Testing for an outlier in a linear model

Testing for an outlier in a linear model

Citation for published version (APA):

Doornbos, R. (1980). Testing for an outlier in a linear model. (Memorandum COSOR; Vol. 8001). Technische

Hogeschool Eindhoven.

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Published: 01/01/1980

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PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memoranaum COSOR M80-01

Testing for an outlier

in a linear model

by

R. Doornbos

Eindhoven, February 1980

The Netherlanas

Testing for an outlier in a linear model

Summary

Several authors have advocated the use of the maximum absolute stud$nti!ed

residual for the detection of a single outlier in a general linear model.

An excellent survey of the work in this field is presented in Chapter 7 of

Barnett and Lewis (1978). Ellenberg (1976) demonstrated that apparently

different approaches based on the maximum reduction in the residual sum of

squares are equivalent to the use of residuals. The main remaining problem

is the determination of critical values. A number of authors use the first

Bonferroni inequality. Ellenberg (1976) showed how to calculate the more

precise second Bonferroni inequality. In this memorandum we show how the

cumbersome computations required for this lower bound correction can be

avoided in a large number of cases occurring in practice. When all the

correlations between the residuals are absolutely smaller than the value

tabulated in table II the approximate test based on the first Bonferroni

inequality will result in a size of the test between (a - ":ta

2

) and a. In

table I critical values are given for a

=

0.10, O.OS, 0.01 and 0.001.

1. Formulation of the problem

We consider the linear model of full rank

y

=

XB

+

e

I (1. 1)

where y is a n dimensional random vector,

B

is a k (k

<

n) dimenSional vector

of unknown parameters and e has a multivariate normal distribution with

2

2

expectation vector

a

and covariance matrix

a I

I

where

a is unknown and I

is

n

n

the n

x

n identity matrix.

When the parameter vector is estimated by the method of least squares the

following vector of residuals is found

e

=

(y';"

y)

=

(y -

XS)

I (1. 2)

where

(1. 3)

The expectation vector of

e

is 0 and the covariance matrix is

The standardized residuals are now defined as

(1. 5)

2

•

~

where

S

=

e'e, the residual sum of squares and m

ii

is the 1-th diaqonal

element of

M.

Now we want to test the hypothesis that the model (1.1) is correct against

the alternative that for one, unknown, value of i

Yi has an expectation

different from xi*S, where xi* 1s the i-th row of

X.

The test procedure one

would like to apply (cf. Ellenberg (1976»

is to reject the null hypothesis

i f

maxi til

~

C

,

i

a

where

C

is determined by

a

Because the exact distribution of maxi til under

So

is intractable the

i

( 1.6)

following approximation is used. Instead of the exact critical value

C

we

a

use c

a

' determined by

prCItii

~

ca'

=

a/n , i

=

1, ••• ,n •

(1.7)

2

The distribution of ti is B(;,(n-k-l)/2). Using a standard procedure of the

Program Library of the Eindhoven University of Technology we calculated c

a

for n

=

5,(1), ••• ,50, k

=

1,(1), ••• ,min{Ck-3),15b a"" 0.10,0.05,0.001 and

0.001. The results are presented in Table I. At several places we checked

our values against the Biometrika Tables, Vol. I, against the related tables

of Lund (1975) and after the necessary transformation against the tables of

Federighi (1959) of the t-distribution. We believe our table to be correct

in the four decimal places presented. The entries in the table of Lund may

be one unit wrong in the last decimal.

Ellenberg (1976) reports already that the approximation (1.7) produces

excellent results in a large number of practical problema. The problem is

however that quite extensive computations are ne.ded to check this in each

case.;

In the next section we derive a condition which can simply be checked by

means of a table and which guarantee. that the approximation defined by

~

3

-2. Derivation of our main result

We denote the

~ctual

size of the test, using the

appr~ximation

(1.7), by

P(l'

Consequently

P

=

Pr(max Itil

~ c )

(l 1

a

and we have the second Bonferroni inequality

n Pr (

I

;i

1

~ c ) -

!

Pr ( Il;i

I

a

iJ'j

~ c ,

Itjl

~ c )

:s:

P

(l (l

a

:s:

n

Pr

(I;i I

~

The bivariate distribution of

~i

and

~j

(Ellenberg (1976»

is

==

0 elsewhere ,

where

P

ij

is the correlation between

8

i

and

8

j ,

to be found from the

covariance matrix M of

e

and where p -

(n-k-2)/2.

(2.l)

(2.2)

(2.3)

Essentially the same distribution was found by Doornbos and Prins (1956) in

a related outlier situation. We introduce now the same transformation as

used in that paper.

After some algebra we find the density function

=

In other words ti and tj are independent and tj has a

B(~,(n-k-2)/2)

distribution. As already remarked by Ellenberg (1976) we can write

(2.4)

(2.6)

~e

remark that

PI

changes into

P

2

when

P

ij

is replaced by

-P

ij

and reversely

and asswne from now on that

P

ij

~

O.

Then we have

Now

c ,

a

This derivative is nonnegative for all ti

~

c

a

if

2

P

_ij

:S:

C

u

Asswning that this is true we can write

because of the independence of ti and t

j

>

*

We now define c by

a

2

Because the ratio of the densities of tiand tj is decreasing with t

(2.7)

(2.8)

(2.9)

(2.10)

extreme percentage points of tj will be larger than the corresponding ones

*

of

t

i

• In other words c

a

> C ,

a

as may be confirmed by inspection of the

entries in adjacent columns of Table I.

Next we calculate for which values of P

ij

After some calculation it is found that (2.11) is equivalent to

2

c

_{a - c}

*2

a

4

+

C

a

*

say. Further it is easily seen, using the inequality c

a

<

c

a

' that

(2.11)

5

-consequently (2.12) implies (2.9). The values of 9 , are presented in

a

table

I I for the same values of n, k and a as those of table I.

Returning to inequality (2.10) we see that, if (2.12) is true, for P

ij

positive

(2.13)

(2.14)

For P

ij

negative (2.14) is trivially fulfilled as can be seen by looking at

the points where (2.9) and (2.12) were introduced. In exactly the same way

it is seen that

i f P

ij

is positive, or negative but -Pij

:s:

9 .

a

combining these results we find that

4

(...!!..)

2

2

pr(ltil

<!: C

a

'

Itjl

<!:C

a

)

=

2P1

+

2P

2

::0

a

I :

-2n

n

(2.15)

i f

(2.16)

Consequently the correction term in the first member in inequality (2.2) has

the following upper bound:

2

2

r

Pr

<lt

i

I

~

c ,

I

f,;j

I

~

c )

:s: en) .!L

<

':Ia

•

i"j

a

a

2

n

This gives us finally the result

i f

2

a -

,,:!a

< P

::oa

a

max IPijl

s

9

•

i,j

a

(2.17)

(2.18)

We note that,

1f

(2.18)

1.

not true, but (2.16)

i .

for all pairs (i,j)

except a few, (2.15) can still be very u •• ful to qet an acceptable lower

bound for P. We will illustrated this point in the first example of the

next section.

