I

national Bureau of Staadard* n r t , Htesr* ^f^Bidg ~ Keference book not to be WW6 1964

/i *

^ecttnic&l r\ot&

library.

238

MISCELLANEOUS STUDIES

IN PROBABILITY AND STATISTICS:

Distribution Theory, Small-Sample Problems,

and Occasional Tables

THE STATISTICAL ENGINEERING LABORATORY

/•'->

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

Technical ^ote 238

ISSUED APRIL 24, 1964

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS:

Distribution Theory, Small-Sample Problems, and Occasional Tables

The Statistical Engineering Laboratory

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price 20 cents
FOREWORD

This note makes generally available the results of a number of special

investigations that have been made at various times, to provide answers to specific

questions raised in connection with consulting work or to supply specific needs for

tables in special forms. Thus, this publication consists of a collection of heretofore unpublished notes, reports, or working papers prepared by various members of

the staff of the Statistical Engineering Laboratory, NBS . Some of the notes, while

complete in themselves, provide only partial answers to the questions which motivated

the studies. They are published now, for convenient reference, since there is no plan for continuation of the studies of whicli they formed a part. Others of the

notes remained unpublished simply because they were believed to be of very limited

interest.

The date of preparation of the original unpublished note is given for each item

included in this publication. We are grateful to E. L. Crow, NBS Boulder Laboratories, for suggesting a number of editorial and technical improvements. The

authors have made at most very minor changes.

March 1964 Churchill Eisenhart

ii
CONTENTS

Page

FOREWORD ..." ii

ABSTRACT iv

DISTRIBUTION OF THE RATIO OF TWO F VARIATES HAVING n-1 AND n DEGREES

OF FREEDOM by J. M. Cameron and Cyrus Derman 1

SOME NOTES ON THE CAUCHY DISTRIBUTION by Cyrus Derman 3

THE BETTER ONE OUT OF TWO by E. P. King 7 VARIANCE OF MEDIANS AND PSEUDO-MEDIANS by Mary G. Natrella 9

PROBABILITY POINTS OF ORDER STATISTICS IN RANDOM SAMPLES OF SIZE n FROM A

UNIFORM DISTRIBUTION OVER (0,1) by Churchill Eisenhart and

Lola S. Deming 13

iii
ABSTRACT

This publication makes available some notes and tables prepared at various

times by the staff of the Statistical Engineering Laboratory.

Contents: (1) Distribution of the ratio of two F variates having n-1 and n degrees of freedom, by J. M. Cameron and Cyrus Derman. (2) Some notes on the

Cauchy distribution, by Cyrus Derman. Includes variance of the sample mean for

truncated Cauchy distribution; the Cauchy distribution whose cumulative

distribution function deviates least from the standard normal c.d.f. (3) The

better one out of two, by E. P. King. Variance of the observation closest to the

population mean. (4) Variance of medians and pseudo-medians, by Mary G. Natrella,

For sample sizes m up to 10, gives the variances (3D) of the median (m odd),

pseudo-median (m even), and average of two values on either side of the median

(ra odd), for tne normal and rectangular distributions and (m £ 6) for the extremevalue distribution. (5) Probability points of order statistics in random samples

of size n from a uniform distribution over (0,1), by Churchill Eisenhart and

Lola S. Deming. Gives probability points (4S) of each order statistic for

probabilities a - .001, .005, .01, .025, .05, .10, .20, .25, .50 and n = 2(1)10.

iv
DISTRIBUTION OF THE RATIO OF TWO F VARIATES

HAVING n-1 AND n DEGREES OF FREEDOM

J. M. Cameron and Cyrus Derman*

Problem: If F, and F_ are independently distributed as F with n-1 and n degrees of

freedom, show that w = VF,/F has the F distribution for 2n-2 and 2n-2 degrees of

freedom.

