Librar9 1964

"Tlote 92c. 2/5

PERCENTAGE POINTS OF THE

BETA DISTRIBUTION

L. E. VOGLER

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical ^ote 215

Issued May 30, 1964

PERCENTAGE POINTS OF THE

BETA DISTRIBUTION

L. E. Vogler

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

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: 25*

Percentage Points of the Beta Distribution

L. E. Vogler

Tables are presented of percentage points

x(p|v->/2, v \l^) of the Beta distribution. The percentage

points are given to six figures correct to within a unit in

the last place for probability levels p : p = OoOOOl, 0.001, 0.005, 0.01, 0.025, 0.05,

0.1, 0.25, 0.5, and for degrees of freedom V£ and v,:

v

2

= 1(0.1)2, 2.2, 2.5(0.5)5, 6(1.0)10, 12,

15, 20, 24, 30, 40, 60, 120,

v = 1(1.0)10, 12, 15, 20, 24, 30, 40, 60, 120.

A table of the Complete Beta function for the above

values of v and v

? is also included.

The present tables of percentage points of the Beta distribution

are the outgrowth of a project started at the Central Radio Propagation

Laboratory of the National Bureau of Standards with the purpose of

extending existing tables of the Fisher-Snedecor F-distribution to

cover lower probability levels p and include non-integer values of

the degrees of freedom v and v_. Values of F for finite v and v

? are most easily obtained through its relationship to the Beta

distribution.

Thompson's excellent five -figure tables of Beta function percentage points [ Thompson, 1941 ] have been the basic compilation of

this statistic since their publication; however, the probability level

does not go below 0.005 and no non-integer values of v or v

? are

included. In certain situations the Beta distribution with non-integer
degrees of freedom may be used to approximate an unknown distribution [Pearson and Hartley, 1956 ]; also in the analysis of variance,

some applications require the use of the F-distribution with noninteger values of v

? [Box, 1954a; 1954b],

It was decided to calculate six-figure percentage points for all

of the probability levels and most of the v 's and v 's given in the

Thompson tables and to add non-integral entries of v between 1 and

5 and extend the range of p down to 0.0001; the percentage points

are tabulated in table II. The six-figure values would assure fivefigure accuracy in the F-distribution and could also serve as a check

on the Thompson tables. A list of the discrepancies encountered is

given in table I

TABLE I p V

l V2 Thompson Table II

0. 5 4 5 0. 56444 0. 564446

0. 5 5 4 0.43556 0.435554

0.5 20 1 0.023052 0.0230514

0.25 24 2 0.023689 0.0236884

0. 1 5 1 0.

2 34818 0.2 348189

0. 1 15 4 0. 064482 0.0644585

0.01 8 1 0. 4 20897 0. 4 208988

0.01 10 7 0.095627 0.0956276

0.01 20 8 0. 069456 0.0694554

0.01 40 9 0.046956 0. 0469567

0.01 60 15 0.075824 0.0758234

0.005 7 6 0.073619 0.0736184

0.005 9 7 0.083707 0. 0837076

0.005 12 8 0.086787 0. 0867877

0.005 120 30 0.098050 0.0980479
The Beta distribution is defined through the cumulative distribution function:

-1 ^x (v /2)-l (v /2)-l

p(x|v

2/2, Vl /Z) = B(v

2/2, Vl /2) j u (1-u) du , (1)

J o

where B(v_/2, v /2) denotes the Complete Beta function given by

w J1 (v /2>-l (v /2)-l T(v /2)T(v /2)

B(v/Z,v/2)s \ u (1-u) du = . (2) o r[(v/2)+(v/2)]

Values of (2) for the v 's and v 's used in the computations of

-L w this paper are shown in table III. Note that B(v /2, v /2) = B(v /2, v /2)

and some of the values in the table are repeated for the convenience of

the reader.

The percentage points x are calculated by using (1) with

given values of p, v , and v . The integral in (1) is expressed in

series form and, since < x < 1 and v and v are finite, rootJL w solving methods are applied to evaluate x. .Unfortunately, no single

series representation is computationally practical for all values of

v and v o Consequently, different methods must be used for different ranges of the degrees of freedom.

After considerable experimentation, two expansions of (1) were

found that could be used in the area covered by the tables. For x not

too near unity, the following relationship was utilized:
v

2

/2 (Vj/2)-!

p(v

2/2) B(v

2/2, V]L /2) = x (1-x) 1 + I

T

k<

X> k=l

(3)

where

T

k<

X> = GrrS") fe) T

k-l<

X)

- T

o<*> = ! •

For x near unity, an alternate expression was derived:

(1 -P) (v

1 /2) B(v

2/2, Vl /2) = (1 -x)

v

L /2

l+(v

1 /2) ^ T^x)

k=l

(4)

where

T

fc(x| .

v +2k-2 \ • v -2k

L If"

v

t +2k 2k -) (1-x) T^x). T^(x) = -2/Vl

Using either (3) or (4) (and occasionally both for checking

purposes), x was obtained through an iteration process, Newton's

method of root-solving being in general the most practical. As a

further check, a third expression was derived for those areas of the

tables containing the change-over from (3) to (4):

<i +l

= Xi{1+(2 '"2>S < Dk- Cfc»<}

k=l

(5)
where

5. + l-kN

v +2k+v -4^ v -2

D, =-l-JH ) D! ,. D =1, C, = -rj^ lC, _, C = -

k / k-1 o k ^ v +2k T

J

k-1 1 v

2 + 2

6.

l in(pv

2

B/2)-(v

2

/2)inx. L/in(l-x.) - (v

]L -2)/2 ,

and the subscript i denotes the number of iterations,,

Equations (3), (4), and (5) were programmed for an electronic computer, at least eight figures being carried in all arithmetic

operations. Successive approximations to the true x were calculated

from the right hand members of (3) or (4) until agreement to at

least seven figures was reached with the expressions on the left. The

final x was then rounded to six figures.

The main portion of the tables was checked (to five figures) bycomparison with the Thompson tables, differences being re-checked

through hand calculations (see table I). For values not given in

Thompson, plots were made of x vs p for a given v and v with the hope of at least avoiding gross errors.

Differencing was not used to check the table entries because of

the irregular intervals in v . In those regions where cumulative

round-off errors were of possible concern, the x-values were checked

by independent runs of at least two (and sometimes all three) of the

generating equations (3), (4), and (5). As a final check most of the

entries were re -calculated using double -precision arithmetic in all

machine operations. To avoid copying errors, the reproduced table

was compared with the original computer print-outs, and it is believed

that all entires are correct to within a unit in the last figure.
The present tables are constructed only up to p = 0.5, thus

giving the lower percentage points of the Beta distribution. Corresponding upper percentage points are easily obtained by interchanging

v and v and replacing p and x by (1-p) and (1-x):

J. c»

p(x|v

2

/2, vJZ) = 1 -p(l-x| Vl /2, v

2 /2) . (6)

Although the Beta distribution may be used indirectly to obtain

a number of important statistical distributions, only two of its applications will be mentioned here.

