLE ’ 9 . 7 SA;.PLE N0.6

Figure 4. Examples of “Good Fit” of randomly generated normal variates

for differing dimensions (v) and sample sizes (n) with the central
0.96 confidence band.
 1.000

No
X
0.100

-
I

0 I 2 3 4 5 6 7 8
v . 1 N=31
S A M P L E N0.5

1.000

- - - - I

-_-..-

- o.io0
X

.- ---._-

0oio 0.010
0 1 2 3 4 5 6 I 8 9 1 0 1 1 1 2 1 3 1 4 1 5
v - 4 N=21 v =4 n:si
SAMPLE N0.3 SAMPLE N O . 1 1

in
Figure 15. Examples of “Bad Fit” of randomly generated normal variates
L
for differing dimensions ( v ) and sample sizes (n) with the central
0.96 confidence band.
 _ -
. I I I

0 1 2 3 4 5 6 7
2
X-

Figure 16. Plot of data from Table 5 (These are the median values of the 2nd and 98th
percentile values of 10 100 random chi-square values of the multivariate one-
dimensional distribution for a sample size of 5. The central line is the median
value for infinite n).

52
 0.600- \\\
\\\
0.400 - \ \ \

0.200 - \\\
0.100 -
i-

\
c

\
b

0.040 -
-
0.0”-
0.02

0.010
0.008
--
-
0.006
1 1 1 I I I I I I I I I

1 2 3 4 5 6 7 8 9 10 11 12 13 14

2
x-

Figure 17. Plot of data from Table 5 (These are the median values of the.2nd and 98th
percentile values of 10 100 random chi-square values of the multivariate two-
dimensional distribution for a sample size of 5 1. The central line is the
median value for infinite n). .
 1.

0.02

.oo I I I
I I I
1 2 3 4 5 6 7

Figure 18. Plot of data from Table 5 (These are the median values of the 2nd and 98th
percentile values of I O 100 random chi-square values of the multivariate five-
dimensional distribution for a sample size of 7. The central line is the
median value for infinite n).
c

54
 1. The maximum absolute differences (MAD) between the theoretical and the
empirical curves. In one dimension this is similar to the Kolmogorov-Smirnov tables,
Beyer (1966). In multivariate form these are similar t o the work of Malkovich and Afifi
(1 973).

2 . The mean sums of squares of the residuals (MSSR) between the theoretical
and empirical curves.

3. Runs of the empirical data above (RAL) and below (RBL) the theoretical
lines of best fit.

4. Runs above (RAM) and below (RBM) the median value. These values are
similar in formation to the one-dimensional tables of Swed and Eisenhardt (1943) and
those found in many handbooks such as Beyers (1 966) and Owens (1962).

5 . Runs of n-tant signs. In the computer instructions quadrant is correct for the
two-dimensional case. Octant will be correct for the three-dimensional case.

Table 6 provides for selected empirical probability levels from 0.01 to 0.99, the
corresponding values of MAD, MSSR, runs above or below the theoretical curve, and runs
above or below the median. Table 6 includes dimensions 1 through 5 at five selected
sample sizes 1 1 through 101.

Table 7 is derived from Table 6. For the same selected probability levels in Table
6, the maximum absolute differences are presented for dimensions 1 through 5 by
sample size.

Table 8 is still another presentation of the MAD shown in Table 6. Table 8

presents for one variate the MAD values for selected sample sizes.

Tables 9 and 10 show presentations analogous to Tables 7 and 8 of the MSSR

taken from Table 6.

Tables 11 and 12 show presentations analogous to Tables 7 and 8 of the runs

above or below the line of chi-square versus the probability.

Tables 13 and 14 show presentations analogous to Tables 7 and 8 of the runs

above or below the median chi-square.

Table 15 further identified by dimensions 2, 3, 4, and 5 and prepared for sample

sizes within each dimension 11, 15, 21, 25, 31, 51, and 101 shows the number of runs to
be expected within each n-tant. Table 15 also gives the number of runs within quadrants
that might be expected on a random sample basis. Though 100 samples of each sample
. size were used to provide a grand sample, only 13 to 14 grand samples were used to

55
I I1 1I I I 1 1
1
1

provide the basis for the maximum, mean, and minimum values shown in the tables.
Thus, if 100 samples are taken, the values given in the tables will be appropriate. If only
one sample is taken, the tabular values should be divided by 100 and rounded to the
nearest integer. For example, in a sample of 11 in two dimensions there could be a
maximum of 10 quadrant sign changes; i.e., each succeeding vector would lie in a
different quadrant. From the 14 grand samples to the nearest integer there will be a
maximum of 8, a mean of 7, and. a minimum of 6 occurrences where the sequential
vectors would shift to another quadrant after the first occurrence. There will be a
maximum of 9, an average of 8, and a minimum of 7 occurrences where one or two
sequential vectors lie in a quadrant followed by a third vector in another quadrant.

This, in essence, is a test of homogeneity, a test of randomness. This test or the

values given do not provide any insight if only two n-tants are involved. If this were the
case, the data set would be oscillatory and would not be homogeneous but would be
heterogeneous. A visual perusal of a sign combination output in the data presented does
not reveal any heterogeneity. However, this does not imply that a future set of data
might not be oscillatory. Hopefully, the authors can investigate this problem in the
future.

The cumulative sum across all runs is not necessarily a constant due to varying
sample sizes and truncation in preparing the tables.

V. APPLICATIONS T O REAL D A T A

A. Temperature and Winds

Cramer (1970) supplied data for this portion of the study. This is a
four-dimensional problem. Computer listings were keyed on magnetic tape of u', v', w',
and T' for three two-level trials (runs 32, 33, and 34 of Cramer's study on the Round
Hill 40 meter tower). The records for each trial cover a 1 hour period and comprise
sequential time series of the discrete values of the four variables at 1.2 second intervals.
Thereare 3000data groups in each run at each level. The units of the wind are meters per
second and the temperature unit is degrees Celsius. The primes indicate deviations from
the run average. The east-west, north-south, and vertical wind components are,
respectively, u, v, and w. These components are positive from the west toward the east,
from the south toward the north, and upwards from below. The temperature symbol is
T. The heights of the two levels were 16 and 40 meters. Cramer (1966) provides the raw
statistics and other information of his data in Volume I11 of the ECOM-65-G1O Report.
Run 32 at 16 meters above the surface was selected for use. The run began at 0834 and
ran to 0934 EST on October 5, 1961.

Figures 19 through 22 illustrate a data subset from run 32. These are separate
sequential plots of u', v', w', and T'. These figures serve to show the variability of the
types of data. Plotted are the 300 points 2701 to 3000 for run 32 at the height 16

56
 4

3- 1
2

CI
1
v)
\
E
Y

n
w 0-
w
n
v)

-1

- 2-

-3-
0 1 60 80
1 I

100 120 1 1 160 180 200 220 240 260 280 300
T I M E I N INCREMENTS OF 1.2 SECONDS

Figure 19. Zonal deviations of the wind (m/s) during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5, 1961,0929-0934 EST.
(The components are positive from the west. The deviations are from the
zonal mean for the hour 0834-0934 EST. The elevation is 16 meters.)
(Cramer, 1970)

57
1lI1l11111111111 11l1I I I1 I I I I I 1l111111111l1111

I hA

I V?IL
Ici-1 ' 1'1
2I . IT
0 20
1 i
I
I
I -
--m
40 60 80 I00 I20 173 I60 1od- 2b0 2h0 240 260 280 300
TIME IN INCREMENTS OF 1.2 SECONDS

Figure 20. Meridional deviations of the wind (m/s) during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5 , 196 1,0929-0934 EST.
(The components are positive from the south. The deviations are from the
meridional mean for the hour 0834-0934 EST. The elevation is 16 meters.)
(Cramer, 1970)

58

. . . ....
 2

CI
YI
<
E
5 .
I

0- dI 5Y
I

w -1
W
n
v)

.- 2

-3 ~

-4. - -
I
-
20 40 60 80 100 120 140 160 180 2 0 2 1 260 280 3 O .
T I M E IN I N C R E M E N T S OF 1.2 SECONDS

Figure 21. Vertical deviations of the wind (m/s) during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5, 1961, 0929-0934 EST.
(The components are positive from below. The deviations are from the
vertical mean for the hour, 0834-0934 EST. The elevation is 16 meters.)
(Cramer, 1970)

59
 0

- 0.1
-0.2

-0.3

-0.4

- -0.5
0
L
-0.6 lh
TI
W
a
3

m
c
u -0.7
a
W

m
n
-0.8
t

-1.1
0 20 40 60
'yy
I
80 100 120 140 160 180 200 220 240 260 280 300
TIME IN I N C R E M E N T S OF 1.2 SECONDS

Figure 22. Temperature deviations in degrees Celsius during the Round Hill
Turbulence Measurements Program at Round Hill, MA, October 5, 196 1,
0929-0934 EST. (The deviations are from the mean for the hour,
0834-0934 EST. The elevation is 16 meters.) (Cramer, 1970)

60
 ~ ..... .- .. .. .... . ... ,..
.... ._ _......._.-.-...-..--- ,, . ,,.,..,,,,,,. ,

meters on October 5, 1961, 0929-0934 EST. The primes indicate deviations of the data
from the 1 hour mean. For example, note that the temperature deviations shown in
Figure 22 are all negative. There has been some smoothing in the drafting of these
figures. Figure 23 is a graphical output of the four-dimensional chi-square ordered
distribution of the same 300 points of run 32. The central line is the expected line of
best fit. This line may be drawn either from a set of chi-square tables such as Table 1 or
as illustrated in Figure 6 . The chi-squares are computed from the data groups per
equation (3).

Figure 24 superposes 0.96 probability confidence bands for a sample size of 101
on Figure 23, although there are 300 data groups or 300 four-dimensional>points
involved. The band actually would be somewhat narrower for the larger sample size of
300. Since none of the chi-square values fall outside the limits shown, the null hypothesis
that the observed distribution is not different from the theoretical multivariate normal
distribution is not rejected. If confidence bands were available for the 300 sample size, it
is recognized that the hypothesis might be rejected. However, the data set appears t o be
well within bounds. The bounds are constructed from data given in Table 2.

Figures 25 and 26 present a similar example for the data subset, points 301 to
400, of the same set used above. Here the sample size is 100 rather than 300. Note that
near the 0.19 probability level the empirical curve does exceed the limits. However, as
the curve draws back within bounds, the multivariate normality hypothesis is not
rejected.

Autocorrelations presumably are inherent in the data sets. These autocorrelations

will not invalidate the tests. In fact, autocorrelations have the tendency to cause rejection
of the null hypotheses. The determination of the first zero crossing for the
autocorrelation and the use of this lag to determine a sampling rate from the data sets I

will minimize the deleterious autocorrelation effects. If the correlation goes from positive
to negative at the sixth lag or reaches a positive minimum, the number 6 provides the
sampling rate. That is, use every sixth observation. Since the hypotheses were not
invalidated here, the autocorrelation effects were not considered at this point.

The nonrejection of the null hypothesis in the two cases above permits the
inference that the distribution of these data, the Round Hill turbulence measurements at
this point in time and space, may be described by the four-dimensional normal
distribution, This inference permits the further inference that each ,of the four
three-dimensional, the six two-dimensional, and the four one-dimensional distributions
may be described by the respective multivariate normal distributions.

The MAD, MSSR, RAL, RBL, RAM, RBM, and runs of n-tant signs tests are not
discussed for these data.

61
 0.001 I ' I ' I ' 1 1 1 1 1 1 I 1-1 1.1 1 1 1 I 11 1 1.1. I

Figure 23. Four-dimensional chi-square ordered distribution of 300 consecutive data

points of wind component deviations (m/s) u', v' and w', and temperature
deviations in degrees Celsius during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5 , 196 1,
0929-0934 EST. (The elevation is 16 meters.)
(Cramer, 1970)

62
Figure 24. Four-dimensional chi-square ordered distribution of 300 consecutive data
points of wind component deviations (m/s), u', v' and w',and temperature
deviations in degrees Celsius during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5, 1961,
0929-0934 EST with the central 0.96 confidence band.
(The elevation is 16 meters.) (Cramer, 1970)

63
11 11l111l1l

I I I I I I 1 1 I 1 I I I I 1 1 I 1 1 I II 1 I 1 I 1 I 1 I I 1 1 1 .
0 2 4 6 8 10 12 14 16 18 20

J-
Figure 25. Four-dimensional chi-square ordered distribution of 100 consecutive data
points of wind component deviations (m/s), u', v' and w', and temperature
deviations in degrees Celsius during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5, 196 1,
0840-0842 EST. (The elevation is 16 meters.)
(Cramer, 1970)

64
Figure 26. Four-dimensional chi-square ordered distribution of 100 consecutive data
points of wind component deviations (m/s), ut, V I and w‘, and temperature
deviations in degrees Celsius during the Round Hill Turbulence
Measurements Program at Round Hill, MA, October 5, 1961,
0840-0842 EST with the central 0.96 confidence bands.
(The elevation is 16 meters. The central line is the
theoretical line of best fit. The outside bounding
lines give the central 0.96 probability confidence
bands for a sample size of 101.)
(Cramer, 1970)

65
II I l l l I I I1 111I1l1111 11ll1l11ll1l1

B. Wind - Surface
Data for examples of trivariate distributions were supplied by Adelfang (1 970).
Wind gust data were obtained a few meters above the desert at Palmdale, California using
a Vector Vane, Adelfang (1970). The data were furnished as run 19, March 3, 1969,
beginning at 1330 PST. The 1 hour period provides 3600 data points in three-dimensional
space. The wind components are longitudinal, lateral, and vertical. The units are in ft/s.
The chi-squares computed are dimensionless in terms of units. Two samples of 300 data
points each were selected.

Figure 27 shows the three-dimensional chi-square order distribution of 300

consecutive data points of wind components in ft/s, u, v, and w in gust research a few
meters above the desert. Longitudinal, lateral, and vertical components are u, v, and w,
respectively. The bounding lines define the central 0.96 confidence region for a sample
size of 101. The region will be a little narrower for the actual sample size of 300. The
theoretical line of best fit is the central heavy line which may be obtained from Table 1
or Figure 6. The null hypothesis of multivariate normality is not rejected.

Figure 28 shows a second subset of the gust data. Again, the subset remains well
within the 0.96 central confidence band. Therefore, the null hypothesis is not rejected
and the usual probability inferences may be drawn for this multivariate distribution.

If the assumption of multivariate normality is valid and if 100 separate

uncorrelated runs of this type were znalyzed, four of the 100 would be expected to
indicate rejection of the null hypothesis.

The MAD, MSSR, RAL, RBL, RAM, RBM, and runs of n-tant signs tests are not
discussed for these data.

C. Wind - Upper Air

The assumption of normality of the two-dimensional distribution of upper air

winds has been made by Brooks et al. (1946), Brooks et al. (1950), and Crutcher (1957).
Groenewoud et al. (1967) provide several examples of applications to geophysical data
and provide extensive tables to provide easy calculation of probabilities over specified
regions.

Figures 29, 30, and 31 show chi-square distributions of upper winds in the
two-dimensional distribution of the zonal and meridional components and shears. The
data are taken from records at the National Climatic Center prepared and placed on
magnetic tape for the National Aeronautics and Space Administration, Marshall Space
Flight Center, Huntsville, Alabama 35812. The data are for Cape Kennedy, Florida. The
altitudes selected are 8, 12, and 16 km above the ground. Figure 29 shows the 8 km

66
Figure 27. Three-dimensional chi-square ordered distribution' of 300 consecutive data
points of wind components, u, v, and w (m/s) in gust research measured 4.7 meters
above the desert at Palmdale, California using a Vector Vane,
March 21, 1969, 1330 PST. (Adelfang, 1970).

67
Il1 I I I1
l1
l1l Ill1 I l I

1.OO(

' 0.m

-n
N2
I
r

o.oia

Figure 28. Three-dimensional chi-square ordered distribution of 300 consecutive data

points of wind components, u, v, and w (m/s) in gust research measured 4.7 meters
above the desert at Palmdale, California using a Vector Vane,
March 3, 1969, 1330 PST. (Adelfang, 1970).

68
 --

0.100 - --
-
-
- -
- -
-
-
-

0.010

0.02

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4

Figure 29. Two-dimensional chi-square ordered distribution of 124 data points

of Cape Kennedy, FL, upper wind zonal and meridional components
u and v (m/s) at 8 km during the month of January; January 1,
1956 to December 3 1, 1967. (The central 0.96 confidence
band is shown.)

69
l Il Il IIllIl IIII 1l11111l11l1l

t
r

Figure 30. Two-dimensional chi-square ordered distribution of 124 data points

of Cape Kennedy, FL, upper wind components u and v (m/s) at 16 km during
the month of January; January 1, 1956 t o December 31, 1967. (The
central 0.96 confidence band is shown.)

70
 2-
Figure 3 1. Two-dimensional chi-square ordered distribution of 124 data points
of Cape Kennedy, FL, upper wind zonal and meridional component shears
(m/s) between altitudes 12 km and 8 km, for the month of January.
(The central 0.96 confidence band is shown.)

71
distribution while Figure 30 shows the 16 km distribution. Figure 31 shows the
distribution of differences (shears) between the 12 km and 8 km winds. The central 0.96
probability confidence band for a sample size of 101 is shown. Tables have not been
prepared for the sample size used here, 124. Figure 29 is included to provide an example
of rejection of the null hypothesis of two-dimensional or bivariate normality. Arrows
point to the regions suggesting rejection. It is possible that the central 0.99 probability
confidence band, i e., an alpha (a)level of rejection of 0.01, would not call for rejection.
Figure 30 illustrates nonrejection of the null hypothesis. Sums and differences of
normally distributed variables also are distributed normally. Figure 3 1 is included to show
an example of nonrejection of wind shears between 8 and 12 km above the ground. A
similar plot of 12 km component winds (not shown here) indicates nonrejection of the
null hypothesis at the 0.04 rejection level, Le., with the central 0.96 confidence band.
Although the plot of 8 km winds in Figure 29 implies rejection for that level, the shears
between 8 and 12 km (Fig. 31) are assumed to be described by the bivariate normal
distribution.

An examination of Figure 29 indicates that four chi-square values between 8 and

11 are one reason for the rejection. Examination of these winds from a quality assurance
point of view might indicate some error. Any change in these winds would change the
value of all chi-squares as there would be a shift in the vector mean. Such a change or
changes more than likely would also materially change the relative magnitude of the
smaller vectors. Thus, the other rejection region near a chi-square value of 2 to 3 might
also be changed. Such a procedure is only suggested if plots such as these are used for
quality assurance or edit checks of data. It may also be pointed out that Figure 29 as a
single presentation also might exemplify a mixture of two bivariate normal distributions
due to the S-shape of the empirical plot. If this is true, its effect is not great because the
shears shown in Figure 31 do not support this hypothesis. The inference that upper air
component winds and shears are distributed normally is not rejected here.

The MAD, MSSR, RAL, RBL, RAM, RBM, and runs of n-tant signs tests are not
discussed for these data.

D. Hurricane Motions

Hurricanes are a part of the general circulation of the atmosphere. As such, they
appear to move under the complete control of the general circulation. At other times
they do not appear to be under such control and seem to move as an entity. These
movements in distances in stated time periods appear to have some consistency in
direction and speed. Hurricane movement statistics are presented in many forms.

Hurricanes develop during the warm season of the year and usually in unison with
the general circulation pattern. The occurrence and frequency of hurricanes vary with the
region. Crutcher (1971), for the North Atlantic, Caribbean, and Gulf regions, showed that

72
 sets of hurricane movements could be described by the two-dimensional no'rmal
distribution. The variates were the components along two sets of orthogonal axes
imposed over the usual latitude-longitude coordinates. The geographical areas selected
were bounded by the 5" longitude and latitude meridians. The squares thus located were
designated by a four digit number. The first two digits gave the southern latitude
boundary of the square, while the second two digits multiplied by five gave the western
boundary. Thus, the identifier 2014 would be a quadrangle bounded on the south by the
20"N latitude line and on the west by the 70"W longitude line.

Sets of the same hurricane data base have been selected to illustrate the tests
developed for the present report. These tests include the x 2 , MAD, MSSR, RAL, RBL,
RAM, RBM, and the quadrant signs (QS) tests at the 0.04 level of rejection. The null
hypothesis being tested. is that the hurricane distributions are not significantly different
from the two-dimensional normal distribution.

The tests are made on the squares of the standardized vector deviations from the
centroid of the distribution. Only these chi-squares are shown in the tables and graphs.

The data are September-October 24 hour hurricane movements for the 5'
quadrangles in the North Atlantic, Caribbean, and Gulf regions, 2514, 2515, 2014, 2516,
301 5, and 201 5 . The sample sizes are 1 1, 2 1, and 3 1. The movements were taken in the
chronological order of their occurrence over the time period.

The tests are summarized in Tables 16 and 17. The first column in the tables
gives the order number of the sample positions and the identification of the test. The
second and last columns give the 0.02 and 0.98 empirical probability levels for the tests.
The remaining columns provide the results of the tests for the quadrangle identified by
the latitude and longitude heading the column. An asterisk indicates where a bound has
been exceeded requiring rejection of the null hypotheses.

Table 16 provides results of tests for quadrangles 25 14, 25 15, and 2014 for sample
sizes 1 1 , 2 1, and 3 1, respectively. Table 17 provides similar results for quadrangles 25 16,
3015, and 2015.

The x2 test devised in this paper implied rejection of the null hypothesis for the
data sets of quadrangles 2014 and 2015. This prompted a .look at the d a t a , since
quadrangle 2015 lies just west of quadrangle 2014.

Prior screening had removed all tropical depressions from the tropical storm. data.
Also, only one 24 hour movement for any storm was retained for the data set. However,
the same storm could appear in two or more data sets for different quadrangles.
Examination of the observations indicated one or two seeming outliers in each set. In one
case this occurred in quadrangles 2014 and 2015. These were extraordinarily fast moving
hurricanes. Also, the hurricanes occurred in the earlier period of the record and

73

I
I I l I I1IIIII I I Il1 I 1
I l1m1lI1 I

because each sample built on the prior chronologically, a rejection in the first sample
would be expected to cause rejection in succeeding samples until a statistically
representative sample was reached. For example, Table 17 shows that a sample size of 31
does not imply the rejection that its subsets, sample sizes 11- and 21, indicate. Sample
sizes 11 and 21 showing rejection were included to illustrate the technique. Ten
quadrangles were tested of which only six are illustrated. Two of the six (2014 and
2015) exhibit rejection which was caused in one quadrangle pair by the same storm.

The MAD test, which is a Kolmogorov-Smirnov test, implied rejection of the null
hypothesis for only quadrangle 2014 at the 0.04 rejection level. The value of 0.1876 is
not significant at the 0.01 rejection level. ! , I

The MAD test supports the x2 rejection for quadrangle 2014.

The MSSR tests imply rejection of the null hypothesis only for squares 2014 and
2015. This test substantiates the x2 test for rejection.

The RAL and RBL imply rejection of square 2514 with a sample size 31 and
square 2014 with sample sizes 21 and 31. The x2 test previously made shows that
rejection for square 2014 is substantiated. This test rejects the null hypothesis for the
three quadrangles 2516, 3015, 2015. There are too few runs above and below the median
line. This implies a mixture of two distributions insofar as magnitude of distance traveled
is concerned. Direction is not considered here. However, these premises are not examined
here.

The RAM and RBM imply no rejection of any of the null hypothesis. This implies
a certain homogeneity in the data that is not implied in the RAL and RBL tests. There
does not seem to be a clustering insofar as time is concerned. However, direction is not a
consideration.

None of the tests involve any more than tests for the magnitude of the vectors in
the multivariate distribution. The last test shown in Tables 16 and 17, the quadrant
sign (QS) test, considers direction of hurricane movement. The Q S test is deficient
because it considers only the direction of the vector radiating from the centroid and not
the direction and magnitude simultaneously. However, it does provide additional
information as one of the necessary, but not sufficient, tests supplementing those shown
previously.

The QS test is based on a grand sample of 14 sets, each set comprised of 100
samples each for sample sizes of 11, 15, 21, 25, 31, 51 , and 101. A summary is
presented in Table 15. The portions of the table are identified by the dimensions 2, 3, 4,
and 5. Only that portion of Table 15 for two dimensions is used for the hurricane
movement examples discussed here.

74
 Referring to Tables 16 and 17, the minimum and maximum Q S values
observed in each run are placed under the 0.02 and 0.98 probability columns. The Q S
values shown in Tables 16 and 17 are those for the runs of length two which is the
second column of ‘the two-dimensional part of Table 15. The empirical bounds given in
Table 15 (for two dimensions) are exceeded in 4 cases out of the 18 examples shown.

The changes of the deviations from one quadrant to another around the centroid
seem to be quasi-random. The 11 random changes in a sample of 11 (Table 17) show
that in no case was a deviation in one quadrant followed by another in the same
quadrant. In a one-dimensional problem this would imply a highly oscillatory process.
This may be the case here, but it would require switching from one quadrant t o another
and back again on a rotating vector. This is possible but not very likely. The fact that it
occurred in only one of the six cases for a sample size of 11 implies that it does not
occur often. When 10 more sequential data were added, the centroid shifted and the
quadrant sign changes fell within bounds. This implies that it was a small sample
problem. Therefore, the rejection of the null hypothesis in this case is discounted. Insofar
as directional changes are concerned, the null hypothesis that the data set is not different
from a bivariate normal distribution is not rejected.

Figures 32 through 37 illustrate the hurricane data shown in Tables 16 and 17.
The 0.02 and 0.98 empirical confidence bands are shown in each figure. Each figure
contains the data for three quadrangles for sample size 1 1, 21, or 3 1. Each quadrangle is
represented by one symbol, an X, box, or asterisk. If 01 percent (ais the rejection level)
or more of the sequential symbols fall outside the 0.02 and 0.98 bounds, the null
hypothesis of bivariate normality is rejected. The decisions reached regarding the null
hypotheses in the discussion of the tables is graphically supported.

Another test for direction not illustrated here is based on the fact that the
multivariate normal distribution is spherical in its standardized and eigenvalue-eigenvector
form. All directions are equally possible and the distribution of the directions from the
centioid is the uniform or rectangular. In higher dimensions this would be hypercubical.
Andrews et al. (1973), as pointed out previously, discuss this in more detail in
Multivariate Analysis, Krishnaiah (1972). This test is a necessary but insufficient test
since it considers only the direction.

The appendix presents graphs for plotting the 0.02, 0.50, and 0.98 probability
levels given in Table 4. These graphs permit plotting of empirical chi-square (squared
radii) values and subsequent testing for multivariate normality for the dimensions and
sample sizes shown. The rejection level, a,is 0.04.

75

I
 I . I
0.9 0.4
0.0 . 8.8
0.7 . 0.7
0.6 . 0.6
0.5 . h.S
O.q - O.Y

0.3 - 0.3
. a x
0.2 - 0.2
* ax I)

-
-x
N
n
, 0.1
0.0E E X
-- 0.1
0.m
0.0E - 0.0a
0.0; - 0.01
0.0t - 0.06
0.0! - 0.05
0.01 - 0.0q

0.0: - 0.03

0.0 - 0.02

0.0 -1
0 I 2 3
I

5 6 7 E 9
-0 0.01

Figure 32. Chi-square (vector deviation square) values of September-October 24 hour

hurricane movements versus the empirical (1 -p(x2)) value of occurrence
(n=l 1). (The 0.02 and 0.98 probability lines provide the central 0.96
confidence band. The symbols represent data from one 5 degree
longitude-latitude-quadrangle.) (See Table 16.)
 -' 0.5
- O.q

- 0.3

0.z -'
W ' x a
- 0.2

.WX E-

--
I
N
x 0.1 - XE .
0.1
h 0.03- 0.09
I 0.00- - 0.08
,- 0.07-, - 0.07
0.06- - 0.06
0.0s- a x
- 0.05
0.0+ - 0.0q

0.03- - 0.03

0.02- - 0.02

0.01 1: I I I I ! 0.01
0 1 2 3 Lf S 6 7 E 9 10 II I2 13 1q IS I6 17 IE 19 20

Figure 33. Chi-square (vector deviation square) values of September-October 24 hour

hurricane movements versus the empirical (1 -p(x2)) value of occurrence
(n=2 1). (The 0.02 and 0.98 probability lines provide the central 0.96
confidence band. The symbols represent data from one 5 degree
longitude-latitude-quadrangle.) (See Table 16.)
 I
0.9 0.9
O.a 0.8
0.7 0.7
0.6 0.6
0.5 0.5
0.q O.q

0.3 I

0.2 -1
n. u
e.x a . - O.z

"x 0 . I -!
0.09- % '
-- 0.1
0.09
I
I 0.0a-1
0.07-!
0.06-1 * E X
-- 0.0a
0.01
0.05:
0 . as--' - 0.05
0 . BY-t - 0.0q
a x
0.03-1
n
- 0.03

0.02-1 '- 0.02

Figure 34. Chi-square (vector deviation square) values of September-October 24 hour

hurricane movements versus the empirical ( 1-p(x2)) value of occurrence
(n=3 1). (The 0.02 and 0.98 probability lines provide the central 0.96
confidence band., The symbols represent data from one 5 degree
longitude-latitude-quadrangle.) (See Table 16.)
 .
I -. -. I
.-.. 0.9
-. 8.8
-. 0.7
'-. 8.6
0.5 -' a 4 E X . '- 0.5
4 E X
O.q 3' '-. 0.q
I t . XE
0.3 -' '-. 8.3
-e E X '

0.2 -' I-. 0.2

X I

N
-
E
0.1 +
0.094.
0.084,
X E -
'-

'-
0.1
0.09
0.06
'
F 0.074 - 0.07
0.064 1- 0.06
0.0s- 1- 0.05
0 . 0q- P 0.0Lt

0.03- 0.03

0.02 0.0z

Figure 35. Chi-square (vector deviation square) values of September-October 24 hour

hurricane movements versus the empirical ( 1-p(x2)) value of occurrence
(n= 1 1). (The 0.02 and 0.98 probability lines provide the central 0.96
confidence band. The symbols represent data from one 5 degree
longitude-latitudequadrangle.) (See Table 17.)
00
0
1
0.9
0.0 0.0
0.7
0.6 - EX
.EX.
.. 0.6
0,s - *ax
ex
.
.
0.5
O.q -'
E. x O.q

0.3 -' 0.3

0.2 -'
XE . 0.2

X E.