Solution : The density function of F "having n-1 and n degrees of freedom is given

by

r(^"fi) , |(n-l) ^(n-3)

g(F) = n-1 n <2

H

i)

[i * (^)f]

*(2n-l) F>0

FsO

If F, and F„ are independent random variables having the above distribution, then

JL £ their joint density function is

r nTl n-l 2 1t(2n-l)

[

1+ (B-1)(F1+F

2

) + (V> F

1F

2 ]

g(F

1 )g(F

2

) F1' F

2>°

= otherwise

Letting F, = yz and F_ = y the joint density function of y and z is

h(y,z)

„2 i(n-3) n-2 K z * y

T n-1 n-1 2 2 n^n-l)

[i + V y(z+1) + (V^ y Z

]

Using formula 5(a), p. 34 of [l], it can be shown that

P oo ../ JKn-3)

J h(y,z)dy = K

n/

J n.l) o (1 + JZ)

^n X)

y,z>0

otherwise

where K

/ _ 1 T(2n-2)

" T(n-l) T(n-l)

By letting w = Jz the density function becomes

g(w) T(2n-2) n-2

T(n-l) T(n-l) (1+w)

2n-2 w>0

otherwise

Present address: Department of Industrial and Management Engineering, Columbia

University, New York 27, New York.

1
It can be seen that w = JF-./F has the distribution of an F variate with 2n-2 and

2n-2 degrees of freedom.

REFERENCES

[l] Grbbner, W. and Hofreiter, N., Integraltaf el , Zweiter Teil, Bestinimte

Integrale, Springer-Verlag, Wien und Innsbruck, 1950,

February 1953
SOME NOTES ON THE CAUCHY DISTRIBUTION*

Cyrus Derman**

1. A critical point of truncation .

It is a well known fact that the distribution of the arithmetic mean of n Cauchy variates is independent of the value of n. For this reason the arithmetic

mean is an unacceptable estimator of the location parameter. The sample median

i is usually used for this purpose. However, the arithmetic mean of variates with

a truncated Cauchy distribution no longer has this undesirable property. One

might then ask the following questions. What is the relative efficiency of the

two estimators? What degree of truncation is necessary in order to make the

arithmetic mean more efficient than the sample median?

The probability density of a Cauchy variate is given by

f (x) = W —~—2" for - od < x < co . (1)

1 + x Let

(x)

'L (x) = 1^- = ±—

j

±—g for - z < x < z , (2)

2 f f(x)dx

J o

2 tan z 1 + x

= otherwise; (3)

i.e., g (x) is the distribution of x truncated at z and -z . We also have

E[x(z)] = (4)

and

2 1 r Z x

2

z - tan 1 z ,. s ° x(z)

=

; I -1 " 2 <* =

7 -1 > (o)

2 tan z -z 1 + x tan z where x(z) is a random variable with a Cauchy distribution truncated at z and -z. The asymptotic variance of the sample median is given by

1 _. (tan" 1 z) 2

2 Taus, since the asymptotic variance of the arithmetic mean is a / \/n, the

(6)

This investigation was suggested by Dr. C. Eisenhart. W. Knight was of

assistance in connection with some of the work with probability tables.

Present address; Department of Industrial and Management Engineering, Columbia

University, New York 27, New York.
efficiency of the arithmetic mean relative to the sample median is

(tan^z) 3

. -1 (7) z - tan z By using the tables of tan" z we can find a value of z that makes (7)

approximately unity. This occurs for z = 3.41. Since (7) is monotonic for z > 0,

the arithmetic mean is more efficient than the sample median for z < 3.41.

(See Note 1.)

The probability that z 3.41 is exceeded in the original population is given

by r - [tan (3.41) ]/n =» .0908. Thus one sees that an appreciable truncation of

the Cauchy distribution is required in order to put the arithmetic mean on an equal basis with the sample median. (See Note 2.)

2

2. The distribution of s and s in samples of two .

If x., ..., x are independent random variables with probability density

function given by (1), the probability density functions of

2 n

(x-i-x) 2 - Xl +

* ** + X

n

s Z -^-J

r

e~- anu s where x = are, in general, unknown. However,

in the case of n=2, it is possible to take advantage of the above-mentioned

property of the arithmetic mean of Cauchy variates, i.e. the distribution of x is

the same as x.