(a) The Fisher-Snedecor variance ratio F with degrees of

freedom v and v and of probability p is related to x

J. C> by

'-^Xtt^-

(b) The relationship to the cumulative sum of the individual

terms of the binomial distribution is given by

n

£ ) x

k

(l-x)

n " k

= k=m

Interpolation within the tables to a high degree of accuracy is

probably most easily accomplished through the method of Lagrangian

interpolation. Tables of Lagrangian coefficients for harmonic interpolation have been developed by Comrie and Hartley [ 1941 ]; a

comprehensive discussion of interpolation is given by Hartley [ 1941 ] and tables to facilitate the p-wise interpolation may be found in a paper

by Salzer [ 1951 ]. For practical applications two or three figures will
often suffice, in which case a first order harmonic interpolation in the

v-directions may be used. Thus, if for a given p: v < v < v ,

1 II 1 I / I I X "/ " ' \

i / ' " \

II 7II II . V 2- V 2-V and X

1 (V

1' V

2

)J X

l

(v

r

V

2

)j X

2

( V

V

2

}

' X

2

( V

V

2

) denote the nearest tabulated values, then

x(v

1 ,v2

) = 6^ + (l-6

2

)x

2

, (9)

where

X

l = 5

1 X

l +(1 - 6

l )xi'

' x

2 = 6

1 X

2 +(1 - 6

1 )X

2 ' (10)

and

I / II . I / II x

6 = , 5 = (11)

v(v-v) v(v-v) l

l 1 V 2V 2 Z'

The function logp may then be used for linear interpolation in the

p-direction„

The author wishes to acknowledge the valuable assistance of

Mr. Judd Payne and Mrs. J. E. Herman who did much of the computer

programming and compilation of values, and Mrs. Marjorie F. Munyon

who carefully typed the many tabulated entries. His thanks also go to

K. A. Norton whose foresight initiated the project and whose guidance

made possible the completion of these tables.
REFERENCES

Box, G. E.Po (1954a), Some theorems on quadratic forms applied in

the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classification, Ann. Math.

Stat. 2_5, 290-302.

Box, G. E. P. (1954b), Some theorems on quadratic forms applied in

the study of analysis of variance problems, II. Effects of inequality of variance and of correlation between errors in the

two-way classification, Ann. Math. Stat. 25, 484-498.

Comrie, L. J. , and H. O. Hartley (1941-1942), Table of Lagrangian

coefficients for harmonic interpolation in certain tables of

percentage points, Biometrika 32, p. 183.

Hartley, H. O. (1941), Methods of interpolation, Biometrika 32,

161-167.

Pearson, E.S., and H. O. Hartley (1956), Biometrika tables for

statisticians, Vol. I, (Cambridge University Press). See

section 8. 4.

Salzer, H. E. (1951), Checking and interpolation of functions tabulated at certain irregular logarithmic intervals, J. Res. Natl.

Bur. Std. 46, No. 1, 74-77.

Thompson, CM. (1941-1942), Tables of percentage points of the

incomplete beta -function, Biometrika 32, p. 151.

8
TABLE II

PERCENTAGE POINTS OF THE BETA DISTRIBUTION
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

vz p = 0.0001 p = 0.00l p = 0.005 p = 0.01 p =0.025 p = 0.05 p= 0.1 p = 0.25 p=0.5

1 (-8)2.46740 (-6)2.46740 (-5)6. 16838 (-4)2.46720 (-3)1. 54133 (-3)6. 15583 (-2)2.44717 (-1)1.46447 (-1)5. 00000

I.I (-7)1. 28058 (-6)8.42537 (-4)1. 57189 (-4)5.54235 (-3)2. 93014 (-2)1. 03081 (-2)3.60442 (-1)1. 81222 (-1)5.42541

1.2 (-7)5. 03764 (-5)2.33825 (-4)3.41820 (-3)1.08496 (-3)4.99017 (-2)1. 57890 (-2)4.95874 (-1)2. 15558 (-1)5. 79267

1.3 (-6)1. 60167 (-5)5.53386 (-4)6.58093 (-3)1. 91093 (-3)7. 81055 (-2)2.25859 (-2)6.47480 (-1)2.48844 (-1)6. 11126

1.4 (-6)4. 30871 (-4)1. 15586 (-3)1. 15160 (-3)3. 09810 (-2)1. 14422 (-2)3. 06241 (-2)8. 11651 (-1)2.80714 (-1)6. 38916

1.5 (-5)1. 01422 (-4)2. 18493 (-3)1.86721 (-3)4. 70126 (-2)1.59001 (-2)3.97876 (-2)9.85004 (-1)3. 10975 (-1)6.63300

1.6 (-5)2. 14216 (-4)3.80898 (-3)2.84592 (-3)6. 76140 (-2)2. 11692, (-2)4.99376 (-1)1. 16454 (-1)3.39547 (-1)6.84818

1.7 (-5)4. 13867 (-4)6.21246 (-3)4. 12258 (-3)9. 30489 (-2)2. 72110 (-2)6. 09264 (-1)1.34772 (-1)3.66428 (-1)7.03915

1.8 (-5)7.42418 (-4)9. 58647 (-3)5. 72461 (-2)1.23442 (-2)3.39705 (-2)7.26073 (-1)1.53243 (-1)3.91663 (-1)7.20954

1.9 (-4)1.25122 (-3)1.41196 (-3)7.67150 (-2)1. 58793 (-2)4. 13822 (-2)8.48411 (-1)1. 71697 (-1)4. 15325 (-1)7. 36234

2 (-4)1.99990 (-3)1.99900 (-3)9. 97500 (-2)1. 99000 (-2)4.93750 (-2)9. 75000 (-1)1. 90000 (-1)4. 37500 (-1)7. 50000

2.2 (-4)4.48703 (-3)3.63681 (-2)1.56637 (-2)2. 93172 (-2)6.68131 (-1)1. 23648 (-1)2.25763 (-1)4. 77771 (-1)7. 73779

2.5 (-3)1. 17884 (-3)7.42762 (-2)2.67998 (-2)4. 64520 (-2)9.55746 (-1)1.63632 (-1)2. 76412 (-1)5.29365 (-1)8.02247

3 (-3)3.81191 (-2)1. 76440 (-2)5. 12365 (-2)8. 08272 (-1)1.46746 (-1)2.28520 (-1)3. 51357 (-1)5.97150 (-1)8. 36806

3.5 (-3)8. 76150 (-2)3.25162 (-2)8. 08201 (-1)1. 19192 (-1)1.97938 (-1)2. 88205 (-1)4. 14731 (-1)6.48611 (-1)8.61240

4 (-2)1. 62854 (-2)5. 11919 (-1)1. 13209 (-1)1. 58745 (-1)2.46642 (-1)3.41628 (-1)4.68123 (-1)6. 88776 (-1)8. 79385

4.5 (-2)2.62922 (-2)7.26186 (-1)1.46647 (-1)1. 97741 (-1)2. 91813 (-1)3. 88943 (-1)5. 13312 (-1)7.20892 (-1)8.93372

5 (-2)3.84789 (-2)9. 58207 (-1)1. 79955 (-1)2. 35204 (-1)3. 33178 (-1)4. 30741 (-1)5.51851 (-1)7.47108 (-1)9. 04474

6 (-2)6. 78050 (-1)1.44552 (-1)2.43556 (-1)3. 03874 (-1)4.05047 (-1)5.00526 (-1)6. 13751 (-1)7. 87255 (-1)9.20967

7 (-1)1.01205 (-1)1.93129 (-1)3. 01262 (-1)3.63705 (-1)4.64417 (-1)5.55933 (-1)6.61037 (-1)8. 16496 (-1)9. 32622

8 (-1)1. 36299 (-1)2. 39415 (-1)3. 52606 (-1)4. 15398 (-1)5. 13779 (-1)6. 00706 (-1)6.98207 (-1)8. 38716 (-1)9.41289

9 (-1)1. 71505 (-1)2. 82511 (-1)3.97990 (-1)4. 60089 (-1)5.55237 (-1)6. 37513 (-1)7.28137 (-1)8. 56159 (-1)9.47985