"x
0 . 1 -'
0.03-
0.08-
X E '. 0. I
0.09
0.0R
I 0.07- 0.07
0.06-
0.0s-
0.0q-4
E X . 0.06
0.05
0.0q

. 0.03-1 0.03

0.02

_t_ -+ 0.01
1 IR 1 20

Figure 36. Chi-square (vector deviation square) values of September-October 24 hour

hurricane movements versus the empirical ( 1-p(x*)) value of occurrence
(n=2 1). (The 0.02 and 0.98 probability lines provide the central 0.96 ' .
confidence band. The symbols represent data from one 5 degree
longitude-latitude-quadrangle.) (See Table 17.)
. . .. _. ., . . . . . .. . / . .. ..,_ ~ ..... . _ . . . . . . . .
~ .-.. 1 - . ,

cL A.9
- ' 0.8
0.7

-- 0.6
O.s
0.q

--0 . 3
0.2 -I - 0.2

xa

--
N
-
E 0.1
;0.09-t -'
x . 0.1
0.09

--
c
0 . HB-! C 0.0B
0.07-' 0.07
0. 06-1 . x
0.06
0 . as4 !- 0.0s
0, a+' - 0.0q
E X .
0.03

i-

Figure 37. Chi-square (vector deviation square) values of September-October 24 hour

hurricane movements versus the empirical ( 1-p(xz )) value of occurrence
(n=3 1). (The 0.02 and 0.98 probability lines provide the central 0.96
confidence band. The symbols represent data from one 5 degree
longitude-latitude-quadrangle.) (See Table 17.)
I
 VI. SUMMARY

Work at the National Aeronatuics and Space Administration, Marshall Space

Flight Center at Huntsville, Alabama, and at the National Oceanic and Atmospheric
Administration, Environmental Data Service, National Climatic Center (for NASA and the
U.S. Navy) required some assessment of the distributions of multivariate or vector-form
data sets.

One of the basic statistical distributions is the multivariate normal. When only one
dimension or variate is considered, it usually is called the normal distribution. Adequate
and well-known tests for testing in the normal higher dimensional forms, such as the
bivariate and trivariate, are not readily available.

We realize that the studies in this report are incomplete, that substantiation by
others is required, and that extensions should be made. We also realize that the number
of samples used in the small sample studies should be increased so that more stable
configurations would be obtained. Also, a larger range of sample sizes should be
processed and the results made available.

The work discussed in this document was initiated approximately 6 years ago to
provide us with statistical tools to test multivariate distributions for normality. We are
reasonably confident that these tools will be useful to others.

George C. Marshall Space Flight Center

National Aeronatuics and Space Administration
Marshall Space Flight Center, Alabama 3581 2, December 1975

82
 . ... . .. . . ..

TABLE 1. FRACTILES (QUANTILES) OF THE x 2 DISTRIBUTION

?
V 0-05 0.1
_c

1 -05333 *05157
2 *0*100 -02200
3 -0153 -02h3
4 -0639 *?go8
5 -158 e210

6 -299 -381 -576 -872 1.24 1.64 2.20 3.07 3-83 4-57
7 .LE5 ,598 -989 1.24 1.69 2-17 2.83 3-32 4-67 5-49
8 -710 45; 1-34 1-65 2.18 2-73 3-49 4-59 5.53 6-42
9 -972 1-15 1-73 2-09 2-70 3.33 4.17 5-38 6-39 7-36
10 1.26 1-48 2-16 2.56 3.25 3-94 4.87 6.18 7.27 . 8-30

11 1-59 1.43 2-60 3-05 3-82 4-57 5.58 6-09 8.15 9-24
12 1-93 2-21 3.G7 3.57 4.40 5-23 6.30 7-61 9-03 10.2
13 2.31 2.62 3-57 4.11 5-51 5-39 7-04 8.63 9.93 11.1
14 2-70 3-04 L *07 4.66 5.63 6.57 7.79 9-47 10.8 12.1
15 3-11 :.La L .c0 5-23 E.26 7.26 8.55 1C.3 11.7 13.0

P r o b a b i l i t y i n P e r Cent P
50.0 90.0 95.0 97.5 99.9 99-95 V
~
- - _ -
-455 -708 1-07 1-64 2-71 3-84 5.02 10.8 12.1 1
1.39 1.83 2-41 3-22 4-61 5.99 7.38 13.8 15-2 2
2.37 2-95 3-67 4-64 6-25 7-81 9.35 16.3 17.7 3
3-36 4-04 4-88 5-99 7 -78 9-49 11.1 18.5 20.0 4
4-35 5-13 6-06 7-29 9-24 11.1 12.8 20.5 22 -1 5

5'35 6.21 7-23 8.56 10.6 12.6 14.4 16.8 18-5 22.5 24-1 6
6-35 7-28 8-38 9-80 12-0 14.1 16.0 18.5 20.3 24.3 26.0 7
7-34 8-35 9-52 11.0 13.4 15.5 17.5 20-1 22.0 26.1 27.9 8
8.34 9-41 10.7 12.2 14.7 16.9 19'0 21.7 23.6 27.9 29.7 9
9 *34 10.5 11.8 13.4 16.0 18.3 20.5 23.2 25'2 29.6 31.4 10

10.3 11.5 12.9 14.6 17-3 19.7 21.9 24-7 26.8 31-3 33.1 11
11-3 12-6 14.0 15.8 18.5 21.0 23'3 26.2 28.3 32.9 34 -8 12
12.3 13.6 15-1 17-0 19.8 22.4 24.7 27-7 29.8 34.5 36-5 13
13-3 14.7 16.2 18.2 21-1 23-7 26.1 29-1 31-3 36.1 38-1 14
14-3 15-7 17-3 19.3 22.3 25-0 27-5 30-6 32.8 37 - 7 39.7 15

Abridged w i t h permission o f :
1. P r o f e s s o r s A. H a l d and A. Sinkbaek and t h e i r p u b l i s h e r s , S k a n d i n a v i s k A k t u e r i e t i d s k r i f t ,
as shown i n S t a t i s t i c a l T a b l e s and Formulas p u b l i s h e d by John W i l e y and Sons, T a b l e V , 1952.
- - 2. P r o f e s s o r E. S. Pearson and t o t h e 8 i o m e t r i k a T r u s t e e s f r o m t h e x 2 t a b l e s ( T a b l e s 7 and 8 )
p r e s e n t e d i n B i o m e t r i k a T a b l e s f o r S t a t i s t i c i a n s , Vol. 1, 1956.

83
 ~

~
~
~
~

TABLE 2. MEDIANS O F SELECTED PERCENTILES O F 10 100 RANDOM CHI-SQUARE VALUES O F THE I

~

01, =
cp
~

MULTIVARIATE NORMAL DISTRIBUTION WITH ZERO MEANS AND COVARIANCES (NID (0, 1 , O ) )
AND THEIR PROBABILITY PLOTTING POSITIONS p = ( 1 - ( ( i - 0.5)/n))
FOR 3 DIMENSIONS

n= 7

Probability 0.929 0.786 0.643 0.500 0.357 0.214 0.071

i
Percentile 1 2 3 4 5 6 7
2 .1104 .6143 1.1836 1.6272 2.5878 3.3996 4.5556
5 .2314 .8786 1.5023 2.2215 3.0202 3.9535 5.2391
10 .3801 1.1350 1.8788 2.6418 3.4505 4.4632 5.8456
50 1.2748 2.3006 3.2372 4.1415 5.1009 6.3326 8.2563
90 2.6101 3.7817 4.7914 5.8407 7.0022 8.5381 11.3032
95 3.0306 4.2132 5.2928 6.3367 7.5058 9.1838 12.2894
98 3.4971 4.7349 5.7669 6.8765 8.1216 9.8958 13.1932

n = 31
Probability 0.984 0.952 0.920 0.887 0.855 0.823 0.790 0.758 0.723
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0400 .2060 .4084 .6117 .8164 1.0313 1.2246 1.4230 1.6059
5 .0804 .2998 .5425 .7638 .9904 1.1881 1.4086 1.5918 1.7865
10 .1393 .4076 .6674 .8954 1.1431 1.3551 1.5841 1.7881 1.9850
50 .4906 .8762 1.1845 1.4668 1.7387 1.9883 2.2096 2.4235 2.6533
90 1.0456 1.4679 1.8270 2.1052 2.3915 2.6463 2.9058 3.1442 3.3589
95 1.2089 1.6763 2.0241 2.3269 2.5962 2.8373 3.0916 3.3199 3.5734
98 1.4113 1.8432 2.2114 2.5249 2.7755 3.0436 3.2960 3.5350 3.7779
 TABLE 2. (Continued)

Probability 0.693 0.661 0.629 0.597 0.565 0.532 0.500 0.468 0.435
i
Percentile 10 11 12 13 14 15 16 17 18
2 1.7752 1.9798 2.1800 2.3679 2.5388 2.7254 2.9408 3.1409 3.3556
5 2.0002 2.1947 2.4024 2.5953 2.7739 2.9787 3.1818 3.3767 3.5971
10 2.1820 2.3733 2.5950 2.7805 2.9782 3.1795 3.3932 3.5873 3.8111
50 2.8638 3.0822 3.2674 3.4939 3.7092 3.9223 4.1422 4.3711 4.5959
90 3.5788 3.8086 4.0202 4.2600 4.486 3 4.6905 4.9174 5.1627 5.3962
95 3.7924 4.0209 4.2564 4.4795 4.6 964 4.9174 5.1465 5.4068 5.6106
98 4.0373 4.2118 4.4615 4.6654 4.9073 5.1546 5.4012 5.6740 5.8791

Probability 0.403 0.371 0.339 0.307 0.277 0.242 0.210 0.177 0.145
i
Percenae 19 20 21 22 23 24 25 26 27
2 3.540 1 3.7882 3.9046 4.1767 4.4399 4.6579 4.9150 5.2103 5.5207
5 3.7867 4.0084 4.2544 4.4649 4.7103 4.9607 5.2520 5.5545 5.8988
10 4.0307 4.2488 4.4640 4.7025 4.9754 5.2531 5.5533 5.8841 6.2251
50 4.8232 5.0660 5.3147 5.5923 5.8871 6.1932 6.5322 6.8931 7.3332
90 5.6616 5.9613 6.2350 6.5306 6.8679 7.2096 7.6148 8.0291 8.5487
95 5.8902 6.1862 6.5246 6.7933 7.1436 7.4972 7.9071 8.3522 8.8911
98 6.1518 6.4698 6.7777 7.0770 7.4097 7.8382 8.2286 8.7019 9.2488
 TABLE 2. (Concluded)

Probability 0.113 0.081 0.048 0.016

i
Percentile 28 29 30 31
2 5.9038 6.3420 6.9085 7.7307
5 6.2985 6.7704 7.3517 8.2260
10 6.6280 7.1068 7.7466 8.7292
50 7.8087 8.4499 9.2524 10.7251
90 9.1481 9.9266 11.0975 13.3151
95 9.5535 10.3608 11.6467 14.1185
98 9.9906 10.8254 12.1919 15.0132
 -
TABLE 3. VALUES O F "c" DERIVED FROM FORMULA c = 1 - (v/4) (4In n/n) WHERE
Y IS THE NUMBER OF DIMENSIONS, n IS THE NUMBER OF
DATA, AND In IS THE NATURAL LOGAHTHM

n 11 15 21 25 31 51 101
V

1 -0.12196 +O. 02785 +O. 17009 +O. 23498 +O. 30690 +O. 44162 +O. 56722

2 -0.31796 -0.22215 -0.07991 -0.01502 +O. 05690 +O. 19162 +O. 31722

3 -0.62196 -0.47215 -0.32991 -0.26502 -0.19310 -0.05838 +O. 06722

4 -0.87196 -0.72215 -0.57991 -0.51502 -0.44310 -0.30838 -0.18278

5 -1.12196 -0.97215 -0.82991 -0.76502 -0.69310 -0.55838 -0.43278

00
TABLE 4. PROBABILITY PLOTTING POSITIONS BASED ON EQUATION 1 - p = 1 - (i - c ) / ( n - 2c + 1)
00
WHERE c = 1 - (v/4) - (4/n) (In n). v IS DIMENSION AND i VARlES FROM 1TO n.

n= 5 n= 7
V
v.

1 2 3 4 5 1 2 3 4 5
1 0.783 0.764 0.748 0.733 0.720 4 0.844 0.825 0.809 0.793 0.780
2 0.641 0.632 0.624 0.617 0.610 2 0.729 0.717 0.706 0.696 0.686
3 0.615 0.608 0.603 0.598 0.593
i 3 0.500 0.500 0.500 0.500 0.500
4 0.359 0.368 0.376 0.383 0.390 i 4 0.500 0.500 0.500 0.500 0.500
5 0.217 0,236 0.252 0.267 0.280 5 0.385 0.392 0.397 0.402 0.407

6 0.271 0.283 0.294 .0.304 0.314

7 0.156 0.175 0.191 0.207 0.220

. n = 11
V

2 3 4 5
0.892 0.878 0.864 0.851
0.814 0.802 0.791 0.781
0.735 0.727 0.718 0.711
0.657 0.651 0.646 0..640
0.578 0.576 0.573 0.570

11 0.092 0.108 0.122 0.136 0.149

 TABLE 4. (Continued)

n = 15 n = 21
V
.*
1 2 3 4 5 1 2 3 4 5
1 0.939 0.926 0.913 0.901 0.890 1 0.962 0.951 0.941 0.932 0.923
2 0.876 0.865 0.854 0.844 0.834 2 0.916 0.906 0.897 0.889 0.880
3 0.814 0.804 0.795 0.787 0,779 3 0.869 0.861 0.853 0.845 0.838
4 0.751 0.743 0.736 0.729 0.723
5 0.688 0.682 0.677 0.672 0.667

11 0.312 0.318 0.323 0.328 0.333 i 11 0.500 0.500 C.500 0.500 0.500
12 0.249 0.257 0.264 0.271 0.277 12 0.454 0.455 0.456 0.457 0.458
13 0.186 0.196 0.205 0.213 0.221 13 0.408 0.410 0.412 0.414 0.415
14 0.124 0.135 0.146 0.456 0.166 14 0.361 0.365 0.368 0.370 0.373
15 0.061 0.074 0.087 0.099 0.110 15 0.315 0.319 0.323 0.327 0.337

16 0.269 0.274 0.279 0.284 0.289

17 0.223 0.229 0.235 0.241 0.246
18 0.177 0.184 0.191 0.198 0.204
19 0.131 0.139 0.147 0.155 0.162
20 0.084 0.094 0.133 0.111 0.120

21 0.038 0.049 0.059 0.068 0.077

W
TABLE 4. (Continued)
0

n = 25 n = 31
V V

1 2 3 4 5 1 2 3 4 5

1 0.970 0.961 0.952 0.944 0.936 I 0.978 0.970 0.963 0.956 0.949
0.923 0.915 0.907 0.900 2 0.946 0.939 0.932 3.926 0.919
2 0.931 7 0.914 0.908 0.901 0.895 0.889
3 0.892 0.884 0.877 0.870 0.863
0.839 0.833 0.827 4 0.882 0.876 0.571 0.865 0.859
4 0.853 0.846 0.834 0.829
5 0.813 0.807 0.802 0.796 0.791 5 0.850 0.845 0.840

0.764 0.759 0.754 6 0.819 0.814 0.809 0.804 0.800

6 0.774 0.769 0.774 0.770
7 0.735 0.731 0.726 0.722 0.718 7 0.737 0.782 0.773
8 0.696 0.692 0.688 0.685 0.682 8 0.755 0.751 0.747 '3.743 0.740
0.648 0.645 9 0.723 0.720 0.716 0.713 0.710
9 0.657 0.654 0.651
10 0.618 0.615 0.613 0.611 0.609 10 0.691 0.688 0.685 0.685 0;680

11 0.578 0.577 0.575 0.574 0.573 11

12 0.659
0.627 0.657
0.655 3.654
3.634 - -~
0.652
0.622 0.650
OI650
12 0.539 0.538 0.538 0.537 0.536
0.500 0.500 0.500 0.500 0.500 13 0.596 0.534 0.593 0.591 0.590
i 13 0.562 3.551 C.550
14 0.461 0.462 0.462 0.463 0.464 14 0.564 0.563
0.422 0.423 0.425 0.426 0.427 15 3.532 0.531 0.531 0.530 0.530
15

0.382 0.585 0.387 0.389 0.391 i 16 0.500 0.500 0.500 0.500 0.500
16 0.469 C.479 0.470
17 0.343 0.346 0.349 0.352 0.355 17 '0.468 0.469
18 0.304 0.308 0.312 0.315 0.378 18 0.435 0.437 0.438 0.439 0.440
0.265 0.269 0.274 0.278 0.282 I9 0.434 0.406 0.407 0.499 0.410
19 20 0.373 0.375 0.376 o m 0.380
20 0.226 0.231 0.236 0.241 0.246 I

21 0.187 0.193 0.198 0.204 0.209 21 3.341 0.343 0.346 0.348 0.350
22 0.147 0.154 0.151 0.167 0.173 22 0.309 0.312 0.325 0.316 0.310
0.108 0.116 0.123 0.130 0.137 23 0.277 9.280 0.284 0.287 0.293
23 0.253 0.257 0.260
24 0.069 0.077 0.085 0.C93 0.100 24 3.245 0.249
0.039 0.048 0.056 0.064 25 0.213 0.31.3 0.722 0.?26 0.230
25 0.030

26 0.181 0.186 0.191 0.196 0.203

27 0.150 0.155 0.150 0.166 Q.171
28 0.118 0.124 0.129 01135 01141
29 9.036 0.092 0.39 0.105 0.111
30 0.054 0.061 0.068 0.074 0.081

31 0.022 0.030 0.037 0.044 0.051

 TABLE 4. (Continued)

n = 35

26 0.273 0.277 0.280 0.283 0.286

27 0.245 0.249 0.252 0.256 0.259
i 28 0.217 0.221 0.225 0.228 0.232
29 0.188 0.193 0.197 0.201 0.205
30 0.160 0,165 0.170 0.174 0.178

i 11 0.698 0.695 0.693 0.690 0.688 31 0.132 0.137 0.142 0.147 0.152
12 0.670 0.668 0.665 0.663 0.661 32 0.104 0.109 0.114 0,120 0.125
13 0.642 0.640 0.638 0.636 0.634 33 0.075 0.081 0.087 0.093 0.098
14 0.613 0.612 0.610 0.609 0.607 34 0.047 0.053 0.059 0.065 0.071
15 0.585 0.584 0.583 0.581 0.580 35 0.019 0.025 0.032 0.038 0.044
 TABLE 4. (Continued)

n = 41
V V

1 2 3 4 5 1 2 3 4 5
1 0.985 0,979 0.974 0.968 0.963 26 0.379 0.380 0.382 0,383 0.384
2 0.961 0.955 0.950 0.945 0,940 27 0.354 0.356 0.358 0.360 0.361
3 0.937 0.931 0.926 0.921 3,916 28 0.330 0.332 0.334 0.336 0.338
4 0.912 0.907 0.903 0.898 0.893 29 0.306 0.308 0.311 0.313 0.315
5 0.888 0.883 0.879 0.874 0.870 30 0.282 0.284 0.287 0.289 0,292

6 0.864 0.860 0.855 0.851 0,847 31 0.257 0.260 0.263 0.266 0.269
7 0.840 0.836 0.83‘2 0,828 0.824 32 0.233 0.236 0.239 0.243 0,246
8 0.815 0.812 0.808 0.804 0.801 i 33 0.209 ’ 0.212 0.216 0.219 0.222
9 0.791 0.788 0.784 0.781 0.778 34 0.185 0.188 0.192 0.196 0.199
10 0.767 0.764 0.761 0.757 0.754 35 0.160 0.164 0.168 0.172 0.176

11 0.743 0.740 0.737 0.734 0.731 36 0.136 0.140 0.145 0.149 0.153
12 0.718 0.716 0.713 0.711 0.708 37 0.112 0.117 0.121 0.126 0.130
i 13 0.694 0.692 0.689 0.687 0.685 38 0.088 0.093 0.097 0.102 0.107
14 0.670 0.668 0,666 0.664 0.662 39 0.063 0.069 0.074 0.079 0.084
15 0.646 0.644 0.642 0.640 0.639 40 0.039 0.045 0.050 0,055 0.060
 TABLE 4. (Continued)

n = 45
V
V
1 2 3 4 5
1 2 3 4 5
0.976 0.971 0.966 26 0.434 0.434 0.435 0.436 0.436
1 0.987 0.982 27 0.411 0.412 0.413 0.414 0.415
2 0.965 0.960 0.955 0.950 0.945
0.928 009~4 28 0.389 0.391 0.392 0.393 0.394
3 0.943 0.938 0.933 0.373
0.911 0.907 0.903 29 0.367 0.369 0.370 0.371
4 ’ 0.921 0.916 30 0.345 0.347 0.348 0-530 0.352
5 0.898 0.894 0.890 0.886 OeW2

6 0.876 0.872 0.868 0.864 0.860 31

32 0.323
0.301 0.325
0.503 0.327
0 0x5 .). i9
,37 0.3%
0.309
7 0.854 0.850 0.846 0.843 0.839 ’ u

8 0.832 0.828 0.825 0.821 0.818 33 0.279 0.281 0.283

0.262 0.286
0.264 0.288
0;267
9 0.810 0.807 0.803 0.800 0.797 34 0.257 0.259
0.779 0.776 35 0.234 0.237 0.240 0.243 0.246
10 0.788 0.785 0.782

0.757 0.754 i 36 0.212 0.215 0.218 9.221 0.224

11 0,766 ,0.763 0.760
0.741 0.738 0.736 0.733 37 0.190 0.193 0.197 0.200 0,203
12 0.743 38 0.168 0.172 0.175 0.179 0.182
i 13 0.721 0.719 0.717 0.714 0.712
0.693 0.691 39 0.146 0.150 0.154 0.157 9.161
14 0.699 0.697 0.695 0.124 0.128 0.132 0.136 0.140
0.673 0.671 0.670 40
15 0.677 0.675

41 0.102 0.106 0.110 0.114 (3.118

16 0.655 0.653 0.652 0.650 42 0.079 0.084 0.089 0.093 0.097
17 0.633 0.631 0.630 0.629 43 0.057 0.062 0.067 0.072 0.076
18 0.611 0.609 0.608 0.607 0.606
19 0.589 0.588 0.587 0.586 0.585 44 0.035 0.040 0.045 0.050 0.055
20 0.566 0.566 0.565 0.564 0.564 45 0.013 0.018 0.024 0.029 0.034
 TABLE 4. (Continued)

_- n = 51
V

1 2 3 4 5 1 2 3 4 5
1 0.989 0.984 O.%O 0.975 0.971 31 0.402 0.403 0.404 0.405 0.406
2 0.970 0.965 0.961 0.956 0.952 32 0.383 0.334 0.385 0.386 0.387
3 0.950 0.946 0.941 0.937 0.933 33 0.363 0.364 0.366 0.367 0.368
4 0.930 0.926 0.922 0.918 0.914 34 0.343 0.345 0.346 0.348 0.349’
5 0.911 0.907 0.903 0.899 0.895 35 0.324 0.326 0.327 0.329 0.331

6 0.891 0.887 0.884 0.880 0.877 36 0.304 0.306 0.308 0.310 0.312
7 0.872 0.868 O.P.55 0.861 0.858 37 0.285 0.287 0.289 0.291 0.293
8 0.852 0.849 0.845 0.842 0.839 38 0.265 0.268 0.270 0.272 0.274
9 0.833 0.829 0.826 0.823 0.820 39 0.246 0.248 0..251 0.253 0.255
10 0.813 0.810 0.807 0.804 0.801 40 0.226 0.229 0.231 0.234 0.236

11 0.793 0.791 0.788 0.785 0.782 i 41 0.207 0.209 0.212 0.215 0.218
12 0.774 0.771 0.769 0.766 0.764 42 0.187 0.190 0.193 0.196 0.199
13 0.754 0.752 0.749 0.747 0.745 43 0.167 0.171 0.174 0.177 0.180
14 0.735 0.732 0.730 0.,728 0.726 44 0.148 0.151 0.155 0,158 0.161
15 0.715 0.713 0.711 0:709 0.707 45 0.128 0.132 .0.135 0,139 0.142

i 16 0.696 0.694 0.692 0.690 0.688 46 0.109 0.113 0.116 0,120 0.123
17 0.676 0.674 0.673 0.671 0.669 47 0.089 0.093 0.097 0,101 0.105
18 0.657 0.655 0.654 0.652 0.651 48 0.070 0.074 0.078 0.W2 0.086
19 0.637 0.636 0.634 0.633 0.632 49 0.050 0.054 0.059 0.063 0.067
20 0.617 0.616 0.615 0.614 0.613 50 0.030 0.035 0.039 0.044 0.048

21 0.598 0.597 0.596 0.595 0.594 51 0.011 0.016 0.020 0.025 0.029
22 0.578 0.577 0.577 0.576 0.575
23 0.559 0.558 0.558 0.557 0.556
24 0.539 0.539 0.538 0.538 0.538
25 0.520 0.519 0.519 0.519 0.519

26 .0.5OO 0.500 0.500 0.500 0.500

7‘2 0.480 0.481 0.481 0.481 0.481
28 0.461 0.461 0.462 0.462 0.462
29 0.441 0.442 0.442 0.443 0.444
30 0.422 0.423 0.423 0.424 0.425
 TABLE 4. (Concluded)

n = 101
V

1 L 3 4 5 1 2' 3 '4
I 0.996 0.993 0.991 0.588 0.986 56 0.450 0.451 0.451 0.451 0.451
2 0.986 0.983 0.981 0.979 0 -976 57 0.441 0.441 0.441 0.441 0.442
3 0.976 0.974 0.971 0.969 0.967 58 0.431 0.431 0.431 0.432 0.432
4 0.966 0 -964 0.961 0.959 0.957 59 0.421 0.421 0.421 0.422 0.422
5 0.956 0.954 0.952 0.949 0.947 60 0.41 I 0.41 I 0.412 0.412 0.413

6
7
8
0.946
0.936
'0.926
0.944
0.934
0.924
0.942
0.932
0.922
0.940
0.930
0.920
0.937
0.928
0 -918
61
62
63
:a
0.3881
0.401
0.391
0.382
0.402
0.392
0.382
0.402
0.393
0.383
0.403
0.593
0.383
' 9 0.916 0.914 0.912 0.910 0.908 64 0.371 0.372 0.372 0.373 0.374
10 0.906 0.904 0.902 0.901 03 9 9 65 0.361 0.382 0.363 0.363 0;34

11 0.897 0 .e95 0.893 0.891 0.889 66 0.351 0.352 0.353 0.353 0.354
12 0 .se7 0 .885 0.883 0.881 0A79 67 0.341 0.342 0.343 0.344 0.344
13 0.877 0 .e75 0.873 0.871 0.869 68 0.331 0.332 0.333 0.334 0.335
14 0.867 0.865 0.863 0.061 0.060 69 0.322 0.322 0.323 .0.324 0.325
15 0.857 0.855 0.853 0.852 0.850 70 0.312 0.313 0.313 0.314 0.315

16 0.847 0.845 0.844 0.842 0.840 71 0.302 0.303 0.304 0.305 0.306
17 0.837 0.835 0.834 0.832 0.831 72 0.292 0.293 0.294
. _ . 0.295
. ~ - -0.296
18 0.827 0.826 0.824 0.822 0.821 73 0.282 0.283 0.284 0.285 0 ;286
19 0317 0 .816 0.814 0.813 0.81 1 74 0.272 0.273 0.274 0.275 0.276
20 0307 a.806 0.804 0.803 0.801 75 0.262 0.263 0.264 0.266 0.267

21 0.797 0.796 0.795 0.793 0.792 76 0.252 0.253 0.255 0.256 0.257
22 0.788 0.786 0.785 0.783 0.782 77 0 ; 242 0.244 0.245 0.246 0.247
23 0.778 0.776 0.775 0.774 0.772 78 0.232 0.234 0.235 0.236 0.238
24 0.768 0.766 0.765 0.764 0.762 i 79 0.222 0.224 0.225 0.226 0.228
25 0.758 0.756 0.75 5 0.754 0.753 80 0.212 0.214 0.215 0.217 0.218

26 0.748 0.747 0.745 0.744 0.743 81 0.203 0.204 0.205 0.207 0.208
27 0.738 0.737 0.736 0.734 0.733 82 0.193 0.194 0.196 0.197 0.199
I 28 0.728 0.727 0.726 0.725 0.724 83 0.183 0.184 0.186 0.187 0.189
29 0.718 0.717 0.716 0.715 0.714 84 0 ; 173 0. i 7 4 0.176 0.178 0.179
30 0.708 0.707 0.706 0.705 0.704 85 0.163 0.165 0.166 0.168 0.169

31 0.698 0.697 0.696 0.695 0.694 86 0.153 0.155 0.156 0.158 0.160
32 0.688 0.687 0.687 0.686 0.6'35 87
~. 0.143 0.145 0.147 0.148 0.150
33 0.678 0.678 0.677 0.676 0.675 88 0.133 0.135 0.137 0.139 0.140
34 0.669 0.668 0.667 0.666 0.665 89 0.123 0.125 0.127 0.129 0.131
35 0.659 0.658 0.657 0.656 :).55G 90 0.113 0.115 0.117 0.119 0;111

36 0.649 0.648 0.647 0.647 0.646 91 0.103 0.105 0.107 0.109 0.111
37 0.639 0.638 0.637 0.637 0.636 92 0.094 0.096 0.098 0.099 0.101
38 0.629 0.628 0.628 0.627 0.626 93 0.084 0.&6 0.088 0.090 0.092
59 0.619 0.618 0.618 0.617 0.617 94 0.074 0.076 0.078 0.080 0.082
40 0.609 0.609 0.608 0.607 0.607 95 0.064 0.066 0.068 0.070 0.072

41 0.599 0.599 0.598 0.598 0.597 96 0.054 0.056 0.058 -0.060 0.063
42 0.589 0.589 0.588 0.588 0.587 97 0.044 0.046 0.048 0.051 0.053
43 0.579 0.579 0.579 0.578 0.578 98 0.034 0.036 0.039 0.041 0.043
44 0.569 0.569 0.569 0.568 0.568 99 0.024 0.026 0.029 0.031 0.033
45 0.559 0.559 0.559 0.559 0.558 100 0.014 0.017 0.019 0.021 0.024

46 0.550 0.549 0.549 0.549 0.549 101 0.004 0.007 0.009 0.012 0.014
47 0.540 0.539 0.539 0.539 0.539
48 0.530 0.530 0.529 0.529 0.529
49 0.520 0.520 0.520 0.520 0.519
50 0.510 0.510 0.510 0.510 0.510

51 0.500 0.500 0.500 0.500 0.500

52 0.490 0.490 0.490 0.490 0.490
53 0.480 0.480 0.480 0.480 0.481
54 0.470 0.470 0.471 0.471 0.471
55 0.460 0.461 0.461 0.461 0.461

95
W
Q,
TABLE 5. MEDIAN VALUES OF' SELECTED PERCENTILE VALUES O F 10 100 RANDOM CHI-SQUARE
VALUES O F THE MULTIVARIATE NORMAL DISTIUBUTION AND THEIR PROBABILITY
PLOTTING POSITIONS FOR DIMENSION ( v) WITH A SAMPLE SIZE (n) .
I/= 1 n= 5