In the case of n*»2 we use the identity

s

2 - ( Xl -x

2

) 2/2 . (8)

Since the Cauchy distribution given by (1) is symmetrical about x = 0, x and -x

are identically distributed. Thus we see that

s - (x

1 -x

2 )/^2 (9)

is distributed as «/2 x which in turn has a distribution given by

g(s) - — ^for s s < oo . (10)

n/2 1 + s /2

2 Thus on making the transformation t - s , the distribution of t is given by

h(t) = —-— for < t < od . (11)

2^2 TT /t(l+t/2)
3. The normal-like Cauchy distribution . A more general form of the Cauchy distribution is given by

f(x,9)-i 1 5. (12)

ne

i + (x/er

In statistical practice, one usually assumes an underlying normal distribution

and then worries about the amount of deviation from this assumption. If we have

an underlying Cauchy distribution, we might then be interested in knowing what

normal distribution is closest to it in some sense. Equivalently, we might want

to know what Cauchy distribution best approximates a given normal distribution.

V/e shall define a distribution from a class of distributions to be closest

in that class to another distribution if the maximum absolute difference of the

cumulative distribution functions is a minimum for all distributions in the class.

In our case this means finding that value of 8 in (12) such that

Max

y

y y

[ f (x,9)dx - J

g(x)dx

-CD

X x

2

(13)

is minimized when g(x) = 1A/2TT e~2"

Since J" f (x,9)dx is essentially tabulated in arc tangent tables and

-co

f y g(x)dx can be found in the normal probability tables, it is possible to find a

-ao value of 8 which approximately satisfies the condition (13). This occurs for

9 = .51.

Thus we say that the normal-like Cauchy distribution is given by (12) where

9 = .51a and where a is the standard deviation of the normal distribution.

(See Note 3.)

September 1952

Referee's Comments:

Note 1. That (7) is mono tonic for z > may be shown by letting y = tan" z. -1 3 Then — —j— = ~2 , where all terms of the power series are

z - tan z 1 + 2y + . .

.

5 positive. Hence the (asymptotic) relative efficiency decreases steadily from

z = to z = cd and approaches an upper bound of 3 as z approaches zero. Note 2. In J. Amer. Stat. Assoc . 55, June 1960, pp. 322-3, P. R. Rider gave the

exact variances of the Cauchy sample median for n = 1, 3, ..., 31. They are uniformly larger than the asymptotic variances. Hence the exact variance of the
truncated Cauchy sample median might be expected to be larger than the asymptotic

variance, and hence the sample mean more efficient than shown by Derman. For

n = 1 the mean and median are identical; hence the relative efficiency for n = 1 is unity for all finite z but undefined for infinite z. Note 3. It would be of interest to know how close the two cumulative distribution

functions can be made. It appears that an error less than about 0.12 cannot be

guaranteed; i.e., the minimax error is 0.12.

The normal-like Cauchy density function (with 9 to 0.62 in contrast to the normal maximum of 0.40.

.51) has its maximum equal
THE BETTER ONE OUT OF TWO

E. P. King*

Problem : Let x.. and x be independent observations on a N(0,1) population. Find

the variance of the observation closer to the true mean, i.e., closer to zero.

Solution : Define

Y = x- when |x | < |x | = x

2 when |

x

x

| > |x 2

| and let F(y) = Pr(Y £ y). Then we obtain

F(y) = Pr(Xl £ y, \x

±

\ < | x

g

| ) + Pr(x

2 £ y, | X]J

> |x2|) . (1)

Clearly Pr (x

1 £ y, | x-j^

| < |x |) = Pr (x £ y, |x.jj > |x |), so that (1) can be

written as F(y) = 2Pr(x

x s y, | X;L | < |x 2

|) . (2)

The appropriate region of integration is composed of two parts of equal probability, one above and one below the x,-axis. Integrating over the appropriate

region above this axis and doubling the result yields,

r r° r°° r y r°° 1 F(y) =

4[_ J d$(X;L )

J d§(x

2

) +

J dMx^ J di(x

2 )J

for y>0

(3)