3. Some examples

As the first example we take the data presented by Mickey, Dunn and Clark

(1967)

on cyanotic heart disease in children, also discussed by Ellenberg

(1976).

This is a simple linear regression case with n

=

21

and k

=

2. The

x values (age of the child in months) range from 7 to 26 with one

exceptional values of 42 months. The response variable y is the Gesell

Adaptive Score. The maximum

t

value is 0.65.

From table I we see that the critical value for a

=

0.05,

which is 0.6398,

is exceeded but that the result is not significant at the 0.01 level

(critical value 0.7081).

From table II we read that 9 , for n

=

21,

k - 2 and a

=

0.05,

is 0.226.

a

This value is exceeded, in absolute value, by three correlations out of

(2;>

210. These correlations occur between the residuals at x

=

42

and

x

=

26

(p "",' -0.556)

and between the residuals at x .. 42 and x

=

20

(2x),

where

p

=

-0.300.

Without hardly any extra calculation

we

can conclude that

the lower bound for

P

is greater than

0.05 - 208.5

x

(~)2

- 1.5

x

~

=

a

0452

21

2 1 "

Therefore

0.045

<

P

O

•

OS

<

0.05

which will be considered as quite satisfactory by most practising

statisticians.

Looking into the case a bit closer we see that for a

=

0.05

we have

(3.1)

Ip

_ij

l

2

_{~ c~}

for all correlations but one. Therefore for two more pairs the

reasoning of section 3 can be pursued up to inequality (2.11). The left hand

2

member of (2.11) gives 0.611 for P

=

0.300.

The tj of (2.10) has in our case

1

17

a

8(2

':2)

distribution and from a standard table of the B-distribution

2

2

(Biometrika Tables, Vol. r) we learn that pr(t

j ~

(0.611) )

<

0.5%,

Therefore we can easily improve (3.1) to

0.05 - 208.5

x

(0;i

5

)2 - I·x O;i

S

0.~05

- 0.5 x

OtiS"

0.0476.

(3.2)

As a second illustration we take the hypothetical data in table 3.1 of a

4

x

5 layout given by Daniel (1960) and discussed by Bross (1961) and

Mickey et ala (1967).

- 7

-Table 3.1

Hypothetical Yields

Levels of B

35

29

25

19

22

Levels

32

29

29

25

20

of A

₃₇

₃₄

30

25

29

40

36

20

35

29

After fitting a linear model

(3.3)

5

with the usual restrictions

in table 2.

a ..

i

r

B

j

-

0, the residuals are as shown

j - l

Table

3.2

Residuals

2

a

2

-4

0

-2

-1

5

1

-3

-1

0

2

-3

2

1

1

-9

6

1

The residual sum of squares is

202~0

with

12

degrees of freedom. The

variance of each residual is

;~

0

2

and therefore the maximum standardized

residual 1s

,

_~

_I..

₉

_{.,. 0.8175 •}

max

\112 •

.E.

202,0'

20

12

In table I en • 20, k

=

B) we find that the critical values for a

==

0.05

and a -

0.01

are

0.7613

and

0.8262

respectively. The result is therefore

- !

for two

3

significant at the 0.05 level. The

1

residua!s in one column,

-5-1

1

correlation. are, -

4=1

==

1

1

1

4.5

=

20 in all other cases. In table II we see that gO.05 is 0.3486.

Therefore (2.16) is satisfied and

0.04875

<

PO.OS

<

0.05

in this case.

In general in a p

x

q layout the maximum correlation will be determined by

min(p,q). As also remarked by Ellenberg (1976) a 2

~

k

layout Is the most

extreme case with correlations equal to -1. Here we have a way out however

by restricting our analysis to k residuals. These are correlated with

p'S

1

1

k-2

equal to - k-l and their squares have the common distribution B(2 ' --2-)'

If we enter therefore table

I

with parameter values k and 1 for nand k

respectively we apply a correct procedure.

References

Barnett, V.D. and Lewis, T. (1978). Outliers in statistical data,

Wiley, Chicester etc.

Biometrika Tables for Statisticians, Vol. I, Third Edition, reprinted 1970,

Cambridge University Press.

Bross, I.D.J. (1961). OUtliers in patterned experiment: a strategic appraisal.

Technometrics 3, 91-102.

Daniel, C. (1960). Locatingoutlierain factorial experiments. Technometrics

2,

149-156.

Doornbos, R. and Prins, H.J. (1956). On slippage tests,

I.

A general type of

slippage test and a slippage test for normal variates. Indagationes

Mathematicae

20,

38-55.

Ellenberg, J.H. (1976). Testing for a single outlier from a general linear

regression. Biometrics 32, 637-645.

Federighi, E.T. (1959). Extended tables of the percentage points of Student's

t-distribution, Journal of the American Statistical Association 54,

683-688.

Lund, R.E. (1975). Tables for an approximate test for outliers in linear

regression. Technometrics 17, 473-476.

Mickey, M.R., Dunn,

O.J~

and Clark,

V.