10 (-1)2. 05858 (-1)3.22169 (-1)4. 38088 (-1)4.98895 (-1)5.90433 (-1)6.68244 (-1)7. 52728 (-1)8. 70211 (-1)9. 53313

12 (-1)2. 70106 (-1)3. 91602 (-1)5.05173 (-1)5.62582 (-1)6.46769 (-1)7. 16537 (-1)7. 90694 (-1)8.91443 (-1)9.61254

15 (-1)3. 53373 (-1)4.74873 (-1)5.81439 (-1)6. 33363 (-1)7. 07564 (-1)7.67535 (-1)8.29959 (-1)9. 12849 (-1)9.69133

20 (-1)4.60835 (-1)5. 74403 (-1)6.67915 (-1)7. 11846 (-1)7. 73052 (-1)8.21313 (-1)8. 70525 (-1)9. 34420 (-1)9. 76949

24 (-1)5.25587 (-1)6. 31114 (-1)7. 15323 (-1)7. 54175 (-1)8.07628 (-1)8.49267 (-1)8.91295 (-1)9.45263 (-1)9.80832

30 (-1)5. 98897 (-1)6.92952 (-1)7.65703 (-1)7.98669 (-1)8.43466 (-1)8. 77941 (-1)9. 12389 (-1)9.56142 (-1)9.84699

40 (-1)6. 81797 (-1)7.60321 (-1)8. 19204 (-1)8.45414 (-1)8. 80593 (-1)9. 07343 (-1)9. 33808 (-1)9.67057 (-1)9.88550

60 (-1)7. 75425 (-1)8.33646 (-1)8. 75981 (-1)8.94493 (-1)9. 19039 (-1)9. 37483 (-1)9. 55550 (-1)9. 78006 (-1)9. 92383

120 (-1)8. 81033 (-1)9. 13380 (-1)9. 36192 (-1)9.45993 (-1)9. 58832 (-1)9.68366 (-1)9. 77614 (-1)9. 88988 (-1)9.96200