,,y r oo = 4 d$(x, ) ! d§(x ) for y^O

s-oo x ±x *

where §(x) denotes the normal distribution function. Letting u = I (x. ) , v = i(x„),

this expression can be integrated readily to give

F(y) = 4f(y) = 2S 2

(y) - 1 for y>0

= 2$ 2 (y) for ysO

V/e differentiate (4) and obtain the following density function for Y

f(y) = 4cp(y)[l-Hy)] y>0

= 4cp(y)§(y) ysO

where cp(y) = -j- $(y) . Clearly E(Y) = 0. The variance of Y becomes

l'° 2 f°° 9 r 00 9 V(Y) = 4 y cp(y)f(y)dy + 4 y cp(y)dy - 4 y c D (y)§(y)dy . (6)

ico J J

(4)

(5)

Present address: Statistical Research Department, Eli Lilly and Co.,

Indianapolis 6, Indiana.
oo „

r 2 Observing that y cp(y)dy = 1/2, and that

J

y cp(y).§(y)dy - y^(y)[l-$ (y) ]dy = ± - I yro(y)§ (y)dy

we can simplify (6) to

T 00 2 V(Y) = 4-8 y cn(y)*(y)dy . (7)

The integral on the right side of (7) can be integrated by parts. Letting

2 U = § (y) and dV = y ep(y)dy, we find

r 00 2 l r°° ry 2

y cp(y)$(y)dy - 4 r cp(y)dy t; cp(t)dt . J o ^ °o J o Substituting this result in (7) yields

,.co y „ V(Y) = 8 co(y)dy t ©(t)dt . (8)

The second integral in (8) can be integrated by parts to give

,y

'o

1

Hence (8) becomes

y J

y2

cp(t)dt = - ycp(y) + #(y) - | .

V(Y) - 8 - yep (y)dy + *(y)cp(y)dy - ± cp(y)dy . (9)

L Jq Jq Z Oq J

The three terms in (9) can be integrated easily. The final result is

V(Y) - 1 - 2/tt

= 0.363

July 1953

(10)
VARIANCE OP MEDIANS AND PSEUDO-MEDIANS

Mary G. Natrella

Problem: When the median is to be used as an estimator, and there is some possibility for choice of the sample size m, is it better to have ra an odd number

or an even number? For m odd, the median is a unique value. For m even, the

median is taken as the average of the twq central observations, and is called the

pseudo-median here. The better estimate is defined as the one with the smaller

variance.

The question would arise when the total number of observations to be made is

fairly large, the average of the medians of small groups is to be used as an

estimator, and the choice between similar small values of m is a matter of

discretion. For example, if a total of 60 observations are to be made, would it

be better to take 10 groups of 6(m-6) or 12 groups of 5(m=5)?

Conclusion : It seems that no general conclusion is possible —the choice of m odd

or m even may be better depending on the circumstances.

Investigation :

a. Normal distribution.

Table 1 was obtained from Table 2 of [1]. Col. 1 of Table 1 ("Variance of

median for m odd") is copied directly from this source. Col. 2 ("Variance of

pseudo-median for m even") is the computed variance of the average of the two

appropriate order statistics. Col. 3 ("Variance of average of two on either side

of median for m odd") was also computed from this source for comparison with

Col. 1. In comparing Col. 1 with Col. 2, it is noted that for m £ 10 the variance

decreases with sample size m and therefore that the larger m is always a better

choice. Col. 3 is also always smaller than Col. 1, but this fact does not affect

choice of sample size.

b. Rectangular distribution.

Table 2 was obtained from Table III of [2] in the same way as described above

for the normal. In comparing Col. 1 with Col. 2, it is noted that the variance for m even is smaller than that for next larger m (odd). This would be expected for

very small samples, since then the average of extremes from the rectangular
distribution is an efficient estimator, and here the two central order statistics

are sufficiently close to the extremes. For m £ 10, the better choice would be m

even.

c. Extreme value distribution.