(1967). Note on the use of stepwise

regression in detecting outliers. Computers and Biomedical Research

It n 1 2 3 4 5 7 8

..,

10 11 12 13 14 15 5 .9587 .9900 6 .9245 .9635 .9917 7 .8907 .9302 _ .9671 .9929 8 .8593 .8965 .9347 .9699 .9938 9 .8306 .8649 .9014 .9385 .9722 .9944 10' .8046 .8359 .8697 .9056 .9417 .9740 .9950 11 .7810 .8095 .8405 .8740 .9092 .9444 .9756 .9955 12 .7594 .7855 .8139 .8446 .8777 .9123 .9468 .9770 .9958 13 .7397 .7636 .7896 .8178 .8483 .8810 .9152 .9489 .9782 .9962 14 .7217 .7436 .7674 .7933 .8213 .8516 .8840 .9177 .9508 .9793 .9964 15 .7050 .7252 .7472 .7709 .7967 .8246 .8547 .8867 .9200 .9525 .9802 .9967 16 .6895 .7083 .7285 .7504 .7741 .7998 .8275 .8574 .8892 .9220 .9540' .9810 .9969 17 .6751 .6926 .7114 .7316 .7534 .7771 .8027 .8303 .8600 .8915 .9239 .9554 .9818 .9971 18 .6618 .67130 .. 6955 .7142 .7344 .7562 .7798 .8053 .8328 .8623 .8936 .9257 .9567 .9825 .9972 19 .6492 .6644 .6807 .6982 .7169 .7371 .7589 .7824 .8078 .8352 .8645 .8955 .9273 .9578 .9831 20 .6375 .6517 .6670 .6U32 .7007 .7194 .7396 .7613· .7848 .8101 .8374 .8666 .8974 .9288 .9589 21 .6264 .6398 .6541 .6693 .6856 .7031 .7218 .7419 •. 7637 .7871 .8123 .8395 .8685 .8991 .9302 22 .6159 .6286 .6420' .6563 .6716 .6879 .7053 .7240 .7442 .7658 .7892 .8144 .8414 .8703 .9006 23 .6061 .6180 .6307 .6442 .•• 6585 .6737 .6900 .7074 .7261 .7463 .7679 .7912 .~63 .8433 .8720 24 .5967 .60BI .6200 .6327 .6462 .6605 .6757 .6920' .7095 .7282 .7482 .7699 .7931 -.8182 .8450 25 .5879 .5986 .6099 .6219 .6346 .6481 .6624 .6777 .6940 .7114 .7301 .7501 .7717 .7950 .8199 26 .5794 • !:'.I396 .6004 .6117 .6237 .6364 .6499 .6642 .6795 .6958 .7132 .7319 .7519 .7735 .7967 27 .5714 .5811 .5913 .6021 .6135 .6:!55 .6382 .6517 .6660. .6813 .6976 .7150 .7336 .7537 .7752 28 .5637 .5730 .5827 .5929 .6037 .6151 .6271 .6399 .6533 .6677 .6830 .6992 .7167 .7353 .7553 29 .5563 .5652 .5745 .5843 .5945 .6053 .6167 .6287 .6415 .6550 .6693 .6846 .700'9 .71B3 .7369 30 .5493 .5~7B .5667 .5760 .5857 .5960 .6068 .6182 .6302 .6430 .6565 .670B .6861 .7024 .7198 31 .5426 .5507 .. ::1592 .5681 .5774 .5972 .5975 .6083 .6197 .6317 .6445 .6580 .6723 .6876 .7039 32 .5361 .5439 .5521 .5606 .5694 .5788 .5886 .5988 .6097 .6211 .6331 .6459 .6594 .6738 .6890 33 ₃₄ .5299 _.5240 .5374 _{.5311 .5386} .5452 .5533 .5619 .5708 .5801 .5899 .6002 .6110 .6224 ;.6345 .6473 .6608 .6752 .5464 .5546 .5631 .5720 .5814 .5912 .6015 .6123 .6238 .6358 .6486 .6621 35 .5182 .5251 .5323 .5398 .5476 .5559 .5643 .5733 .5826 .5924 .6027 .6136 .6250 .6371 .6499 36 ₃₇ .5127 _.5073 .5193 _.5138 .5263 _{.5204 .5274} .5335 .5410 .5488 .5570 .5655 .5744 .5638 .5936 .6040 .6148 .6263 .6383 .5:546 .5421 .5499 .5581 .5667 .5756 .5850 .5948 .6051 .6160 .6275 38 .5022 .5084 .5148 .5215 .5284 .5357 .5432 .5510 .5592 .5678 .5767 .5861 .5959 .6063 .6171 39 .4972 .5f)32 .5094 .5158 .5225 .5295 .5367 .5442 .5521 .5603 .5688 .5778 .5872 .5970 .6074 40 .4924 .4981 .5041 .5t04 .5168 .5235 .5305 .5.377 .5453 .5531 .5613 .5699 .5789 .5882 .5981 41 .4877 .4933 .4991 .5051 .5113 .5178 .5245 .5314 .5387 .5462 .5541 .5623 .5709 .5799 .5893 42 .4832 .4886 .4942 .5000 .5060 .5122 .5187 .5254 .5324 .5397 .5472 .5551 .5633 .5719 .5809 43 .478S .4840 .4895 .4951 .5009 .5069 .5132 .5196 .5263 .5333 .5406 .5482 .5560 .5643 .5729 44 _~5 .4745 _.4704 .4796 _.4753 .4849 .4'/03 .4959 .S018 .5078 .5140 .5205 .5272 .5342 .5415 .5491 .5570 .5652 .41104 .4857 .4912 .4968 .5026 .5086 .5149 .5214 .5281 .5351 .5424 .5500 .5579 46 .4664 .4712 .4761 .4812 .4865 .4920 .4976 .5034 .5095 .5157 .5222 .5290 .5360 .5433 .5508· 47 .4625 .4<',72 .4720 .4769 .4820 .4873 .4928 .4984 .5042 .5103 .5166 .5231 .5298 .5368 .5441 48 .4587 .4<>32 .4679 .4727 .4777 .4828 .4881 .4935 .4992 .5050 .5111 .5174 .5239 .5306 .5376 49 .4550 .4594 .4640 .4686 .4735 .4784 .4835 .4S88 .4943 .5000 .5058 .5119 .5181 .5246 .5314 50 .4514 .4~57 .4601 .4647 .4694 .4742 .4791 .4843 .4896 .4950 .5007 .5066 .5126 .5189 .5254