/ 00 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

/ 1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

10
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Vx -2

vz p = 0.0001 p = 0.00l p = 0.005 p = 0.0l p =0.025 p=0.05 p=0.l p = 0.25 p=0.5

1

I.I

1.2

1.3

(-8)1. 00000

(-8)5. 33670

(-7)2. 15443

(-7)7. 01704

(-6)1. 00000

(-6)3.51119

(-5)1. 00000

(-5)2.42446

(-5)2. 50000

(-5)6. 55102

(-4)1.46201

(-4)2.88372

(-4)1. 00000

(-4)2. 31013

(-4)4.64159

(-4)8. 37678

(-4)6.25000

(-3)1.22226

(-3)2. 13747

(-3)3.42999

(-3)2. 50000

(-3)4. 31014

(-3)6. 78604

(-3)9.96358

(-2)1. 00000

(-2)1. 51991

(-2)2. 15443

(-2)2.89427

(-2)6.25000

(-2)8. 04166

(-2)9.92126

(-1)1. 18510

(-1)2. 50000

(-1)2. 83578

(-1)3. 14980

(-1)3.44252

1.4

1.5

1.6

1.7

1.8

(-6)1.93070

(-6)4.64159

(-5)1. 00000

(-5)1.96842

(-5)3.59381

(-5)5. 17947

(-4)1. 00000

(-4)1. 77828

(-4)2.95521

(-4)4.64159

(-4)5. 16196

(-4)8. 54988

(-3)1. 32957

(-3)1.96293

(-3)2. 77524

(-3)1.38950

(-3)2. 15443

(-3)3. 16228

(-3)4.43669

(-3)5.99484

(-3)5. 14450

(-3)7. 31004

(-3)9.94088

(-2)1.30384

(-2)1.65933

(-2)1. 38480

(-2)1.84202

(-2)2. 36435

(-2)2.94698

(-2)3.58436

(-2)3.72759

(-2)4.64159

(-2)5.62341

(-2)6.66085

(-2)7.74264

(-1)1. 38011

(-1)1. 57490

(-1)1. 76777

(-1)1.95747

(-1)2. 14311

(-1)3.71499

(-1)3. 96850

(-1)4.20448

(-1)4.42433

(-1)4.62937

1.9

2

2.2

2.5

3

(-5)6. 15848

(-4)1.00000

(-4)2.31013

(-4)6. 30957

(-3)2. 15443

(-4)6.95193

(-3)1.00000

(-3)1.87382

(-3)3.98107

(-2)1.00000

(-3)3. 78323

(-3)5.00000

(-3)8.09384

(-2)1.44270

(-2)2.92402

(-3)7.84760

(-2)1.00000

(-2)1.51991

(-2)2.51189

(-2)4.64159

(-2)2. 05883

(-2)2.50000

(-2)3.49608

(-2)5.22820

(-2)8. 54988

(-2)4.27066

(-2)5.00000

(-2)6. 56516

(-2)9. 10282

(-1)1. 35721

(-2)8.85867

(-1)1.00000

(-1)1.23285

(-1)1.58489

(-1)2. 15443

(-1)2.32409

(-1)2.50000

(-1)2. 83578

(-1)3.29877

(-1)3. 96850

(-1)4.82088

(-1)5. 00000

(-1)5. 32521

(-1)5.74349

(-1)6.29961

3.5 4

4.5

5

6

(-3)5. 17947

(-2)1. 00000

(-2)1.66810

(-2)2.51189

(-2)4.64159

(-2)1.93070

(-2)3. 16228

(-2)4.64159

(-2)6. 30957

(-1)1. 00000

(-2)4.84313

(-2)7. 07107

(-2)9.49118

(-1)1.20112

(-1)1. 70998

(-2)7. 19686

(-1)1. 00000

(-1)1.29155

(-1)1.58489

(-1)2. 15443

(-1)1.21489

(-1)1. 58114

(-1)1. 94077

(-1)2.28653

(-1)2.92402

(-1)1. 80532

(-1)2.23607

(-1)2.64098

(-1)3.01709

(-1)3.68403

(-1)2.68270

(-1)3. 16228

(-1)3. 59381

(-1)3.98107

(-1)4.64159

(-1)4.52862

(-1)5. 00000

(-1)5.40030

(-1)5. 74349

(-1)6.29961

(-1)6. 72950

(-1)7. 07107

(-1)7. 34867

(-1)7. 57858

(-1)7.93701

7

8

9

10

12

(-2)7. 19686

(-1)1. OOOOC

(-1)1.29155

(-1)1.58489

(-1)2. 15443

(-1)1. 38950

(-1)1. 77828

(-1)2. 15443

(-1)2.51189

(-1)3. 16228

(-1)2.20071

(-1)2.65915

(-1)3.08078

(-1)3.46572

(-1)4. 13519

(-1)2.68270

(-1)3. 16228

(-1)3. 59381

(-1)3.98107

(-1)4.64159

(-1)3.48553

(-1)3.97635

(-1)4.40541

(-1)4. 78176

(-1)5.40742

(-1)4.24891

(-1)4. 72871

(-1)5- 13904

(-1)5.49280

(-1)6.06962

(-1)5. 17947

(-1)5.62341

(-1)5.99484

(-1)6. 30957

(-1)6.81292

(-1)6. 72950

(-1)7. 07107

(-1)7. 34867

(-1)7.57858

(-1)7.93701

(-1)8.20335

(-1)8.40896

(-1)8.57244

(-1)8. 70551

(-1)8. 90899

15

20

24

30

40

(-1)2.92864

(-1)3.98107

(-1)4.64159

(-1)5.41170

(-1)6. 30957

(-1)3.98107

(-1)5.01187

)-l)5. 62341

(-1)6. 30957

(-1)7.07946

(-1)4.93396

(-1)5.88704

(-1)6.43054

(-1)7.02422

(-1)7.67270

(-1)5.41170

(-1)6. 30957

(-1)6.81292

(-1)7. 35642

(-1)7.94328

(-1)6. 11494

(-1)6.91503

(-1)7. 35352

(-1)7.81981

(-1)8. 31567

(-1)6. 70702

(-1)7.41134

(-1)7. 79078

(-1)8. 18964

(-1)8.60892

(-1)7.35642

(-1)7. 94328

(-1)8.25404

(-1)8. 57696

(-1)8. 91251

(-1)8.31238

(-1)8. 70551

(-1)8. 90899

(-1)9. 11722

(-1)9.33033

(-1)9. 11722

(-1)9. 33033

(-1)9.43874

(-1)9. 54842

(-1)9.65936

60

120

00

(-1)7. 35642

(-1)8.57696

1.00000

(-1)7.94328

(-1)8.91251

1.00000

(-1)8.38106

(-1)9. 15481

1.00000

(-1)8.57696

(-1)9.26119

1.00000

(-1)8.84297

(-1)9.40371

1.00000

(-1)9.04966

(-1)9.51297

1.00000

(-1)9.26119

(-1)9.62351

1.00000

(-1)9.54842

(-1)9. 77160

1.00000

(-1)9.77160

(-1)9.88514

1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

11
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

ZA = 3^2 p = 0.0001 P

= 0.00l p = 0.005 p = 0.01 p =0.025 p=0.05 p=0.l p = 0.25 p=0.5

1

I.I

1.2

1.3

(-9)6. 16850

(-8)3.32319

(-7)1.35370

(-7)4.44703

(-7)6. 16850

(-6)2. 18644

(-6)6.28333

(-5)1.53650

(-5)1.54213

(-5)4.07941

(-5)9. 18654

(-4)1.82765

(-5)6. 16863

(-4)1.43860

(-4)2.91673

(-4)5.30961

(-4)3.85581

(-4)7.61294

(-3)1.34361

(-3)2. 17518

(-3)1.54292

(-3)2.68627

(-3)4.26959

(-3)6.32652

(-3)6. 18124

(-3)9.49366

(-2)1.35948

(-2)1. 84456

(-2)3.90631

(-2)5.09155

(-2)6.36104

(-2)7.69123

(-1)1.63194

(-1)1.88195

(-1)2.12294

(-1)2.35408

1.4

1.5

1.6

1.7

1.8

(-6)1.23364

(-6)2.98914

(-6)6.48841

(-5)1.28641

(-5)2. 36493

(-5)3.30952

(-5)6.44001

(-4)1. 15386

(-4)1.93140

(-4)3.05465

(-4)3.29861

(-4)5.50690

(-4)8.62887

(-3)1.28327

(-3)1.82713

(-4)8.88067

(-3)1.38798

(-3)2.05298

(-3)2.90176

(-3)3.94902

(-3)3.29032

(-3)4. 71394

(-3)6.46165

(-3)8.54063

(-2)1.09509

(-3)8.87149

(-2)1. 19029

(-2)1.54069

(-2)1.93608

(-2)2.37357

(-2)2.39876

(-2)3.01521

(-2)3.68664

(-2)4.40582

(-2)5. 16583

(-2)9.06251

(-1)1.04589

(-1)1. 18677

(-1)1.32785

(-1)1.46834

(-1)2.57508

(-1)2. 78597

(-1)2.98699

(-1)3. 17850

(-1)3.36093

1.9

2

2.2

2.5

3

(-5)4.07960

(-5)6.66678

(-4)1.55881

(-4)4.32869

(-3)1.51437

(-4)4.60570

(-4)6.66778

(-3)1.26474

(-3)2. 73261

(-3)7.03688

(-3)2.50774

(-3)3.33612

(-3)5.46843

(-3)9.91860

(-2)2.06325

(-3)5.20543

(-3)6.67783

(-2)1.02808

(-2)1.72979

(-2)3.28335

(-2)1.36864

(-2)1.67369

(-2)2.37244

(-2)3.61573

(-2)6.08303

(-2)2.84993

(-2)3.36175

(-2)4.47797

(-2)6.33473

(-2)9.73082

(-2)5.96021

(-2)6.78302

(-2)8.49310

(-1)1. 11551

(-1)1.56476

(-1)1.60761

(-1)1.74518

(-1)2.01383

(-1)2.39744

(-1)2.98014

(-1)3.53474

(-1)3.70039

(-1)4.00905