Table 3 was obtained from Table II of [3], As in the normal, the larger m is

the better choice. In the extreme value distribution, the extremes (especially

the larger one—i.e. sample maximum) also have a large weight, but the effect is

opposite to that of the rectangular, i.e. the extremes have larger variance.

Note : For the rectangular distribution (Pearson curve with coefficients

Pi = 0, 6„ = 1.8) and for the normal distribution (Pearson curve with coefficients

Pi = 0, (3„ = 3.0), standard errors of the median are given in [4] and [5] as multiples of the standard error of the mean (o/Jxi) for sample sizes from 2 to 20.

REFERENCES

[1] Godwin, H. J., "Some Low Moments of Order Statistics", Annals of Math. Stat . vol. XX No. 2, June 1949, p. 279 (Table 2).

[2] Hastings, C. Jr., Mosteller, F., Tukey, J. W., and Winsor, C. P., "Low

Moments for Small Samples: A Comparative Study of Order Statistics",

Annals of Math. Stat, vol. XVIII No. 3, Sept. 1947, p. 413 (Table III).

[3] Lieblein, Julius, A New Method of Analyzing Extreme Value Data , NACA

Technical Note 3053, January 1954, (Table II).

[4] Pearson, E. S. and Adyanthaya, N. K. , "The Distribution of Frequency

Constants in Small Samples from Symmetrical Populations", Biome trika , vol. XXA, 1928, pp. 356-360.

[5] Pearson, Karl, Tables for Statisticians and Biome tricians Part II,

Cambridge University Press, 1931. (See CXIX of Introduction and Table

XXIII).

10
Table 1 Variance of Median and Pseudo-Median for

the Normal Distribution

Sample Size

m Variance of

median

(m odd)

Variance of

pseudo-med ian

(m even)

Variance of

avg. of 2 on either side

of median

(m odd)

23

4

5

67

8

9

10

.44867

.28083

.21045

.16610

. 50000

.29820

.21474

.16818

.13832

.36217

.23073

.17466

.14162

Table 2 Variance of Median and Pseudo-Median for

the Rectangulai Distribution

Sample Size

m Variance ofmedian

(m odd)

Variance of

pseudo-median

(m even)

Variance of

avg. of 2 on either side

of median

(m odd)

2

3

4

5

67

8

9

10

.60000

.42857

.33333

.27273

. 50000

.40000

.32143

.26667

.22727

. 30000

.28571

.25000

.21818

11
Table 3

Variance of Median and Pseudo-Median for the

Extreme Value Distribution

Sample

m Size Variance ofmedian

(m odd)

Variance

pseudo-med

(m even)

of

ian

Variance of

avg. of 2 on either side

of median

(m odd)

2 .82247

3 .65852 .64524

4 .43544

5 .40598 .34553

6 .30295

March 1954

12
PROBABILITY POINTS OF ORDER STATISTICS

IN RANDOM SAMPLES OF SIZE n FROM A UNIFORM

DISTRIBUTION OVER (0,1)

Churchill Eisenhart and Lola S. Deming

Let H , (x) denote the cumulative distribution function of the r-th order r/n

statistic x (x.. £ x„ £ ... £ x ) in random samples of size n from the uniform

distribution with probability density function f (x) =1, £ x £ 1, then the

t a-probability level of x s x . is, by definition, the value of x for which

H , (x) = a, r/n ' '

and is given implicitly by

n a

= 1 (g)xS (l-x) n_S

= I x

(r, n-r+1), (1) s=r where I (p,q) is the incomplete beta-function ratio. See Curtiss [l].

Values of x for which I (p,q) = .005, .01, .025, .05, .10, .25, are given by

Catherine M. Thompson [2] in terms of the argument v, = 2q = 1(1)10, 12, 15, 20,

24, 30, 40, 60, 120; v

g = 2p = 1(1)30, 40, 60, 120, ao . The table which appears below gives the values of x for which H . (x) = a, where a = ,001, .005, .01, .025, .05, .10, .20, .25, .50; n = 2(1)10; r - l(l)n.