Table·I.2 Approximate critical values c

8 .8209 .8624 .9073 .9521 .9875 9 .7911 .8281 .8690 .9127 .9557 .9889 10 .7646 .7977 .8343 .8745 .9172 .9587 .9900 11 .7409 .7706 .8035 .8398 .8794 .9211 .9613 .9909 12 • 71'96 .7464 .7759 .8086 .8445 .8836 .9245 .9635 .9917 13 .7002 .7245 .7513 .7807 .8132 .8488 .8873 .9275 .9654 .9923 14 .6826 .7048 .7290 .7557 .7850 .8173 .8526 .8907 .9302 .9671 .9929 15 .6664 .6868 .7089 .7332 .7598 .7890 .8211 .8561 .8938 .9326 .9685 .9933 16 .6515 .6703 .6906 .7127 .7370 .7635 .7926 .8245 .8593 .8965 .9347 .9699 .9938 17 .6377 .6551 .6738 .6942 .7163 .7405 .7669 .7959 .8277 .8622 .8991 ,9367 .9711 .9941 18 .6249 .6410 .6584 .6771 .6975 .7196 .7437 .7701 .7990 .8306 .8649 .9014 .9385 .9722 .9944 19 .6130 .6280 .6441 .6614 .6802 .7005 .7227 .7468 .7731 .8019 .8334 .8674 .9036 .9402 .9731 20 .6018 .6158 .6308 .6470 .6643 .6831 .7034 .7255 .7496 .7759 .8046 .8359 .8697 .9056 .9417 21 .5912 .6044 .6185 .6335 .6497 .6671 .6858 .7062 .7282 .7523 .7785 .8071 .8383 .8719 .9074 22 .5813 .5937 .6069 .6210 .6361 .6522 .6696 .6884 .7087 .7308 .7548 .7810 .8095 .8405 .8740 23 .5720 .5837 .5961 .6093 .6234 .6385 .6547 .6721 .6908 .7112 .7332 .7572 .7833 .8118 .8426 24 .5632 .5742 .5859 .5983 .6116 .6257 .6408 .6570 .6744 .6932 .7135 .7355 .7594 .7855 .8139 25 .5548 .5652 .5763 .5880 .6005 .6137 .6278 .6430 .6592 .6766 .6954 .7157 .7377 .7616 .7876 26 .5.%'8 .5567 .5672 .5783 .5900 .6025 .6158 .6299 .6450 .6613 .6787 .6975 .7178 .7397 .7636 27 .5392 .5487 .5586 .5691 .5802 .5920 .6045 .6177 .6319 .6470 .6632 .6B07 .6995 .7198 .7417 28 .53'20 .5410 .5504 .5604 .5709 .5920 .5938 .6063 .6196 .6338 .6489 .6652 .6826 .7014 .7217 29 .5251 .5337 .5427 .5522 .5621 .5727 .5838 .5956 .6081 .6214 .6356 .6509 .6670 .6844 .7032 30 .5185 .5267 .5353 .5443 .5538 .5638 .5744 .5855 .5973 .6098 .6231 .6373 .6525 .6688 .6862 31 .5121 .5200 .5282 .5368 .5459 .5554 .5654 .5760 .5871 .5989 .6115 .6248 .6390 .6542 .6704 32 .5061 .5136 .5215 .5297 .5383 .5474 .5569 .5669 .5775 .5887 .6005 .6131 .6264 .6406 .6558 33 .5003 .5075 .5150 .5229 .5311 .5398 .5489 .5584 .5684 .5790 .5902 .6021 .6146 .6280 .6422 34 .4946 .5016 .5088 .5164 .5243 .5325 .5412 .5503 .5598 .5699 .5805 .5917 .6035 .6161 .6295 35 .4893 .4959 .5029 .5101 .5177 .5256 .5339 .5425 .5516 .5612 .5713 .5819 .5931 .6049 .6175 36 .4841 .4905 .4972 .5041 .5114 .5190 .5269 .5352 .5438 .5529 .5625 .5726 .5832 .5944 .6063 37 .4791 .4852 .4917 .4984 .5053 .5126 .5202 .5281 .5364 .5451 .5542 .5638 .5739 .5845 .5958 38 .4742 .4802 .4864 .4928 .4995 .5065 .5138 .5214 .5293 .5376 .5463 .5555 .5651 .5752 .5858 39 .4695 .4753 .4813 .4875 .4939 .5007 .5076 .5149 .5225 .5305 .5388 .5475 .5567 .5063 .5764 40 .4650 .4706 .4764 .4824 .4886 .4950 .5018 .5088 .5161 .5237 .5316 .5400 .54B7 .557a .5674 41 .4607 .4660 .4716 .4774 .4834 .4896 .4961 .5028 .5098 .5171 .5248 .5327 .5411 .5498 .5590 42 .4564 .4616 .4670 .4726 .4784 .4844 .4906 .4971 .5039 .5109 .5182 .5258 .5338 .5422 .5509 43 .4523 .4574 .4626 .4680 .4736 .4794 .4854 .4916 .4981 .5049 .5119 .5192 .5269 .5349 .5432 44 .4483 .4532 .4583 .4635 .4689 .4745 .4803 .4863 .4926 .4991 .5059 .. 5129 .5202 .5279 .5359 45 .4445 .4492 .4541 .4592 .4644 .4698 .4754 .4813 .4873 .4935 .5000 .5068 .5139 .5212 .5289 46 .4407 .4453 .4501 .4550 .4601 .4653 .4707 .4763 .4822 .4882 .4945 .5010 .5077 .5148 .5222 47 .4371 .4415 .4462 .4509 .4558 .4609 .4662 .4716 .4772 .4830 .4891 .4954 .5019 .5087 .5157 48 .4335 .4379 .4423 .4470 .4517 .4567 .4617 .4670 .4724 .4781 .4839 .4900 .4962 .5028 .5096 49 .4301 .4343 .4387 .4431 .4478 .4525 .4575 .4626 .4678 .4733 .4789 .4847 .4908 .4971 .5036 50 .4267 .4308 .4351 .4394 .4439 .4485 .4533 .4583 .46J3 .4686 .4741 .4797 .4856 .4916 .4979

Ie n _{2 3 4 5 6 7 8 9 10 11 12 13 14 15} 5 .9859 .9980 6 .9665 .9875 .9983 7 .9433 .9690 .9888 .9986 8 .9190 .9462 .9710 .9897 .9988 9 .8951 .9222 .9487 .9727 .9905 .9989 10 .8721 .8983 .9249 .9509 .9741 .9911 .9990 11 .8504 .8752 .9011 .9273 .9527 .9753 .9917 .9991 12 .8300 .8534 .8780 .9036 .9294 .9544 .9763 .9922 .9992 13 .8109 .8329 .8562 .8806 .9058 .9313 .9558 .9773 .9926 .9992 14 .7931 .8137 .8355 .8586 .8828 .9078 .9330 .9571 .9781· .9929 .9993 15 .7763 .7956 .8162 .8379 .8609 .8849 .9097 .9346 .9583 .9788 .9932 .9993 16 .7606 .7787 .7980 .8185 .8401 .8630 .8868 .9114 .9360 .9594 .9795 .9935 .9994 17 .7458 .7628 .7810 .8002 .8206 .8422 .8649 .8886 .9129 .9373 .9604 .9801 .9938 .9994 18 .7319 .7479 .7650 .7831 .8022 .8226 .8441 .8667 .8902 .9144 .9385 .9613 .9807 .9940 .9994 19 .7187 .7339 .7500 .7670 .7850 .8042 .8244 .8458 .8683 .8917 .9157 .9396 .9621 .9812 .9942 20 ₂₁ .7063 .7207 .7358 .7519 .7689 .7869 .8060 .8262 .8475 .8699 .8932 .9170 .9406 .9629 .9817 .6946 .7081 .7225 .7376 .7537 .7706 .7886 .8077 .8278 .8491 .8713 .8945 .9182 .9416 .9636 22 .6834 .6963 .7099 .7242 .7393 .7554 .7723 .7903 .8093 .8294 .8505 .8727 .8958 .9193 .9425 23 .6729 .6851 .6980 .7115 .7258 .7410 .7570 .7739 .7918 .8108 .8308 .8519 .8740 .8970 .9203 24 .6628 .6744 .6867 .6995 .7131 .7274 .7425 .7585 .7754 .7933 .8122 .8322 .8533 .8753 .8981 25 .6532 .6643 .6759 .6881 .7010 .7146 .7289 .7440 .7600 .7769 .7947 .8136 .8335 .8545 .8764 26 ₂₇ .6440 .6546 .6657 .6773 .6896 .7024 .7160 .7303 .7454 .7614 .7782 .7961 .8149 .8348 .8557 .6353 .6454 .6560 .6671 .6787 .6909 .7038 .7174 .7317 .7467 .7627 .7795 .7973 .8162 .8360 28 .6269 .6366 .6467 .6573 .6684 .6800 .6923 .7051 .7187 .7330 .7480 .7640 .7808 .7986 .8174 29 .6189 .6282 .6378 .6479 .6585 .6696 .6813 .6935 .7064 .7199 .7342 .7493 .7652 .7820 .7998 30 .6112 .6201 .6294 .6390 .6492 .6597 .6708 .6825 .6947 .7076 .7211 .7354 .7505 .7664 .7831 31 .6038 .6123 .6212 .6305 .6402 .6503 .6609 .6720 .6837 .6959 .7088 .7223 .7366 .7516 .7675 32 .5967 .6049 .6134 .6223 .6316 .6413 .6514 .6620 .6731 .6848 .6970 .7099 .7234 .7377 .7527 33 .5899 .5978 .6060 .6145 .6234 .6327 .6424 .6525 .6631 .6742 .6859 .6981 .7110 .7245 .7387 34 .5833 .5909 .5988 .6070 .6155 .6244 .6337 .6434 .6535 .6642 .6753 .6869 .6991 .7120 .7255 35 .5769 .5842 .5919 .5998 .6080 .6165 .6254 .6347 .6444 .6546 .6652 .6763 .6879 .7002 .7130 36 .5708 .5779 .5852 .5928 .6007 .6089 .6175 .6264 .6357 .6454 .6555 .6661 .6773 .6889 .7011 37 .5649 .5717 .5788 .5861 .5937 .6016 .6099 .6184 .6273 .6366 .6463 .6565 .6671 .6782 .6899 38 ·.5591 .5657 .5726 .5796 .5870 .5946 .6025 .6108 .6,93 .6282 .6375 .6473 .6574 .6680 .6791 39 .5536 .5600 .5666 .5734 .5805 .5878 .5955 .6034 .6116 .6202 .6291 .6384 .6481 .6583 .6689 40 .5482 .5544 .5608 .5674 .5742 .5813 .5887 .5963 .6042 .6125 .6210 .6300 .6393 .6490 .6592 41 .5430 .5490 .5552 .5616 .5682 .5751 .5821 .5895 .5971 .6051 .6133 .6219 .6308 .6401 .6498 42 .5380 .5438 .5498 .5560 .5624 .5690 .5758 .5829 .5903 .5979 .6059 .6141 .6227 .6316 .6409 43 .5331 .5387 .5445 .5505 .5567 .5631 .5697 .5766 .5837 .5911 .5987 .6066 .6149 .6235 .6324 44 ₄₅ .5284 _.5238 .5338 .5395 .5453 .5513 .5575 .5639 .5705 .5773 .5844 .5918 .5995 .6074 .6157 .6242 .5291 .5345 .5402 .5460 .5520 .5582 .5646 .5712 .5781 .5852 .5925 .6002 .6081 .6164 46 .5193 .5244 .5297 .5352 .540B .5467 .5527 .5589 .5653 .5719 .5788 .5859 .5933 .6009 .6089 47 .5149 .5199 .5251 .5304 .5359 .5415 .5473 .5533 .5595 .5660 .5726 .5795 .5866 .5940 .6016 48 .5107 .5151> .5206 .5257 .5310 .5365 .5422 .5480 .5540 .5602 .5666 .5733 .5801 .5873 .5946 49 .5066 .5113 .5162 .5212 .5264 .5317 .5372 .5428 .5486 .5546 .5609 .5673 .5739 .5808 .5879 50 .5025 .5072 .5119 .5168 .5218 .5270 .5323 .5378 .5434 .5493 .5553 .5615 .5679 .5746 .5814