(-1)4.42179

(-1)5.00000

3.5

4

4.5

5

6

(-3)3.71794

(-3)7. 31190

(-2)1.23989

(-2)1. 89484

(-2)3.59193

(-2)1.38848

(-2)2.31842

(-2)3.46210

(-2)4. 77978

(-2)7. 78110

(-2)3.49658

(-2)5.20987

(-2)7. 12092

(-2)9. 15930

(-1)1.34079

(-2)5.21263

(-2)7.39602

(-2)9. 73164

(-1)1.21419

(-1)1.69786

(-2)8.86091

(-1)1. 17863

(-1)1.47479

(-1)1. 76736

(-1)2.32591

(-1)1.32823

(-1)1.68250

(-1)2.02652

(-1)2.35534

(-1)2.95987

(-1)2.00123

(-1)2.41363

(-1)2. 79773

(-1)3. 15290

(-1)3. 78160

(-1)3.49355

(-1)3.94477

(-1)4. 34217

(-1)4.69360

(-1)5.28481

(-1)5.47196

(-1)5.86373

(-1)6. 19379

(-1)6.47548

(-1)6.93053

7

8

9

10

12

(-2)5.68970

(-2)8. 05085

(-1)1.05620

(-1)1.31384

(-1)1.82653

(-1)1. 10555

(-1)1.44174

(-1)1. 77496

(-1)2. 09827

(-1)2.70194

(-1)1. 76557

(-1)2. 17454

(-1)2. 56037

(-1)2.92037

(-1)3.56323

(-1)2. 16358

(-1)2.59968

(-1)3.00243

(-1)3. 37189

(-1)4.01907

(-1)2.83752

(-1)3.29926

(-1)3. 71374

(-1)4.08549

(-1)4. 72022

(-1)3.49293

(-1)3.96068

(-1)4. 37155

(-1)4.73377

(-1)5. 34024

(-1)4. 31507

(-1)4. 77003

(-1)5. 16097

(-1)5.49964

(-1)6.05554

(-1)5. 76094

(-1)6. 15157

(-1)6.47732

(-1)6. 75286

(-1)7. 19307

(-1)7.28193

(-1)7. 56136

(-1)7. 78880

(-1)7.97751

(-1)8.27249

15

20

24

30

40

(-1)2.54647

(-1)3.56193

(-1)4.21858

(-1)5.00145

(-1)5.93619

(-1)3.48790

(-1)4.51517

(-1)5. 14322

(-1)5.86357

(-1)6.69080

(-1)4.35690

(-1)5.34001

(-1)5.9176C

(-1)6.5622C

(-1)7.28232

(-1)4.79986

(-1)5.74472

(-1)6.29027

(-1)6.89192

(-1)7.55605

(-1)5.46281

(-1)6.33391

(-1)6.82531

(-1)7. 3587C

(-1)7.93812

(-1)6.03327

(-1)6.82706

(-1)7.26692

(-1)7. 73857

(-1)8.24466

(-1)6.67576

(-1)7. 36913

(-1)7. 74639

(-1)8. 14593

(-1)8.56930

(-1)7.66843

(-1)8. 18249

(-1)8.45528

(-1)8. 73930

(-1)9. 03514

(-1)8.58268

(-1)8.90923

(-1)9.07901

(-1)9.25336

(-1)9.43244

60

120

CD

(-1)7. 05402

(-1)8. 39307

1.00000

(-1)7.64198

(-1)8. 73718

1.00000

(-1)8.08767

(-1)8.98934

1.00000

(-1)8.28985

(-1)9. 10144

1.00000

(-1)8.56806

(-1)9.25350

1.00000

(-1)8. 78809

(-1)9. 37203

1.00000

(-1)9.01819

(-1)9.49442

1.00000

(-1)9. 34341

(-1)9.66480

1.00000

(-1)9.61645

(-1)9.80556

1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

12
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Z/, = 4

p = 0.0001 p=0.00l p = 0.005 p = 0.0l p=0.025 p=0.T)5 p=0.l p = 0.25 p=0.5

I.I

1.2

1.3

-9)4.44444

-8)2.40555

-8)9. 84313

-7)3.24761

7)4.4444

6)1. 5827

6)4.56879

5)1. 12209

(-5)1. 1111

(-5)2.9529

(-5)6.6798

(-4)1. 33474

5)4.4445?

4)1.04138

-4)2. 12092

4)3.87783

4)2.77829

4)5.51138

4)9. 77159

-3)1.58899

-3)1. 11194

3)1. 94526

3)3. 10642

-3)4.62427

3)4.45768

3)6. 88163

3)9.90437

2)1. 35055

•2)2. 83095

2)3. 71331

•?H. 66821

2)5.67912

1)1.20615

1)1.40257

1)1. 59486

1)1. 78205

1.4

1.5

1.6

1.7

1.8

-7)9. 04694

-6)2.20098

-6)4. 79635

-6)9.54553

-5)1. 76130

-5)2.42705

5)4. 74199

5)8.52963

•4)1.43318

•4)2.27506

(-4)2.4191

(-4)4.0551

(-4)6. 37934

(-4)9.5237

(-3)1.36108

•4)6.51345

3)1. 02220

3)1.51801

3)2. 15400

3)2.94256

-3)2.41405

-3)3.47322

-3)4. 78067

-3)6.34447

-3)8. 16728

3)6.51387

3)8. 77860

2)1. 14125

•2)1.44029

2)1. 77319

(-2)

(-

2)1. 76499

2)2.22933

2)2. 73878

2)3.28843

2)3.87346

2)6. 73197

2)7.81500

2)8. 91847

•1)1. 00344

1)1. 11562

(-

(-

1)1.96360

1)2. 13923

1)2. 30887

1)2.47253

1)2.63031

1.9

2

2.2

2.5

3

-5)3.04916

-5)5.00013

-4)1. 17689

-4)3.29851

-3)1. 17015

•4)3.44257

•4)5. 00125

4)9.54989

3)2.08284

3)5.44068

(-3)1.87491

(-3)2.50313

(-3)4. 13127

(-3)7. 56647

(-2)1. 59763

•3)3.89319

3)5.01256

3)7. 77145

2)1. 32073

•2)2.54583

2)1.02473

2)1.25791

2)1.79634

2)2. 76687

2)4. 73157

2)2. 13787

-2)2.53206

2)3.39948

2)4.86347

2)7.60095

(-2)

(-2)

(-

2)4.48929

2)5. 13167

2)6.48064

2)8.61580

1)1.23099

1)1.22787

1)1. 33975

1)1.56109

1)1. 88370

1)2.38845

1)2. 78239

1)2.92893

1)3.20627

1)3.58610

1)4. 13627

3.5

4

4.5

5

6

-3)2.90870

-3)5. 78467

-3)9.90940

-2)1.52855

-2)2. 94587

•2)1.08740

2)1. 83703

2)2. 77271

2)3.86578

2)6.40381

(-2)2. 74442

(-2)4. 14002

(-2)5. 72294

(-2)7.43779

(-1)1. 10885

•2)4.09878

2)5.89031

2)7.84117

2)9. 88773

1)1.40868

-2)6.99504

-2)9.42993

-1)1. 19434

-1)1.44711

-1)1.94120

1)1.05374

•1)1. 35350

1)1.65077

1)1. 94034

1)2.48605

(-1)

(-

1)1.60021

1)1. 95800

1)2.29885

1)2.62044

1)3.20461

•1)2.84802

1)3.26352

1)3.63859

1)3.97749

1)4.56322

<-

(-

1)4.60178

1)5. 00000

1)5. 34419

1)5.64446

1)6. 14272

7

8

9

10

12

-2)4. 73331

-2)6. 78131

-2)8.99421

-1)1. 12969

-1)1.59637

•2)9.23581

1)1.22014

1)1.51921

1)1. 81386

1)2.3747

(-1)1.48299

(-1)1. 85097

(-1)2.20461

(-1)2.53993

(-1)3. 15088

1)1.82355

1)2.22072

1)2.59453

1)2.94314

1)3.56635

-1)2.40629

-1)2.83582

-1)3.22901

-1)3.58765

-1)4.21277

1)2.98110

1)3.42592

1)3.82452

1)4. 18197

1)4. 79297

1)3. 71508

1)4. 16110

1)4.55219

1)4.89684

1)5.47435

1)5.04943

1)5.45819

1)5. 80601

1)6. 10521

1)6.59290

1)6.53914

1)6.86190

1)7. 12972

1)7. 35550

1)7. 71510

15

20

24

30

40

60

120

CD

-1)2.26817

1)3.24365

1)3.88989

1)4.67473

1)5.63088

1)3. 12430

1)4. 13359

1)4.76597

1)5.50478

1)6.37018

(-1)3.92551

(-1)4.91435

(-1)5.50975

(-1)6. 18641

(-1)6. 9571l|

1)4.33867

1)5. 30184

1)5.87173

1)6.51162

1)7.23155

-1)4.96408

-1)5.87220

-1)6.39703

-1)6.97679

-1)7.61840

-1)5.51019(-1)6. 13604

-1)6.35641 (-1)6.89757

-1)6.83660(-1)7. 32164

-1)7.36043(-

-1)7. 93275 (-1)8. 27065

1)7. 13315

1)7. 73373

1)8.05946

1)8.40389

1)8. 76847

1)8. 10232

1)8.52037

1)8. 74209

1)8.97297

1)9.21356

•1)6. 79968

1)8.23367

1.00000

1)7.38672

1)8. 58439

1.00000

(-1)7. 83703

(-1)8.84417 1.00000

-1)8 04331

-1)8.96070

1.0000C

1)8.32979

1)9. 12012

1.00000

1)8.55910

1)9.24578

1.00000

(-1)

1)8.80231

1)9. 37726

1.00000

-1)9. 15481

-1)9.56467

1.00000

•1)9.46448

1)9. 72639

1.00000

The numbers in parentheses indicate the the number following is to be multiplied, e.g, power of ten by which

, (-1)1.23456 = 0. 123456.