Probability 0.783 0.641 0.500 0.359 0.217

i
Percentile 1 2 3 4 5

2 . 0000 .0043 .0502 . 4361 1.2120

5
10
50
...
0002
0007
0286
.0119
.0265
. 1859
.0959
.1476
.4523
..5334
6548
1.0 886
1.3152
1.4324
2.0683
90
95
.1835 . 5251 .8340 1.5487 2.7982
.2493 .6084 .9188 1.6601 2.9343
98 .3187 . 6941 . 9750 1.7762 3.0365

v= 1 n= 7
Probability 0.844 0.729 0.615 0.500 0.385 0.271 0.156
i
Percentile 1 2 3 4 5 6 7
2 .oooo
.. .0019 .0147
. .0679 .2094 ..6400 1.4942
5
10
50 .
0001
0004
0142
.0049
.0112
.0920
0337
.0614
. 2416
.1189
.
1766
.
4956
..
.3088
4070
8632
. 8047
9490
1.4630
1.6465
1.8043
2.5553
90
95
-1105
..
1603
.
.
3079
3819
.5362
. 6160
..
8597
9573
1.2671
1.3611
2.0361
2.1983
3.7323
3.9965
98 2189 . 4659 .6990 1.0392 1.4508 2.3691 4.3132
 TABLE 5. (Continued)

v= 1 n = 11
Probability 0.908 0.827 0.745 0.663 0.582 0.500 0.418 0.337 0.255
i
Percentile 1 2 3 4 5 6 7 8 9
. .
..
2 0000 0006 .0050 .ON7 .0465 .0999
5 0000 .0020 1938 .3415 .5891

10
50
.0001
.0059
.0043
.0354
-0107
.0192
.0311
.0500
.0759
.lo94
-1487
.2013
2669
.3509
.4680
.5731
.7625
.go97
.0915 .1782 .3043 .4712
90 .0504 . 1412 .2580 .4005 .5822 .7846
-6926
1.0502
.9915 1.4062

95 .0789 .1848
1.4002 1.9017

98 .1138 .2386
e3167
.3856
.4697
.5471
.6564
.7427
.8666
.9528
1.1462
1.2438
1.4980
1.6094
2.0347
2.1632

Probability 0.173 0.092

i
Percentile 10 11
2 1.1157 1.9410
5 1.3060 2.1274
10 1.4714 2.3541
50 2.0687 3.3666
90 2.8175 4.9138
95 3.0372 5.4303
98 3.2875 5.9654
 TA BLE 5. ( Continued)

v= 1 n = 15
Probability 0.939 0.876 0.814 0.751 0.688 0.625 0.563 0.500 0.437
i
Percentile 1 2 3 4 5 6 7 8 9
2 .oooo .0003 .0026 .0094 .0235 .0450 .0803 .1333 ,2067 ~

I
5 .oooo .OOll .0055 .0158 .0369 .0663 .1136 .1848 .‘2727 I
10 .OOOl .0023 .0094 .0251 .0526 .0929 .1526 .2360 .3435 1
-
I
-
=

50 .0032 .0191 .0500 .0957 .1583 .2391 .3386 .4667 .6280 ~

90 .0296 .0810 .1505 .2335 .3377 .4498 .5884 .7529 .9384

95 .0447 .lo95 .1914 .2879 .3985 .5133 .6596 .8325 1.0239
98 .0652 .1435 .2363 .3466 .4549 .5805 .7336 .9089 1.1036

Probability 0.375 0.312 0.249 0.186 0.124 0.061

i
Percentile 10 11 12 13 14 15
2 .3116 .4397 .6983 1.Q064 1.4983 2.2787
5 .4057 .5850 .8383 1.1729 1.7077 2.4859
10 .4877 .6898 .9652 1.3193 1.8732 2.7247
50 .8225 1.0737 1.3987 1.8353 2.5152 3.8443
90 1.1678 1.4534 1.8346 2.3919 3.3594 5.6407
95 1.2581 1.5503 1.9609 2.5647 3.6269 6.2559
98 1.3453 1.6486 2.1069 2.7375 3.9581 6.9673
 TABLE 5. (Continued)

v= 1 n = 21

Probability 0.962 0.916 0.869 0.823 0.777 0.731 0.685 0.639 0.592

i
Percentile 1 2 3 4 5 6 7 8 9
2 .0000 .0001 .0011 .0037 .0106
5 .0000 .0004 .0025 .0078 .0172
.0194
.0303
.0335
.0487
.0516
.0743
.0810
.1130
10 .oooo .OOlO .0047 .0124 .0250 .0433 .0676 .0992 .1413
50 .0017 .0099 .0256 .0478 .0773 .1158 .1633 .2229 .2931
90 .0161 .0451 .Of322 .1258 ,1804 -2429 -3153 ,3967 .4898
95 .0245 .0611 .lo60 .1545 .2198 .2877 .3597 .4466 .5447
98 .0382 .0833 .1331 .1889 .2609 .3313 .4138 .SO88 -6147

Probability 0.546 0.500 0.454 0.408 0.361 0.315 0.269 0.223 0,177

i
Percentde 10 11 12 13 14 15 16 17 18
2 .1164 .1573 .2268 .2845 .3883 .5230 .6526 .8385 1,1080
5 .1551 .2092 ,2780 .3649 .4759 .6108 .7626 .9913 1.2540
10 .1937 .2544 .3355 .4319 .5497 .6861 .8689 1.1019 1.3884
50 .3695 .4660 .5749 .7035 .8534 1.0287 1.2529 1.5225 1.8659
90 .5936 .7085 .8437 .9919 1.1585 1.3679 1.6226 1.9323 2.3419
95 .6546 ,7739 ,9124 1.0674 1.2410 1.4549 1.7124 2.0552 2.5087
98 .7203 .8435 .9901 1.1480 1.3227 1.5512 1.8017 2.1867 2.6769
CI TABLE 5. (Continued)
0
0
Probability 0.131 0.084 0.038
i
Percentile 19 20 21
2 1.4201 1.9394 2.6386
5 1.6224 2.1334 2.8924
10 1.7792 2.3002 3.1324
50 2.3295 3.0259 4.4030
90 2.9655 4.0309 6.4939
95 3.1805 4.3283 7.2760
98 3.3760 4.6725 8.0819

v= 1 n = 31

Probability 0.978 0.973 0.914 0.882 0.850 0.819 0.787 0.755 0.723
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0000 .0001 .0005 .0018 .0041 .0080 .0130 .0221 .0320
5 .oooo .0002 .0011 .0031 .0070 .0125 .0206 .0319 .0455
10 .0000 .0004 .0020 ' .0053 .0109 .0184 .0296 .0421 .0590
50 .0007 .0046 .0120 .0221 .0362 .0535 .0745 .lo01 .1288
90 .0072 .0210 ,0395 .0611 .0881 .1183 .1536 .1896 .2324
95 .0121 .0298 .0515 .0784 .1108 .1447 .1770 .2206 .2651
98 .0180 .0402 .0694 .0969 .1319 .1662 .2038 .2531 .3020
 TABLE 5. (Continued)

Probability 0.691 0.659 0.627 0.596 0.564 0.532 0.500 0.468 0.436
i
Percentile 10 11 12 13 14 15 16 17 18
2 .0449 .0622 .OB16 .lo47 .1297 .1647 .1984 .2451 .3029
5 .0621 .OB21 .lo59 .1333 .1667 .2010 .2495 .2959 .3537
10 .0786 .lo37 .1297 .1623 .1987 .2399 .2866 .3418 .4059
50 .1605 .1970 .2397 .2894 .3418 .3984 .4663 .5344 .6187
90 .2780 .3291 .3829 .4467 .5121 .5845 .6606 .7433 .8403
95
98
.3121
.3500
.3691
.4145
.4271
.4768
.4918
.5401
.5622
.6081
.6351
.6834
.7110
.7618
.BO33
.a573
.9073
.9655

Probability 0.404 0.373 0.341 0.309 0.277 0.245 0.213 0.181 0.150
i
Percentile 19 20 21 22 23 24 25 26 27
2 .3446 .4145 .4897 .5919 .7041 .8238 .9829 1.1394 1.3715
5 .4109 .4906 .5743 .6844 .7974 .9254 1.0970 1.2823 1.5359
10 .4729 .5595 .6569 .7600 .8788 1,0386 1.1950 1.3994 1.6519
50 .7061 .BO60 .9215 1.0502 1.1941 1.3688 1.5648 1.7898 2.0807
90 .9432 1.0582 1.1864 1.3357 1.5023 1.6904 1.9087 2.1968 2.5458
95 1.0063 1.1245 1.2549 1.4090 1.5830 1.7708 2.0131 2.3093 2.6969
98 1.0786 1.1908 1.3270 1.4849 1.6668 1.8657 2.1164 2.4392 2.8575

I
 TABLE 5. (Continued)

Probability 0.118 0.086 0.054 0.023

i
Percentile 28 29 30 31
2 1.6588 1.9837 2.4391 3.0930
5 1.7989 2.1625 2.6486 3.3991
10 1.9413 2.3211 2.8361 3.6938
50 2.4410 2.9300 3.6650 5.0676
90 3.0146 3.6869 4.8173 7.3689
95 3.1917 3.9109 5.2021 8.2211
98 3.3909 4.1388 5.6107 9.2171

v= 1 n = 51

Probability 0.989 0.970 0.950 0.930 0.911 0.891 0.872 0,852 0.833
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0000 .oooo .0002 .0006 .0014 .0027 .0044 .0070 .0108
5 .oooo .0001 .0004 .OO11 .0025 .0042 .0069 .0101 .0149
10 .oooo .0002 .0007 .0019 .0037 .0063 .0097 ,0142 .0198
50 .0003 .0017 .0043 .0082 .0133 .0195 .0272 .0361 .0461
90 .0028 .OO8O .0156 .0244 .0346. .0463 .0603 .0736 .0905
95
98
.0049
.0075
.0115
.0168
.0208
.0277
.0312 .0427 .0577 .0729 .0885 .lo71
.0387 .0539 .0702 .0864 .lo73 .1240
 TABLE 5. ( C ontinued)

Probability 0.813 0.793 0.774 0.754 0.735 0.715 0.696 0.676 0.657
i
Percentile 10 11 12 13 14 15 16 17 18
2 .0143 .0192 .0297 .0309 .0391 .0477 .0575 .0700 .0826
5 .0199 .0265 .0339 .0419 .0506 .0613 .072l, .0881 .1031
10 .0261 .0336 .0422 .0518 .0621 .0743 .0881 .1044 .1215
50 .0576 .0707 .0847 .lo06 .1178 .1364 .1559 .1788 ,2010
90 .lo84 .1273 .1488 .1714 .1945 ,2204 .2485 .2734 ,3042
95 .1267 .1466 .1707 .1945 .2184 .2457 .2764 .3034 .3356
98 .1476 .1695 .1947 .2215 .2480 .2761 .3070 .3370 .3690

Probability 0.637 0.617 0.598 0.578 0.559 0.539 0.520 0.500 0.480
i
Percentile 19 20 21 22 23 24 25 26 27
2 .0963 .1123 .1308 .1502 .1694 .1945 .2186 .2439 ,2745
I
5 .1200 .1368 .1585 .1813 .2003 .2268 -2546 .2811 ,3145
10 .1412 .1603 .1804 .2037 .2280 .2578 .2880 .3188 .3542
50 .2268 .2528 .2819 .3128 .3459 .3841 .4207 .4605 ,5027
90 .3393 .3698 .4070 .4415 .4826 .5297 .5703 -6189 ,6663
95 .3666 .4047 .4415 .4799 .5206 .5670 .6126 .6625 .7144
98 .4037 .4408 .4731 .5145 .5604 .6068 .6571 .7068 .7560

w
0
W
 TABLE 5. (Continued)

Probability 0.461 0.441 0.422 0.402 0.383 0.363 0.343 0.324 0.304
i
Percentile 28 29. 30 31 32 33 34 35 36
2 .3084 .3449 .3817 .4225 .4613 .5106 .5634 .ti175 .6832
5 .3515 .3891 .4299 .4751 .5303 .5753 .6372 .6943 .7096
10 .3940 .4352 .4817 .5278 .5824 -6345 .6973 .7642 .8291
50 .5532 .6045 .6556 .7084 .7707 .8417 .9093 .9848 1.0672
90 .7187 .7791 .8387 .9015 .9650 1.0412 1.1169 1.2058 1.2971
95 .7669 .8253 .8844 .9471 1.0201 1.0917 1.1817 1.2663 1.3588
98 .8127 .8709 .9362 1.0098 1.0757 1.1583 1.2418 1.3257 1.4838

Probability 0.285 0.265 0.246 0.226 0.207 0.187 0.167 0.148 0.128
i
Percentile 37 38 39 40 41 42 43 44 45
2 .7541 .8304 .9195 1.0122 1.1189 1.2417 1.3584 1.5192 1.6709
5 .8387 .9166 1.0063 1.1056 1.2203 1.3376 1.4733 1.6212 1.7987
10 .9149 .9931 1.0852 1.1908 1.3050 1.4252 1.5597 1.7240 1.8987
50 1.1509 1.2461 1.3512 1.4669 1.5911 1.7380 1.8934 2.0742 2.2811
90 1.3931 1.4922 1.6154 1.7397 1.8839 2.0542 2.2360 2.4586 2.7036
95 1.4560 1.5641 1.6841 1.8092 1.9627 2.1404 2.3472 2.5529 2.8341
98 1.5242 1.6272 1.7476 1.8982 2.0555 2.2320 2.4390 2.6689 2.9538
 TABLE 5. (Continued)

Probability 0.109 0.089 0.070 0.050 0.030 0.011

i
Percentile 46 47 48 49 50 51
2 1.8718 2.0926 2.3477 2.6680 3.0925 3.7021
5 1.9995 2.2380 2.5165 2.8513 3.3254 4.0509
10 2.1178 2.3555 2.6501 3.0246 3.5433 4.3488
50 2.5357 2.8388 3.1968 3.7205 4.4853 5.9592
90 3.0023 3.3976 3.8860 4.6137 5.8202 8.5327
95 3.1783 3.5691 4.1079 4.9267 6.2516 9.4304
98 3.3202 3.7726 4.3521 5.2872 6.7889 10.5413

v= 1 n = 101

Probability 0.996 0.976 0.956 0.936 0.916 0.897 0.877 0.857 0.837
i
Percentile 1 3 5 7' 9 11 13 15 17
2 .oooo 0000 .0003 .0010 .0024 .0045 .0074 .0109 .0158
5 .0000 .0001 .DO06 .0017 .0035 .0061 .0096 .0143 .0197
10
50
..0001
0000 .0002
.0011
.0009
.0033
.0024
.0068
.0048
.0113
.0080
.0177
.0122
.0249
.0177
.0335
.0234
.0430
90 .0008 .0042 .0090 .0158 .0235 .0338 .0448 .0580 .0719
95 .0013 .0056 .0117 .0192 ,0291 .0403 .0524 .0670 .0813
98 .0020 .0073 .0149 .0236 -0342 .0474 .0607 .OW2 .0930
 TABLE 5. (Continued)

Probability 0.817 0.797 0.778 0.758 0.738 0.718 0.698 0.678 0.659
i
Percentile 19 21 23 25 27 29 31 33 35
3 .0213 .0282 .0347 .0425 .0539 .0659 .0793 0918 .1069
.
Y

5 .0262 .0336 0429 .0523 .0648 .0776 .0908 .lo78 .1236

10 .0314 .0403 .0506 .0611 .0747 .0883 .lo45 .1202 .1385
50 .0544 .0673 .0819 .0974 .1140 .1327 .1538 .1742 .2001
90 .0882 .lo50 .1230 .1437 .1664 .1876 .2122 .2388 .2682
95 .0988 .1163 .1374 .1590 .1813 .2064 .2320 .2593 .2874
98 .1106 .1319 .1522 .1747 .1986 .2243 .2512 .2791 .3081

Probability 0.639 0.619 0.599 0.579 0,559 0.540 0.520 0.500 0.480
i
Percentile 37 39 41 43 45 47 49 51 53
2
5
.1251
.1425
.1440
.1645
..
1652
1863
.1877
.2113
.2135
.2371
.2387
.2670
.2677
.2951
.2980
-3280
3328
.3644
10
50
.1585
.2229
.1809
.2499
.2053
.2800
.
.
2315
3122
.2586
.3445
.2894
.3806
.3214
. 4181
.
.
3564
4572
.3950
.5043
90
95
.2979
,3205
. 3298
.3531
.3626
.3875
.3990
.4265
.4389
.4676
. 4798
.5087
.5227
5557
.5685
6022
,6178
.6496
98 .3435 .3778 .4159 .4499 ,4943 .5397 . 5827 .6311 .6880
 TABLE 5. (Continued)

Probability 0.460 0.441 0.441 0.421 0.401 0.381 0.361 0.341 0.322
i
Percentile 55 57 59 61 63 65 67 . 69 ‘71
2 .3699 .4048 .4471 .4932 .5434 .6045 .6605 .7293 .7995
5
10
.4015
.4344
.4434
.4776
,4888
.5232
.5358
.5772
.5883
.6307
-6466
.6894
.708-1
.7548
.7814
..
8283
.8555
.9004
50
90
.5502
.6716
.5983
.7248
.6526
.7865
.7102
.8495
.7725
.9127
.8389
.9862
.9113
1.0636
9865
1.1478
1.0694
1.2346
.95 .7048 .7611 .8253 .8855 .9551 1.0220 1.1047 1.1901 1.2813
98 .7417 .7954 ,8589 ,9288 9933 1.0622 1.1452 1.2396 1.3299

Probability 0.282 0.262 0.242 0.222 0.203 0.183 0.163 0.143 0.123
i
Percentile 73 75 77 79 81 83 85 87 89
2 .8721 .9585 1.0509 1.1506 1.2743 1.3869 1.5429 1.7158 1.8913
5 .9351 1.0216 1.1155 1.2292 1.3368 1.4751 1.6165 1.7932 1.9779
10 .9871 1.0769 1.1724 1.2879 1.4038 1.5324 1.6857 1.8641 2.0646
50 1.1583 1.2617 1.3658 1.4888 1.6195 1.7652 1.9341 2.1281 2.3524
90 1.3337 1.4391 1.5589 1.6867 1.8367 2.0067 2.1891 2.4082 2.6729
95 1.3792 1.4882 1.612? 1.7457 1.8974 2.0667 2.2618 2.4846 2.7531
98 1.4323 1.5414 1.6689 1.8026 1.9636 2.1297 2.3525 2.5694 2.8554
 TABLE 5. (Continued)

Probability 0.103 0.084 0.064 0.044 0.024 0.004

i
..
Percentile 91 93 95 97 99 10 1 <.

2 .2.1110 2.3580 2.6836 3.0882 3.6098 4.5764

5 2.2030 2.4810 2.8013 3.2335 3.8040 4.9862
10 2.2992 2.5721 2.9163 3.3810 4.0134 5.3912
50 2.6154 2.9482 3.3640 3.9351 4.8586 7.1110
90 2.9745 3.3713 3.8787 4.6598 5.9743 9.9689
95 3.0735 3.4970 4.0367 4.8723 6.3168 10.9848
98 3.1866 3.6247 4.2129 5.1104 6.7353 12.1836

v= 2 n= 5

Probability 0.764 9.632 0.500 0.368 0.236

i
Percentile 1 2 3 4 5
2 .0108 .1746 .9105 1.5599 2.0830
5 .0308 .2949 1.0438 1.6852 2.2118
10 .0657 .4245 1.1816 1.8232 2.3379
50 .3682 .9140 1.5915 2.2699 2.8266
90 .8447 1.3586 1.9979 2.7655 3.1309
95 .9632 1.4716 2.1209 2.8817 3.1636
98 1.0548 1.5981 2.2239 2.9733 3.1833
 TABLE 5. (Continued)

v= 2 L1= 7

Probability 0.825 0.717 0.608 0.500 0.392 0.283 0.175

i
Percentile 1 2 3 4 5 6 7
2 .0065 .0795 :2788 .6176 1.2532 1.9390 2.5208
5 .0170 .1467 .3954 .7969 1.4028 2.0794 2.6 990
10 .0375 .2142 .5238 .9744 1.5637 2.2262 2.8567
50 .2318 .5941 1.0148 1.4792 2.0683 2.7646 3.6641
90 .6227 1.0522 1.4531 1.9325 2.5759 3.4944 4.4924
95 .7606 1.1610 1.5687 2.0622 2. 70811 3.7383 4.6802
98 .9002 1.2768 1.6695 2.1784 2.8604 3.9878 4.8180

v= 2 n = 11

Probability 0.892 0.814 0.735 0.657 0.578 0.500 0.422 0.343 0.265
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0038 .0434 .1357 .2748 .4453 .6673 .9598 1.3795 1.8699
5 .0104 .0776 .1954 .3585 .5745 .8449 1.1444 1.5656 2.0851
10 .0216 .1179 .2663 .4470 .6834 .9798 1.2912 1.7224 2.2286
50 .1334 .3472 .5767 .8273 1.1212 1.4374 1.8181 2.2550 2.8040
90 .4070 .6956 .9728 1.2476 1.5502 1.8843 2.2637 2.7632 3.4219
95 .4990 .7861 1.0678 1.3666 1.6528 1.9988 2.3880 2.9057 3.6509
98 .5964 .8957 1.1878 1.4686 1.7761 2.1073 2.4964 3.0618 3.8671
w TABLE 5. (Continued)
w
0
Probability 0.186 0.108
i
Percentile 10 11
2 2.5097 3.1223
5 2.6841 3.3818
10 2.8576 3.6299
50 3.5564 4.7520
90 4.5189 6.1970
95 4.8943 6.6230
98 5.2501 7.0900

v= 2 n = 15

Probability 0.926 0.865 0.804 0.743 0.682 0.622 0.561 0.500 0.439
1

Percentile 1 2 3 4 5 6 7 8 9
2 .0020 .0320 .0901 .1741 .2718 .4160 .5562 .7443 .9789
5 .0070 .0530 .1290 .2332 .3561 .5065 .6826 .8793 1.1231
10 .0136 .0818 .1737 .2957 .4290 .6093 .7804 1.0044 1.2609
50 .0986 .2494 .4044 .5754 .7639 .9581 1.1812 1.4248 1.6956
90 .2984 .5115 .7113 .9110 1.1191 1.3333 1.5762 1.8337 2.1228
95 .3647 .5963 .8070 1.0175 1.2087 1.4390 1.6754 1.9383 2.2248
98 .4439 .6791 .8974 1.1050 1.3154 1.5355 1.7826 2.0345 2.3450
 TABLE 5. (Continued)

Probability 0.378 0.318 0.257 0.196 0.135 0.074

i
Percentile 10 11 12 13 14 15
2 1.1977 1.5395 1.9199 2.4024 2.9527 3.5661
5 1.3996 1.7192 2.1110 2.5706 3.1646 3.8758
10 1.5371 1.8723 2.2667 2.7484 3.3512 4.1499
50 2.0093 2.3721 2.8051 3.3621 4.1444 5.4021
90 2.4610 2.8686 3.3660 4.1012 5.2263 7.3268
95 2.5738 3.0049 3.5250 4.3450 5.6033 7.9399
98 2.7094 3.1545 3.7297 4.5676 6.0180 8.4897

v= 2 n = 21

Probability 0.951 0.906 0.861 0.816 0.771 0.726 0.681 0.635 0.590

\ i
Percentile 1 2 3 4 5 6 7 8 9
2 .0019 .0213 .0616 .1123 .1843 .2543 .3478 .4476 .5556
5 .0047 .0338 .0878 .1554 .2309 .3178 .4140 .5276 .6547
10 .0098 .0530 .1167 .1964 .2802 .3795 .4803 .6181 .7431
50 .0683 .1712 .2790 .3946 .5091 .6457 .7744 .9180 1.0799
90 .2165 .3620 .5117 .6549 .7933 -9387 1.0807 1.2388 1.4033
95 .2635 .4288 .5829 .7248 .8585 1.0100 1.1706 1.3304 1.4931
98 .3285 .5026 .6648 .8128 .9529 1.1040 1.2439 1.4068 1.5809
 TABLE 5. (Continued)

Probability 0.545 0.500 0.455 0.410 0.365 0.319 0.274 0.229 0.184

Percentile 10 11 12 13
i
14 15 16 17 18
'=
2 .6865 .8204 .9642 1.1393 1.3559 1.5815 1.8562 2.1404 2: 5244
5 .7846 .9450 1.0972 1.2831 1.4991 1.7431 2.0147 ,2.3284 2.7136
10 .8895 1.0481 1.2059 1.4137 1.6280 1.8742 2.1534 2.4769 2.8568
50 1.2384 1.4188 1.6073 1.8242 2.0593 2.3265 2.6372 2.9753 3.4200
90 1.5829 1.7794 1.9967 2.2119 2.4677 2.7564 3.1326 3.5480 4.1041
95 1.6709 1.8764 2.0948 2.3201 2.5807 2.8773 3.2527 3.7167 4.3166
98 1.7734 1.9724 2.1936 2.4309 2.6905 3.0152 3.3822 3.8829 4.5287

Probability 0.139 0.094 0.049

i
Percentile 19 20 21
2 2.9358 3.4594 4.1453
5 3.1288 3.6668 4.4266
10 3.3079 3.8736 4.7382
50 4.0046 4.7925 6.1702
90 4.8517 6.0625 8.3238
95 5.1274 6.4608 9.1264
98 5.4153 7.0227 10.0008
 TABLE 5. (Continued)

v=2 n=31

Probability 0.970 0.939 0.908 0.876 0.845 0.814 0.784 0.751 0.720
i
Percentile 1 2 3 4 5 6 7 8 9
2 .OOll .0135 .0374 .0735 .I109 .1562 .2122 .2616 ..3165
5
10
.0032
.0069
.0222
.0341
.0591
.0775
.(I976
.1265
.1439
.1811
.1980
.2382
,2529 .'3174 .3858
.3044 .3675 .4393
50 .0457 .1132 .1%47 .2548 .3312 .4075 .4923 .5768 .6670
90 ,1470 .2485 .3388 .4365 .5326 .6269 .7212 .8134 .9166
95 .1873 .2946 .3959 .4890 .5893 .6839 .7826 .8756 .9746
98 .2298 .3465 -4459 .5492 .6544 .7426 .8464 .9462 1.0557

Probability 0.688 0.657 0.625 0.594 0.563 0.531 0.500 0.469 0.437
i
Percentile 10 11 12 13 14 15 16 17 18
2 .38?3 .4551 .5268 .6126 .7055 .7970 .8884 .9950 1.1286
5 .4567 .5285 .6129 .6964 .7938 .8922 1.0032 1.1206 1.2410
10 .5157 .5944 .6871 .7764 .8732 .9795 1.0951 1.2067 1.3391
50 .7568 .8495 .9459 1.0512 1.1590 1.2808 1.4042 1.5338 1.6763
90 1.0191 1.1226 1.2209 1.3415 1.4572 1.5806 1.7129 1.8467 2.0010
95 1.0836 1.1968 1.3063 1.4165 1.5368 1.6679 1.7942 1.9343 2.0818
98 1.1684 1.2765 1.3798 1.4936 1.6093 1.7366 1.8732 2.0201 2.1723
 TABLE 5. (Continued)

Probability 0.406 0.375 0.343 0.312 0.280 0.249 0.218 0.186 0.155
i
Percentile 19 20 21 22 23 24 25 26 27
2 1.2599 1.3902 1.5476 1.7408 1.8924 2.0703 2.2944 2.5674 2.8517
5 1.3892 1.5220 1.6798 1.8561 2.0394 2.2451 2.4653 2.7337 3.0252
10 1.4742 1.6290 1.7956 1.9582 2.1643 2.3670 2.5937 2.8784 3.1694
50 1.8278 1.9918 2.1692 2.3578 2.5718 2.7955 3.0497 - 3.3588 3.7165
90 2.1593 2.3464 2.5258 2.7348 2.9682 3.2331 3.5372 3.8894 4.3468
95 2.2524 2.4333 2.6179 2.8419 3.0731 3.3515 3.6872 4.0500 4.5279
98 2.3486 2.5411 2.7154 2.9443 3.1920 3.4906 3.8412 4.2423 4.7526

Probability 0.124 0.092 0.061 0.030

i
Percentile 28 29 30 31
2 3.1373 3.5742 4.0384 4.6812
5 3.3513 3.7610 4.2960 5.0960
10 3.5160 3.9796 4.5383 5.4352
50 4.1689 4,7384 5.5653 7.0698
90 4.9078 5.7424 6.9634 9.5566
95 5.1353 6.0653 7.5163 10.4307
98 5.4034 6.3860 7.9955 11.5191

i
i
 TABLE 5. ' ( C ontinued)

v= 2 n = 51

Probability 0.984 0.965 0.946 0.926 0.907 0.887 0.868 0.849 0.829
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0008 .0080 .0234 .0410 .0649 .0905 .1190 .1452 .1755
5 .0019 .0145 .0338 .0589 .0840 ,1141 .1453 .1761 .2143
10 .0041 .0213 .0452 .0726 .lo34 .1356 .1714 .2077 .2468
50 .0274 .0683 .lo81 .1512 .1953 .2404 .2847 .3321 .3776
90 .0894 .1544 .2121 .2638 .3176 .3721 .4288 .4829 .5345
95 .1143 .1856 .2391 .3005 .3563 .4106 .4715 .5278 .5889
98 .1440 .2166 .2774 .3454 .3981 .4564 .5185 .5773 .6317

Probability 0.810 0.791 0.771 0.752 0.732 0.713 0.694 0.674 0.655
i
Percentile 10 11 12 13 14 15 16 17 18
2 .2139 .2499 .2849 .3189 .3607 .4015 .4435 .4834 .5427
5 .2507 .2896 .3285 .3695 .4072 .4565 .4995 .5488 .6024
10 .2855 .3247 .3670 .4131 .4553 .5036 .5534 .6055 .6580
50 .4288 .4771 .5281 .5763 .6295 .6860 .7390 .8023 .8619
90 .5879 .6502 .7066 .7650 .8205 .8826 .9451 1.0025 1.0691
95 .6416 .7007 .7581 .8138 .8762 .9347 1.0011 1.0680 1.1267
98 .6934 .7569 .8099 .8727 .9280 .9907 1.0534 1.1275 1.1937
 TABLE 5. (Continued)

Probability 0.636 0.616 0.597 0.577 0.558 0.539 0.519 0.500 0.481
i
6 .