The original version of this table was prepared in 1952 from several sources. Entries for the probability levels .005, .01, .025, .05, .10, .25, .50 were taken

from the Thompson table, either directly or through interpolation. Many of the

entries for the .20 level were taken from an unpublished table prepared by

Horace W„ Norton III [3]; others at this level and most of the entries at the .001

level were computed on the SEAC by Lambert S. Joel and Alan J. Hoffman, National

Bureau of Standards. The Tables of Fractional Powers [4] prepared under the

sponsorship of the National Bureau of Standards provided values and checks for

x. and x . In 1963 the table was checked by Roy H. Wampler against a manuscript

copy of H„ Leon Harter's 7-S table, "Percentage Points of the Beta Distribution"

[5], which is to be published shortly. Wherever discrepancies were found, Harter's

values were used in the present table. Some entries were obtained by direct

calculation, using formula (1) above. These were cases where either it was impossible to determine the fourth significant digit from Harter's table or the

entry was originally obtained by interpolation and Harter's table did not provide

13
a check. The table, as it now stands, is believed to be accurate to within half

a unit in the last digit. Credit is due to Mrs. Marion T. Carson for carrying out

the direct calculations and assisting in the checking.

It will be noted that Table 1 of Robert E. Clark, "Percentage Points of the

Incomplete Beta Function", 1953 [6], for samples of size n •> 10(1)50, gives

directly to 4 significant figures the .005, .01, .025, and .05 probability points

of all order statistics x , 1 £ r £ n.

r'

REFERENCES

[1] Curtiss, J. H. , "Convergent Sequences of Probability Distributions",

American Mathematical Monthly , 50 , 1943, pp. 103-5.

[2] Thompson, Catherine M., "Tables of Percentage Points of the Incomplete BetaFunction", Biometrika , XXX 1

1

, Oct. 1941, pp. 151-181.

[3] Norton, Horace W., Ill, "Table of the Upper Limit of the Incomplete Beta

Integral, for Area 0.2", dated 26 March 1950, unpublished.

[4] National Bureau of Standards, Tables of Fractional Powers , Columbia

University Press, 1946.

[5] Harter, H. Leon, "Percentage Points of the Beta Distribution", manuscript

copy, Oct. 1963, to be published shortly.

[6] Clark, Robert E., "Percentage Points of the Incomplete Beta Function",

Journal of the American Statistical Assn. , 48 , 1953, pp. 831-843.