Table I.3 Approximate critical values C

8 9 10 11 12 13 14 15 16 17 18 19

~!

I

23 24 26 27 28 29 30 31

HI'

_35, 361 37 38 39 40 41 421 43 ! 44 : 451 46

i

471

48, 49 ; 50i .9628 .9464 .9294 .9122 .8952 .8787 .8627 .8474 .8327 .8186 .8051 .7923 .7800 .7682 .7570 .7462 .7358 .7259 .7164 .7073 .6985 .6900 .6819 .6740 .6664 .6391 .6520 .6452 .6386 .6322 .6260 .6200 .6141 .6085 .6029 .5976 .5924 .~873 .3824 .5776 .5729 .5684 .5639 .9902 .9787 .9643 .9480 .9311 .9139 .8969 .8804 .8644 .8490 .8342 .8201 ;8066 .7937 .7813 .7695 .7582 .7473 .7370 .7'270 .7175 .7('83 .6994 .6910 .6828 .6749 .6673 .6599 .6~28 .6460 .6393 .6329 .6266 .6206 .6147 .60"'1 .60,55 • ~:;Y82 .3930 .3879 .5829 .5781 .5734 .5689 .9976 .9909 .9797 .9655 .9495 .9326 .9154· .8985 .8819 .8659 .8505 .8357 .8215 .8079 .7949 .7825 .7707 .7593 .7485 .7380 .7280 .7184 .7092 .7004 .6919 .6836 •. '.>757 .6681 .6607 .6536 .6467 • 64()0 .6336 .6273 .6213 .6154 .6097 .6041 .5988 .5935 .~;884 .58]5 .5786 .5739 .9999 .9978 .9914 .9806 .9666 .9507 .9339 .9169 .8999 .8834 .8673 .8519 .8370 .8228 .8092 .7962 .7837 .7718 .7604 .7495 .7391 .7290 .7194 .7101 .7012 .6927 .6844 .6765 .6688 .6614 .6543 .6474 .6407 .6342 .6279 .6219 .6160 .6103 .6047 .5993 .5940 .5890 .5840 .5791 .9999 .9980 .9918 .9813 .9676 .9519 .9352 .9182 .9013 .8847 .8686 .8532 .8383 .8240 .8104 .7973 .7849 .7729 .7615 .7505 .7400 .7300 .7203 .7110 .7021 .6935 .6852 .6773 .6696 .6622 .6550 .6481 .6413 .6349 .6286 .6225 .6166 .6108 .6052 .5999 .5946 .5895 .5845 .9999 .9981 .9922 .9819 .9684 .9529 .9363 .9194 .9025 .8859 .8699 .8544 .8395 .8252 .8115 .7984 .7859 .7740 .7625 .7515 .7410 .7309 .7212 .7119 .7029 .6943 .6860 .6780 .6703 .6629 .6557 .6487 .6420 .6355 .6292 .6231 .6171 .6114 .6058 .6004 .5951 .5900