Pe rckntile 19 20 21 22 23 24 25 26 27
2 .5955 .6452 .6947 .7440 .8131 .8776 .9422 1.0129 1.0878
5 .6600 .7183 .7672 .8269 .8877 .9528 1.0203 1.0835 1.1592
10 .7181 .7714 .8300 .8900 .9511 1.0138 1.0820 1.1548 1.2230
50 .9247 .9804 1.0461 1.1118 1.1795 1.2523 1.3270 1.4003 1.4796
90 1.1371 1.2030 1.2767 1.3459 1.4196 1.4979 1.5754 1.6566 1.7408
95 1.1986 1.2630 1.3352 1.4123 1.4865 1.5596 1.6 344 1.7155 1.8019
98 1.2600 1.3259 1.4067 1.4788 1.5521 1.6282 1.7027 1.7829 1.8620

Probability 0.461 0.442 0.423 0.403 0.384 0.364 0.345 0.326 0.306
i
Percentile 28 29 30 31 32 33 34 35 36
2 1.1549 1.2224 1.3084 1.3911 1.4759 1.5709 1.6605 1.7658 1.8820
5 1.2321 1.3144 1.3948 1.4850 1.5708 1.6642 1.7660 1.8670 1.-9829
10 1.3106 1.3841 1.4695 1.5550 1.6468 1.7465 1.8516 1.9554 2.0693
50 1.5656 1.6540 1.7452 1.8344 1.9337 2.0364 2.1435 2.2572 2.3795
90 1.8272 1.9132 2.0025 2.1006 2.1991 2.3201 2,4324 2.5581 2.6927
95 1.8918 1.9850 2.0690 2.1768 2.2740 2.3936 2.5110 2.6395 2.7790
98 1.9605 2.0583 2.1471 2.2543 2.3574 2.4621 2.5999 2.7157 2.8580
 TABLE 5. (Continued)

Probability Oi287 0.268 0.248 0.229 0.209 0.190 0.171 0.151 0.132
i

Percentile 37 38 39 40 41 42 43 44 45
2 1.9959 2.1015 2.2348 2.3682 2.5119 2.6815 2.8359 3.0240 3.2436
5 2.0931 2.2188 2.3466 2.4909 2.6427 2.8008 2.9755 3.1759 3.3958
10 2.1894 2.3124 2.4408 2.5848 2.7472 2.9181 3.0964 3.2924 3.5307
50 2.5068 2.6426 2.7899 2.9475 3.1224 3.3169 3.5225 3.7480 4.0190
90 2.8314 2.9745 3.1374 3.3126 3.5081 3.7247 3.9722 4.2372 4.5781
95 2.9128 3.0623 3.2376 3.4314 3.6188 3.8466 4.0989 4.3946 4.7393
98 3.0099 3.1545 3.3376 3.5217 3.7299 3.9698 4.2468 4.5452 4.9300

Probability 0.113 0.093 0.074 0.054 0.035 0.016

i

Percentile 46 47 48 49 50 51

2 3.4841 3.7466 4.0181 4.3589 4.8428 5.5143

5 3.6261 3.9056 4.2477 4.6136 5.1202 5.9228
10 3.7659 4.0651 4.4143 4.8294 5.3964 6.3085
50 4.3295 4.6875 5.1258 5.7089 6.5511 8.0794
90 4.9729 5.4333 6.0245 6.8670 8.1091 10.8130
95 5.1646 5.6459 6.2825 7.2279 8.6231 11.8215
98 5.3690 5,8904 6.6023 7.5673 9.2162 12.8615
 TABLE 5. (Continued)

v= 2 n = 101

Probability 0.993 0.974 0.954 0.934 0.914 0.895 0.875 0.855 0.835
i
Percentile 1 3 5 7 9 11 13 15 17
2 .0003 .0105 .0301 .0550 .0803 .1107 .1442 .1783 .2144
5 .0010 .0162 .0390 .0678 .0979 .i3io .1652 .2052 .2430
10 .0020 .0222 .0503 .0814 .1145 .1505 .1872 .2271 .2680
50 .0137 .0542 .0961 .1383 .1813 .2266 .2692 .3168 .3649
90 .0462 .lo54 .1603 .2131 .2661 .3148 .3687 .4193 .4736
95 .0581 .1217 .1800 .2377 .2912 .3436 .3974 .4499 .5038
98 .0725 .1414 .2044 .2601 .3173 .3728 .4272 .4812 .5376

Probability 0.816 0.796 0.776 0.756 0.737 0.717 0.697 0.678 0.658
i
Percentile 19 21 23 25 27 29 31 33 35
2 .2524 .2905 .3323 .3757 .4195 .4631 .5109 .5648 .6125
5 .2829 .3238 .3646 .4132 .4567 .5070 .5526 .6055 86617
10 .3119 .3535 .4000 .4466 .4939 .5412 .5929 .6435 .7017
50 .4149 .4628 .5147 .5658 .6165 .6736 .7300 .7861 .8448
90 .5259 .5820 .6362 .6986 .7522 .8106 .8683 .9362 .9960
95 .5595 .6152 .6730 .7349 .7842 .8481 .9070 .9741 1.0365
98 .5927 .6505 .7075 .7679 .8283 .8842 .9543 1.0169 1.0822
 TABLE 5. (Continued)

Probability 0.638 0.618 0.599 0.579 0.559 0.539 0.520 0.500 0.480
i
Percentile 37 39 41 43 45 47 49 51 53
2 .6625 .7161 .7805 .8359 .8990 .9623 1.0281 1.1019 1.1662
5 .7146 .7697 .8323 .8879 .9582 1.0230 1.0915 1.1644 1.2391
10 .7564 .8147 .8721 .9327 1.0036 1.0722 1.1415 1.2174 1.2927
50 .9062 .9703 1.0346 1.1029 1.1734 1.2439 1.3203 1.3960 1.4769
90 1.0620 1.1295 1.1958 1.2679 1.3398 1.4170 1.4932 1.5739 1.6612
95 1.1015 1.1720 1.2456 1.3091 1.3885 1.4666 1.5392 1.6269 1.7106
98 1.1509 1.2158 1.2889 1.3571 1.4312 1.5085 1.5905 1.6705 1.7626

Probability 0.460 0.441 0.421 0.401 0.382 0.362 0.342 0.322 0.303
i

Pe rc entile 55 57 59 61 63 65 67 69 71
2 1.2431 1.3330 1.4154 1.4998 1.5960 1.6861 1.7987 1.9009 2.0298
5 1.3190 1.3969 1.4791 1.5683 1.6646 1.7650 1.8695 1.9852 2.1074
10 1.3717 1.4501 1.5407 1.6314 1.7311 1.8312 1.9400 2.0492 2.1720
50 1.5617 1.6496 1.7403 1.8368 1.9375 2.0446 2.1591 2.2703 2.3974
90 1.7522 1.8356 1.9330 2.0312 2.1417 2.2463 2.3635 2.4949 2.6235
95 1.7966 1.8897 1.9840 2.0839 2.1973 2.3011 2.4233 2.5556 2.6881
98 1.8538 1.9464 2.0385 2.1458 2.2504 2.3577 2.4773 2.6111 2.7502
 TABLE 5. (Continued)

Probability 0.283 0.263 0.244 0.224 0.204 0.184 0.165 0.145 0.125
i
Percentile 73 75 77 79 81 83 85 87 89
2 2.1447 2.2786 2.4031 2.5482 2.7108 2.8829 3.0673 3.2935 3.5352
5 2.2306 2.3547 2.4948 2.6452 2.8080 2.9841 3.1809 3.3943 3.6433
10 2.2982 2.4247 2.5671 2.7181 2.8873 3.0721 3.2662 3.4935 3.7415
50 2.5315 2.6717 2,8223 2.9937 3.1763 3.3705 3.5931 3.8376 4.1199
90 2.7673 2.9155 3.0855 3.2618 3.4585 3.6805 3.9201 4.2046 4.5325
95 2.8279 2.9911 3.1599 3.3397 3.5409 3.7705 4.0159 4.3092 4.6599
98 2.8898 3.0548 3.2246 3.4284 3.6254 3.8483 4.1248 4.4228 4.7944

Probability 'j 0.105 0.086 0.066 0.046 0.026 0.007

i
/

Percentilk 91 93 95 97 99 101
2 3.7803 4.0957 4.4648 4.9092 5.5550 6.7349
5 3.9174 4.2222 4.6172 5.1102 5.8280 7.1887
10 4.0161 4.3612 4.7659 5.2960 6,0619 7.6049
50 4.4498 4.8480 5.3533 6.0315 7.0859 9.5396
90 4.9214 5.4031 6.0467 6.9222 8.3732 12.6119
95 5.0554 5.5844 6.2297 7.1649 8.8023 13.8253
98 5.2014 5.7695 6.4413 7.5072 9.8122 15,1859 - .- -
 TABLE 5. (Continued)

v= 3 n= 5

Probability 0.748 0.624 0.500 0.376 0.252

i
Percentile 1 2 3 4 5
2 .1479 1.3683 2.1842 2.4523 2.8357
5 .2681 1.4976 2.2556 2.5519 2.9200
10 .4147 1.6295 2.3455 2.6487 2.9894
50 1.1481 2.0961 2.7209 2.9964 3.1687
90 1.7576 2.5002 3.0366 3.1696 3.1990
95 1.8857 2.6063 3.0891 3.1845 3.1997
98 1.9562 2.7029 3.1363 3.1934 3.1999

v= 3 n= 7

Probability 0.809 0.706 0.603 0.500 0.397 0.294 0.191

i
Percentile 1 2 3 4 5 6 7
2 .OlOl .4607 1.0144 1.7405 2.3725 2.8733 3.3451
5 .1465 .6202 1.2127 1.9339 2.5193 3.0243 3.5302
10 .2496 .7882 1.4160 2.0782 2.6413 3.1630 3.6959
50 .7867 1.4295 2.0067 2.5597 3.1170 3.7042 4.3474
90 1.4319 1.9544 2.4675 3.0228 3.7083 4.3328 4.8971
95 1.5796 2.0980 2.5889 3.1557 3.8772 4.4932 4.9852
98 1.7274 2.2131 2.7025 3.2718 4.0 540 4.6349 5.0433

-
 TABLE 5. (Continued)

v= 3 n = 11

Probability 0.878 0.802 0.727 0.651 0.576 0.500 0.424: 0.349 0.273
1
Percentile 1 2 3 4 5 6 7 8 9
2 .0459 .2655 .5432 .8489 1.2552 1.6556 2.0850 2.5308 3.0102
5 .0965 .3780 .7112 1.0537 1.4283 1.8564 2.2793 2.7229 3.1743
10 .1648 ,4902 .8486 1.2131 1.6010 2.0110 2.4296 2.8714 3.3300
50 .5296 .9759 1.3737 1.7504 2.1310 2.5180 2.9239 3.3769 3.9055
90 1.0647 1.5091 1.8771 2.2511 2.6006 2.9816 3.4133 3.9320 4.6206
95 1.2133 1.6700 1.9982 2.3674 2.7179 3.1080 3.5380 4.0967 , 4.8450
98 1.3694 1.8053 2.1256 2.4792 2.8207 3.2236 3.6847 4.2745 5.0858

Probability 0.198 0.122

i
Percentile 10 11
2 3.5081 4.0498
5 3.6988 4.3195
10 3.8526 4.5531
50 4.5951 5.6114
90 5.5440 6.9004
95 5.8420 7.2747
98 6.1603 7.6230
 TABLE 5. (Continued)

u= 3 n = 15

Probability 0.913 0.854 0.795 0.736 0.677 0.618 0.559 0.500 0.441
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0246 .1964 .3912 .6444 .8892 1.1434 1.4246 1.7272 2.0290
5 .0717 .2828 .5067 .7731 1.0411 1.3242 1.5890 1.8981 2.2144
10 .1224 .3715 .6261 .go68 1.1740 1.4561 1.7316 2.0470 2.3690
50 .4104 .7694 1.0811 1.3717 1.6687 1.9327 2.2128 2.5232 2.8364
90 .8202 1.2331 1.5498 1.8364 2.0975 2.3698 2.6544 2.9455 3.2868
95 .9952 1.3619 1.6650 1.9589 2.2092 2.4771 2.7567 3.0628 3.3835
98 1.1357 1.4965 1.8106 2.0606 2.3457 2.5996 2.8782 3.1721 3.5073

Probability 0.382 0.323 0.254 0.205 0.146 0.087

i
Percentile 10 11 12 13 14 15
2 2.3817 2.7444 3.1258 3.5267 3.9760 4.5749
5 2.5551 2.8996 3.3048 3.7174 4.1950 4.8511
10 2.6925 3.0472 3.4385 3.8936 4.3930 5.1355
50 3.1736 3.5516 3.9785 4.5113 5.2001 6.3528
90 3.6489 4.0901 4.6159 5.2718 6.2585 8.0016
95 3.7960 4.2469 4.7939 5.5377 6.6097 8.5107
98 3.9284 4.3939 4.9872 5.7667 6.9894 9.0045
 TABLE 5. (Continued)

v= 3 n = 21

Probability 0.941. 0.897 0.853 0,809 0.765.- 0.721 0.671 0.632 0.588
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0331 .1560 .2919 .4620 .6412 .8075 1.0263 1.2113 1.4065
5 .0600 .2218 .3964 .5890 .7703 .9552 1.1541 1.3573 1.5407
10 .0959 .2893 .4910 .6837 .8744 1.0713 1.2671 1.4799 1.6744
50 .3254 .6047 .8437 1.0638 1.2635 1.4781 1.6850 1.8879 2.0981
90 .6930 .9921 1.2396 1.4652 1.6744 1.8851 2.0870 2.2811 2.4849
95 .8106 1.1019 1.3582 1.5759 1.7828 1.9857 2.1827 2.3916 2.6052
98 .9340 1.2264 1.4813 1.6897 1.8944 2.0982 2.2786 2.4997 2.6988

Probability 0.544 0.500 0.456 0.412 0.368 0.323 0.279 0,235 0.191
i
Percentile 10 11 12 13 14 15 16 17 18
2 1.6408 1.8391 2.0537 2.3016 2.5596 2.7954 3.0978 3.4063 3.7345
5 1.7786 1.9969 2.1968 2.4342 2.7044 2.9648 3.2394 3.5596 3.8957
10 1.8912 2.1134 2.3360 2.5716 2.8355 3.0979 3.3889 3.6979 4.0484
50 2.3115 2.5327 2.7588 3.0008 3.2683 3.5458 3.8655 4.2211 4.6331
90 2.7033 2.9244 3.1539 3.4158 3.6949 4.0320 4.3765 4.8102 5.3131
95 2.8032 3.0195 3.2639 3.5296 3.8156 4.1641 4.5377 4.9756 5.5049
98 2.9075 3.1209 3.3815 3.6593 3.9501 4.3008 4.6827 5.1550 5.7392
 TABLE 5. (Continued)

Probability 0.147 0,103 0.059

i
Percentile 19 20 21
2 4.0978 4.4888 5.0732
5 4.2925 4.7450 5.3865
10 4,4352 4.9530 5.6974
50 5.1358 5.8328 7.0195
90 6.0141 7.0077 8.9299
95 6.2816 7.3985 9.5586
98 6.5524 7.8060 10.2951

'v= 3 n = 31

Probability 0.963 0.932 0.901 0.871 0.840 0.809 0.778 0.747 0.716
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0179 .0904 .1851 .2831 .3847 .4986 .5950 .7079 .8285
5 ,0352 .1338 .2483 .3572 .4656 .5833 .6959 .8110 .9318
10 .0571 .1809 .2994 .4226 .5400 .6580 .7819 .8989 1.0291
50 .2053 .3784 .5417 .6853 .8227 ,9573 1.0940 1.2309 1.3672
90 .4548 .6659 .8282 ,9891 1.1382 1,2889 1.4267 1.5605 1.7148
95 .5408 .7415 .9119 1.0710 1.2271 1.3756 1.5176 1.6591 1.8049
98 .6250 ,8334 1.0036 1.1683 1.3056 1.4652 1.6152 1.7652 1.9067
 TA BLE 5. ( Continued)

Probability 0.685 0.654 0.624 0.593 0.562 0.531 0.500 0.469 0.438
i
Percentile 10 11 12 13 14 15 16 17 18
2 .9425 1.0884 1.2103 1.3321 1.4683 1.6106 1.7773 1.9249 2.0983
5 1.0619 1.1889 1.3159 1.4397 1.5870 1.7466 1.8926 2.0544 2.2323
10 1.1496 1.2871 1.4125 1.5619 1.7038 1.8478 2.0076 2.1770 2.3493
50 1.5030 1.6380 1.7824 1.9293 2.0791 2.2411 2.3999 2.5739 2.7508
90 1.8563 2.0007 2.1451 2.3007 2.4452 2.6085 2.7745 2.9551 3.1303
95 1.9613 2.0935 2.2448 2.3911 2.5430 2.7029 2.8805 3.0551 3.2265
98 2.0532 2.1910 2.3272 2.4890 2.6516 2.7999 2,9655 3.1451 3.3482

Probability 0.407 0.376 0.346 0.315 0.284 0.253 0.222 0.191 0.160
i
Percentile 19 20 21 22 23 24 25 26 27
2 2.2632 2.4486 2.6233 2.8396 3.0644 3.3175 3.5792 2.8450 4.1492
2.4021 2.5835 2.7849 3.0078 3.2303 3.4623 3.7383 4.0157 4.3424
10 2.5283 2.7152 2.9212 3.1354 3.3564 3.5912 3.8726 4.1797 4.5123
50 2.9373 3.1416 3.3500 3.5719 3.8199 4.0879 4.3884 4.7501 5.1584
90 3.3289 3.5413 3.7696 4.0191 4.3012 4.6072 4.9816 5.4013 5.9108
95 3.4375 3.6551 3.8941 4.1483 4.4398 4.7637 5.1254 5.6090 6.1224
98 3.5370 3.7782 4.0 170 4.2603 4.5592 4.9340 5.3240 5.8065 6.3816
 TABLE 5. (Continued)

Probability 0.129 0.099 0.068 0.037

i
Percentile 28 29 30 31
2 4.4904 4.9051 5.4188 6.1690
5 4.7138 5.1620 5.6801 6.5468
10 4.9063 5.3634 5.9738 6.9044
50 5.6326 6.2631 7.1256 8.6727
90 6.5338 7.3592 8.6508 11.1334
95 6.8071 7.7366 9.1474 12.0331
98 7.1212 8.1195 9. 6680 12.8834

v= 3 n = 51

Probability 0.980 0.961 0.944 0.923 0.903 0.884 0.865 0.845 0.826
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0128 .0623 .1312 .1911 .2573 .3178 .3878 .4509 .5303
5 .0268 .3906 .le56 .238i .3075 .3759 .4567 .5331 .g009
10 ,0400 .1203 .2046 .2772 .3575 .4310 .5145 .5898 .6625
50 .1467 .2655 .3724 .4674 .5616 .6471 .7312 .8146 .8966
90 .3237 .4724 .5852 .6973 .7907 .E3832 .9723 1.0611 1.1481
95 .3864 .5294 .6437 .7611 .8537 .9512 1.0472 1.1383 1.2215
98 .4565 .5998 .7230 .8345 .9361 1.0209 1.1264 1.2219 1.3021
 TABLE 5. (Continued)

Probability 0.807 0.788 0.769 0.749 0.730 0.711 0.692 0.673 0.654
i
Percentile 10 11 12 13 14 15 16 17 18
2 .6048 .6732 .7431 .8116 .8843 .9512 1.0244 1.1087 1.1915
5 .6747 .7409 .8151 .8989 . 9710 1.0439 1.1239 1.1996 1.2787
10 .7296 .8098 .8892 .9641 1.0441 1.1207 1.1982 1.2812 1.3632
50 .9771 1.0608 1.1456 1.2299 1.3099 1.3918 1.4754 1.5576 1.6456
90 1.2363 1.3254 1.4106 1.4982 1.5783 1.6673 1.7446 1.8367 1.9288
95 1.3132 1.3965 1.4894 1.5687 1.6640 1.7405 1.8216 I.. 9153 2.0071
98 1.3944 1.4743 1.5651 1.6467 1.7338 1.8158 1.9055 1.9940 2.0908

Probability 0.634 0.615 0.596 0.571 0.558 0.538 0.519 0.500 0.481
i
Percentile 19 20 21 22 23 24 25 26 27
2 1.2596 1.3435 1.4258 1.5036 1.5932 1.7005 1.7840 1.8692 1.9664
5 1.3604 1.4486 1.5227 1.6115 1.7086 1.8035 1.8930 1.9852 2.0875
10 1.4449 1.5254 1.6110 1.7018 1.7972 1.8925 1.9815 2.0763 2.1750
50 1.7305 1.8195 1.9087 1.9982 2.1007 2.1922 2.2927 2.3931 2.5017
90 2.0143 2.1115 2.2039 2.2900 2.3906 2.4879 2.5923 2.6925 2.8097
95 2.0995 2.1845 2.2742 2.3764 2.4710 2.5678 2.6716 2.7796 2.8840
98 2.1772 2.2738 2.3624 2.4537 2.5519 2.6601 2.7454 2.8626 2.9656
 TABLE 5. (Continued)

Probability 0.462 0.442 0.423 0.404 0.385 0.366 0.346 0.327 0.308
i
Percentile 28 29 30 31 32 33 34 35 36
2 2.0678 2.1599 2.2744 2.4920 2.5003 2.6266 2.7338 2.8827 3.0038
5 2.1794 2.2877 2.3901 2.5920 2.6157 2.7471 2.8707 3.0000 3.1321
10 2.2785 2.3807 2.4908 2. 6016 2.7271 2.8494 2.9784 3.0998 3.2398
50 2.6046 2.7147 2.8326 2.9488 3.0707 3.1974 3.3305 3.4721 3.6154
90 2.9123 3.0301 3.1583 3.2737 3.4025 3.5358 3.6810 3.8245 3.9812
95 2.9959 3.1231 3.2305 3.3608 3.490 8 3,6231 3.7721 3.9280 4; 0895
98 3.0877 3.2090 3.3265 3.4573 3.5770 3.7287 3.8767 4.0298 4.1897

Probability 0.289 0.270 0.251 0.231 0.212 0.193 0.174 0.155 0.135
i
Percentile 37 38 39 40 41 42 43 44 45
2 3.1457 3.3043 3.4573 3.6327 3.7890 3.9660 4.1458 4.3820 4.5850
5 3.2726 3.4217 3.5859 3.7567 3.9325 4.1222 4.3095 4.5341 4.7633
10 3.3883 3.5322 3.7141 3.8780 4.0481 4.2445 4.4446 4.6886 4.9240
50 3,7755 3.9376 4.1055 4.2807 4.4889 4.7034 4.9425 s. 212i 5.5051
90 4.1484 4.3229 4.5451 4.7322 4.9548 5.2177 5.5044 5.8123 6.1846
95 4.2510 4.4400 4.6447 4.6567 5.1099 5.3623 5.6624 5.9847 6.3565
98 4.3686 4.5768 4.7824 4.9896 5.2606 5.5434 5.8431 6.1856 6.5824
w TABLE 5. (Continued)
W
0

Probability 0.116 0.097 0.078 0.059 0,039 0.020

i
Percentile 46 47 48 49 50 51
2 4.8396 5.1449 5.4643 5.8622 6.3607 7.1004
5 5.0378 5.3338 5.6952 6.1518 6.6884 7.5662
10 5.2038 5.5290 5.9059 6.3866 7.0211 7.9817
50 5.8480 6.2443 6.7487 7.3898 8.2948 9.9261
90 6.5895 7.1153 7.7568 8.6679 9.9853 12.8678
95 6.8247 7.3712 8.0951 9.0620 10.6058 13.8407
98 7.0523 7.6576 8.4479 9.5102 11.2605 14.8702

v= 3 n = 101

ProbabiLj 0,991 0.971 0.952 0.932 0.912 0.893 0.873 0.853 0.834
i'
Percentile 1 3 5 7 9 11 13 15 17
2 .0077 .0768 .1545 .2325 .3091 .3787 .4518 .5285 .6027
5 .0155 .lo06 .1868 .2711 .3453 .4254 .5016 .5792 .6518
10 .0265 .1243 .2189 .3035. .3851 .4692 .5441 .6258 .7030
50 .0928 .2288 ,3378 .4360 .5278 .6154 ,7027 .7874 ,8723
90 .2015 .3613 .4821 .5882 .6898 ,7822 .8726 .9619 1.0446
95 .2377 .4014 .5217 ,6367 .7370 .8310 .9189 1.0076 1.0992
98 .2762 .4430 .5660 .6872 .7844 .8812 .9729 1.0609 1.1522
 TABLE 5. (Continued)

Probability 0.814 0.795 0.775 0.755 0.736 0.716 0.696 0.677 0.657
i
Percentile 19 21 23 25 27 29 31 33 35
2 .6753 .7493 .8197 ,8976 .9731 1.0443 1.1241 1.2111 1.2913
5 .7311 .8054 .8804 ,9536 1.0 296 1.1136 1.1937 1.2763 1.3591
10 .7751 .8556 .9339 1.0075 1.0880 1.1718 1.2520 1.3360 1.4206
50 .9548 1.0381 1.1196 1.2093 1.2869 1.3721 1.4545 1.5392 1.6259
90 1.1332 1.2231 1.3076 1.4902 1.4759 1.5689 1.6513 1.7439 1.8323
95 1.1863 1.2747 1.3564 1.4409 1.5259 1.6169 1.7068 1.7921 1.8894
98 1.2329 1.3286 1.4072 1.4955 1.5835 1.6688 1.7640 1.8533 1.9404

Probability 0.637 0.618 0.598 0.579 0.559 0.539 0.520 0.500 0.480
i
Percentile 37 39 41 43 45 47 49 51 53

2 1.3755 1.4645 1.5454 1.6343 1.7184 1.8120 1.9005 1.9928 2.1016

5 1.4520 1.5295 1.6129 1.7044 1.7960 1.8884 1.9870 2.0808 2.1853
10 1.5022 1.5911 1.6799 1.7704 1.8617 1.9585 2.0546 2.1522 2.2569
50 50 1.7146 1.8064 1.8986 1.9901 2.0879 2.1828 2.2827 2.3828 2.4899
90 1.9179 2.0113 2.1059 2.1992 2.3018 2.4013 2.5029 2.6038 2.7110
95 1.9761 2.0664 2.1574 2.2551 2.3555 2.4637 2.5632 2.6700 2.7819
98 2.0344 2.1167 2.2184 2.3197 2.4205 2.5249 2.6203 2.7379 2.8491
 TABLE 5. (Continued)

Probability 0.461 0.441 0.421 0.402 0.382 0.363 0,343 0.323 0.304
i
Percentile 55 57 59 61 63 65 67 69 71
2 2.2003 2,3038 2,4313 2.5445 2.6652 2.7803 2.9269 3.0629 3.2095
5 2.2862 2.3994 2.5173 2.6329 2.7590 2.8751 3.0162 3.1511 3.2920
10 2.3618 2.4706 2.5821 2.7089 2.8249 2.9548 3.0883 3.2259 3.3715
50 2.5949 2.7096 2.8317 2.9539 3.0778 3.2084 3.3501 3.4943 3.6440
90 2.8295 2.9396 3.0647 3.1924 3.3253 3.4584 3.6009 3.7587 3.9153
95 2.8893 3.0052 3.1277 3.2591 3.3916 3.5273 3.6733 3.8320 3.9861
98 2.9594 3,0701 3.1987 3.3203 3.4537 3.5984 3.7450 3.9008 4.0725

Probability 0.284 0.264 0.245 0.225 0.205 0.186 0.166 0.147 0.127
i
Percentile 73 75 77 79 81 83 85 87 89
2 3.3502 3.4978 3.6758 3.8493 4.0299 4.2309 4.4568 4.6915 4.9550
5 3.4440 3.6060 3.7648 3.9476 4.1250 4.3472 4.5762 4.8236 5.0969
10 3.5290 3.6914 3.8478 4.0370 4.2232 4.4457 4.6742 4.9312 5.2123
50 3.8041 3.9780 4.1586 4.3549 4.5654 4.8015 5.0527 5.3392 5.6586
90 4.0899 4.2748 4.4758 4.6855 4.9191 5.1761 5.4587 5.7765 6.1513
95 4.1899 4.3525 4.5639 4.7772 5.0271 5.2677 5.5868 5.9043 6.3137
98 4.2531 4.4561 4.6540 4.8870 5.1199 5.4028 5.7169 6.0370 6.4666
 TABLE 5. (Continued)

Probability 0.107 0.088 0.068 0.048 0.029 0.009

i
Percentile 91 93 95 97 99 101
2 5.2471 5.5980 5. 9910 6.4957 7.2023 8.4081
5 5.3933 5.7494 6.1782 6.7345 7.4666 8.9158
10 5.5373 5.9009 6.3701 6.9482 7.7621 9.4285
50 6.0346 6.4837 7.0292 7.7778 8.8828 11.5020
90 6 . 6002 7.1128 7.8283 8.7897 10.3687 14.6263
95 6.7750 7.3270 8.0656 9.0942 10.8869 15.7611
98 6.9395 7.5614 8.3065 9.4403 11.4272 17.1230

l.J= 4

Probability 0.793 0.696 0.508 0.500 0.402 0,304 0.207

i
Percentile 1 2 3 4 5 6 7
2 .2579 1.0410 2.1762 2.9226 3.4160 3.8139 4.1969
5 .4293 1.3443 2.3831 3.0584 3.5433 3.9427 4.3585
10 .6200 1.5772 2.5414 3.1826 3.6672 4.0680 4.4931
50 1.4551 2.3425 3.0666 3. G420 4.1356 4.5463 4.9066
90 2.2410 2.9007 3.5632 4.1615 4.6102 4.9141 5.0997
95 2.4031 3.0112 3.6892 4.3064 4.7179 4.9717 5.1198
98 2.5665 3.1248 3.8367 4.4247 4.8159 5.0382 5.1318

I
w
w
 TABLE 5. (C6ntinued)

v= 4 n = 11

Probability 0,864 0.791 0.718 0.646 0.573 0.500 0.427 0.354 0.282
i
Percentile 1 2 3 4 5 6 7 8 9
2 .1449 .5657 1.0485 1.4669 1.9648 2.4972 3.0532 3.5879 4.0734
5 .2627 .7414 1.2128 1.7119 2.2150 2.7320 3.2556 3.7759 4.2539
10 . 3733 .9143 1.4273 1.9063 2.4259 2.9204 3.4254 3.9426 4.4251
50 .9534 1.5775 2.0992 2.5752 3.0241 3.4911 3.9840 4.5008 5.0856
90 1.6779 2.2249 2.6 913 3.1240 3.5616 4.0195 4.5509 5.1741 5.8759
95 1.8821 2.3974 2.8558 3.2579 3.6916 4.1541 4.7093 5.3781 6.1019
98 2.0365 2.5738 3.0021 3.3765 3.8096 4.2927 4.8675 5.5821 6.3399

Probability 0.209 0.136

i
Percentile 10 11
2 4,5809 5.1796
5 4.7942 5.4589
10 4.9844 5.7086
50 5.7847 6.7221
90 6.7129 7.7950
95 6.9643 8.0430
98 7.2579 8.2843
 TABLE 5. (Continued)

v= 4 n = 15

Probability 0.901 0.844 0.787 0.729 0.672 0.615 0.557 0.500 0.443
i
Percentile 1 2 3 4 5 6 7 8 9
2 .1151 .4329 .7533 1.0778 1.4333 1.7654 2.1227 2.4828 2.9092
5 .1971 .5627 .9408 1.2952 1.6218 1.9655 2.3144 2.7066 3.0 970
10 .2871 .7013 1.0850 1.4449 1.7990 2.1359 2.4978 2.8762 3.2819
50 .7542 1.2629 1.6598 2.0288 2.3827 2.7331 3.0860 3.4485 3.8408
90 1.3649 1.8398 2.2345 2.5851 2.9191 3.2499 3.5958 3.9661 4.3689
95 1.5329 2.0190 2.3937 2.7332 3.0475 3.3620 3.7333 4.1052 4.5244
98 1.7299 2.1688 2.5285 2.8482 3.1742 3.5085 3.8577 4.2507 4.6777