January 1964

14
Probability Level a

n T .001 .005 .01 .025 .05 .10 .20 .25 .50

2 1 .03 5001 .0 2 2503 .0 2 5013 .01258 .02532 .05132 .1056 .1340 .2929

2 . 03162 .07071 .1000 .1581 .2236 .3162 .4472 .5000 .7071

3 1 ,03 3334 .0 2 1669 .0 2 3345 .0 2 8404 .01695 .03451 .07168 .09144 .2063

2 .01837 .04140 .05890 .09430 .1354 .1958 .2871 .3264 .5000

3 .1000 .1710 .2154 .2924 .3684 .4642 .5848 .6300 .7937

4 1 .03 2501 .0 2 1252 .0 2 2509 ,0 2 6309 .01274 .02600 .05426 .06940 .1591

2 .01302 .02944 .04200 .06759 .09761 .1426 .2123 .2430 .3857

3 .06404 .1109 .1409 .1941 .2486 .3205 .4175 .4563 .6143

4 .1778 .2659 .3162 .3976 .4729 .5623 .6687 .7071 .8409

5 1 .03 2001 .0 2 1002 .0 2 2008 .0 2 5051 .01021 .02085 .04365 .05591 .1294

2 .01010 .02288 .03268 .05274 .07644 .1122 .1686 .1938 .3138

3 .04755 .08283 .1056 .1466 .1893 .2466 .3266 .3594 .5000

4 .1220 .1851 .2221 .2836 .3426 .4161 .5098 .5458 .6862

5 .2512 .3466 .3981 .4782 .5493 .6310 .7248 .7579 .8706

6 1 .03 1667 .03 8351 .0 2 1674 .0 2 4211 .0 2 8512 .01741 .03651 .04682 .1091

2 .02 8255 .01872 .02676 .04327 .06285 .09260 .1399 .1612 .2644

3 .03792 .06628 .08473 .1181 .1532 .2009 .2686 .2969 .4214

4 .09395 .1436 .1731 .2228 .2713 .3332 .4146 .4468 .5786

5 .1814 ,2540 .2943 .3588 .4182 .4897 .5776 .6105 .7356

6 .3162 .4135 .4642 .5407 .6070 .6813 .7647 .7937 .8909

7 1 .03 1429 .0 3 7158 .0 2 1435 .0 2 3610 .0 2 7301 .01494 .03137 .04026 .09428

2 „0 2 6982 .01584 .02267 .03669 .05338 .07882 .1195 .1380 .2285

3 .03156 .05530 .07080 .09899 .1288 .1696 .2283 .2531 .3641

4 .07665 .1177 .1423 .1841 .2253 .'2786 .3501 .3788 .5000

5 .1438 .2030 .2363 .2904 .3413 .4038 .4832 .5139 .6359

6 .2375 .3151 .3566 .4213 .4793 .5474 .6291 .6593 .7715

7 .3728 .4691 .5179 .5904 .6518 .7197 .7946 .8203 .9057

8 1 .03 1251 .03 6264 .0 2 1256 ,0 2 3160 ,0 2 6391 .01308 .02751 .03532 .08300

2 .0 2 6049 .01374 .01966 .03185 .04639 .06863 .1044 .1206 .2011

3 .02704 .04746 .06084 .08523 .1111 .1469 .1986 .2206 .3205

4 .06483 .09987 .1210 .1570 .1929 .2397 .3032 .3291 .4402

5 .1196 .1697 .1982 .2449 .2892 .3446 .4163 .4445 .5598

6 .1927 .2578 .2932 .3491 .4003 .4618 .5379 .5668 .6795

7 .2887 .3685 .4101 .4735 .5293 .5938 .6696 .6973 .7989

8 .4217 .5157 .5623 .6306 .6877 .7499 .8178 .8409 .9170

9 1 .03 1112 .0 3 5568 .0 2 1116 .0 2 2809 .0 2 5683 .01164 .02449 .03146 .07413

2 .02 5337 .01212 .01736 .02814 .04102 .06077 .09263 .1072 .1796

3 .02366 .04158 .05335 .07485 .09775 .1295 .1757 .1955 .2862

4 .05621 .08679 .1053 .1370 .1688 .2104 .2675 .2910 .3931

5 .1025 .1461 .1710 .2120 .2514 .3010 .3661 .3920 .5000

6 .1629 .2191 .2500 .2993 .3449 .4006 .4709 .4980 .6069

7 .2388 .3074 .3437 .3999 .4504 .5099 .5823 .6095 .7138

8 .3349 .4150 .4560 .517 5 .5709 .6316 .7022 .7277 .8204

9 .4642 .5550 .5995 .6637 .7169 .7743 .8363 .8572 .9259

10 1 .03 1000 .03 5011 .0 2 1005 .0 2 2529 ,0 2 5116 .01048 .02207 .02836 .06697

2 ,0 2 4774 .01085 .01554 .02521 .03677 .05453 .08326 .09640 .1623

3 .02104 .03701 .04751 .06674 .08726 .1158 .1576 .1756 .2586

4 .04963 .07677 .09321 .1216 .1500 .1876 .2394 .2609 .3551

5 .08981 .1283 .1504 .1871 .2224 .2673 .3268 .3507 .4517

6 .1413 .1909 .2183 .2624 .3035 .3542 .4191 .4445 .5483

7 .2046 .2649 .2971 .3475 .3934 .4483 .5163 .5423 .6449

8 .2815 .3518 .3883 .4439 .4931 .5504 .6191 .6446 .7414

9 .3763 .4557 .4956 .5550 .6058 .6632 .7290 .7526 .8377

10 .5012 .5887 .6310 .6915 .7411 .7943 .8513 .8706 .9330

15

* U. S. GOVERNMENT PRINTING OFFICE : 1964 O - 727-472