7able I.4 Approximate critical values c

a' a

=

0.001 .9999 .9982 .9925 .9825 .9692 .9538 .9373 .9205 .9036 .8871 .8710 .85~5 .8406 .8263 .8126 .7995 .7870 .7749 .7635 .7524 .7419 .7318 .7220 .7127 .7037 .6951 .6868 .6788 .,!>710 .6636 .6563 .6494 .6426 .6361 .6298 .6236 .6177 .6119 .6063 .6009 .5956 .9999 .9983 .9928 .9830 .9699 .9.547 .9383 .9215 .9047 .8882 .8721 .8566 .8417 .8274 .8137 .8005 .7879 .7759 .7644 .7533 .7428 .7326 .7229 .7135 .7045 .6959 .6875 .6795 .6717 .6642 .6570 .6500 .6432 .6367 .6303 .6242 .6182 .6125 .6068 .6014 .9999 .9984 .9931 .9835 .9705 .9554 .9392 .9225 .9057 .8892 .8732 .8577 .8427 .8284 .8146 .8015 .7889 .7768 .7653 .7542 .7436 .7334 .7237 .7143 .7052 .6966 .6882 .6802 .6724 .6649 .6576 .6506 .6438 .6373 .6309 .6248 .6188 .6130 .6073 .9999 .9985 .9933 .9839 .9711 .9562 .9400 .9234 .9067 .8902 .8742 .8586 .8437 .8293 .8156 .8024 .7898 .7777 .7661 .7550 .7444 .7342 .7244 .7150 .7060 .6973 .6889 .6808 .6730 .6655 .6582 .6512 .6444 .6378 .6314 .6253 .6193 .6135 .9999 .9985 .9935 .9843 .9717 .9569 .9408 .9242 .9076 .8911 .8751 .8596 .8446 .8303 .8165 .8033 .7907 .7786 .7670 .7559 .7452 .7350 .7252 .7157 .7067 .6980 .6895 .6815 .6736 .6661 .6588 .6518 .6450 .6384 .6320 .6258 .6198 .9999 .9986 .9937 .9847 .9722 .9575 .9415 .9250 .9084 .8920 .8760 .8605 .8455 .8311 .8174 .8042 .7915 .7794 .7678 .7566 .7460 .7357 .7259 .7164 .7074 .6986 .6902 .6821 .6742 .6667 .6594 .6524 .6455 .6389 .6325 .6263 .9999 .9987 .9939 .9850 .9727 .9581 .9422 .9258 .9092 .8928 .8768 .8613 .8464 .8320 .8182 .8050 .7923 .7802 .7685 .7574 .7467 .7364 .7266 .7171 .7080 .6993 .6908 .6827 .6748 .6673 .6600 .6529 .6461 .6394 .6330 .9999 .9987 .9941 .9853 .9731 .9587 .9429 .9265 .9100 .8936 .8776 .8621 .8472 .8328 .8190 .8058 .7931 .7809 .7693 .7581 .7474 .7371 .7273 .7178 .7087 .6999 .6914 .6833 .6754 .6679 .6605 .6535 .6466 .6400 .9999 .9988 .9942 .9856 .9735 .9592 .9435 .9272 .9107 .8944 .8784 .8629 .8480 .8336 .8198 .8065 .7938 .7817 .7700 .7588 .7481 .7378 .7279 .7184 .7093 .7005 .6920 .6839 .6760 .6684 .6611 .6540 .6471 " '

k ") 3 4 5 6 7 8 9 10 11 12 13 14 15 5 .6298 .8522 6 .5191 .6628 .8752 7 .4488 .5440 .6890 .8921 8 .3988 .4675 .5648 .7104 .9049 9 .3605 .4132 .4837 .5827 .7283 .9150 10 .3299 .3720 .4260 .4979 .5983 .7435 .9232 11 .3048 .3393 .3823 .4375 .5106 .6120 .7567 .9299 12 ;2837 .3126 .3478 .3917 .4478 .5220 .6242 .7682 .9355 13 .2656 .2902 .3197 .3556 .4002 .4572 .5324 .6352 .7784 .9404 14 .2500 .2712 .2962 .3262 .3627 .4081 .4659 .5418 .6452 .7875 .9445 15 .2363 .2548 .2763 .3018 .3323 .3694 .4154 .4739 .5505 .6543 .7957 .9481 16 .2241 .2404 .2593 .2812 .3070 .3380 .3756 .4221 .4813 .5586 .6627 .8031 .9513 17 .2133 .2278 .2444 .2635 .2857 .3119 .3433 .3813 .4285 .4882 .5660 .6704 .8098 .9541 ! 18 .2036 .2166 .2313 .2481 .2674 .2900 .3165 .3483 .3868 .4344 .4947 .5730 .6776 .8160 .9565 19 .1948 .2065 .2197 .2346 .2516 .2712 .2940 .3209 .3530 .3919 .4400 .5007 .5795 .6843 .8217 20 .1869 .1975 .2093 .2226 .2377 .2549 .2747 .2978 .3250 .3574 .3967 .4452 .5064 .5856 .6905 21 .1796 .1892 .1999 .2119 .2254 .2406 .2580 .2781 .3014 .3289 .3616 .4013 .4502 .5118 .5914 22 .1729 .1817 .1915 .2023 .2144 .2280 .2434 .2610 .2813 .3049 .3326 .3657 .4057 .4550 .5170 23 .1668 .1749 .1838 .1936 .2046 .2168 .2305 .2461 .2639 .2844 .3082 .3361 .3695 .4098 .4595 24 .1611 .1686 .1767 .1857 .1957 .2067 .2191 .2330 .2487 .2666 .2873 .3113 .3395 .3732 .4138 25 .1559 .1628 .1703 .1785 .1876 .1976 .2088 .2213 .2353 .2512 .2693 .2901 .3143 .3428 .3767 26 .1510 .1574 .1643 .1719 .1803 .1894 .1995 .2108 .2234 .2375 .2535 .2718 .2928 .3172 .3459 27 .1464 .1524 .1588 .1658 .1735 .1819 .1911 .2014 .2127 .2254 .2397 .2558 .2742 .2954 .3200 28 .1421 .1477 .1537 .1602 .1673 .1750 .1835 .1928 .2031 .2146 .2273 .2417 .2580 .2765 .2979 29 .1381 .1433 .1489 .1550 .1616 .1687 .1765 .1850 .1944 .2048 .2163 .2292 .2437 .2601 .2788 30 .1344 .1393 .1445 01501 .1562 .1629 .1701 .1779 .1865 .1960 .2064 .2181 .2310 .2456 .2621 31 .130B .1354 .1403 .1456 .1513 .1575 .1641 .1714 .1793 .1880 .1975 .2080 .2197 .2328 .2475 32 .1275 .1318 .1364 .1414 .1467 .1524 .1586 .1653 .1726 .lS06 .1893 .1989 .2096 .2213 .2345 33 .1243 .1284 .1328 .1374 .1424 .1478 .1535 .1598 .1665 .1739 .1819 .1907 .2004 .2110 .2229 34 .1213 .1252 .1293 .1337 .1384 .1434 .1488 .1546 .1609 .1677 .1751 .1832 .1920 .2017 .2125 35 .1185 .1222 .1260 .1302 .1346 .1393 .1443 .1498 .1556 .1619 .1688 .1762 .1844 .1933 .2031 36 .1158 .1193 .1230 .1269 .1310 .1354 .1402 .1453 .1507 .1566 .1630 .1699 .1774 .1855 .1945 37 .1133 .1166 .1200 .1237 .1276 .1318 .1363 .1411 .1462 .1517 .1576 .1640 .1709 .1785 .1867 38 .1108 .1140 .1173 .1208 .1245 .1284 .1326 .1371 .1419 .1471 .1526 .1585 .1650 .1719 .1795 39 .1085 .1115 .1146 .1179 .1215 .1252 .1292 .1334 .1379 .1427 .1479 .1535 .1595 .1659 .1729 40 .1063 .1091 .1121 .1153 .1186 .1222 .1259 .1299 .1342 .1387 .1435 .1488 .1543 .1604 .1669 41 .1042 .1069 .1097 .1127 .1159 .1193 .1228 .1266 .1306 .1349 .1395 .1443 .1496 .1552 .1612 42 .1021 .1047 .1074 .1103 .1133 .1165 .1199 .1235 .1273 .1313 .1356 .1402 .1451 .1504 .1560 43 .1002 .1027 .1053 .1080 .1109 .1139 .1171 .1205 .1241 .1279 .1320 .1363 .1409 .1459 .1511 44 .0983 .100)1 .1032 .1058 .1086 .1115 .1145 .1177 .1211 .1248 .1286 .1327 .1370 .1416 .1466 45 .0965 .0988 .1012 .1037 .1063 .1091 .1120 .1151 .1183 .1217 .1254 .1292 .1333 .1377 .1423 46 .0948 .0970 .0993 .1017 .1042 .1068 .1096 .1125 .1156 .1189 .1223 .1260 .1298 .1340 .1383 47 .0932 .0953 .0975 .0998 .1022 .1047 .1073 .1101 .1131 .1162 .1194 .1229 .1266 .1304 .1346 48 .0916 .0936 .0957 .0979 .1002 .1026 .1052 .1078 .1106 .1136 .1167 .1200 .1234 .1271 .1310 49 .0900 .0920 .0940 .0961 .0983 .1007 .1031 .1056 .1083 .1111 .1141 .1172 .1205 .1240 .1277 50 .0886 .0904 .0924 .0944 .0965 .0988 .1011 .1035 .1061 .1088 .1116 .1146 .1177 .1210 .1245