Probability 0.385 0.328 0.271 0.213 0.156 0.099

i
Percentile 10 11 12 13 14 15
2 3.3521 3.7622 4.2099 4.6670 5.1696 5.8486
5 3.5205 3.9324 4.3981 4.8834 5.4268 6.1732
10 3.6932 4.1164 4.5739 5.0661 5.6519 6.4956
50 4.2540 4.6973 5.2206 5.8182 6.6017 7.7876
90 4.8338 5.3520 5.9769 6.7703 7.7791 9.4530
95 4.9881 5.5143 6.1889 7.0621 8.1357 9.8619
98 5.1791 5.7523 6.4319 7,3901 8.5200 10.3852
 TABLE 5. (Continued)

v= 4 n = 21

Probability 0.932 0.889 0.845 0.802 0.759 0.716 0.673 0.630 0.586
i
Percentile 1 2 3 4 5 6 7 8 9
2 .0850 .3436 .5771 .8112 1.0540 1.3015 1.5290 1.7841 2.0482
5 .1487 .4477 .7153 .9761 1.2131 1.4544 1.7145 1.9768 2.2207
10 .2254 .5528 .8342 1.0946 1.3582 1.6005 1.8597 2.1101 2.3631
50 .6031 1.0025 1.3119 1.5959 1.8619 2.1266 2.3828 2.6391 2.9032
90 1.1259 1.5197 1.8306 2.0 995 2.3691 2.6161 2.8516 3.1155 3.3838
95 1.2695 1.6664 1.9685 2.2397 2.4934 2.7488 2.9943 3.2400 3.5080
98 1.4233 1.7986 2.0977 2.3703 2.6182 2.8779 3.1209 3.3499 3.6363

Probability 0.543 0.500 0.457 0.414 0.370 0.327 0.284 0.241 0.198
i
Percentile 10 11 12 13 14 15 16 17 18
2 2.2812 2.5592 2.8642 3.1606 3.4681 3.7939 4.1483 4.4771 4.9157
5 2.4717 2.7492 3.0279 3.3371 3.6422 3.9737 4.3152 4.7046 5.1418
10 2.6322 2.8988 3.1881 3.4894 3.7901 4.1167 4.4807 4.8651 5.3103
50 3.1599 3.4368 3.7129 4.0072 4.3236 4.6739 5.0580 5.4918 6.0176
90 3.6391 3.9031 4.2011 4.5085 4.8687 5.2534 5.7132 6.2496 6.8811
95 3.7510 4.0289 4.3177 4.6575 5.0031 5.4017 5.9042 6.4482 7.1285
98 3.4892 4.1657 4.4684 4.7975 5.1640 5.5975 6.0849 6.6816 7.4332
 TABLE 5. (Continued)

Probability 0.155 0.111 0.068

i
Percentile 19 20 21
2 5.3709 5.8676 6.6351
5 5.5859 6.1606 6.9574
10 5.7999 6.4316 7.2866
50 6.6507 7.5095 8.8874
90 7.7252 8.9088 11.0005
95 a. 0688 9.3003 11.6164
98 8.4500 9.7815 12.2477

v= 4 n = 31

Probability 0.956 0.926 0.895 0.865 0.834 0.804 0.774 0.743 0.713
i

Percentile 1 2 3 4 5 6 7 8 9
2 .0726 ,2627 .4327 .6370 .7824 .9677 1.1275 1.2860 1.4463
5 .1238 .3426 .5455 .7315 .8995 1.0933 1.2438 1.4291 1.5846
10 .1800 .4216 .6409 .8349 1.0122 1.1824 1.3674 1.5393 1.7108
50 .4771 .7819 1.0171 1.2257 1.4184 1.6110 1.7823 1.9653 2.1441
90 -8896 1.2014 1.4385 1.6435 1.8440 2.0191 2.2030 2.3924 2.5696
95 1.0011 1.3196 1.5537 1.7605 1.9495 2.1337 2.3242 2.4977 2.6676
98 1.1262 1.4187 1.6582 1.8749 2.0688 2.2337 2.4394 2.5851 2.7814
P TABLE 5. (Continued)
w
01,

Probability 0.682 0.652 0.622 0.591 0.561 0.530 0.500 0.470 0.439
i
Percentile 10 11 12 13 14 15 16 17 18
2 1.6321 1.7765 1.9374 2.1311 2.3060 2.4772 2.6790 2.8512 3.0495
5 1.7600 1.9207 2.0976 2.2658 2.4656 2.6504 2.8257 3.0081 3.2162
10 1.8800 2.0517 2.2251 2.3993 2.5910 2.7830 2.9677 3.1499 3.3466
50 2.3226 2.4892 2.6588 2.8373 3.0224 3.2090 3.4053 3.6032 3.8109
90 2.7386 2.9087 3.0768 3.2650 3.4417 3.6259 3.8242 4.0218 4.1511
95 2.8512 3.0180 3.1974 3.3665 3.5526 3.7475 3.9382 4.1346 4.3612.
98 2.9595 3.1399 3.3099 3.4729 3.6523 3.8539 4.0570. 4.2608 4.4729

Probability 0.409 0.378 0.348 0.318 0.287 0.257 0.226 0.196 0.166
i
Percentile 19 20 21 22 23 24 25 26 27
2 3.2531 3.4815 3.7016 3.9153 4.1665 4.4596 4.7308 5.0289 5.3526
5 3.4227 3.6243 3.8731 4.0963 4.3476 4.6125 4.9084 5.2149 5.5656
10 3.’5470 3.7734 4.0084 4.2442 4.4996 4.7795 5.0644 5.3977 5.7514
50 4.0217 4.2487 4.4863 4.7505 5.0264 5.3283 5.6562 6.0305 6.4828
90 4.4762 4.7135 4.9842 5.2557 5.5628 5.9166 6.3195 6.7496 7.2985
95 4.5992 4.8406 5.1252 5.3838 5.7208 6.0717 6.5006 6.9732 7.5317
98 4.6994 4.9771 5.2536 5.5640 5.8891 6.2512 6.6992 7.1717 7.7934
 -
r-L TABLE 5. (Continued)
rp
0
Probability 0.804 0.785 0.766 0.747 0.728 0.709 0.690 0.671 0.652
i
Percentile 10 11 12 13 14 15 16 17 18
2 1.0987 1.2133 1.3046 1.4095 1.5138 1.6110 1.7190 1.8204 i.9092
5 1.2049 1.3118 1.4229 1.5271 1.6262 1.7219 1.8331 1.9364 2.0394
10 1.3029 1.4014 1.5168 1.6149 1.7209 1.8300 1.9289 2.0242 2.1325
50 1.6263 1.7383 1.8410 1.9504 2.0533 2.1638 2.2641 2.3809 2.4825
90 1.9621 2.0709 2.1839 2.2904 2.3935 2.4979 2.6058 2.7124 2.8148
95 2.0557 2.1632 2.2714 2.3839 2.4809 2.5828 2.6975 2.7932 2.9133
98 2.1278 2.2636 2.3563 2.4690 2.5653 2.6755 2.7812 2.8938 2.9872

Probability .
0 6 33 0.614 0.595 0.576 0.557 0.538 0.519 0.500 0.481
i
Percentile 19 20 21 22 23 24 25 26 27
2 2.0165 2.1245 2.2470 2.3491 2.4563 2.5658 2.6850 2.8007 2.9073
5 2.1385 2.2576 2.3665 2.4754 2.5827 2.6969 2.8196 2.9220 3.0420
10 2.2399 2.3543 2.4602 2.5724 2.6844 2.7973 2.9200 3.0324 3.1522
50 2.5877 2.7027 2.8114 2.9300 3.0343 3.1474 3.2694 3.3940 3.5111
90 2.9303 3.0381 3.1455 3.2594 3.3746 3.4915 3.6154 3.7294 3.8556
95 3.0245 3.1255 3.2381 3.3455 3.4695 3.5847 3.6941 3.8304 3.9538
98 3.1149 3.2210 3.3265 3.4270 3.5608 3.6850 3.7998 3.9186 4.0659
 TABLE 5. (Continued)

Probability 0.462 0.443 0.424 0.405 0.386 0.367 0.348 0.329 0.310
i
Percentile 28 29 30 31 32 33 34 35 36

2 3.0 287 3.1516 3.2721 3.4089 3. 5452 3.6931 3.8321 3.9995 4.1404
5 3.1685 3.2773 3.4152 3.5474 3.6955 3.8328 3.9734 4.1245 4.2661
10 3.2655 3.3915 3.5246 3.6582 3.7931 3.9459 4.0 843 4.2327 4.3850
50 3.6335 3.7668 3.8984 4.0331 4.1834 4.3189 4.4801 4.6430 4.8060
90 3.9927 4.1282 4.2540 4.4095 4.5539 4.7062 4.8764 5.0415 5.2274
95 4.0892 4.2181 4.3613 4.5014 4.6564 4.8114 4.9837 5.1519 5.3374
98 4.1771 4.3055 4.4577 4.6137 4.7563 4.9231 5.0919 5.2788 5.4514

Probability 0.291 0.272 0.253 0.234 0.215 0.196 0.177 0.158 0.139
i
Percentile 37 38 39 40 41 42 43 44 45

2 4.3031 4.4525 4.6 360 4.8127 4.9989 5.1926 5.4101 5.6488 5.8897
5 4.4457 4.5981 4.7743 4.9631 5.1791 5.3569 5.5892 5.8272 6.0851
10 4.5585 5.7219 4.9069 5.0978 5.2974 5.5070 5.7339 5.9883 6.2551
50 4.9797 5.1683 5.3582 5.5643 5.7804 6.0196 6.2844 6.5772 6.9031
90 5.4154 5.6211 5.8388 6.0608 6.3089 6.5928 6.9105 7.2397 7.6594
95 5.5370 5.7463 5.9757 6.2063 6.4566 6.7408 7,0711 7.4584 7.8683
98 5.6776 5.8736 6.1045 6.3642 6.6335 6.9352 7.2746 7.6630 8.1124
w TABLE 5. (Continued)
Ip
N
Probability 0.120 0.101 0.082 0.063 0.044 0.025
i
Percentile 46 47 48 49 50 51
2 6.1632 6.4687 6.8317 7.2214 7.7723 8.5786
5 6.3838 6.7078 7.0920 7.5380 8.1283 9.0191
10 6.5507 6.8929 7.3250 7.7910 8.4959 9.5007
50 7.2718 7.6914 8.2380 8.9064 9.8663 11;5328
90 8.1054 8.6723 9.3705 10.2782 11.66 86 14.5028
95 8.3598 8.9354 9.7687 10.70 16 12.2827 15.5050
98 8.6 194 9.2328 10.1347 11.1283 12.9519 16.7038

v= 4 n = 101

Probability 0.988 0.969 0.949 0.930 0.910 0.891 0.871 0.852 0,832
i
Percentile 1 3 5 7 9 11 13 15 17
2 .0394 .2233 .3838 5205 .6491 .7643 .8875 1.0089 1.1021
5 ,0640 .2698 .4436 .5800 .7216 .8464 .9567 1.0735 J..1826
10 .0 949 .3195 .4923 .6406 .7783 .9033 1.0188 1.1325 1.2469
50 .2505 .5104 .6929 .8519 .9936 1.1230 1.2476 1.3664 1.4810
90 .4642 .7332 .9185 1.0781 1.2195 1.3513 1.4731 1.5924 1.7142
95 .5288 7969 .9827 1.1392 1.2780 1.4224 1.5394 1.6580 1.7776
98 .5942 .8590 1.0441 1.2083 1.3453 1.4752 1.6014 1.7238 1.8359
 TABLE 5. (Continued)

Probability 0.813 0.793 0.774 0.754 0.734 0.715 0.695 0.676 0.656
i
Percentile 19 21 23 25 27 29 31 33 35

/I
2 1.2126 1.3130 1.4169 1.5208 1.6231 1.7265 1.8324 1.9490 2.0483
5 1.2907 1.3938 1.5051 1.6038 1.7120 1.8253 1.9260 2.0270 2.1409
10 1.3593 1.4707 1.5737 1.6754 1.7816 1.8924 1.9972 2.1048 2.2096
50 1.5909 1.6996 1.8129 1.9205 2.0 242 3.1333 2.2406 2.3524 2.4589
90 1.8219 1.9379 2.0476 2.1564 2.2664 2.3783 2.4879 2.5882 2.6926
95 1.8933 2.0042 2,1162 2.2200 2.3346 2.4417 2.5486 2.6588 2.7648
98 1.9658 2.0701 2.1731 2.2810 2.3967 2.5028 2.6158 2.7272 2.8352

Probability 0.637 0.617 0.598 0.578 0.559 0.539 0.520 0.500 0.480
i
Percentile 37 39 41 43 45 47 49 51 53
2 2.1683 2.2734 2.3749 2.4847 2.6024 2.7098 2.8247 2.9354 3.0654
5 2.2438 2.3581 2.4587 2.5720 2.6891 2.8019 2.9253 3.0387 3.1631
10 2.3164 2.4255 2.5384 2.6486 2.7626 2.8786 2.9938 3.1195 3.2409
50 2. 5689, 2.6797 2.7936 2.9043 3.0226 3.1397 3.2581 3.3832 3.5091
90 2.8153 2.9233 3.0371 3.1537 3.2766 3.3939 3.5131 3.6325 3.7602
95 2.8814 2.9885 3.1045 3.2229 3.3386 3.4582 3.3785 3.7077 3.8242
98 2.9364 3.0550 3.1698 3.2842 3.4087 3.5204 3.6448 3.7729 3.8980

P
rt
w
 TABLE 5. (Continued)

Probability 0.461 0.441 0.422 0.402 0.383 0.363 0.344 0.324 0.305
i
Percentile 55 57 59 61 63 65 67 69 71
2 3.1917 3.3046 3.4446 3.5689 3.7242 3.8694 3.0046 4.1704 4.3150
5 3.2869 3.4063 3.5395 3.6848 3.8204 3.9784 4.1235 4.2675 4.4321
10 3.3637 3.4911 3.6 227 3.7653 3.9063 4.0592 4.2069 4.3523 4.5176
50 3.6371 3.7671 3.9069 4.0469 4.1934 4.3420 4.5033 4.6654 4.8407
90 3.8986 4.0334 4.1706 4.3120 4.4760 4.6277 4.7922 4.9708 5.1460
95 3.9692 4.1019 4.2452 4.3982 4.5456 4.6979 4.8657 5.0526 5.2309
98 4.0334 4.1653 4.3157 4.4632 4.6 249 4.7800 4.9528 5.1303 5.3164

Probability 0.285 0.266 0.246 0.226 0.207 0.187 0.168 0.148 0.129
i
Percentile 73 75 77 79 81 83 85 87 89
2 4.5039 4.6799 4.8606 5.0644 5.2691 5.4929 5.7181 6.0119 6.3010
5 4.6037 4.7920 4.9740 5.1937 5.3900 5.6288 5.8738 6.1463 6.4507
10 4.6922 4.8823 5.0755 5.2885 5.5038 5.7134 5.9920 6.2769 6.5933
50 5.0232 5.2186 5.4202 5.6503 5.8827 6.1469 6.4210 6.7361 7.0937
90 5.3500 5.5508 5.7746 6.0162 6.2773 6.5557 6.8729 7.2308 7.6443
95 5.4375 5.6442 5.8872 6.1378 6.3887 6.6782 6.9995 7.383.9 7.7956
98 5.5368 5.7574 5.9768 6.2368 6.4968 6.7991 7.1400 7.5373 7.9816
 TABLE 5. (Continued)

Probability 0.109 0.090 0.070 0.051 0.031 0.012

i
Percentile 91 93 95 97 99 101
2 6.5832 6.9889 7.4272 7.9869 a. 7251 10.0342
5 6.7863 7.1637 7.6403 8.2125 9.0332 10.5852
10 6.9291 7.3390 7.8249 8.4565 9.3356 11.0882
50 7.5059 7.9771 8.5817 9.3721 10.5790 13.3698
90 8.1224 8.7277 9.4507 10.4612 12.1743 16.7198
95
98
8.3023
8.5260
8.9397
9.1668
9.7020
9.9624
.
10 7906
11.1430
12.6368
13.1559
17.8425
19.3843

u= 5 n= 7

Probability 0.780 0.686 0.593 0.500 0.407 0.314 0.220

i
Percentile 1 2 3 4 5 6 7
2 .7027 2.6928 3.6509 4.0501 4.4032 4.6336 4.8819
5 1.0464 2.9076 3.7519 4.1632 4.5110 4.7488 4.9693
10 1.4117 3.1039 3.8615 4.2625 4.6050 4.8369 5.0265
50 2.5329 3,6586 4.2764 4.6392 4.8990 5.0502 5.1270
90 3.2990 4.1674 4.7270 4.9541 5.0794 5.1304 5.1424
95 3.4397 4.2943 4.8121 5.0149 5.1043 5.1367 5.1427
98 3.5411 4.4222 4.9068 5.0555 5.1186 5.1397 5.1428
e TABLE 5. (Continued)
t
b
a

v= 5 n = 11

Probability 0.851 0,781 0.711 0.640 0.570 0,500 0.430 0.360 0.289
i
Percentile 1 2 3 4 5 6 7 8 9
2 .3532 1.0725 1.7047 2.3496 2.9804 3.5956 4.1258 4.6201 5.0789
5 .5522 1.3049 1.9747 2.6304 3.2251 3.8024 4.3235 4.8005 5.2557
10 ,7505 1.5480 2.2176 2.8276 3.4307 3.9872 4.4825 4.9567 5.4290
50 1,5785 2.3642 3.0022 3.5471 4.0499 4.5346 5.0267 5.5363 6.0775
90 2.4834 3.1333 3.6519 4.1199 4.5875 5.0817 5.6260 6.2170 6.8339
95 2,6839 3.3003 3.7951 4.2572 4.7208 5.2456 5.8147 6.4274 7.0470
98 2.8983 3.4608 3.9397 4.3849 4.8642 5.4002 6.0032 6.6451 7.2541

Probability 0.219 0 .J49

i
Percentile 10 11
2 5.5617 6.1058
5 5,7554 6,3583
10 5.9305 6.6017
50 6.6879 7.4850
90 7.5048 8.3299
95 7.7186 8.4975
98 7.9546 8.6779
 TABLE 5. (Continued)

v= 5 n= 15

Probability 0. 890 0.834 0.779 0,723 0.667 0.611 0.556 0.500 0.444
i
Percentile 1 2 3 4 5 6 7 8 9
2 .2876 .8227 1.3197 1.7516 2.1863 2.6193 3.0216 3.4012 3.8815
5 ,4296 1.0128 1.5346 1.9823 2.4138 2.8498 3.2759 3.6936 4.1212
10 .5860 1.1980 1.7242 2.1741 2.6158 3.0334 3.4556 3. 8733 4.3113
50 1.256 1 1.9271 2.4288 2.8607 3.2780 3.6748 4.0703 4.4785 4.8774
90 2.0686 2.6249 3.1033 3,4838 3.8559 4.2493 4.6204 5.0085 5.4450
95 2.2718 2.8149 3.2681 3.6574 4.0103 4.3877 4.7548 5.1502 5.5880
98 2,4445 3.0176 3.4290 3.7967 4.1609 4.5341 4.9101 5.2864 5.7579

Probability 0.389 0.333 0.277 0.221 0.166 0.110

i
Percentile 10 11 12 13 14 15
2 4.3540 4.8146 5.2445 5.7065 6.2196 6.8528
5 4.5751 5.0172 5.4571 5.9153 6.4761 7.2046
10 4.7436 5.1688 5.6389 6.1246 6.6977 7.5350
50 5.3126 5.7915 6,3016 6.9018 7.6754 8.8017
90 5.9141 6,4803 7.1107 7.8506 8.8450 10.2645
95 6.0966 6.7013 7.3658 8.1570 9.1650 10.7476
98 6.2824 6.9151 7,6355 8.4759 9.5611 11.1266
w
I&
TABLE 5. (Continued)
00

v= 5 n = 21

Probability 0.923 0.880 0.838 0.796 0.754 0.711 0.669 0.627 0,585
i
Percentile 1 2 3 4 5 6 7 8 9
2 .2313 .6486 1.0321 1.3313 1.6813 1.9433 2.2643 2.5564 2.8698
5 .3547 .8158 1.1900 1.5423 1.8847 2.1457 2.4841 2.7894 3,0943
10 .4752 .9599 1.3586 1.7158 2.2084 2.3685 2.6591 2.9689 3.2623
50 1.0591 1.5668 1.9688 2.3194 2.6419 2.9412 3.2444 3.5329 3.8390
90 1.7149 2.2123 2.5740 2,9145 3.2143 3,5117 3.8061 4.0757 4.3526
95 1.9190 2.3846 2.7440 3.0792 3.3722 3.6442 3.9410 4.2143 4.4770
98 2.0854 2.5608 2,8997 3.2378 3.5031 3.7705 4.0 883 4.3360 4.6135

Probability 0.542 0,500 0.458 0,415 0.373 0.331 0,289 0.246 0.204
i
Percentile 10 11 12 13 14 15 16 17 18
2 3.1914 3.4929 3.8250 4.1930 4.4977 4.8519 5.1987 5.6170 6.0362
5 3.3877 3.6920 4.0199 4.3690 4.7062 5.0588 5.4014 5.7936 6.2268
10 3.5565 3.8717 4.1874 4.5105 4.8597 5.2182 5.5787 5.9916 6.4152
50 4.1311 4.4277 4.7416 5.0766 5.4226 5.7957 6.2053 6.6556 7.1928
90 4,6416 4.9414 5,2720 5,6104 6.0085 6.4333 6.8910 7.4482 8.1344
95 4.7766 5.0894 5.4212 5.7696 6.1763 6.6212 7.1064 7.7108 8.4086
98 4.8965 5.2080 5.5652 5.9323 6.3500 6.8039 7,3408 7.9403 8.7029
 TABLE 5. (Continued)

Probability 0.162 0.120 0.077

i
Percentile 19 20 21
2 6.4675 7.0260 7.7390
5 6.7192 7.3087 8.0972
10 6.9171 7.5596 8.4883
50 7.8307 8.7083 10.1093
90 8.9701 10.1251 12.0978
95 9.3008 10.5160 12.6600
98 9.6678 11.0111 13.2939

v= 5 n = 31

Probability 0.949 0.919 0,889 0.859 0.829 0.800 0.770 0.740 0.710
i
Percentile 1 2 3 4 5 6 7 8 9
2 .1789 .4995 .7904 1.0534 1.2823 1.4975 1.7039 1.9150 2.1266
5 .2683 .6308 .9340 1.2117 1.4474 1.6728 1.8825 2.0753 2.3022
10 .3767 .7619 1.0743 1.3444 1.5911 1.8168 2.0167 2.2416 2.4518
50 .8418 1.2632 1.5778 1.8462 2.0878 2.3252 2.5386 2.7550 2.9599
90 1.4023 1.7907 2.0973 2.3639 2.5785 2.8129 3.0359 3.2336 3.4439
95 1.5487 1.9427 2.2382 2,4928 2.7155 2.9527 3.1626 3.3705 3.5604
98 1.7191 2.0829 2.4023 2.6469 2.8602 3.0889 3.2769 3.5044 3.6957
 TABLE 5. (Continued)

Probability 0.680 0.650 0.620 0.590 0.560 0.530 0.500 0.470 0.440
i
Percentile 10 11 12 13 14 15 16 17 18
2 2.3324 2.5487 2.7395 2.9482 3.1656 3.3808 3.5943 3.8130 4.0561
5 2.5107 2.7050 2.9210 3.1449 3.3360 3.5376 3.7585 3.9939 4.2205
10 2.6659 2.8629 3.0646 3.2793 3.4870 3.6980 3.9013 4.1262 4.3580
50 3.1616 3.3756 3.5795 3.7728 3.9811 4.2005 4.4186 4.6284 4.8593
90 3.6397 3.8445 4.0511 4.2369 4.4381 4: 6620 4.8739 5.0991 5.3356
95 3.7651 3.9630 4.1557 4.3581 4.5652 4.7812 4.9927 5.2270 5.4656
98 3.9013 4.0904 4.2709 4.4842 4.6858 4.9015 5.1178 5.3526 5,5921"

Probability 0.410 0.380 0.350 0.320 0.290 0.260 0.230 0.200 0.171

Percentile 19 20 21 22 23 24 25 26 27

2 4.2676 4.5127 4.7593 5.0166 5.2974 5.5681 5.8641 6.1729 6.5656

5 4.4327 4.6825 4.9409 5.2023 5.4866 5.7700 6.0608 6.3793 6.7470
10 4.5862 4.8349 5.0915 5.3451 5.6238 5.9189 6.2318 6.5646 6.9488
50 5.1037 5.3434 5.6043 5.8822 6.1826 6.5009 6.8598 7.2577 7.7235
90 5.5811 5.8561 6,1356 6.4285 6.7581 7.1512 7.5566 8.0517 b 8.6277
95 5.7236 5.9824 6.2766 6.5852 6.9205 7.3214 7.7683 8.2926 8.9054
98 5.8610 6.1354 6.4385 6.7487 7,1247 7,5506 7.9659 8,5198 9.1672
 TABLE 5. (Continued)

Probability 0.141 0.111 0.081 0.051

i
Percentile 28 29 30 31
2 6.9085 7.3453 7.9361 8.7318
5 7.1441 7.6118 8.2420 9.1631
10 7.3866 7.9174 8.5567 9.6074
50 8.2690 8.9591 9.8739 11.4460
90 9.3595 10.2529 11.5567 13.9791
95 9. G739 10.6lll 12.1099 14.9425
98 10.0425 11.0984 12.7178 15.8378

11 =5 n = 51

Probability 0.971 0.958 0.933 0.914 0.895 0.877 0.858 0.839 0.820
i
Percentile 1 2 3 4 5 G 7 8 9
2 .1385 .4015 .6124 .8026 .9399 1.1144 1.2771 1.4396 1.5767
5 . %171 .4989 .7293 .9130 1.0 887 1.2632 1.4123 1.5593 1.6973
10 .2967 . 6040 .8305 1.0210 1.2013 1.3727 1.5143 1.664G 1.8105
50 .6534 .9803 1.2178 1.4222 1.6020 1.7665 1.9212 2.0611 2.1972
90 1.0939 1.4088 1.6423 1.8304 1.9973 2.1631 2.3161 2.4588 2.5954
95 1.2110 1.5229 1.7630 1.9355 2.1153 2.2824 2,4237 2.5624 2.6962
98 1.3483 1. Belo 1.8836 2.0602 2.2316 2.3841 2.5316 2.6841 2.8091
C’L
en TABLE 5. (Continued)
E3

Probability 0.801 0.782 0.764 0.745 0.726 0.707 0,688 0.669 0.651
i
Percentile 10 11 12 13 14 15 16 17 18
2 1.6932 1.8214 1.9601 2.0787 2.1971 2.3309 2.4647 2.5817 2.7098
5 1.8263 1,9527 2.0 832 2,2120 2,3476 2. M 8 1 2.5945 2,7303 2.8512
10 1.9377 2.0650 2.1899 2.3302 2.4617 2.5861 2.7084 2.8341 2.9598
50 2.3333 2.4740 2.6027 2.7255 2.8547 2.9846 3.1092 3.2319 3.3599
90 2.7328 2.8618 2.9851 3.1145 3.2275 3.3700 3.4875 3.6101 3.7406
95 2.8360 2.9631 3.0893 3.2035 3.3298 3.4563 3.5909 3.7206 3,8367
98 2.9475 3.0622 3.1865 3.3133 3.4484 3.5640 3.7000 3.8182 3.9419

Probability 0.632 0.613 0.594 0.575 0.556 0.538 0.519 0.500 0.481
i
Percentile 19 20 21 22 23 24 25 26 27
2 2.8404 2.9448 3.0608 3.2063 3.3226 3.4656 3.6010 3.7151 3.8542
5 2.9613 3.0927 3.2158 3.3557 3.4693 3.6070 3.7408 3.8578 4.0037
10 3.0840 3.2128 3.3402 3.4624 3.5824 3.7173 3.8475 3.9956 4.1372
50 3.4836 3.6148 3.7353 3.8616 3.9908 4.1242 4.2524 4.3962 4.5324
90 3.8628 3.9858 4.1083 4.2438 4.3738 4.5154 4.6431 4.7880 4.9276
95 3.9640 4.0821 4.2110 4.3440 4.470 1 4.6158 4.7413 4.8842 5.0373
98 4.0625 4.1808 4.3241 4,4437 4.5704 4.7052 4.8506 4.9868 5.1290
 TABLE 5. (Continued)

Probability 0.462 0.444 0.425 0.406 0.387 0.368 0.349 0.331 0.312
i
Percentile 28 29 30 31 32 33 34 35 36
2 3.9927 4.1365 4.2810 4.4366 4.5841 4.7226 4.9026 5.0484 5.2247
5 4.1436 4.2876 4.4273 4.5761 4.7371 4.9012 5.0567 5.1974 5.3813
10 4.2609 4.4040 4.5550 4.6999 4.8606 5.0189 5.1753 5.3373 5.5071
50 4. 6797 4.8157 4.9621 5.1139 5.2654 5.4315 5.5999 5.7804 5.9545
90 5.0684 5.2188 5.3634 5.5196 5.6744 5.8508 6.0394 6.2124 6.4065
95 5.1717 5.3300 5.4659 5.6306 5.7842 5.9656 6.1553 6.3295 6.5384
98 5.2602 5.4216 5,5586 5.7422 5.8973 6.0792 6.2690 6.4801 6.6722

Probability 0.298 0.274 0.255 0.236 0.218 0.189 0.180 0.161 0.142
i
Percentile 37 38 39 40 41 42 43 44 45
2 5.3878 5.5506 5.7598 5.9473 6.1616 6.3952 6.5803 6.8448 7.1099
5 5.5458 5.7257 5.9271 6.1146 6.3565 6.5675 6.7824 7.0532 7.3196
10 5. 6821 5.8726 6.OtiG2 6.2586 6.4853 6.7142 6.9544 7.2121 7.5200
50 6.1416 6.3437 6.5471 6.7732 7.0081 7.2729 7.5534 7.8731 8.2092
90 6. 6271 6 . 8422 7.0801 7.3334 7.6138 7.9187 8.2322 8.5916 9. o m
95 6.7465 6.9872 6. 2329 7.4756 7.7727 8.1076 8.4169 8.8247 9.2644
98 6.8839 7.1364 7.3625 7.6585 7.9387 8.2732 8.6165 9.0283 9.5292
w TA BLE 5. ( Continued)
u1
P P

Probability 0.123 0.105 0.086 0.067 0.048 0.029

i
Percentile 46 47 48 49 50 51
2 7.3985 7.7457 8.1588 8.5566 9.0740 9.9692
5 7.6512 7.9915 8.3677 8.8362 9.4324 10.4095
10 7.8330 8.1915 8.6133 9.1430 9.8351 10.8510
50 8.6128 9.0696 9.6122 10.2944 11.2788 13.0004
90 9.5165 10.0895 10.8880 11.7474 13.2284 16.0655
95 9.7771 10.3976 11.1788 12.2448 13.7944 17.1642
98 10.1198 10.7200 11.5841 12.7129 14.5153 18.2737

y= 5 n = 101

Probability 0.986 0.967 0.947 0.928 0.908 0.889 0. 69 0.85 0.831

i
Percentile 1 3 5 7 9 11 13 15 17
2 .lo49 .4287 .6796 .8923 1.0606 1.2381 1.3778 1.5310 1.6798
5 .1568 ,5171 .7671 .9690 1.1587 1.3243 1.4815 1.6232 1.7774
10 .2171 .5913 .8471 1.0455 1.2310 1.4002 1.5643 1.7115 1.8558
50 .4835 .8708 1,1310 1.3405 1.5197 1.6883 1.8431 1.9896 2.1353
90 .7916 1.1736 1.4256 1.6314 1.8105 1.9742 2.1225 2.2733 2.4209
95 .8938 1.2644 1.5005 1.7136 1.8927 2.0465 2.2053 2.3490 2.4899
98 .9854 1.3493 1.5772 1.7964 1.9701 2.1249 2.2831 2.4337 2.5686
 TABLE 5. ( C ontinued)