13
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

V, = 5^2 p = 0.0001 p=0.00l p = 0.005 p = 0.01 p =0.025 p=0.05 p=o.i p = 0.25 p=0.5

1 (-9)3.46978 (-7)3.46978 (-6)8.67453 (-5)3.4699C (-4)2. 16908 (-4)8.68199 (-3)3.48189 (-2)2.21735 (-2)9.55258

I.I (-8)1.88320 (-6)1.23902 (-5)2.31175 (-5)8. 15255 (-4)4. 31487 (-3)1.52319 (-3)5.39149 (-2)2.91930 (-1)1. 11637

1.2 (-8)7.72638 (-6)3.58628 (-5)5.24341 (-4)1.66486 (-4)7.67105 (-3)2.43923 (-3)7. 78302 (-2)3.68364 (-1)1.27557

1.3 (-7)2.55585 (-6)8.83081 (-4)1.05045 (-4)3.05196 (-3)1.25074 (-3)3.64112 (-2)1.06446 (-2)4.49786 (-1)1.43196

1.4 (-7)7. 13794 (-5)1.91492 (-4)1.90874 (-4)5. 13940 (-3)1.90516 (-3)5. 14300 (-2)1.39523 (-2)5.35113 (-1)1.58498

1.5 (-6)1. 74083 (-5)3. 75063 (-4)3.20752 (-4)8.08583 (-3)2.74811 (-3)6.94982 (-2)1. 76748 (-2)6.23429 (-1)1. 73429

1.6 (-6)3.80271 (-5)6.76264 (-4)5.05810 (-3)1.20373 (-3)3.79219 (-3)9.05916 (-2)2. 17772 (-2)7. 13965 (-1)1.87969

1.7 (-6)7.58573 (-4)1.13895 (-4)7.56920 (-3)1.71214 (-3)5.04521 (-2)1. 14630 (-2)2.62232 (-2)8.06079 (-1)2.02110

1.8 (-5)1.40288 (-4)1.81213 (-3)1.08425 (-3)2.34445 (-3)6.51070 (-2)1.41493 (-2)3.09765 (-2)8.99237 (-1)2. 15850

1.9 (-5)2.43406 (-4)2.74819 (-3)1.49697 (-3)3. 10904 (-3)8. 18863 (-2)1.71032 (-2)3.60025 (-2)9.92995 (-1)2.29192

2 (-5)4.00012 (-4)4.00120 (-3)2.00301 (-3)4.01206 (-2)1.00760 (-2)2.03083 (-2)4. 12685 (-1)1.08699 (-1)2.42142

2.2 (-5)9.45458 (-4)7.67259 (-3)3.32018 (-3)6.24792 (-2)1.44563 (-2)2. 74015 (-2)5.24003 (-1)1.27452 (-1)2.66904

2.5 (-4)2.66562 (-3)1.68348 (-3)6. 11887 (-2)1.06862 (-2)2.24181 (-2)3.94858 (-2)7.02107 (-1)1.55155 (-1)3.01369

3 (-4)9.54251 (-3)4.43853 (-2)1. 30458 (-2)2.08065 (-2)3.87476 (-2)6.24125 (-1)1.01537 (-1)1.99369 (-1)3.52452

3.5 (-3)2. 39163 (-3)8.94707 (-2)2.26131 (-2)3.38123 (-2)5. 78523 (-2)8. 74291 (-1)1. 33453 (-1)2.40543 (-1)3.96787

4 (-3)4. 79240 (-2)1. 52347 (-2)3.43991 (-2)4.90140 (-2)7.87055 (-1)1. 13378 (-1)1.64929 (-1)2. 78522 (-1)4. 35554

4.5 (-3)8.26707 (-2)2.31640 (-2)4.79234 (-2)6.57742 (-1)1.00527 (-1)1.39492 (-1)1.95398 (-1)3. 13422 (-1)4.69705

5 (-2)1.28351 (-2)3.25178 (-2)6.27368 (-2)8.35630 (-1)1.22754 (-1)1.65280 (-1)2.24569 (-1)3.45464 (-1)5.00000

6 (-2)2.50279 (-2)5.45389 (-2)9.47591 (-1)1.20653 (-1)1.66955 (-1)2. 14770 (-1)2.78583 (-1)4.01981 (-1)5.51335

7 (-2)4.06312 (-2)7.95192 (-1)1.28181 (-1)1.58006 (-1)2.09417 (-1)2.60634 (-1)3.26847 (-1)4.50008 (-1)5.93155

8 (-2)5.87480 (-1)1.06068 (-1)1.61587 (-1)1.94368 (-1)2.49327 (-1)3.02601 (-1)3.69818 (-1)4.91174 (-1)6.27868

9 (-2)7.85603 (-1)1.33198 (-1)1.94151 (-1)2.29097 (-1)2.86423 (-1)3.40802 (-1)4.08107 (-1)5.26776 (-1)6.57136

10 (-2)9.94020 (-1)1.60247 (-1)2.25419 (-1)2.61906 (-1)3.20714 (-1)3.75528 (-1)4.42317 (-1)5.57830 (-1)6.82144

12 (-1)1.42256 (-1)2. 12538 (-1)2.83315 (-1)3.21529 (-1)3.81489 (-1)4.35898 (-1)5.00623 (-1)6.09304 (-1)7.22621

15 (-1)2.05183 (-1)2.83892 (-1)3.58331 (-1)3.97056 (-1)4.56176 (-1)5.08363 (-1)5.68934 (-1)6.67584 (-1)7.67121

20 (-1)2.98775 (-1)3.82389 (-1)4.56539 (-1)4.93664 (-1)5.48768 (-1)5.96046 (-1)6.49538 (-1)7.33951 (-1)8. 16264

24 (-1)3.62052 (-1)4.45402 (-1)5. 16925 (-1)5.52037 (-1)6.03408 (-1)6.46843 (-1)6.95346 (-1)7. 70656 (-1)8.42805

30 (-1)4.40136 (-1)5.20205 (-1)5. 86649 (-1)6. 18619 (-1)6.64712 (-1)7.03107 (-1)7.45411 (-1)8. 10016 (-1)8. 70803 40 (-1)5. 36959 (-1)6.09372 (-1)6.67443 (-1)6.94816 (-1)7.33694 (-1)7.65589 (-1)8.00254 (-1)8.52303 (-1)9.00378

60 (-1)6.57659 (-1)7. 16139 (-1)7.61424 (-1)7.82325 (-1)8. 11561 (-1)8.35174 (-1)8.60480 (-1)8.97822 (-1)9. 31665

120 (-1)8.09005 (-1)8.44604 (-1)8. 71199 (-1)8.83212 (-1)8.99753 (-1)9. 12900 (-1)9.26789 (-1)9.46922 (-1)9.64816

CD 1.00000 1.00000 1.00000 1.00000 1.0000C 1.00000 1.00000 1.00000 1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0. 123456.