8 .4813 9 .4344 10 .3967 11 .3657 12 .3395 13 .3172 14 .2979 15 .2810 16 .2660 17 .2527 18 .2408 19 .23tH 20 .2203 21 .2114 22 ,;2033 23 .1958 24 .1889 25 .1825 26 .1766 27 .1710 28 .1659 29 .t610 30 .156~ 31 .. 1522 32 • 141:J2 33 .1444 34 .1408 35 .1374 36 .1342 37 .1312 38 .. 1282 39 -,1255 40 .1228

4

4~_1 .1203 ..: .117S· 43 .115:;' 441 .1133 45 .1112 46 .1092 47 .1072 48 .1053 49 .1035 50 .1017 .55B6 .4947 .4452 .4057 .3732 .3459 .3226 .3026 .2851 .2697 .2560 .2437 .2326 .2226 .21~5 .2052 .191'6 .1905 .1840 .1780 .1723 .1671 .16:'·~~ .1576 .1532 .1491 .1453 .141",' • 138:,! .1349 .13Hl .12H9 .1261 .1234 .1208 .1184 .116.1 .li38 .1117 • 1 \)')16 .1076 .1057 .1039 .6627 .5730 .5064 .4550 .4138 .3800 .3517 .3277 .3070 .2890 .2731 .2590 .2464 .23':·1 .2249 .2156 .2071 .1993 .1921 .1855 .1193 .1736 .1682 .1632 .1586 .1542 .1~OO .1461 .1424 .1390 .1357 .1325 .1.295 .1267 .1::-40 .1214 .1189 .1166 .11.43 .1121 .110Q .1080 .1061 .8031 .6776 .5856 .5170 .4637 .4212 .3863 .3572 .3324 .3111 .2926 .2763 .2619 .2490 .2375 .:2270 .217~ .2089 .2009 .19:~6 .1869 .ta06 .1743 .1694 .1641 .15'16 .1551 .1~09 .1470 .1432 .1397 .1363 .1332 .1301 .1273 .1245 .1219 .1194 .1170 .1148 .1126 .1105 .1085 .9513 .8160 .6905 .5968 .5264 .4718 .4280 .3922 .3622 .3368 .3150 .2960 .2794 .2647 .2515 .2397 .2291 .2194 .2106 .2025 .1951 .1882 .1819 .1760 .1705 .1653 .1605 .1560 .1517 .1477 .1440 .1404 .1370 .1338 .1308 .1279 .1251 .1225 .1199 .1175 .1152 .1130 .1109 .9565 .8269 .7018 .6069 .5351 .4791 .4343 .3976 .3670 .3410 .3187 .2993 .2823 .2673 .2539 .2419 .2310 .2212 .2122 .2040 .1965 .1895 .H131 .1771 .1715 .1663 .1614 .1569 .1526 .1485 .1447 .1411 .1377 .1344 .1313 .1284 .1256 .1230 .1204 .1180 .1157 .1134 .9608 .8363 .7118 .6159 .5429 .4859 .4402 .4027 .3714 .3449 , .3222 .3024 .2851 .2698 .2561 .2439 .2329 .2229 .2138 .2055 .1978 .1908 .1842 .1782 .1725 .1673 .1623 .1577 .1534 .1493 .1454 .1418 .1383 .1350 .1319 .1290 .1262 .1235 .1209 .1185 .1161

Table 11.2 Upper bounds for correlations ga' n

=

0.05

.9643 .8445 .7208 .6242 .5502 .4922 .4457 .4075 .3756 .3486 .3255 .3054 .2877 .2722 .2583 .2459 .2347 .2246 .2153 .2069 .1991 .1920 .1854 .1792 .1735 .1682 .1632 .1585 .1541 .1500 .1461 .1424 .1389 .1356 .1325 .1295 .1267 .1240 .1214 .1189 .9672 .8517 .7290 .6318 .5569 .4981 .4508 .4120 .3796 .3522 .3286 .3082 .2903 .2745 .2604 .2478 .2365 .2262 .2168 .2083 .2004 .1932 .1865 .1803 .1745 .1691 .1641 .1593 .1549 .1507 .1468 .1431 .1396 .1362 .1330 .1300 .1272 .1244 .1218 .9697 .a581 .7364 .638a .5632 .5036 .4557 .4163 .3834 .3555 .3316 .3109 .2927 .2767 .2624 .2496 .2381 .2277 .2183 .2096 .2016 .1943 .1875 .1813 .17S4 .1700 .1649 .1601 .1556 .1514 .1475 .1437 .1401 .1368 .1336 .1306 .1277 .1249 .9718 .8638 .7431 .6453 .5691 .5088 .4602 .4203 .3870 .3587 .3345 .3134 .2950 .2788 .2643 .2514 .2398 .2292 .2196 .2109 .2028 .1954 .1886 .1822 .1763 .1708 .1657 .1609 .1564 .1521 .1481 .1443 .1407 .1373 .1341 .1311 .1281 .9737 .8690 .7494 .6513 .5745 .5137 .4646 .4242 .3904 .3618 .3372 .3159 .2973 .2808 .2662 .2531 .2413 .2307 .2210 .2121 .2040 .1965 .1896 .1832 .1772 .1717 .1665 .1616 .1571 .1528 .1487 .1449 .1413 .1379 .1346 .1316 .9753 .8737 .7551 .6570 .5797 .5183 .4687 .4278 .3937 .3647 .3398 .3183 .2994 .2828 .2680 .2548 .2429 .2321 .2223 .2133 .2051 .1976 .1906 .1841 .1781 .1725 .1673 .1624 .1578 .1534 .1494 .1455 .1419 .1384 .1351 .9768 .8779 .7605 .6622 .5846 .5227 .4726 .4313 .3968 .3675 .3424 .3206 .3015 .2847 .2697 .2564 .2443 .2334 .2235 .2145 .2062 .1986 .1915 .1850 .1789 .1733 .1680 .1631 .1584 .1541 .1500 .1461 .1424 .1389 .9780 .8819 .7654 .6672 .5892 .5269 .4763 .4347 .3998 .3702 .3448 .3228 .3035 +2865 .2714 .2579 .2458 .2348 .2248 .2157 .2073 .1996 .1925 .1859 .1798 .1741 .1688 .1638 .1591 .1547 .1506 .1466 .1430