Probability 0.811 0.792 0.772 0.753 0.733 0.714 0.694 0.675 0.656
i
Percentile 19 21 23 25 27 29 31 33 35
2 1. a202 1.9547 2.0879 2,2058 2.3342 2.4656 2.6051 2.7130 2.8472
5 1.9056 2.0449 2.1750 2.3084 2.4360 2.5658 2. m a 7 2.8208 2.9413
10 1.9927 2.1239 2.26123 2.3882 2.5195 2.6456 2.7656 2.9025 3.0287
50 2.2790 2.4122 2.5422 2.6780 2. a029 2.9328 3.0641 3.1843 3.3172
90 2.5598 2.6919 2.8249 2.9565 3.0832 3.2132 3.3408 3.4669 3.5936
95 2.6339 2.7686 2.9049 3.0375 3.1585 3.2848 3.4141 3.5329 3.6641
98 2.7093 2. a373 2.9734 3.1078 3.2342 3.3614 3.4802 3.6215 3.7407

Probability 0.636 0.617 0.597 0.578 0.558 0.539 0.519 0.500 0.481
i
Percentile 37 39 41 43 45 47 49 51 53
2 2.9744 3.1028 3.2287 3.3571 3.4866 3.6273 3.7642 3.8936 4.0367
5 3.0719 3.2002 3.3290 3.4611 3.5951 3.7244 3.8543 3.9931 4.1341
10 3.1518 3.2868 3.4149 3.5437 3.6775 3.8166 3,9480 4,0850 4.2295
50 3.4494 3.5754 3.7105 3.8403 3.9724 4.1034 4.2443 4.3838 4.5222
90 3.7249 3.8476 3.9801 4.1148 4.2576 4.3891 4.5305 4.6671 4.8082
95 3.7943 3.9220 4.0618 4.1958 4.3332 4.4646 4.6165 4.7507 4.8864
98 3.8650 3.9996 4.1373 4.2761 4.4079 4.5473 4.6885 4.8197 4.9706
 ~~

F
u1 TA BLE 5. ( Continued)
0,

Probability 0.461 0.442 0.422 0.403 0.383 0.364 0.344 0,325 0.306
i
Percentile 55 57 59 61 63 65 67 69 71
2 4.1712 4.3228 4.4690 4.6294 4.7851 4.9334 5.1097 5.2837 5.4617
5 4.2746 4.4269 4.5748 4.7342 4.8972 5.0429 5.2293 5.3947 5.5767
10 4.3692 4.5173 4.6674 4.8227 4.9790 5.1461 5.3104 5.4828 5.6713
50 4.6664 4.8169 4.9756 5.1306 5.2959 5.4636 5,6396 5.8294 6.0214
90 4.9557 5.1111 5.2688 5.4384 5.6039 5.7725 5.9629 6.1590 6.3630
95 5.0287 5.1823 5.3545 5.5165 5.6818 5.8577 6.0533 6.2446 6.4634
98 5.1151 5.2676 5.4273 5.6007 5.7697 5.9522 6.1385 6.3376 6.5619

Probability 0.286 0.267 0.247 0.228 0.208 0.189 0.169 0.150 0.131
i
Percentile 73 75 77 79 81 83 85 87 89
2 5.6456 5.8440 6.0240 6.2416 6.4744 6.7340 6.9918 7.2639 7.5841
5 5.7653 5.9556 6.1673 6.3844 6.6322 6.8790 7.1502 7.4187 7.7563
10 5.8632 6,0605 6.2807 6.5014 6.7501 7.0045 7.2720 7.5586 7.9137
50 6.2166 6.4288 6.6649 6,9030 7.1599 7.4316 7.7455 8.0818 8.4615
90 6.5765 6.7957 7,0520 7.3107 7.5902 7.8823 8.2220 8.6338 9.0475
95 6.6750 6.8911 7.1601 7.4356 7.7041 8.0119 8.3637 8.7758 9.2327
98 6.7844 7.0226 7.2647 7,5628 7,8192 8.1492 8.5273 8.9497 9.4292
 TABLE 5. (Concluded)

Probability 0.111 0.092 0.073 0.053 0.033 0.014

i
Percentile 91 93 95 97 99 101
2 7.9329 8.3301 8.8038 9.3655 10.2155 11.5337
5 8.1359 8.5389 9.0412 9.6756 10.4997 12.1101
10 8.3022 8.7090 9.2551 9.9231 10.8147 12.6454
50 8.9077 9.4101 10.0375 10.9013 12.1630 14.9807
90 9.5837 10.1899 11.0004 12.0587 13.7662 18.4422
95 9.7979 10.4497 11.2754 12.3882 14.1558 19.6405
98 10.0162 10.6897 11.5586 12.7606 14.7997 21.0941
 -~

TABLE 6. SELECTED PERCENTILE VALUES O F SELECTED

STATISTICS FOR TESTING NORMALITY

From the expected line of best f i t these are:

1. The maximum absolute difference.

2. The mean sum of squares of residuals.
3. The number of runs above and below the line.
From the computed Chi-square values in each sample, these are:

4. The runs above and below the median X 2 value.

v= 1 n = 11
Runs Runs
Above o r Above o r
Below Below
Frob . MAD NSSR Line Median
0.01 0.0648 0.0241 1 7
0.02 Q. 0693 0.0266 1 1

0.05 0.0834 0.0347 1 3

0.10 0.0936 0.0450 2 2

0.70 0.1104 0.0598 7 2

0.30 0.1263 0.0750 2 3

0.40 0.1413 0.0968 2 3
0.50 0.1544 0.1252 3 3
0.60 0.1694 0.1576 3 3
0.70 0.1868 0.2222 7 '3
0 .E30 0.2037 0.3426 3 4

0.90 0.2302 0.5619 4 4

0.95 0.2567 0.mr37 4 A

0 .?8 0.273 1 0.9987 5 4

0.99 0.78P0 1.2486 5 4

158
 TABLE 6. (Continued)

v= 1 n = 21

RUZM Runs
Above o r Above or
q elow Belqw

I
Frob . KSSR Line Median
n nrl
Q.015g z
0.9191 3

0.WTO 7

n. 0348 3
0.043? A

0.0648 A

0.m27 5
0.50 0.?014 5
0.60 0.1347 5
c) .?O 0.1A69 6
9,so 0.3062 6

0.90 0.7946 7

'0.95 0,6208 7

(3.98 0.8860 7
0.99 1.9c)?jE3 8

159
 TABLE 6. (Continued)

u= 1 n = 31

Runs
Above o r
Frob . MAD MSSR
Below
Line
0.01 .
0 0487 0.01 53 3
0.02 0.0539 0,0196 3
0*05 0,06?4 0.0251 4
0. I O 0.0703 .
0 0342 4
0.20 0.0809 0.0464 5
0.30 .
0 0899 0.0594 6
0.40 0.1004 0.0777 6
0.50 0.1080 0.0941 7
0.60 0.1168 0.1719 7
0.70 0.1263 0.1403 8

0.80 0.1429 0.1973 8

0.90 0.1660 0.3297 9

0.95 0. I828 0.4999 10

0.98 0.1980 0.7333 10

0.99 0.2138 0.9393 10

160
 TABLE 6 . (Continued)

v= 1 n = 51

Runs Runs
Above or Above o r
Below Below
Prob . MAD MS %
6 Line Median
0.01 .
0 0448 0.0133 5 8
0 002 0.0481 0.0154 6 9
0.05 0.0533 0.0210 7 10

0.10 0.0598 0.0292 8 11

0.20 0.0676 .
0 0388 9 12

0.30 0 -0747 0.0495 10 12

0.40 0,0821 0.0610 11 13

Q.50 .
0 0900 0.0746 11 13

0.60 0.0951 0.0903 12 13

0,70 0.1076 0.1122 13 14

0.50 0.119s 0,142’3 14 15

0.90 0,1357 0 . I991 15 15

“095 0,1685 0.2K882 15 IC;

0.98 0.1639 0.4267 16 17

0.?9 0.1796 0.5136 16 17

 TABLE 6. (Continued)

v= 1 n = 101

Rllr! s
Above o r
prob . M 47) MS SR
Relow
Line
0.07 0.0320 0.0078 8
0.03 0.9755 0.01 24 IO
0.05 .
0 04.07 0.0156 12
0. I Q n. 0460 0.0216 15
0.20 0.0512 0.0712 18
0.70 0,0551 0. onm 30

n. An 0.0599 0.0469 21
0.50 0.0648 0.06l)l 23
0.60 0.0708 0.0719 24
0.70 0,0778 0.9R9'7 25
0.m 0.0860 n.1140 36;

O8 q o 0.0947 A. 1531 27
O8'5 n, I O 5 0 0.2047 ?8

fl. 1186 0.2689 39

n. 99 n.17?9 0.3314 30

162
 TABLE 6. (Continued)

v= 2 n = 11

91m5
Above o r
Below
Frob. MAD F'ISSR T, i n e

0.01 0,0298 i

9on2 0,0686
0.05
9, I O .m97 n, 0 7 d A 2

0.70 0.0999 7

0.7JO n.1120 0.1315 2

0.40 Q.1270 9. I 4 6 2 7

Q.50 0.1353 0.4718 3

0.60 0,1490 0.2131 7
9.70 fl. 1646 .
0 267s 3
0.80 0.17?3 0.3508 1.
0.90 0,3068 0.5175 4

0.95 0.2260 0.6752 A

Q.98 0,3545 0.945f; 5

0.S9 0.2700 I , 1528 5
 TABLE 6. (Continued)

v= 2 n = 21

Rims
Above OT
Below
WAD Tjine

0.0533 2 3
0.0568 ’3 7
0.0669 ’3 4
0.0766 4
0. W76 4 5
n.0qm 4 5
n, 1067 5 5

0.1141 5 6
0.72”52 5 6
0.1367 6 6
0.1527 6 6
0.1761 7 7
0.1976 7 7
0.2166 7 8

0.77j-l7 8 8
 TABLE 6. (Continued)

v= 2 n = 31

Runs r1n5
Above o r Above o r
Below R e 1o w
Prob . MAD M SSR Line Me d i a?.?.

0.01 .
0 0485 0.0280 3 5
0.02 0.0539 0.0308 4 5

0.05 0.0625 0.04’35 5 6

fl.10 0.0692 0.0554 6 6

0.70 9. 0796 0.0764 6 7

0.70 0.0971 .
0 0986 7 7

0.40 0. 0958 0.1184 7 8

0.50 0.1033 0.1409 7 8

0.60 0.1127 0.1672 a 8

0.70 0,1245 0.1945 a 9

0.80 0.1359 0.2544 9 9
0.9 0.1524 .
0 3727 10 10

0.95 0.1676 0.5813 10 10

0.98 0.1860 0.9384 10 11

0.99 0.1902 1.0859 11 I?

 TABLE 6. (Continued)

y= 2 n = 51

Runs
Above o r
Prob . MAD MS SR
Below
Line

0 -01 0.0447 0.0176 6

0.02 0.0478 0.0301 7

0.05 0.0525 0.0365 8

0.10 0.0573 0.0455 9

0.20 .
0 0662 0.0599 10

0.30 0.0717 0.0773 11

0.40 0.0774 0.0924 11

0.50 0.0848 0.1161 12

0.60 0.0901 0.1367 12

0.70 .
0 0976 0,1693 13

0.80 0.1089 0.2074 14

0.90 0.1249 0,3509 15

0.95 0.1’195 0.4701 15
0.98 0.1?87 .
0 6287 16
0.99 0.1700 .
0 7902 16

166
 TABLE 6 . (Continued)

v= 2 n = 101

Prob . wsS?
0.n1 O.(-US5

0.02 0.0230
Q.05 0.Q273
3.10 Q.9344
0.20 0,(3452

0.30 0.9554

'1.42 0.0682

0.50 (3.9809

0.m 0.0977
0.70 9.1 I 7 1

n.w n. lAS6

0.99 0. ?flA5

n.95 ?* 79fi5
n. Q q 0 '3977

f?,9 0. 1
 TABLE 6. (Continued)
V= 3 n = ,11

Runs Runs
Above o r Above o r
3elow Below
Prob. MAD MS SR Line Median
0.01 (3.0576 0.0392 I 1
0.03 0.0667 1
0.05 0.077’3 2

0.10 0.0877 3

0.20 0.1000 0.1261 2 2

0.30 0. I108 0.1561 3 ’5

0,40 0.1189 0.1804 3 3
0.50 0.1272 0.2156 7 3
0.60 0.1’374 0.2603 7 7
0.70 Q. 1504 0,3185 4 3
0.80 0.1641 0.3876 4 4
0.90 0,1874 0,5340 4 4
0.95 0.7107 0.6700 4 4
0.98 0,2292 0.92OO 5 5
0.99 0.24d 4 1.0202 5 5
 TAaLE 60 (Continued)

v= 3 n = 21
Runs Runs
Above o r Abore o r
Below Below
Prob . MAD r4ssR Line Median
0.01 0.0526 .
0 0406 2 3
0.m 0.9590 0.0508 3 3
(7.05 0.0668 0 9684 3 4

Q. 10 .0709 0,0877 4 4

0,20 0 ; 0827 0.11w 4 5

?. 39 0.0?26 0. j373 5 5
0.40 0.1O35 9.1696 5 5
? * 50 0.1113 0 ,7097 5 6
17 .60 fl. 1186 0,2612 6 6

0 70 . (?. I 3 8 4 n.3438 6 6

fi.81) no Id39 Q.441 I 5 6

0.90 0,1637 0,6489 7 ?

0.?5 O.I817 .
0 7800 7 7
0.98 .
0 2004 1.1000 7 8

0.99 0.2187 .
I 2000 8 8

169
 .

Rims Runs
Above or Above o r
Below Below
Prob, MAD MSSR Line Median
0.01 0.0482 0.0323 4 5
0 002 0.0530 0.0499 5 5
0.05 0.0594 0.0661 5
0. I O 0.0657 0.05’32 6
0,20 0.0759 0. I039 6
0.30 0.0839 0.1511 7
0.40 0.0925 0.15’34 7
0.50 0.0983 0.1894 s
0.60 0.1057 .
0 ?344 8 8
0.70 0 . q148 0.2920 9 9
0.80 0.1355 0.3674 9 9
0,90 0.1445 0,5384 10 IO

0.95 0.1569 .
0 7285 10 -lo
0.98 0,1784 1,0554 10 71
0.99 .
0 1836 I,I781 11 11

170
 TABLE 6. (Continued)

v= 3 n = 51

Runs Runs
Above o r Above o r
Below Below
MAT) M3 SR Line Median
0.0446 0.0283 7 9
0.0466 0.0’365 8 9
0.0512 0.0514 9 10

0.0572 0.@637 10 11
0.0648 0.0521 10 12
0.0703 0.1023 11 72
0.0758 0.7266 12 13
OaOs27 0.1477 13 13
0,0882 0.1764 17 113
0.0941 0.2118 74 14
0.1049 0.3733 14 15
0.1172 0.387 75 15
0.1716 (3.5992 16 16

0.1517 0.7507 76 17
0.1618 0.841 1 97 17
TABLE 60 (Continued)

v= 3 n = 101

Runs
Above o r
Re low
ine
IJ

??

15
18

A?

?I
??

23
24

34

25
26

28

75

39

30
 TABLE 6. (Continued)

v= 4 n = ll

Runs
Above or
Below
Prob, NAD MSSFi Line
0.01 0.0520 .
0 0485 1

0.02 0,0648 0.0602 2

0.05 0.0741 0.m10 2

0.10 .
0 0787 0.1087 2

0.20 0.0905 .
0 1497 2

9.30 0.1929 0.1783 3

r) .no 0.1122 r) 2350 '3

q.5@ 0.1237 9.7684. 7

0.m n, 1735 0 3305 3
0.70 0, I A A 6 0.375 1 4
0.m .
n -t609 0,4500 A

0.90 Q. 1809 'I.55364 d

' 8 95 I), 1989 9.7376 4

0.99 0.?208 0.939 3 5
0.99 0.2303 .
-l 0078 5
 TABLE6. (Continued)

v= 4 n = 21

Runs Runs
Above o r Above o r
Below Below
Psob . MAD MSSR Line Median
0.01 0.0488 .
0 0444 3 3
0.02 0.0561 0.0708 3 3
0.05 0.0651 0.0953 A. 4
0.10 0.0746 0.1164 4 4
.
0 2.0 0.0842 0.1540 5 5
0.'70 0.0979 0.1930 5 5
0.40 0.1012 0.2256 ' . 5 5
0.50 0.1106 .
0 2798 5 6
0.60 0.118fj 0.'7278 6 6

0.70 0.1297 .
0 3994 6 6
0.80 0.1408 0.4948 7 6
0.90 0.1596 .
0 6788 7 7
0.95 0.1757 .
0 8309 7 7

0.98 0. j 9 8 2 I.2311 8

9.99 0.2198 .
1 3964 8
 TABLE 6. (Continued)

v= 4 n = 31

Runs Runs
Above o r Above o r
Below Relow
Prob. MAD MSSR Line Median
0.01 0.0480 0.0550 4 5
0.02 0.0518 0.0614 5 5
0.05 0.0584 0.0797 5 6
0.10 0.0654 0. ?074 6 6
0.30 0.0750 0.1355 6 7
0.30 0.C879 0 . I669 7 7
9.40 0.0894 0.1987 7 8

0.50 .
0 0970 .O. 2346 8 8

0.60 0.1044 0.2817 8 8

0.70 0.1120 0.75 54 9 9
0.80 0.1227’ 0.4456 9 9
0.90 0.1416 0.6983 10 lU
0.95 0.1568 0.9180 10 10

0.98 0.7654 1.1949 10 11

0.99 0.1762 7.7856 I1 71

17 5
 Proh . MAD

0.01 0.Oh I 8
0.02 0.0453
0.05 0.0494
0. I O 0.0544
0.20 0.0627
0.70 0.0fjp2

0.40 0.0745

0.50 0.0791
0.60 0.0858

(3.70 0.0941
0.80 0.1028

0.90 0.1164

0.95 0.1289

0.98 0.1500

0.99 0.1581

176
TABLE 6. (Continued)

v= 4 n = 101
 TABLE 6. (Continued)

v= 5 n = 11

Piin s
9bova or
RPI ow
Woh. MAP rww JIi n e
9.01 o.nF;or, n. 0687 1

n.02 0.9583 0.m73 7

n. 05 0.0705 0.1704- 2
0.I O 0,0780 n. IA-’lA 2

0.30 0.9898 0.1941 2

O.m 0.100’; 0.7583 3

0.40 0.1090 0.7791 3
0.50 0.1184 0.3166 3
0.60 0.1298 0 7794 3
0.70 0.IT79 0 4497 A

0.80 I). 1504 0,5678 4

O.?O 9.171 1 0.6883 4

0.95 0.1876 0.8678 5

0.98 0.706n .
7 0428 5
n.99 0.77 14 I.1’767 5

178
TABLE 6. (Continued)

v= 5 n = 21

Rlli-!S
Above o r
R e 1.o?:
?!?SF Line
0.0714 3
9.090d 3
n. ID68 4

9.16(3?3 4
0.1913 5
9.2121 5
q.7500 5
0.7880 5
(3.35’37 6

0.4238 6

n. 5676 7
0.7500 7

0,9515 7
1.2848 8

7.4530 8
 TABLE 6. (Continued)

v= 5 n = 31

Runs
Above o r
Prob . MAD MS SR
Relow
Line
0.01 0.0674 0.0569 4
0.03 0.0515 0.07 75 5
0.05’ .
0 0580 0.0971 5
0.10 0.0653 0.1262 6
Q,20 0.1554 7
n o 30 0.079’7 0.1936 7
0.40 .
0 0854 0.21325 8
0.50 0.2721 8
O.h@ 0.0996 0.3397 8

0.70 0.1075 0.3982 9

0.80 0.7783 0.5254 9
0090 0.1’326 0.7851 10
0.95 0.1451 1.028’3 10
0.98 0,1604 1 * 5767 11
0 099 .
0 1708 1.8290 1.1

180:
TABLE 6. (Continued)

v= 5 n = 51

m1ns Ins
?.I
Above 0'1: Above or
BeLow Below
TLne Median
7 9
8 9
9 10

10 11

I1 77

72 I?
72 1'3

75 77

17 13
14 14

14 15
15 15
?6 IC;

16 17
17 17
 TABLE 6 . (Concluded)
,

v= 5 n = 101

Runs Runs
Above o r Above o r
Re 1.0w
Prob . MAD MSSQ ’I,i n e
R e low
Median
0.91 0,0322 (-1 .9375 16 19
n. 02 .
0 0342 0.0434 17 20

0.95 0.9385 0,9543 I? 21

o..l0 o,m14 0,0669 30 22
0.70 0.0477 O.r)8?8 37 ?S
Q. 70 0.0529 0,I ? 0 6 ?3 24
n,AO 0.(3566 n..r’175 ?4 35

0,w 0 .0609 0.15A1 24 26

0,60 0,0654 0,1832 35 26

0.70 .
0 0708 0.3312 ?6 37
0.80 .
0 0760 0.7989 27 27
0.90 0.0860 0.3960 29 28

0.95 0.0914 0,5609 29 29

0.98 0.1011 0.6627 30 30
0.99 0.1047 0.7989 30 71

18 2
 TABLE 7.' MAXIMU31 ABSOLLTE DIFFEEEKCE
AR,R.AYED BY DIhIENSIONS
n = 11

Prob . 1 Dim,
0.01 0.0648

0.02 0.0693
0.05 0.Q374
0.10 0.0936

0.20 0. 1104

0.70 0.1263

0.40 0.1413

0.50 0.1544
0.60 0.1694

0,79 0.1868

0 .BO 0,2037

0.90 0.2'302

0.95 9,2567
I?. 98 0,273 1
0.99 .
0 289.c,
 TABLE 7. (Continued)

n = 21

Prob. 1 Dim. 2 Dim. 3 Dim. 4 Dim. 5 Dim.

0.01 0.0519 0.0532 0.0526 0.0488 0.0523

0.02 0.0605 0.0568 0.0590 0.0561 0.0564

0.05 0.0736 0.0669 0.0668 0.0651 0.0647

0. I O 0.0788 0.0766 .
0 0709 0,0746 0.0701

0.20 0.0953 0 .O876 O.oS27 O.CB42 0.0799

0.30 0.1054 0.0960 0.0926 0.0939 0.0873

0.40 0.1140 0.1062 0.1035 0.1012 .

0 0946

0.50 0.1233 0.1141 0.11 13 0.1106 0.1031

0.60 0.1333 0.1232 0.1186 0.1186 0.1120

0.70 0 1444 0.1367 0.1284 0.1297 0.1219

0.80 0.1620 0.1527 0.1439 0.1408 0.1364

0.90 0.1839 0.1761 0.1637 .

0 1596 0.1514

0.95 0.2087 0.1976 0.1812 0.1757 0.1671

0.98 .
0 2350 0,2166 .
0 2004 0.1982 0.17~8

0099 0.2485 0.2317 0.2187 0.2108 0.1846

184
 TABLE 7. (Continued)

n = 31

-.
- I
7 -.
Prob. 1 Dim. 2 Dim, 3 Dim. A Dim. ~ I - - -

0.01 0.0487 0.0485 0.0482 0.0480 T‘,f- t--

0.02 0.0539 0,0539 0.0530 0.0519 ? ,’

-.---
0.05 0.0624 0.0625 0,0594 e.0534 1 I --

0.13 0.0703 0.0692 0.0657 0.0554 3.26~‘

0.20 o.mo9 0,0796 0.0759 0.0750 \?.17”15
m

0.30 6.9899 0.0371 0.0339 0.0829 r.?7”-7

0.40 0.1004 0.0958 0.0925 0.0894 T: .(’??A

0.50 0.1080 0.1033 0.0983 0.0970 G . L ? ? ~3

0.60 0.1168 0.1127 0.1057 0.1044 O.

- J:G~
C ?

0.70 0.1263 0.1245 0.1148 0.1120 ‘I.$.C:75

0 -80 0.1429 0.1359 0.7255 0. 1227 1?5?

0.90 0.1660 0.1524 (3.1445 0.1416 PJ326

0.95 0. 1 9 2 8 0.1675, 0.1569 0.1568 f g e 4 ‘ - ) I r i;

0.98 0.1980 0.1860 0.1784 0.1654 O , I * 7:

0.2138
- ,- 7

0.99 0.1902 0. 1 8 3 6 5.1762 -

I ’

3 ,j
 TABLE 7, (Continued)

n = 51

€'rob. 1 Dim. 2 Dim. 3 Dim. 4 Dim. 5 Dim.

0.01 0.0448 0.0447 0.0446 0.0418 0.0390

0.02 0.0481 0.0478 0.0466 0.0453 0.0451

0.05 0.0533 0.0525 0.0512 0.0494 0.0497

0.10 0.0598 .
0 0573 0.057 2 0.0544 0.0545

0.20 0.0676 0.0662 0.0648 0.0627 0.0619

0.30 0.0747 0.0717 0.0703 0.0692 0.0674

0.40 0.0821 0.0774 0.0758 0.0748 0.0735

0.50 0.0900 0.0848 0.0827 0.0791 .

0 0798

0.60 0.0951 o.ogo1 .

0 0882 0.0858 0.0859

0.70 0.1074 0.0976 0.0941 0.0941 0.0923

0.80 0.1193 0. IO89 0.1049 0.1328 0. I021

0.90 0.1357 0.1249 0.1172 0.1 164 0.1131

0.95 0.1485 0.1395 0.1316 .

0 1289 0.1227

0.98 0.1639 0.1587 0.1517 0.1500 0.1363

0 099 0.1796 .
0 1700 0.1618 0.1581 0.1429 ,

186
 TABLE 7. (Concluded)

n = 101

2 Dim. 5 Dim. 4 Dim. 5 Dim.

0.0320 0.0324 0.0329 0.039 0.0322
0.0355 0.0339 0.0339 0.0342 0.0342
0.0407 0 -039 1 0.0378 0.0390 0.0385
O.OA6O 0.0435 0.0438 0.0422 0.0d14

0.05 12 0.0493 0.0488 0,0476 0.0472

0.0551 0.0525 0.0530 0.0525 0.05 29
0.0599 000575 0.0573 0.0576 0.0566
0.0648 0.0623 0.0618 O.O6l8 0,nc;og
0.0708 0.(3669 0,0666 0.0667 9.0654
0.07 78 0.0726 0.07 17 0.0719 0 0708
0

0.0860 0.m03 0.0790 0.0774 0.0760

0.094? 0.0924 0.0893 0.0852 O.@I60

0. 1050 0.0995 0.0963 13.0924 0.09j4

0,1186 0.1067 0.1052 0,1022 0.107 1
0,1229 0.1151 0.1104 0. IO61 0.1047
TABLE 8. MAXIMUM ABSOLUTE DIFFERENCE
ARRAYED BY SAMPLE SIZE
v= 1

21 31 51 10 I

0.0645 0.0519 0.0487 0.0448 0.0320

0.0697 0.0605 0 0579 0.0481 0.0355

0.0874 0.0736 0.0624 0.0533 0.0407
0,0976 0.0788 0.0703 0,0598 0,0460
0,1104 0.0953 0.0809 0.0676 0.0512
0.1?63 0,1054 @om99 0,0747 0.0551
0.141'3 0.1140 0.1004 0.0821 0.0599
0. I544 0.1233 .
0 1080 .
0 0900 .
0 0648
0.1694 0. 1333 0.1168 0.0951 0 0708
0.1868 0.1444 0.1263 0.1074 0.0778
0.3037 0.1620 0.1429 0.1 193 0.0860
0.2702 0.1839 0.1660 0.1357 .
0 0947

0.2567 0.208 7 0.1828 0.1485 0.7050

0.2771 0.2350 0.1980 0.7639 9.1186
0.2890 0,2485 0.21'38 0.1796 0 1225,
 TABLE 8. (Continued)

y= 2

21 31 51 101

0.0606 0.0532 0 a485 0 ,0447 0.0324

0.0686 0.0568 0.0539 0.0478 0.Q339

0.0806 0,0669 0.0625 0.0525 0.0391

0.0897 0.0766 0.0692 0.0573 0.0435

0.1026 0.E376 .
0 0796 0.0662 (3.0493

0.1120 0.0960 0.0871 0.0717 0.0525

0.1230 0. I O 6 2 0.0958 0.0774 0.0575

0.13C;T 0.1141 0.1033 0,0848 0.0623

0.1490 0.1732 0.112’7 0.0901 0.0669

0.1546 0.1367 0.1245 0.0976 0.0726

Q. 1793 0.1527 0.1359 0.1089 0.m03

0.2068 0.1761 0.1524 0.1249 0.0924

0.2260 0,1976 0.1676 0.1395 0.0995

0.2545 0.2166 0.1860 0.1587 0.1067

0.2700 0.2317 0.1902 0.1700 0.1151

 TABLE 8. (Continued)

v= 3

Prob. I1 21 31 51 101

0,OI 0.0576 0.0526 0.0482 0.0446 0.0329

0.02 .
0 0667 0.0590 0.0530 .
0 0466 .
0 0339
(3.05 0.0773 0,0668 0.0594 0.0512 0.0378

0.10 0.0877 .
0 0709 0.0657 0.0572 0.0438
0.30 0.1000 0.0527 0.0759 0.0648 0.0488

(3.30 0.1108 0.0926 0.0839 0.0703 0.0530

0.40 0.1189 0.1035 0.0925 0.0758 0.0573

.
0 50 0.1272 0.1 113 0.0983 0.0827 0.0618

O.%O 0.1’374 0.1186 0.1057 0.0882 0.0666

0.70 0.1504 0.1284 0.1148 0.0941 0.0717

0.80 0 . 1 6 ~ 00 1439 0.1255 0. IO49 0.0790

0.90 0,1874 0.1%37 0.1445 0.1172 o.os93

0. 95 0.2107 0. If312 0.1569 0.1316 0.0963

0.?8 0.2292 .
0 2004 0.1784 0.1517 0.1052

0.99 0.2444 0.2187 0.1836 0.1618 0.1104

 TABLE 8. (Continued)

v= 4

21 31 51 10 1

0.0520 0.0488 0.0480 0.0418 0 ,03F?

0.9648 0.0561 0.0518 9.0453 0.0342

0.074 I 0.0651 .
0 0584 .
0 0494 0.0390
0.0787 0.0746 0.0654 0.0544 0.0422
0.0905 0.0842 0.0750 0.0627 0.0476
0.1029 0.0939 0.0829 0.0692 0.0525
0.1122 0,1012 .
0 0894 .
0 0748 0.0576
0.1231 0.1106 0.0970 0.0791 0.0618

0.1325 0.1186 0.1044 0.0858 0.0667

0.1446 0.1297 0.1120 0.0941 0.0715
0.1608 0.1408 0.1227 0.1028 0.077.1.