ih
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Z/, = 6

^2 p = 0.0001 p=0.00l p = 0.005 p = O.OI p =0.025 p = 0.05 p=0.l p = 0.25 p=0.5

1

I.I

1.2

1.3

[-9)2.84444

(-8)1.54660

(-8)6.35664

(-7)2. 10639

(-7)2.84445

(-6)1.01756

(-6)2.95050

(-6)7.27787

[-6)7. 11118

[-5)1.89856

:-5)4. 31388

(-5)8.65734

(-5)2.84455

(-5)6.69545

(-4)1.36973

(-4)2.51533

(-4)1.77820

(-4)3.54379

(-4)6.31157

(-3)1.03091

(-4)7. 11786

-3)1.25112

;-3)2. 00725

(-3)3.00179

-3)2.85530

[-3)4.43003

[-3)6.40778

(-3)8.78111

-2)1.82151

[-2)2.40402

[-2)3.04090

[-2)3. 72213

[-2)7.90328

[-2)9.26681

[-1)1.06225

[-1)1. 19624

1.4

1.5

1.6

1.7

1.8

(-7)5.89265

(-6)1.43950

(-6)3. 14957

(-6)6.29278

(-5)1. 16557

(-5)1.58085

(-5)3. 10142

(-5)5.60114

(-5)9.44831

(-4)1.50561

(-4)1.57576

(-4)2.65239

(-4)4. 18952

(-4)6.27946

(-4)9.00924

(-4)4.24297

(-4)6.68669

(-4)9.97081

(-3)1.42052

(-3)1.94825

(-3)1.57305

(-3)2.27297

(-3)3. 14189

(-3)4. 18709

(-3)5.41235

(-3)4.24769

(-3)5. 75036

(-3)7.50911

(-3)9.51860

(-2)1. 17700

[-2)1. 15325

(-2)1.46383

(-2)1.80713

(-2)2. 18031

(-2)2.58051

[-2)4.43902

[-2)5. 18412

(-2)5.95113

(-2)6. 73473

(-2)7.53047

[-1)1.32813

[-1)1.45757

[-1)1. 58435

(-1)1.70833

(-1)1.82944

1.9

2

2.2

2.5

3

(-5)2.02538

(-5)3.33344

(-5)7.90157

(-4)2.23696

(-4)8.05925

(-4)2.28682

(-4)3.33445

(-4)6.41264

(-3)1.41292

(-3)3. 74960

(-3)1.24578

(-3)1.66945

(-3)2. 77554

(-3)5. 13727

(-2)1. 10282

(-3)2.58770

(-3)3.34451

(-3)5.22427

(-3)8.97517

(-2)1. 75990

(-3)6.81846

(-3)8.40376

(-2)1.20960

(-2)1.88464

(-2)3.28198

(-2)1.42522

(-2)1.69524

(-2)2.29522

(-2)3.32406

(-2)5.29625

(-2)3. 00498

(-2)3.45106

(-2)4.39840

(-2)5.92560

(-2)8.64344

(-2)8.33460

(-2)9. 14397

(-1)1.07683

(-1)1.31909

(-1)1. 71128

(-1)1.94766

(-1)2. 06299

(-1)2.28516

(-1)2.59797

(-1)3.06947

3.5 4

4.5

5

6

(-3)2.03179

(-3)4.09365

(-3)7.09785

(-2)1. 10728

(-2)2. 17821

(-3)7.60454

(-2)1.30229

(-2)1.99078

(-2)2.80889

(-2)4. 75519

(-2)1.92392

(-2)2.94448

(-2)4. 12565

(-2)5.43009

(-2)8.28292

(-2)2.87912

(-2)4. 19986

(-2)5.66941

(-2)7.24286

(-1)1.05640

(-2)4.93499

(-2)6. 75860

(-2)8.68633

(-1)1.06687

(-1)1.46633

(-2)7.47483

(-2)9.76115

(-1)1.20875

(-1)1.44085

(-1)1.89255

(-1)1. 14510

(-1)1.42559

(-1)1. 70039

(-1)1.96641

(-1)2.46636

(-1)2.08258

(-1)2.43022

(-1)2.75402

(-1)3. 05496

(-1)3.59436

(-1)3.48650

(-1)3.85728

(-1)4. 18874

(-1)4.48665

(-1)5.00000

7

8

9

10

12

(-2)3.56417

(-2)5. 19012

(-2)6.98524

(-2)8.89021

(-1)1.28533

(-2)6.99126

(-2)9.39539

(-1)1. 18782

(-1)1.43774

(-1)1.92700

(-1)1. 13027

(-1)1.43596

(-1)1. 73735

(-1)2.02972

(-1)2.57831

(-1)1.39586

(-1)1.73070

(-1)2.05430

(-1)2.36324

(-1)2.93230

(-1)1.85619

(-1)2.22778

(-1)2.57744

(-1)2.90421

(-1)3.49144

(-1)2.31820

(-1)2.71338

(-1)3.07771

(-1)3.41261

(-1)4.00311

(-1)2.92101

(-1)3.33194

(-1)3. 70292

(-1)4. 03820

(-1)4.61785

(-1)4.06136

(-1)4.46802

(-1)4.82448

(-1)5. 13903

(-1)5.66795

(-1)5.42634

(-1)5. 78593

(-1)6.09323

(-1)6. 35884

(-1)6. 79481

15

20

24

30

40

(-1)1.87696

(-1)2.77495

(-1)3.39283

(-1)4. 16608

(-1)5. 14018

(-1)2.60642

(-1)3.56432

(-1)4. 18831

(-1)4.93961

(-1)5.84937

(-1)3.30216

(-1)4.27050

(-1)4.87693

(-1)5.58708

(-1)6.42287

(-1)3.66662

(-1)4.62657

(-1)5.21736

(-1)5.90077

(-1)6.69500

(-1)4.22678

(-1)5. 15862

(-1)5. 71871

(-1)6.35591

(-1)7.08387

(-1)4. 72548

(-1)5.61895

(-1)6. 14610

(-1)6. 73807

(-1)7.40533

(-1)5. 31002

(-1)6. 14478

(-1)6.62786

(-1)7. 16298

(-1)7. 75776

(-1)6.27824

(-1)6.98821

(-1)7.38781

(-1)7.82189

(-1)8.29472

(-1)7.28296

(-1)7.83314

(-1)8. 13526

(-1)8.45782

(-1)8.80296

60

120

CO

(-1)6.37631

(-1)7.95788

1.00000

(-1)6.95795

(-1)8.31817

1.00000

(-1)7.41189

(-1)8.58924

1.00000

(-1)7.62272

(-1)8. 71237

1.00000

(-1)7.91931

(-1)8.88281

1.00000

(-1)8. 16057

(-1)9.01916

1.00000

(-1)8.42125

(-1)9. 16430

1.00000

(-1)8.81129

(-1)9.37742

1.00000

(-1)9. 17310

(-1)9.57104

1.00000

The numbers in parentheses indicate the the number following is to be multiplied, e.g.power of ten by which

, (-1)1.23456 = 0. 123456.

15
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

V, = 7

"z p = 0.0001 p=0.00l p = 0.005 p = o.oi p =0.025 P

= 0.05 P=0.l p = 0.25 p=0.5

i (-9)2.40957 (-7)2.40957 (-6)6.02399 (-5)2.40967 (-4)1.50636 (-4)6.02999 (-3)2.41931 (-2)1.54525 (-2)6. 73783

i.i (-8)1.31183 (-7)8.63098 (-5)1.61037 (-5)5.67912 (-4)3.00593 (-3)1.06130 (-3)3.75886 (-2)2.04295 (-2)7.91884

1.2 (-8)5.39850 (-6)2.50577 (-5)3.66366 (-4)1.16328 (-4)5.36048 (-3)1.70496 (-3)5.44462 (-2)2.58866 (-2)9.09825

1.3 (-7)1.79110 (-6)6.18849 (-5)7.36152 (-4)2. 13886 (-4)8.76667 (-3)2.55306 (-3)7.47173 (-2)3. 17410 (-1)1.02691

1.4 (-7)5.01668 (-5)1. 34585 (-4)1.34153 (-4)3.61235 (-3)1.33937 (-3)3.61741 (-3)9.82672 (-2)3. 79202 (-1)1. 14265

1.5 (-6)1.22697 (-5)2.64352 (-4)2.26082 (-4)5.69971 (-3)1.93771 (-3)4.90345 (-2)1.24907 (-2)4.43618 (-1)1.25672