It n 1 2 3

"

5 6 7 8 9 10 11 12 13 14 15 5 .9001 .9841 6 .8018 .9107 .9868 7 .7159 .814" .9188 .9886 8 .6452 .7277 .8246 .9253 .9901 9 .5868 .6555 .7377 .8333 .9306 .9912 10 .5381 .5957 .6646 .7465 .8407 .9350 .9920 11 .... 970 .5458 .6037 .6726 .7541 .8471 .9388 .9928 12 .4618 .5036 .5527 .6109 .6798 .7610 .8528 .9421 .9934 13 .4313 .4675 .5096 .5591 .6174 .6863 .7671 .8579 .94"9 .9939 14 .4048 .4364 .4728 .5152 .56"8 .6233 .6923 .7727 .8624 .9475 .9943 15 .3814 .4092 .4410 .4777 .5203 .5702 .6288 .6977 .7778 .8666 .9497 .9947 16 .3607 .3853 .4133 ."453 .4822 .5251 .5751 .6339 .7027 .7825 .8703 .9517 .9950 17 .3422 .3641 .3889 '''171 .4494 .4865 .5296 .5798 .6386 .7074 .1868 .8737 .9536 .9953 18 .3255 .3453 .3674 .3924 .4207 .4532 .4905 .5337 .5841 .6430 .7118 .7908 .8769 .9552 .9956 19 .3105 .3283 .3482 .3705 .3956 .4241 .4568 .4943 .5377 .5882 .6471 .7158 .7945 .8798 .9568 20 .2969 .3130 .3310 .3510 .3734 .3987 .4274 .4601 .4978 .5414 .5920 .6510 .7196 .7980 .8825 21 .2845 .2992 .315" .3335 .3536 .3761 .4016 .4304 .4634 .5012 .5449 .5956 .6541 .7232 .8013 22 .2131 .2866 .3014 .3177 .3359 .3561 .3788 .4043 .4333 .4664 .5044 .5482 .5990 .6581 .7266 23 ₂₄ .2627 _{.2530 .2644} .2750 .2886 .3035 .3199 .3381 .3585 .3813 .4070 .4361 .4693 .5074 .5514 .6023 .6614 .2768 .2905 .3054 .3220 .3403 .3608 .3837 .4095 .4387 .4721 .5103 .5544 .6054 25 .2441 .2546 .2661 .2786 .2923 .3073 .3240 .3424 .3630 .3860 .4119 .4413 .4747 .5131 .5572 26 .2358 .2456 .2562 .2677 .2803 .2940 .3092 .3259 .3444 .3651 .3882 .4142 .4437 .4773 .5157 27 .2281 .2372 .2470 .2577 .2693 .2819 .2957 .3109 .3277 .3463 .3671 .3903 .4164 .4460 .4797 28 .2210 .2294 .2385 .2484 .2591 .2708 .2834 .2974 .3126 .3295 .3482 .3690 .3923 .4186 .4483 29 .2142 .2221 .2307 .2398 .2498 .2605 .2722 .2849 .2989 .3143 .3312, .3500 .3709 .3943 .4206 30 .2079 .2154 .2233 .2319 .2411 .2510 .2618 .2736 .2864 .3004 .3158 .3328 .3517 .3727 .3962 31 .2020 .2090 .2164 .2244 .2330 .2423 .2523 .2631 .2749 .2878 .3019 .3174 .3344 .3534 .3744 32 .1964 .2030 .2100 .2175 .2255 .2341 .2434 .2535 .2644 .2762 .2891 .3033 .3188 .3360 .3550 33 .1912 .1974 .2039 .2110 .2185 .2265 .2352 .2446 .2547 .2656 .2775 .2905 .3047 .3203 .3375 34 .1862 .1921 .1983 .2049 .2119 .2195 .2276 .2363 .2456 .2558 .2668 .2787 .2917 .3060 .3216 35 .1815 .1870 .1929 .1991 .2058 .2128 .2204 .2285 .2373 .2467 .2569 .2679 .2799 .2930 .3073 36 .1771 .1823 .1878 .1937 .2000 .2066 .2137 .2214 .2295 .2383 .2477 .2580 .2690 .2811 .2942 37 .1729 .1778 .1831 .1886 .19"5 .2008 .2075 .2146 .2223 .2304 .2393 .2487 .2590 .2701 .2822 38 .1688 .1736 .1785 .1838 .1894 .1953 .2016 .2083 .2155 .2231 .2313 .2402 .2497 .2600 .2711 39 .1650 .1695 .1742 .1792 .1845 .1901 .1960 .2024 .2091 .2163 .2240 .2322 .2411 .2506 .2610 40 .1614 .1656 .1701 .1749 .1799 .1852 .1908 .1968 .2031 .2099 .2171 .2248 .2331 .2420 .2516 41 .1519 .1620 - .1663 .1708 .1755 .1805 .1859 .1915 .1975 .2038 .2106 .2179 .2256 .2339 .2429 42 .1546 .1585 .1625 .1668 .1714 .1761 .1812 .1865 .1922 .1982 .2046 .2114 .2186 .2264 .2347 43 ₄₄ .1514 _.1484 .1551 _.1519 .1590 .1631 .1674 .1720 .1768 .1818 .1872 .1928 .1989 .2053 .2121 .2194 .2272 .1556 .1596 .1637 .1680 .1725 .1773 .1824 .1878 .1935 .1995 .2060 .2128 .2201 45 .1'\55 .1489 .1524 .1562 .1601 .1642 .1685 .1731 .1779 .1830 .1884 .1941 .2002 .2066 .2135 46 ₄₇ .1427 _.1400 .1459 _{.1431 .1464} .1493 .1529 .1567 .1606 .1647 .1691 .1737 .1785 .1836 .1890 .1947 .2008 .2073 .1498 .1534 .1572 .1611 .1652 .1696 .1742 .1791 .1842 .1896 .1953 .2015 48 .1374 .1404 .1436 .1468 .1503 .1539 .1576 .1616 .1657 .1701 .1747 .1796 .1847 .1902 .1959 49 .1349 .1378 .1408 .1440 .1473 .1507 .1543 .1581 .1621 .1662 .1706 .1753 .1801 .1853 .1907 ~, 50 ,1326 .1353 .1382 .1413 .1444 .1477 .1512 .1548 .1586 .1626 .1667 .1711 .1758 .1807 .1858

Table 11.3 Upper bounds for correlations 9