0.1809 0.1596 0.1416 0.1164 O.cB52

0.1989 0.1757 0.1568 o.im 0.092L

0.2208 0.1382 0.1654 0.1500 0.1022

0.2303 0.2108 0.1762 0.1581 0.106 I

 TABLE 8. (Concluded)

v= 5

Prob . 11 21 31 51 101

0.01 0.0500 0.0523 0.0474 0.0390 0.0322

0.02 0.0582 0.0564 0.0516 0.0451 0.0342 i

0.05 0,0703 0.0647 0.0580 .
0 0497 .
0 0385

0.10 .
0 0780 0.0701 0.0653 0.0545 0.0414 1
0.20 0.0898 0.0799 0.0725 0.0619 .
0 0472

0.30 0.1003 0.0873 0.0793 0.0674 0.0529

0.40 0.1090 0.0946 0.m54 0.0735 0.0566

0.50 0.1 I S 4 0.1031 0.0930 0.0798 .

0 0609

0.60 0.1298 0.1120 0.0996 0.0859 0.0654

0.70 0.1379 0.1219 0.1075 0.0923 .

0 0708

0.80 0.1504 0.1364 0.1183 0.1021 0,0760

0.90 0.1711 0.1514 0.1’326 0.1?31 0.0860

0.95 0.1876 0.1671 0.7451 0.1227 0.0914

0.98 0.2060 0.1788 0.1604 0.1763 0.1Oll

0.99 0.21 14 .
0 1846 0.1708 0.1429 0.1047

192

I
 TABLE 9. (Continued)

Prob. 1 Dim. 2 13S.m. 3 !limo 4 Dim.

0.01 0.0150 0.0226 0.0406 0.0444
0.02 0.0191 0.0313 .
0 0508 .
0 0708

0.05 0.0250 0.0465 0.0684 O.Og53

0.10 0.0348 0.0589 0.0833 0.1164

0.20 0.0482 0 .O798 0.1158 0.1540

0.30 0.0648 0.1094 0.1373 0.1930

0.40 0.0823 0.1368 0.1696 0.2256

0.50 0.1014 0.1636 0.2097 .

0 2798

0.60 0.1243 0.1941 0.2612 0.3278

0.70 0.1469 0.2433 0.3438 .

0 3994

0.80 0.2062 0.3140 0.4411 0.4948

0.90 0.346 0.4711 0.6489 0.6788

0.95 .
0 6208 0.71 16 .
0 7800 0.8309

0.98 0.8860 1.0265 1. 1000 1.2311

0.99 .
1 00’78 1.1209 1. 2 0 0 0 .
1 3964

194
 TABLE 9. (Continued)

Frob. 1 Dim. 2 Dim. 3 Dim. 4. D i m . 5 Dim.

0.01 0.0153 0.0280 0.0323 0.0550 0.0569

0.02 0.0196 0,0308 .

0 0499 0.0614 0.0775
0.05 0.025 1 0.0435 0.0661 0.0797 0.0971
0.10 0.0342 0.05 54 0.0832 0.1076 0.1262.

0.20 0.0464 0.0764 0. 1 0 3 9 0.1355 0.1554

0.30 0.0594 0.0986 0.1311 0.1669 0.1936

0.40 0.0777 0.1184 0.1534 0.1987 0.2325

0.50 0.0941 0.1409 0.1894 0.2346 0.2721

0.60 0.1119 0.1672 0.2344 0.2817 0.3397

0.70 0.1403 0.1945 0.2920 0.3554 0.3982

0.80 0.197’3 0.254d 0.7674 0.4456 0.5254

0.90 0.3297 0.3727 0 5384

0 0.6983 0.7851

0.95 0.4999 0.5813 0.7385 0.9180 1.0283

0.98 0.7373 0.9884 1.0554 1.1949 1.5767

0.99 0.939’3 1.0850 1.1781 1.3856 1 .E3290

19 5
 ; TABLE
.,. , 9. ,
(Continued)
,. -

n = 51
..

Proh. 1 Dim. 2 .Dim. 3 Dim. 4 nim. 5 Dim.

0.01 0.0173 0.0176 0.0283 0.0378 0.0537
-9
0.02 0.0154
., 0.0301 0.0365 0,0466 0.0635
I t . . I

0.0’5 0.0210 0.0363 0.0514 0.0645 0.0758

0.10 0.0292 0.0455 0.0637 0.0786 0.1002
0.20 0.0788 0.0599 0.0871 0.1018 0.1394
0.30 0.0495 0.0777 0.102’; 0.1251 0.1617
0.40 0.0610 0.0924 0.1266 0.1515 0.1903
0.50 0.0746 0.1161 0.1477 0.1790 0.2253
0.60 0.0903 0.1367 .
0 1764 0.2187 .
0 2734
0.70 0.1122 0.1693 0.2118 0.2648 0.3401
0.80 0.1423 0.2074 0.2773 0.3522 0.4719
0.90 0.1991 0.3509 0.3887 0.5517 0.6605

0.95 0.2582 0.4701 0.5992 0.7519 0.8283

0.98 0.4267 0.6287 0 7507 1.0198 1.1094

0.99 0.5136 0 7902 0.8411 1.0927 1.3017

196
 TABLE 9. (Concluded)

n = 101

Prob. 1 Dim. 2 Dim. 3 Dim. 4 Dim. 5 Dim.

0.01 0.0078 0.018S 0.0276 .

0 0298 0.0375
0.02 0.0124 0.0230 0.0306 0.0365 0.0434

0.05 0.0156 0.027’3 0.0365 0.0430 0.0543

0.10 0.0216 0 . 0344 0.0440 0.0529 0.0669

0.20 0.0’312 0.0452 0.0561 0.0694 .

0 0878

0.30 0.0400 0.0554 0.. 0 7 2 3 .

0 0874 0.1106

0.40 0.0469 0.0682 0.0867 n.1038 0.1325

0. SO 0.0601 o.wo9 0.1021 0.1276 0.1541

0.60 0.0719 0.0977 0.1216 0.1512 0.1837

0.70 0.oF393 0.1171 0.1580 0.1840 0.2312

0.80 0.1140 0.1456 0.2048 .

0 2328 .
0 2989

0 -90 0.1521 0.2065 0.3041 0.3529 0.3960

0.95 0.2042 0.2865 0.3876 0.4650 0.5609

0.98 0.2689 0.3837 0.6191 0.6746 0.6627

0.99 0.3314 0.4421 0.7258 .

0 7690 0 7989

197
 TABLE 10. MEAN SUM SQUARES RESIDUALS
ARRAYED BY SAMPLE SIZE
v= 1

Prob . 11 21 31 51 101
0.01 0.0241 0.0?50 0.0153 0.0133 0.0078

0.02 0.0266 0.0191 0.0196 0.0154 0.0124

0.05 0.0347 0.0250 0.0251 0.0210 0.0156

0.10 0.0450 0.0348 0.0342 0.0292 0.0216
0.20 0.0598 0.0482 0.0464 0.0388 0.0312
0.30 0.0750 0.0648 0.0594 0.0495 0.0400
0.40 0.0968 0.0823 0.0777 0.0610 0.0469
0.50 0.1252 0.1014 0.0941 0.0746 0.060 1
0.60 0.1576 0.1243 l 119
0
: 0.0903 0.0719
0.70 0.2222 0.1469 0.1403 0.1122 0.0893
0.80 0.3426 0.2062 0.1973 0.1423 0.1140
0.90 0.5619 0.3846 0.3297 0.I991 0.1521

0.95 0.8082 0.6208 0.4999 0.2582 0.2042

0.98 0.9987 0.8860 0.7333 0.4267 0.2689
0.99 1.2486 1.0038 0.9393 0.5136 0.3314

198
 TABLE 10. (Continued)

v= 2

Prob . 11 21 31 51 10 1

0.Orl 0.0298 0.0226 0.0280 0.0176 0.0185

0.02 0.0448 0.0313 0.0308 0.0301 0.0230
0.05 0.0589 0.0465 0.0435 0.0363 0.0273
0.10 0.0744 0.0589 0.0554 0.0455 0.0344
0.20 0.0999 0.0798 0.0764 0.0599 0.0452
0.30 0.1215 0.1094 0.0986 0.0773 0.0554
0.40 0.1462 0.1368 0.1184 0.0924 0.0682
0.50 0.1718 0.1676 0.1409 0.1161 0.0809
0.60 0.2131 0.1941 0.1672 0.1367 0.0977
0.70 0.2679 0.2433 0,1945 0.1697 0.1171
0.80 0.'3508 0.3140 0.2544 0.2074 0.1456
0.90 0.5175 0.4711 0.3727 0.3509 0.2065
0.95 0.6752 0.7116 0.5813 0.4701 0.2865
0.98 0.9456 1.0265 0.9884 0.6287 0.3833
0.99 1.1528 1.1209 1.0850 0.7902 0.4421

199

I
 TABLE 10. (Continued)

v= 3

Prob. 11 21 31 51 101
0.01 0.0392 0.0406 0.0323 0.0283 0.0276
0.02 0.0493 0.0508 0.0499 0.0365 0.0306
0.05 0.0725 0.0684 0.0661 0.0514 0.0365
0.10 0.0932 0.0833 0.0832 0.0637 0.0440
0.20 0.1261 0.1158 0.1039 0.0821 0.0561
0.30 0.1561 0.1373 0.1311 0.1023 0.0723

0.40 0.1804 0.1696 0.1534 0.1266 0.0867

0.50 0.2156 0.2097 0.1894 0.1477 0.1021
0.60 0.2602 0.2612 0.2344 0.1764 0.1216
0.70 0.3185 0.3478 0.2920 0.2118 0.1580
0.80 0.3876 0.441 1 0.3674 0.2733 0.2048
0.90 0.5340 0.6489 0.5384 0.3887 0.3041
0.95 0.6700 0.7800 0.7285 0.5992 0.3876
0.98 0.9200 .
1 1000 1.0554 0.7507 0.6191
0.99 1.0202 1.2000 1.1781 0.8411 0.7258

200
 TABLE 10. (Continued)

v= 4

Frob. 11 21 31 51 10 1

0.01 0.0485 0.0444 0.0550 0..O378 0.0298

0.02 0.0602 0.0708 0.0614 0.0466 0.0365
0.05 0.0810 0.0953 0.0797 0.0645 0.0430
0.10 0.1087 0.1164 0.1074 0.0786 0.0529
0.20 0.1497 0.1540 0.1355 0.1018 0.0694
0.30 0.1783 0.1930 0.1669 0.1251 0.087LF
0.40 0.2250 0.2256 0.1987 0.1515 0.1038
0.50 0.2684 0.2798 0.2346 0.1790 0.1276
0.60 0,3208 0.3278 0.2817 0.2187 0.1512

0.70 0.376 1 0.3994 0.3554 0.2648 0.1840

0.80 0.4500 0.4948 0.4456 0.3522 0.2328

0.90 0.5964 0.6788 0.6987 0.5517 0.3539

0.95 0.7376 0.8309 0.9180 0.7519 0.4650
0.98 0.9293 1.3311 1.1949 1 .Ol98 0.6746
0.99 1.0078 1.3964 1.3856 1.0927 0.7690

201
 TABLE 10. (Concluded)

v= 5

Frob . 11 21 31 51 10 1

0.01 0.0685 0.0714 0.0569 0.0537 0.0375

0.02 0.087'3 0.0904 0.0775 0.0675 0.0434

0.07 0,1204 0.1068 0.0971 0.0758 0.0543

0.10 0.1454 0.1405 0.1263 0.1002 0.0669

0.20 0.1941 0.1813 0.1554 0.1294 .

0 0878

0.30 0.2382 0.2121 0.1976 0.1617 0.1106

0.40 0.2791 0.2500 0.2325 0.1903 0.1325
0.50 0.3166 0.2880 0.2721 0.2253 0.1541
0.60 0.3794 0.3537 0.3397 0.2734 0.1832
0.70 0.4497 0.4238 0.3982 0.3401 0.2312
0.80 0.5678 0.5676 0.5254 0.4719 0.2989

0.90 0.6882 0.7500 0.7851 0.6605 0. '3960

0.95 0.8638 0.9515 1.0283 0.8283 0.5609

0.98 1A 4 2 8 1.2848 1.5767 1.1094 0.6627

0.99 1.1367 1.4530 1.8290 1.3017 0.7989

202
 TABLE 11. RUNS ABOVE OR BELOW LINE OF CHI-SQUARE VERSUS PROBABILITY (1-p) I
ARRAYED BY DIMENSIONS
n = 11 n = 21

I
Prob. 1 Dim. 2 Dim. 3 Dim. 4 Dim. 5 Dim. Prob. 1 Dim. 2 Dim. 3 Dim. 4 Dim. 5 Dim. !

0.01 1 1 1 1 1 0.01 2 2 2 3 3
I
0.02 1 1 2 2 2 0.02 2 3 3 3 3
0.05 1 2 2 2 2 0.05 3 3 7 4 4 I
I

0.10 2 2 2 2 2 0.10 3 4 4 4 4
I
0.20 2 2 2 2 2 0.20 4 4 4 5 5
i
0.30 2 2 3 3 3 0.30 4 4 5 5 5 i
0.40 2 3 3 3 3 0.40 5 5 5 5 5
:
0.50 3 3 3 3 3 0.50 5 5 5 5 5 i
0.60 3 3 3 3 3 0.60 5 5 6 6 6
I
9.70
0.80
3
3
3
4
4
4
4
4
4
4
0.70

0.80
6
6
6
6 -
6
6
6
7
6
7
i
i
0.90 4 4 4 4 4 0.90 7 7 7 7 7
0.95 4 4 4 4 5 0.95 7 7 - 7 7 7
0.99 5 5 5 5 5 0.98 7 7 7 8 8
to
0
w 0.99 5 5 5 5 5 0.99 8 8 8 8 8
I
1!
 N TABLE 11. (Continued)
0
A

n = 31 n = 51

?rob. 1 Dim. 2 Dip. 3 Dim. 4 Dim. 5 Dim. %-ob. 1 Dim. 2 Dim. 7 Vim. 4 Dim. 5 Dim.
0.01 3 3 4 4 4 0.01 5 6 7 7 7
0.92 3 A 5 5 5 0.07 6 7 8 8 8

0.05 4 5 5 5 5 0.05 7 8 9 51 9
0.10 d 6 6 6 6 0.10 8 9 10 10 10

0.70 5 6 6 6 7 0.20 9 IO 10 11 11

0.30 6 7 7 7 7 0.30 10 11 11 11 12

0.40 6 7 7 7 8 0.40 11 11 12 12 12

0.50 7 7 8 8 8 0.50 11 12 13 13 13
0.60 7 8 8 8 8 0.60 12 12 17, 1'; 13

8 8 9 9 9 0.70 13 13 14 14 14
0.80 14 14 14 14 14
8 9 9 9 9
9 10 10 10 10 0.90 15 15 15 15 15

0.95 10 10 10 10 10 0.95 15 15 16 16 16

0.98 10 10 10 10 11 0.98 16 16 16 16 16
10 I1 1I 11 11 0.99 16 16 I? 17 I?
 TABLE 11. (Concluded)

n = 101

0.01 8 I1 12 14 16
0.02 10 13 15 16 17
0.05 12 15 18 18 19
0.10 15 18 I9 20 70

0.m 18 30 21 32 73

0.70 ?O 21 22 77, 37

0.40 31 13 23 24 24-
0.50 23 34 26 21 3 4-

0.60 24- 2A 24 35 25
0.70 25 25 25 26 36
0.80 26 26 36 27 ?7
0.90 27 ?8 38 38 ?9
(3.95 28 28 28 29 39
0.9s 29 29 39 30 '30
0.w 530 70 70 70 30

20 5

_. ... . ~ .. ... ~
 TABLE 12, RUNS ABOVE OR BELO’S,’ LIKE O F CHI-SQUARE VERSUS PROBABILITY (I-p)
ARRAYED BY SAMPLE SIZE
v= 1 v= 2
n n

Prob. 11 21 31 51 101 Prob. 11 21 31 51 101

0.01 1 2 3 5 8 0.01 1 2 3 6 11

0.02 1 2 3 6 10 0.02 1 3 4 7 13

0.05 1 3 4 7 12 0.05 2 3 5 8 15

0.10 2 ’3 4 8 15 0.10 2 4 6 9 18
0.20 2 4 5 9 18 0.20 2 4 6 10 20
0.30 2 4 6 IO 20 0.30 2 4 7 11 21.
0.40 3 5 G 11 21 0.40 3 5 7 11 23
0.50 ’3 5 7 11 23 0.50 3 5 7 12 24
0.60 3 5 7 12 24 0.GO 3 5 8 12 24
0.70 3 6 8 13 25 0.70 3 6 8 13 25
0.80 3 6 8 14 26 0.80 4 6 9 14 26
0.90 4 7 9 15 27 0.90 4 7 10 15 28

0.95 4 7 10 15 28 0.95 4 7 10 15 28
0.98 5 7 10 16 29 0.98 5 7 10 16 29
0.99 5 8 10 16 30 0 -99 5 8 11 16 30
 TABLE 12. (Continued)

v= 3 v= 4
n n
Prob. 11 21 31 51 101 Prob. 11 21 31 51
0.01 1 2 4 7 12 0.01 1 3 4 7
0 002 2 3 5 8 15 0.02 2 3 5 8

0.05 2 3 5 9 18 0.05 2 4 5 9
0.10 2 4 6 10 19 0.10 2 4 6 10 20
0.20 2 4 6 10 31 0.20 2 5 6 11 22
0.30 7 5 7 11 22 0.30 3 5 7 11 23
0.40 3 5 7 12 23 0.40 3 5 7 12 24

0.50 3 5 8 13 24 0.50 3 5 8 13 24

0.60 3 6 8 13 24 0.60 3 6 8 13 25
0.70 4 6 9 14 25 0.70 4 6 9 14 26
0.8Q 4 6 9 14 26 0.80 4 7 9 14 27
0.90 4 7 10 15 28 0.90 4 7 10 15 28 -

0.95 4 7 10 16 28 0.95 4 7 10 16 29 .
0
t\3 0.98 5 7 10 16 29 0.98 5 8 10 I6 30 !
4 i
i
0.99 5 8 11 17 30 0.99 5 8 11 17 30 I
i
 TABLE 12. (Concluded)

v= 5

Prob . 11 21 31 51 101

0.01 1 3 4 7 16

0.02 2 3 5 a 17
0.05 2 4 5 9 19
9.10 2 4 6 10 30

0.30 2 5 7 11 22

0.70 3 5 7 12 23

0.40 3 5 8 12 24

0.50 3 5 8 13 24

0.60 3 6 8 I3 25

0.70 4 6 9 14 26

0.80 4 7 9 14 27

0 090 4 7 10 15 29
0.95 5 7 10 16 29
0.98 5 8 11 16 30
0.99 5 8 11 17 30

208
 TABLE 13. RUNS AEOVL OR BELOW MEDIAN CHI-SQUARE ARRAYED BY DIMENSIONS

n = 11 n = 21

Frob. 1 Dim. 2 Dim. 3 Dim. 4 Dim. 5 Dim. ’ Frob. 1 Dim. 2 Dim. 3 Dim. 4 Dim. 5 Dim.
0.01 1 1 1 1 1 0.01 3 3 3 3 3
0.02 1 1 1 1 1 0.02 3 3 . 3 3 3
0.05 2 2 2 2 2 0.05 4 4 4 4 4
0.10 2 2 2 2 2 0.10 4 4 4 4 ..- 4
0.20 2 2 2- -2 ’ 3 0.20 5 5 5’ 5 :’ 5
0.30 3 ’1 3 3 3 0.70 5 5 5 5 . 5
0.40 3 3 3 3 3. 0.40. 5 5 5 5 - 5
0.50 3 3 3 3 3 0.50 5 6 6 6 6
0.60 3 3 3 3 3 0.60 6 6 . 6 6 6
0.70 3 3 3 3 3 0.70 6 6 6 6 6
0 .83c) 4 A 4 4 4 0.80 6 6 6 6 6
0.90 4 4 4 A 4 n.90 7 7 _.’ 7 7 7
0.95 4 4 4 A 4 0.9 7 7 7 7 7
c.3
g 0.98 fi 4 5 5 5 0.qQ 8 8 8 8 8
-
ci I I

5 -1 ,4? c: e 5 s ici
1.3 TABLE 13. ( Continued)
+
c
n = 31 n = 51

Prob. 1 Dim. 3 Dim. 7 Dim. 4 Dim. 5 Dim. Prob. 1 gin. 3 Dim. 3 Dim. A Dim. 5 Dim,
9.01 5 5 5 5 5 r>.o? 8 9 9 9 9
0.02 5 5 5 5 5 0.w 9 10 9 9 9

0 095 6 6 6 6 6 9.05 10 10 10 I9 10

0.10 6 6 6 6 6 0.10 11 11 11 11 17

0.70 7 7 7 7 7 0.70 12 12 12 12 12

0.70 7 7 7 7 7 0. '30 12 12 12 22 22

0.40 8 8 8 8 8 0.40 13 13 13 13 13

n,50 8 8 8 8 8 0.50 13 13 13 13 13

0.60 8 8 8 8 8 0.60 13 13 13 13 13

0.70 9 9 9 9 9 0 070 14 14 14 14 14

0.m 9 9 9 9 9 0.80 15 15 15 15 25

0.90 in -lo 10 10 10 0.90 15 15 15 15 15

0.s5 I0 10 IO 10 10 0.95 16 16 16 I6 16

0.98 11 I1 I1 11 11 0.98 17 17 17 17 17

0 .99 11 11 11 11 11 0.99 17 17 17 17 17
 TABLE 13. (Concluded)

n = 101

Frob . I Dim. 2 Dim. 7 Dim. 4 Dim. 5 Dim.

0.91 19 19 19 ,t9 19

0.93 70 20 20 20 20
n.05 21 71 21 21 21
0.10 72 22 22 22 22
'1.20 73 77 77 '7 23

Q. 30 74 74 34 ?d ?4
Q.dO 75 75 35 75 75
0.50 26 76 26 76 ?6

0.60 26 76 36 76 76

0.70 27 ?7 37 27 27

0.m 28 28 28 27 27

0.90 29 29 29 28 28

0.95 TO 70 so 29 29
0.98 70 70 70 x, '30

0.99 '31 31 31 71 31
N
w
N

v= 1 v= 2
n n

Proh . 11 21 31 51 101 Prob. 11 21 31 51 10 1

0.01 1 3 5 8 19 0.01 1 3 5 9 IC

0.02 I 7 5 9 20 0.02 1 3 5 10 20
0.05 3 4 6 10 21 0.05 3 4 6 10 21
0.10 7 4 6 11 ?2 0.10 2 4 6 11 22
0.20 3 5 ? 12 23 0.20 2 5 7 12 23
0.70 3 5 7 12 24 0.30 3 5 7 12 24
0.40 7 5 8 13 25 0.40 7 5 8 13 25
0.50 3 5 8 13 26 0.50 3 6 a 13 26
0.60 3 6 8 13 26 0.60 3 6 8 13 26
0.70 3 6 9 14 27 0.70 3 6 9 14 27
0.80 4 6 9 15 28 0.80 4 6 9 15 28
0.90 n 7 70 15 29 0.90 4 7 10 15 29
0.95 4 7 IO 16 30 0.95 4 7 10 16 30
0.98 4 8 11 17 '30 0.93 4 8 11 17 30

0.99 4 8 I1 17 31 0.99 5 8 11 17 31
 TABLE 14. (Continued)

v= 3 v= 4
n n
Prob . 11 21 31 51 10 1 Prob . 11 21 31 51 10 1

0.01 1 3 5 9 19 0.01 1 3 5 9 19
0.02 1 T 5 9 20 0.02 1 3 5 9 20
0.05 2 4 6 10 21 0.05 2 4 6 10 21

0.10 2 4 6 11 22 0.10 2 4 6 11 22

0.20 2 5 7 12 23 0.20 3 5 12 23

0.30 3 5 7 12 74 0.70 3 5 12 24

0.40 7 5 8 13 25 0.40 3 5 13 25

0.50 3 6 8 13 26 0.50 6 8 13 36
0.60 3 6 8 13 26 0.60 6 8 13 26
0.70 '3 6 9 14 27 0.70 6 9 14 27
0.80 4 6 9 15 28 0.80 6 9 15 37
0.90 4 7 10 15 29 0.90 7 10 15 28

0.95 4 7 10 16 30 0.95 7 10 16 29

'5 8 11 17 30 0.98 8 11 17 30
w
0.99 5 8 11 17 31 0.99 8 11 17 31
 TABLE 14. (Concluded)

v= 5

Prob . 11 21 31 51 10 1

0.01 1 3 5 9 19
0.02 1 3 5 9 20
0.05 2 4 6 10 21
0.10 2 4 6 11 22
0.20 2 5 7 12 23

0.30 3 5 7 12 24
0.40 3 5 8 13 25
0.50 3 6 8 13 26
0.60 3 6 8 13 26
0.70 3 6 9 14 27

0.80 4 6 9 15 27
0.90 4 7 10 15 28
0.95 4 7 10 16 29
0.98 5 8 I1 17 '50
0.99 5 8 11 17 31

214
TABLE 15. NUMBER O F n-tant RUNS TO BE EXPECTED FOR THE x OR X 2
VECTOR FROM MULTIVARIATE NOR;MAL (Wishart) DISTRIBUTIONS.
(The number of grand samples is 14. The number of samples in
each grand sample is 100. The number in each sample is n.
M x is the maximum noted. Mn is the minimum noted.
R is the length of runs. M is the mean. v is dimension.
Percentage of total possible is cumulative.)

v= 2
R 1 2 3 4 5
n 11
Mx 8 9 10 10 10
M 7 8 9 9 9
Mn 6 7 7 7 7

n 15
Mx 10 13 14 14 14
M 9 12 12 12 12
Mn 8 9 70 70 10

n 21
Mx 13 18 19 20 20
M 12 15 16 16 16
Mn 11 13 13 13 13

n 25
Mx 16 21 22 22 23
M 14 18 19 19 19
Mn 13 16 16 16 16

n 31
Mx 21 27 29 30 31
M 18 22 23 23 23
Mn 16 19 19 19 19

n 51
Mx 33 43 45 46 47
M 30 36 38 38 38
Mn 26 31 31 31 31

n 101
Mx 60 80 86 88 89
M 56 71 74 75 75
Mn 46 48 49 49 49

21 5
 TABLE 15. (Continued)

'v= 3

R 1 2 3 4 5
n 11
Mx 9 11 11 11 11
M 9 10 10 10 10
Mn 8 9 9 9 9

n 15
Mx 12 14 15 15 15
M 12 13 13 13 13
Mn 11 11 11 11 11

n 21
Mx 17 20 20 21 21
M 16 18 18 18 18
Mn 15 16 16 16 16

n 25
Mx 20 23 24 24 24
M 19 21 22 22 22
Mn 19 20 20 20 20

n 31
Mx 25 28 29 29 29
M 24 26 26 26 26
Mn 22 23 23 23 23

n 51
Mx 41 47 48 49 49
M 39 43 44 44 44
Mn 34 39 39 39 39

n 101
Mx 82 95 97 98 98
M 76 85 86 86 86
Mn 71 75 75 75 75

216

-_.--. -,..m-111,-- ,II I I 11111 I II 111111I I I 1111 I I 1.11 I I II II I II I1111111111

 TABLE 15. (Conthued)

v= 4

R 1 2 3 4
L
5
n 11
Mx 10 11 11. 11 11
M 10 10 10 10 10
Mn .9 9 9 9 9

n 15
Mx 14 15 15 15 15
M 13 14 14 14 14
Mn 13 12 12 12 12

n 21
Mx 19 20 21 21 21
M 18 19 19 19 19
Mn 17 18 18 18 18

n 25
Mx 23 24 24 24 24
M 22 23 23 23 23
Mn 21 21 21 21 21

n 31
Mx 27 29 30 30 30
M 27 28 28 28 28
Mn 26 27 27 27 27

n 51
Mx 45 51 52 52 52
M 43 46 46 46 46
Mn 41 42 42 42 42

n 101
Mx 88 97 99 99 99
M 85 91 91 91 91
Mn 82 85 85 85 85

217
 TABLE 15. (Concluded)

v= 5

R 1 2 3 4 5
n 11
Mx 10 11 11 11 11
M 10 10 10 10 10
Mn 10 10 10 10 10

n 15
Mx 14 15 15 15 15
M 14 14 14 14 14
Mn 13 13 13 13 13

n 21
Mx 19 20 20 20 20
M 19 19 20 20 20
Mn 19 19 19 19 19

n 25
Mx 23 25 25 25 25
M 23 23 23 23 23
Mn 21 21 22 22 22

n 31
Mx 28 31 31 31 31
M 28 29 29 29 29
Mn 26 27 27 27 27

n 51
Mx 47 50 51 51 51
M 46 47 47 47 47
Mn 44 44 44 44 44

n 101
Mx 93 99 99 99 99
M 91 93 93 93 93
Mn 87 88 88 88 88

218

... .
TABLE 16. TESTS O F BIVARIATE NORMALITY O F 24 HOUR HURRICANE MOVEMENTS FOR SELECTED
5 DEGREE LATITUDE-LONGITUDE QUADRANGLES. (The null hypothesis is that the data set is
not different from the bivariate normal. The probability rejection level is 0.04. This
provides the central 0.96 probability confidence band. The latitude is the
southern boundary while the longitude is the western
boundary of the quadrangle.)

n
Prob
0.02
. 25
70
N
W
25 N
75 w
20
70
N
W
Prob.
0.98
1 0.0038 0.1000 0.O5OO 0.0700 0 5964
2
3
4
0.0434
0 . 1357
0,2748
0.1400
0.3900
0.4300
..
0.1800
0 4800
1 0700
.
0.1300
0 2700
0.33OO
.
0 -8957
1 1878
1.4686
5
6
0.4453
0.6673 .
0.9700
1 9500
1.7100
1.7300 ..
0.40oO*
0 7300
1.7761
2.1073
7
8
0.9598
1.3795 .
2.2400
2 7800
..
2.0100
2 6400
0 8800*

.
1.3600*
2 04964
3.0618
10
11
9 1.8699
2. 5097
3.1223
.
3.0200
3 0700
4.9200
2 7300

.
2.9800
4 4600
2 3700

.
3 5400
9 9600*
3.8671
5,2501
7.0900

MAD 0.0686 0.1420 0.1570 0.2364 0.2545

MSSR 0.0448 0.1170 0.1040 2.7642* 0 9456
RAL 1 2 1 1 5
RBL 1 2 2 1 5
, RAM 1 3 4 1 4
RBM 1 3 4 2 4
,x2
1 ( 5 Lf.)
0 7500
0.0200
8.0000
0.1562
14 .oooo*
0.0156*
10.0000
0.0752
13.3900
0.9800
I

MN Mx
QS 7 9 8 8 9
 .-
TABLE 16. (Continued)

n
Prob.
0.02
25 N
79 w
25 N
75
20 N
70 W
Prob
0.98
.
1 0.0019 0.0700 0.0500 0.1100 0.3285

3
4
2 0.0213
0.0616
0.1123
0.1100
0.1600
0.1600
0.1000
0.1400
0.4200
.
0.2200
0 2300
0.2500
0.5026
0.6648
0.8128
5
6
' 0,1843
0.2543
0 , 1900
0.3500
0.5500
0.5900
0.2700
0.2900 .
0.9529
1 1048
7.
8
,- 0.3478
0.4476
0.3500
0.7400
0.8300
0.8400 ..
0.3500
0 5000
1.2439
1.4068
9 0.5556 0.8100 0.9200
. ..
0 7200
0 7500
1.5809

..
10 0.6865 0.9800 0.9600 1.7734
11
12
0.8204

.
0.9642
1 0800
1 1300
0 9800

.
1.0100 .
0 7800*
0 9200"
1 9724
2.1938
13
14
1 1393
1 3559
2.1700
..
2.6600
1 3800
1,8900
..
1.0300*
1 2400* .
2 04309
2 6905
15
16
1.5815
1.8562
2 6900
2 8000
2.0100
2.2100
1 3700*
1 ,6300* .
3.0152
3 3822
3.8829
...
1.7200*
17
18 .
2.1404
2 5244 ..
2.9400
3 0900
3 2200
..
2.5900
3 4400
5 2000
2 2200*
3 0700
4.5287
5.4153
19
20
21
2.9358
3.4594
4.1453
6 3900
7.8700
. 6.6500
7.2900
4 6500
17.7100*
7.0227
10.0008

MAD 0.0568 0.1582 0.1488 0.1951 0.2166

MSSR 0.0313 0.3696 0,3563 6.7677* 1.0265
RAL 3 3 3 2* 7
RBL 3 2" 3 l* 7
RAM 3 6 6 5 8
RBM 3 5 5 5 8

X2
( 5 d.f.)
0.7800
0.0200
.
13 5000*
0.0191*
4.5000
0.4799
13.0000
0.0233
13.3900
0.9800

mT Mx
18
QS 13 15 15 11*

220
 .._.. ... . ...
I

TABLE 16. (Concluded)

n
Prob
0.02
. 25
70 W
N 25
75 'VJ
N 20
70 Id
N Prob.
0.98
1
2
0.0011
0.0135
0.1100
0.1300
0.01 00
0.0300 .
0.1200
0 2700
0 ..2800
0.2298
0.3465
0.4459
,

3
4
0.0374
0.0'735
0.1400
0.1600
0,1600
0.1900 0,2800 .
0 5492
5
6
0.1109.
0.1562
0.2000
0.2200 .
0.3700
0 4000
0.4900
.
0.2goo
0 3000
0.3100
.
0.6544
0,7426
0.8464
7
8
0.2122
0.2616
0.2400
0.3000 .
0 5600 0.3200 0.9462
9
10
0.3165
0.3873
0.3508
0.4500
0.7400
0.8600 ..
0.3400
0 4900 .
1.0557
1 1684
I1
12
3.4551
0.5268
0.6700
0.6800 .
0.8900
1 0400
0 5600
0.7000
1.2765
1 3798
13
14
0.6126
0. 7055
0.7900
0.9100 .
1.0900
1 2300
0.8000
0.8300
1,4936
1.6093
15
16 .
0.7973
0 8884
0.9900
.
1.1600 ..
1 3000
1 3200
0.8600

.
0.9600
1.7366

.
1.8732
17
18 .
0.9950
I 1286 1.5900
1 4200
1.
..
1 3600
1 5200
0 9600*
1.0300*
2 0201

.
2.1723
19
20
1 2599

1.
1.3902
.
6400
2.1500 .
1 6800
1 7500 .
1.0800*
1 1500*
2 3486
2.5411
21
22 . 5476
1 7408
2 2900
2.2900
1.7700
..
2.2900 .
1.4600*
1 4900"
2.7154
2.9443
23
24
1.8924
2.0703 .
2.3100
2 4200
2 3200
2 4800
1.6100*
1.8900* .
3.1920
3 4906
25
26
2.2944
2.5674
2.6600
2.9100 .
2.6000
3 7200 ..
2.0400*
2 4500*
3.8412
4.2423
27
28
2.8517
3.1373 .
3.8600
4 5800
3 7900
3.8900 .
3 6200
3 9200
4.7526
5 4034
29
30
31
3.5722
4.03 4
4.6812
5. 5900
9 .0400
7.8200 ::2::2
8.1500
3:t:::
.
18 6 7 0 0 "
$:
11.5191
3%?
MAD
MSSR
0.0539
0.0308
0.12 10
0.3653
0.0688
0.1140
0.1876*
4.6122"
0 1860
0 9884
..
RAI; 4 3" 4 3* 10
RBL 4 2" 4 2* IO
RAM 5 8 8 6 11
RBM 5 7 8 7 11

X2
(5 d . f . )
0.7500
0.0200
.
9 3300
0.0965
3.3300
0.6487
.
15 3 300*
0.0090*
13.3900
0.9800

f4N MX
QS 19 23 22 22 27

221

I
TABLE 17. TESTS O F BIVARIATE NORMALITY O F 24 HOUR HURRICANE MOVEMENTS FOR
SELECTED 5 DEGREE LATITUDE-LONGITUDE QUADRANGLES. (The null hypothesis
is that the data set is not different from the bivariate normal. The probability
rejection level is 0.04. This provides the central 0.96 probability
confidence band. The latitude is the southern boundary while the
longitude is the western boundary of the quadrangle.)

n
Prob
0.02
. 80
25 N
W
30 N
75 w
20 N
75 TnT
Prob
0.98
. .

1 0.0038 0.2300 0.0500 0.2000 0.5964

2 0.0434 0.4800 0.2200 0.3000 0.8957
3 0.1357 0.6600 0.3400 0.4100 1.1878
4 0.2748
..
0.8700
1 5900
0.6700 0.6900

.
1.4686

...
5 0.4453 0.8400 0.8200 1.7761
6 0.6673
.
1 6300 1 3400 0 9000 2.1073

..
7 0.9598 1 8800 1 7300 0.9400* 2 4964
8
9
1.3795
1.8699 .
2.1400
3 4000
2 2600
2.9800
1 0700*
1 9400
3.0618
3.8671
10
11
2.5097
3.1223
3.4500
3.6900
3.7500
5.8100 .
5.0500
7 6900*
5. 250 1
7.0900

MAD
MSSR
RAL
0.0686
0.0448
1
0.1307
0.1640
2
0.0968
0.1262'
1 2
.
0.2503
1 2925*
0.2545
0.9456
5
RBL 1 2 1 1 5
RAM 1 3 4 3 4
RBM 1 2 3 3 4
X2
( 5 d.f.)
0.7500
0.0200
10 .oooo
0.0752
.
4 0000
0 5444
.
12 0000
0.0348
13 3900
0.9800

MN
QS 7 ll* 7 9
 ..-

TABLE 17. (Conthued)

n
Prob
0.02
. 25 N
80 W
30N
75 w
20 N
75 w Pr3b*
0. 8

1
2
0.0019
0.0213
0.0700
o.ogo0 ..
0.1000
0 3000
.
0 0900
0.1600
.
0 3285
0.5026
3
4
0.06 16
0.1123
0.1800
0.4200
0 3800
0.3800 .
0.1900
0 3700
0.6648
0.8128
0.4900
0.6100
7
5
6
0.1843
0.2543
0.3478
.
0 5.500
0 5900
'0.8000
0.7500
0.7800
0.4300
0.4900
0.9529
h, 1048
1 2439
0.6300 .
8 0.4476 1.0500 0.8500 0.5700 1.4068
9
10
11
0.5556
0.6865
0.8204
1.1200
1.1700
1.2500
0 .9700
0.9600
0.9900
0.7300
0.7300*
1 5809
1.7734
1 9724
12
13
14
.
0.9642
1 1393
1.3559
1.2500
1.4100
1.8300
1
1
..oooo*
0.9900
6800
1.0600
1.3300*
1.2000
2.1938
2.4309
2.6905
15
16
1.5815
1.8562 .
2.2300
2 2900 3.2400
2.1300
3.2900
1.3800*
.
1.8800
3.0152
3.3822
17
18
19
2.1404
2. 5244
2.9358
.
3.3800
3 5200
3. 9400
.
3 7700
5.1800
1 9700"
4.3200
5 4500"
3.8829
4.5287
5.4753
9.3000
20 3.4594 5.1400 5.6600 7.2300" 7.0227
21 4.1453 7.6600 5.9900 10.0008

MAD
MS SR
RAL
0.0568
0.0313
3
0.0848
0.1464
5
.
0.1972
0 2056
3
0.1942
1.1197*
2*
0.2166
1.0265
7
RBL 3 4 3 1" 7
RAM 3 5 7 4 8
RBM 3 4 6 4 8

X*
(5 d.f.)
0.7500
0.0200
.
7 0000
0.2206
.
1 1 5000
0.0423
10.5000
0.0622
13.3900
0.9800
MN Mx
QS 13 11* 12* 16 18

223

I
 TABLE 17. (Concluded)

Prob. 25 N 30 N 20 N Prob.
n 0.02 80 W 75 w 75 w 0.98
1 0.0011 0.0100 0.1100 .
0 1900 . 0 2298
2
3
4
0.0135
0.0374
0.0735
0.0700
o.ogo0
0.1600
0.1500
.
0.3900
0 4000 ...
0.2400
0 3000
0 3800
;
..
0.3465
0 4459
0 5492

. ...
5 0.1109 0.2000 0.4500 0 5900 0.6544
6 0.1562 0.3800 0 5200 0 6500 0.7426
7 0.2122 0.4700 0.7200 0 6500 0.8464
8
9
0.2616
0.31 65
0.5200
0.5600
0.7200
.
0 7700 .
0 6900
0 7600
0.9462

.
1.0557

..
10 0.3873 0.9500 0.8600 0.9500 1 1684

..
1 1800
11
12
0.4551
0.5268 1 2500
0.8700
.
0.9800
0.9800
1 0000 .
1.2765
1 3798
13
14
0.6126
0. 7055 ..
1.2800
1 3300
1 0700
1.1500
1 0500
..
1.0600 .
1.4936
1 6093
15
.
0.7973
0 8884
1 4800
. 1.1500 1 0900 1.7366

...
1 2800
16
17
18 .
0.9950
1 1286
1 4900

..
1.5100
1 6200
1.2700
1 2300
1 2400
..
1 3500
1 4400
1.8732
2.0201

19 1 2599
.
1 6800 1 4700
. 1.4900 .
2.1723
2 3486
20
21
22
1.3902
.
1 5476
1 7408
.
1 8200
1 9600
2.2600
1 4900
1 5700
2.0000
1.5700
1.9100
2.1300
2.5411
2.7154
.
2 9443
23 1.8924 2.3100 2.0100 2.3100 3.1920
24 2.0703 3.3200 2.6300 2 5400 3.4906
25
26
2.2944
2.5674
3 5900
3.6600 .
2 6900
3 8000 .
2.7100
2 7800 .
3.8412
4 2423
27
28
2.8517
3.1373 .
3.6700
4 0000 .
3.9700
4 2400
3.0300
4.8900
4.7526
5.4034
29
30
31
3.5742
4.0364
4.6812
4.1300
5.1800
7.9400
.
6.1400
6 9500
7.1100
.
5.1500
6 4400
8.4100
6.3863
7.9955
11.5191

MAD 0.0539 0.1017 0.1122 0.0988 0.1860

r m SR 0.0308 0.0827 0.1940 0.1666 0.9884
RAL 4 3* 2* 2* 70
RBL 4 3* I* l* 10
RAT4 5 8 8 7 11
RBM 5 7 8 8 11

x2 -
( 5 d.f.)
0.7500
0.0200
10.8000
0.0752
10.0000
0.0752
.
6 6700
0.2466 .
13.3900
0 9800
m MX
QS 19 24 19 22 27

224
 REFERENCES

Adelfang, Stanley I., 1970: “Standard Deviation of Turbulence Velocity Components Over
Flat AridTerrain,” Journal of Geophysical Research, Vol. 75, No. 33, pp. 6864-6867.

Adelfang, Stanley I., 1970: Personal communications providing data of three-dimensional

winds.

Anderson, R. L. and T. A. Bancroft, 1952: Statistical Theory in Research, McGraw-Hill

> I
Book Company, Inc., New York, NY.

Anderson, T. W., 1958: An Intoduction. to Multivariate Statistical Analysis, John Wiley

and Sons, Inc., New York, NY.

Anderson, T. W., 1966: “Some Non-Parametric Multivariate Procedures Based on

Statistically Equivalent Blocks,” Multivariate Analysis, Edited by P. R. Krishnaiah,
Academic Press, New York, NY., pp. 5-27.

Andrews, D. F., R. Gnanadesikan, and J. L. Warner, 1973: “Methods for Assessing

Multivariate Normality,” Multivariate Analysis- 1 1 1, Proceedings of the Third
International Symposium on Multivariate Analysis held at Wright State University,
Dayton, Ohio, June 19-24, 1972, Edited by P. R. Krishnaiah, Academic Press, New
York, NY, pp. 95-1 15.

Barlett, M. S., 1934: “The Vector Representation of a Sample,” Proceedings of the

Cambridge Philosophical Society, Vol. 30, pp. 327-340.

Bates, Carl B., 1966: “The Chi-square Test of Goodness of Fit for a Bivariate Normal
Distribution,” U. S. Naval Weapons Laboratory, Dahlgren, Virginia, December.

Bendat, Julius S. and Allan G. Piersol, 1966: Measurement and Analysis of Random
Data, John Wiley and Sons, Inc., New York, NY.

Bertrand, J., 1888a: “Calcul des probabilities: Note sur la probabilite du tir a la cible,”
Comptes Rendus, Acadamie des Sciences, Paris, France, pp. 232-234.

Bertrand, J., 1888b: “Calcul des probabilities: Trisieme note sur la probabilitd du tir a la
cible,” Comptes Rendus, Academie des Sciences, Vol. 106, pp. 387-391, 521-522.

Beyer, William H., 1966: Handbook of Tables for Probability and Statistics, The
Chemical Rubber Company, Cleveland, Ohio.

225

.
 REFERENCES (Continued)

Blom, Gunnar, 1958: Statistical Estimates and Transformed Beta-Variables, John Wiley
and Sons, Inc., New York, NY, Chapter 6, pp. 60-76.

Box, G . E. P. and D. R. Cox, 1964: “An Analysis of Transformations,” Journal of the

Royal Statistical Society, England, B 26, pp. 21 1-252.

Box, G. E. P. and Mervin E. Muller, 1958: “A Note on the Generation of Random

Normal Deviates,” Annals of Mathematical Statistics, Institute of Mathematical
Statistics, California State College, Hayward, CA, Vol. 29, pp. 6 10-61 1.

Bravais, August, 1846: “Analyse MathCmatique sur les probabilities des erreurs de
situation d’un point,” Memoirs Presentes par Divers Savants, 2nd Series, Vol. 9,
Institut de France, Acadamie des Sciences, Paris, France, pp. 255-332.

Brooks, C. E. P. and N. Carruthers, 1953: Handbook of Statistical Methods in

Meteorology, Her Majesty’s Stationery Office, London, England.

Brooks, C. E. P., C. S. Durst, and N. Carruthers, 1946: “Upper Winds Over the World,
Part 1, The Frequency Distribution of Winds at a Point in the Free Air,” Quarterly
Journal of the Royal Meteorological Society, England, Vol. 72, pp. 55-73.

Brooks, C. E. P., C. S . Durst, N. Carruthers, D. Dewar, and J. S. Sawyer, 1950: “Upper

Winds Over the World,” Her Majesty’s Stationery Office, London, England.

Cochran, William G., 1952: “The x2 Test of Goodness of Fit,” Annals of Mathematical
Statistics, Institute of Mathematical Statistics, California State College, Hayward,
CA, Vol. 23, pp. 315-345.

Cochran, William G., 1954: “Some Methods for Strengthening the Common x2 Tests,”
Biometrics, Vol. 10, North Carolina State University, Institute of Statistics, Raleigh,
NC, pp. 417-45 1.

Cramer, Harald, 1946 : Mathematical Methods of Statistics, Princeton University Press,

Princeton, NJ.

Cramer, Harrison E., F. E. Record, and J. E. Tillman, 1966: “Round Hill Turbulence
Measurements,” Technical Report ECOM-65-Gl0, U.S. Army Electronics Command,
Atmospheric Sciences Laboratory, Research Division, Fort Huachuca, Arizona, Vols.
1 through 5.

Cramer, Harrison E., 1970: Personal communication providing four-dimensional

distributions of wind and temperature.

226
 REF E RENCES (Continued)

Crutcher, Harold L., 1957: “On the Standard Vector-Deviation Wind Rose,” Journal of
Meteorology, American Meteorological Society, 45 Beacon Street, Boston, MA, Vol.
14, No. 1, February, pp. 28-33.

Crutcher, Harold L., 1962: “Computations from Elliptical Wind Distribution Statistics,”
Journal of Applied Meteorology, American Meteorological Society, Boston, MA, Vol.
1, No. 4, December, pp. 522-536.

Crutcher, Harold L., 197 1 : “Atlantic Tropical Cyclone Statistics,” NASA Contractor
Report CR-6 1355, National Aeronautics and Space Administration, George C.
Marshall Space Flight Center, Huntsville, AL.

Czuber, Emanuel, 1891 : Theorie der Beobachtungsfehler, B. G. Teubner, Leipzig,

Germany, pp. 4 0 0 4 1 1 .

Dahiya, Ram C. and John Gurland, 1973: “How Many Classes in the Pearson Chi-square
Test?,” Journal of the American Statistical Association, American Statistical
Association, Washington, D.C., Vol. 68, No. 343, September, pp. 707-71 2.

De Moivre, 1733: “Approximatio ad Summam Terminorum Binomii (a + b)n in Seriem

Expansi,” Reprinted in ISIS, 8, 1926, by R. C . Archibald, entitled, “A Rare
Pamphlet of Moivre and Some of His Discoveries.”

Edgeworth, F. Y., 1916: “On the Mathematical Representation of Statistical Data,”

Journal of the Royal Statistical Society, England, Vol. 79, pp. 455-500.

Elderton, William Palin and Norman Lloyd Johnson, 1969: Systems of Frequency Curves,
Cambridge University Press, London, England.

Feller, William, 1966: An Introduction to Probability Theory and Its Applications, Vol.
1, John Wiley and Sons, Inc., New York, NY.

Fisher, R. A., 1924: “The Conditions Under Which x2 Measures the Discrepancy Between
Observation and Hypothesis,” Journal of the Royal Statistical Society, Bracknell,
Berkshire, England, Vol. 87, pp. 442-450.

Fisher, R. A., 1950f Contributions t o Mathematical Statistics, especially Papers 4, 5, and

6, Edited by Walter A. Shewhart, John Wiley and Sons, Inc., New York, NY.

Freund, J. E., 1962: Mathematical Statistics, Prentice-Hall, Inc., Englewood Cliffs, N.J.

Gauss, C. F., 1823: Theory of the Combination of Observations, Gottingen, Germany,

Band IV, p. 1.

227

I
 REFERENCES (Continued)

Geary, R. C., 1947: “Testing for Normality,” Biometrika, University College, London,
England, Vol. 34, pp. 209-242.

Gnanadesikan, R. and J. R. Kettenring, 1972: “Robust Estimates, Residuals’and Outlier

Detection with Multi-Response Data,” Biometrics, North Carolina State University,
Raleigh, NC, Vol. 28, pp. 81-124.

Gringorten, Irving, I., 1963: “‘A Plotting Rule for Extreme Probability Paper,” J o h k a l of
Geophysical Research, American Geophysical Union, Washington, D.C., Vol. 63, No.
10, pp. 813-814.

Groenewoud, C., D. C. Hoaglin, J. A. Vitalis, and H. L. Crutcher, 1967: Bivariate Normal

Offset Circle Probability Tables with Offset Ellipse Transformations, Vols. I, 11, and
111, CAL No. XM-246443-1, Cornel1 Aeronautical Laboratory, Inc., of Cornel1
University, Buffalo, NY, Vols. I through 111, “Applications to Geophysical Data.”

Hald, A., 1952a: Statistical Theory with Engineering Applications, John Wiley and Sons,
Inc., New York, NY.

Hald, A., 1952b: Statistical Tables and Formulas, John Wiley and Sons, Inc., New York,
NY.

Hald, A. and S. A. Sinkbaek, 1950: “A Table of Percentage Points of the x2

Distribution,” Skandinavisk Aktuarietidskrift, pp. 168-175.

Hamdan, M. A., 1963: “The Number and Width of Classes in the Chi-square Test,”
Journal of the American Statistical Association, Washington, D.C., Vol. 58, No. 303,
September, pp. 678-689.

Healy, M. J. R., 1968: “Multivariate Normal Plotting,” Applied Statistics, Royal

Statistical Society, London, England, Vol. 17, pp. 157-16 1,

Hotelling, Harold, 193 1: “The Generalization of Student’s Ratio,” Annals of

Mathematical Statistics, Vol. 2, pp. 360-378.

Hotelling, Harold, 195 1 : “A Generalized T Test and Measure of Multivariate Dispersion,”

Proceedings of the Second Berkeley Symposium on Mathematical Statistics and
Probability, July 3 1-August 12, 1950, University of California Press, Berkeley and
Los Angeles, CAYpp. 23-4 1.

Hull, T. E. and A. P. Dobell, 1962: “Random Number Generators,” SIAM Review,

Society for Industrial and Applied Mathematics, Philadelphia, PA, Vol. 4, No. 3,
July, pp. 230-253.

228
 R E F ER ENCES (Continued)

Joshi, S. W., 1970: “Construction of Certain Bivariate Distributions,” The American

Statistician, American Statistical Association, Washington, D.C., Vol. 24, No. 2,
April, p. 32.

Kac, M., J. Kiefer, and J . Wolfowitz, 1955: “On Tests of Normality and Other Tests of
Goodness of Fit Based on Distance Methods,” Annals of Mathematical Statistics,
Institute of Mathematical Statistics, California State College, Hayward, CA. Vol. 26,
pp. 189-21 1.

Kempthorne, Oscar, 1967: “The Classical Problem of Inference - Goodness of Fit,”

Proceedings of the Fifth Berkeley Symposium On Mathematical Statistics and
Probability, Vol. 1, Edited by Lucien M. Le Cam and Jerzy Neyman, University of
California Press, Berkeley and Los Angeles, CAYpp. 235-249.

Kendall, Maurice G. and Alan Stuart, 1968: The Advanced Theory of Statistics, Vol. 3,
“Design and Analysis, and Time Series,” Second Edition, Hafner Publishing
Company, New York, NY.

Kessel, D. L., and K. Fukunaga, 1972: “A Test for Multivariate Normality With
Unspecified Parameters,” Unpublished Report, Purdue University School of
Electrical Engineering, Lafayette, IN.

Kluyver, J. D., 1906: “A Local Probability Problem,” Proceedings of the Royal Academy
of Amsterdam, Holland, pp. 341-350.

Knuth, Donald E., 1969: The Art of Computer Programming, Vol. 2, “Semi-Numerical
Algorithms,” Addison Wesley Publishing Company, Reading, MA.

Kolmogorov, A. N., 1933: “Sulla Determiniazione Empirica di una Leggi di

Distribuzione,” Giornale dell’Istituto Italian0 Degli Attuari, Rome, Italy, pp. 83-9 1.

Kowalski, Charles J., 1970: “The Performance of Some Rough Tests for Bivariate
Normality Before and After Coordinate Transformations to Normality,”
Technometrics, Vol. 12, No. 3, pp. 517-544. ..

Kowalski, Charles J., 1973: “Non-Normal Distributions With Normal Marginals,” The
American Statistician, The American Statistical Association, Washington, D.C., Vol.
27, No. 3, June, pp. 103-106.

Krishnaiah, P. R., 1972: Proceedings of the Third International Symposium on

Multivariate Analysis, held at Wright State University, Dayton, OH, June 19-24,
1972, Academic Press, New York, NY.

229
 REFERENCES (Continued) .

Lancaster, H. O., 1969: The Chi-square Distribution, John Wiley and Sons, Inc., New
York, NY.

LaPlace, P. S. de, 1878/1912: Oeuvres Complites de LaPlace, publiees sous les auspices
de 1’Acade”ie des Sciences par M. M. les Secritaires perpgtuels, Vols. 1 through 14,
Gauthrer-Villar, Paris, France.

Malkovich, J. F. and A. A. Afifi, 1973: “On Tests for Multivariate Normality,” Journal
of the American Statistical Association, Washington, D.C., Vol. 68, No. 341, March,
pp. 176-179.

Mann, H. B. and A. Wald, 1942: “On the Choice of the Number of Class Intervals in the
Applications of the Chi-square Test,” Annals of Mathematical Statistics, Institute of
Mathematical Statistics, California State College, Hayward, CAY Vol. 13, pp.
306-3 17.

Marsaglia, G. K., 1972: “The Structure of Linear Congruential Sequences,” Applications

of Number Theory to Numerical Analysis, Academic Press, Inc., New York, NY, pp.
249-285.

Marsaglia, G. K., N. Anathanarayanan, and N. Paul, 1972: “Random Number Generator

Package, ‘Super-Duper’ Uniform, Normal and Random Number Generator,” McGill
University, School of Computer Science Computing Center, Montreal, Canada.

Mauchly, J. W., 1940a: “Significance Test for Sphericity of a Normal N-Variate

Distribution,” Annals of Mathematical Statistics, Institute of Mathematical Statistics,
California State College, Hayward, CA, Vol. 11, pp. 204-209.

Mauchly, J. W., 1940b: “A Significance Test for Ellipticity in the Harmonic Dial,” Terr.
Magn. Atmos. Electr., Washington, D.C. Vol. 45, pp. 145-148.

Maxwell, John C., 1859: “Illustrations of the Dynamical Theory of Gases, Part 1, On the
Motions and Collisions of Perfectly Elastic Spheres. “ Philosophical Magazine,
London, England, Vol. 39, Series 4, July-December, pp. 19-32.

Mills, Frederick C., 1955: Statistical Methods, Henry Holt and Company, New York, NY.

Milton, Roy C., 1970: “Computer Evaluation of the Multivariate Normal Integral,”
Technical Report No. 19, University of Wisconsin Computing Center, Madison, WI,
September.

Owen, D. B., 1956: “Tables for Computing Bivariate Normal Probabilities,” Annals of
Mathematical Statistics, Institute of Mathematical Statistics California State College,
Hayward, CA, Vol. 27, pp. 1075-1090.

230
 REFERENCES’(Continued)

Owen, D. B., 1962: Handbook of Statistical Tables, Addison-Wesley Publishing Company,

Inc., Reading, MA. .

Panofsky, Hans A. and Glenn W. Brier, 1968: Some Applications of Statistics to

Meteorology, Pennsylvania State University, University Park, PA.

Pearson, Karl, 1900: “On the criterion that a given system of deviations from the
probable in the case of a correlated system of variables is such that it can be
reasonably supposed to have arisen from random samplings,” the London, Edinburgh
and Dublin Philosophical Magazine, and Journal of Science, England, Vol. 50, Fifth
Series, pp. 157-175.

Roscoe, John T. and Jackson A. Byars, 1971: “An Investigation of the Restraints With
Respect to the Sample Size Commonly Imposed on the Use of the Chi-square
Statistic,” Joumal of the American Statistical Association, Washington, D.C.,
December, Vol. 66, No. 336, pp. 755-759.

Sarhan, Ahmed, and B.ernard G. Greenberg, 1962: Contributions of Order Statistics, John
Wiley and Sons, Inc., New York, NY.

Smirnov, N., 1948: “Table for Estimating the Goodness of Fit of Empirical
Distributions,” Annals of Mathematics of Statistics, Institute of Mathematical
Statistics, California State College, Hayward, CAYVol. 19, pp. 279-281.

Smith, W. B. and R. R. Hocking, 1972: “Wishart Variate Generator,” Paper of the

Institute of Statistics, Texas A&M University, College Station, TX.

Strutt, John William (Lord Rayleigh), 1919: “On the Problem of Random Vibrations and
of Random Flights in One, Two or Three Dimensions,” Philosophical Magazine and
Journal of Science, London, England, 6th Series, Vol. 37, No. 4, pp. 321-347.

Student (W. S. Gosset), 1908: “The Probable Error of a Mean,” Biometrika, Cambridge
University Press for the Biometrika Trustees, England, Vol. VI, March, pp. 1-25.

Student (W. S. Gosset), 1925: “New Tables for Testing the Significance of Observations,”
Metron., Vol. 5, No. 3, pp. 105-120.

Sturges, H. A., 1926: “The Choice of a Class Interval,” Journal of the American
Statistical Association, Washington, D.C., Vol. 21, No. 153, March, pp. 65-66.

Swed, Frieda S. and C. Eisenhart, 1943: “Tables for Testing Randomness of Grouping in
a Sequence of Alternatives,” Annals of Mathematical Statistics, Institute of
Mathematical Statistics, Califomia State College, Hayward, CAYVol. 14, pp. 66-87.
 REFERENCES (Concluded)

Tate, M. W. and L. A. Hyer, 1973: “Inaccuracy of X2 Test of Goodness of Fit When

Expected Frequencies are Small,” Journal of the American Statistical Association,
Washington, D.C., Vol. 68, No. 344, December, pp. 836-842.

Thomas, M. A. and J. R. Crigler, 1974: “Tolerance Limits for the p-Dimensional Radial
Error Distribution,” Communications in Statistics, Marcel Dekker, Inc., New York,
NY, Vol. 3, NO. 5 , pp. 477-484.

U.S. Department of Commerce, 1959: Tables of the Bivariate Normal Distribution

Function and Related Functions, Applied Mathematics Series, U.S. Department of
Commerce, National Bureau of Standards, available from the Superintendent of
Documents, Government Printing Office, Washington, D.C.

Vessereau, A., 1958: “Sur les conditions d’application du criterium x2 de Pearson,”

Bulletin, Institute of International Statistics, The Hague, Holland, Vol. 36, pp.
87-101.

Votaw, David., 1948: “Testing Compound Symmetry in a Normal Multivariate

Distribution,” Annals of Mathematical Statistics, Institute of Mathematical Statistics,
California State College, Hayward, CAYVol. 19, pp. 447-473.

Weiss, Lionel, 1958: “A Test of Fit for Multivariate Distributions,” Annals of

Mathematical Statistics, Institute of Mathematical Statistics, California State College,
Hayward, CAYVol. 29, pp. 595-599.

Whittaker, Sir Edmund and G. Robinson, 1954; The Calculus of Observations, Blackie and
Son Limited, London.

Williams, C. Arthur, Jr., 1950: “On the Choice of the Number and Width of Classes for
the Chi-square Test of Goodness of Fit,” Journal of the American Statistical
Association, Washington, D.C:, Vol. 45, pp. 77-86.

Wishart, John, 1928: “The Generalized Product Moment Distribution in Samples From a
Normal Multivariate Population,” Bometrika, Univesity College, London, England,
Vol. 20A, No. 32, p. 424.

Wishart, John, 1948: “Proofs of the Distribution Law of the Second Order Moment
Statistics,”’ Bometrika, University College, London, England, Vol. 35, No. 55, p.
422.

Yule, G. Udny and M. G. Kendall, 1940: An Introduction t o the Theory of Statistics,

Charles Griffin and Company, Ltd., London, England.

232
 ?

APPEND IX

PLOTTING DIAGRAMS WITH CONFIDENCE BAND (CENTRAL 0.96

PROBABILITY) FOR MULTIVARIATE NORMAL DISTRIBUTIONS
FOR VARIOUS DIMENSION AND SAMPLE SIZE

233

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