1.6 (-6)2.68767 (-5)4.77973 (-4)3.57522 (-4)8.50918 (-3)2.68174 (-3)6.41145 (-2)1.54418 (-2)5. 10128 (-1)1.36891

1.7 (-6)5. 37605 (-5)8.07194 (-4)5.36492 (-3)1.21371 (-3)3.57821 (-3)8. 13763 (-2)1.86569 (-2)5.78282 (-1)1.47907

1.8 (-6)9.96883 (-4)1.28773 (-4)7. 70590 (-3)1.66653 (-3)4.63088 (-2)1.00753 (-2)2.21124 (-2)6.47697 (-1)1.58710

1.9 (-5)1. 73417 (-4)1. 95805 (-3)1.06676 (-3)2.21605 (-3)5.84096 (-2)1.22156 (-2)2.57855 (-2)7. 18053 (-1)1.69297

2 (-5)2.85 724 (-4)2.85816 (-3)1.43113 (-3)2.86741 (-3)7.20756 (-2)1.45484 (-2)2.96544 (-2)7.89076 (-1)1. 79665

2.2 (-5)6. 78707 (-4)5.50837 (-3)2.38451 (-3)4.48902 (-2)1. 03986 (-2)1.97467 (-2)3. 78985 (-2)9.32237 (-1)1.99744

2.5 (-4)1.92726 (-3)1.21741 (-3)4.42754 (-3)7. 73725 (-2)1.62580 (-2)2.87042 (-2)5. 12633 (-1)1. 14726 (-1)2.28264

3 (-4)6.97649 (-3)3.24647 (-3)9.55296 (-2)1.52515 (-2)2.84709 (-2)4.60072 (-2)7. 52571 (-1)1.49913 (-1)2. 71807

3.5 (-3)1.76661 (-3)6.61434 (-2)1.67464 (-2)2.50761 (-2)4.30394 (-2)6.52994 (-1)1. 00304 (-1)1.83647 (-1)3. 10886

4 (-3)3.57412 (-2)1. 13764 (-2)2.57480 (-2)3.67545 (-2)5.92426 t-2)8. 57275 (-1)1.25578 (-1)2. 15599 (-1)3.46086

4.5 (-3)6.22131 (-2)1.74626 (-2)3.62353 (-2)4.98408 (-2)7.65054 (-1)1.06691 (-1)1.50571 (-1)2.45676 (-1)3. 77925

5 (-3)9.74136 (-2)2.47355 (-2)4.78913 (-2)6.39480 (-2)9.43903 (-1)1.27776 (-1)1. 74976 (-1)2.73904 (-1)4.06845

6 (-2)1.92948 (-2)4.21811 (-2)7.36184 (-2)9.40145 (-1)1.30809 (-1)1.69269 (-1)2.21392 (-1)3.25157 (-1)4.57366

7 (-2)3. 17693 (-2)6.24276 (-1)1.01159 (-1)1.25112 (-1)1.66808 (-1)2.08897 (-1)2.64206 (-1)3. 70213 (-1)5. 00000

8 (-2)4.65261 (-2)8.44004 (-1)1.29326 (-1)1.56118 (-1)2.01512 (-1)2.46129 (-1)3.03387 (-1)4. 09965 (-1)5. 36446

9 (-2)6.29451 (-1)1. 07289 (-1)1.57358 (-1)1. 86374 (-1)2.34501 (-1)2.80818 (-1)3.39147 (-1)4.45207 (-1)5.67954

10 (-2)8.04945 (-1)1.30511 (-1)1.84783 (-1)2. 15513 (-1)2.65613 (-1)3. 13006 (-1)3. 71778 (-1)4. 76616 (-1)5.95459

12 (-1)1. 17363 (-1)1. 76455 (-1)2.36819 (-1)2.69805 (-1)3.22186 (-1)3. 70445 (-1)4.28886 (-1)5.30087 (-1)6.41163

15 (-1)1. 73177 (-1)2.41213 (-1)3.06555 (-1)3.40980 (-1)3.94182 (-1)4.41873 (-1)4.9821° (-1)5.92817 (-1)6.93176

20 (-1)2.59391 (-1)3.34203 (-1)4. 01621 (-1)4.35814 (-1)4.87188 (-1)5.31942 (-1)5. 83466 (-1)6.67199 (-1)7.52890

24 (-1)3. 19633 (-1)3.95750 (-1)4.62128 (-1)4.95136 (-1)5.44012 (.1)5.85960 (-1)6.33607 (-1)7.09734 (-1)7. 86185

30 (-1)3.95976 (-1)4. 70802 (-1)5.33894 (-1)5.64637 (-1)6.09481 (-1)6.47383 (-1)6.89839 (-1)7.56494 (-1)8.22136

40 (-1)4.93534 (-1)5.62993 (-1)6. 19563 (-1)6.46556 (-1)6.85325 (-1)7. 17577 (-1)7.53187 (-1)8.08089 (-1)8.61071

60 (-1)6. 19382 (-1)6.77163 (-1)7.22563 (-1)7.43759 (-1)7.73719 (-1)7.98235 (-1)8.24904 (-1)8.65253 (-1)9.03376

120 (-1)7.83464 (-1)8. 19847 (-1)8.47385 (-1)8.59954 (-1)8.77427 (-1)8.91482 (-1)9.06534 (-1)9.28866 (-1)9.49506

CO 1.0000C 1.00000 1.0000C 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

16
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Z/| = 8vz p = 0.0001 p = 0.00l p = 0.005 p = 0.0l p =0.025 p = 0.05 p=0.l p = 0.25 p=0.5

1

I.I

1.2

1.3

(-9)2.08980

(-8)1. 13882

(-8)4.69093

(-7)1.55778

(-7)2.08980

(-7)7.49271

(-6)2. 17735

(-6)5.38234

(-6)5.22454

(-5)1.39799

(-5)3. 18348

(-5)6.40260

(-5)2.08988

(-5)4.93017

(-4)1.01082

(-4)1.86027

(-4)1.30646

(-4)2.60956

(-4)4.65806

(-4)7.62511

(-4)5.22996

(-4)9.21402

(-3)1.48166

(-3)2.22086

(-3)2.09858

(-3)3.26396

(-3)4.73273

(-3,)6.50164

(-2)1. 34159

[-2)1. 77597

(-2)2.25329

(-2)2.76649

(-2)5.87108

(-2)6.91226

(-2)7.95548

(-2)8.99451

1.4

1.5

1.6

1.7

1.8

(-7)4.36713

(-6)1. 06905

(-6)2.34381

(-6)4.69229

(-6)8. 70831

(-5)1. 17159

(-5)2.30329

(-5)4. 16823

(-5)7.04533

(-4)1. 12491

(-4)1. 16784

(-4)1.96988

(-4)3. 11788

(-4)4.68273

(-4)6. 73184

(-4)3. 14470

(-4)4.96633

(-4)7.42 093

(-3)1.05942

(-3)1.45596

(-3)1. 16605

(-3)1.68853

(-3)2.33905

(-3)3. 12383

(-3)4.04651

(-3)3. 14978

(-3)4.27373

(-3)5.59349

(-3)7. 10634

(-3)8.80695

(-3)8.55991

(-2)1.08920

(-2)1.34797

(-2)1.63035

(-2)1.93437

(-2)3.30937

(-2)3.87659

(-2)4.46357

(-2)5.06642

(-2)5.68184

(-1)1.00250

(-1)1. 10439

(-1)1.20491

(-1)1.30391

(-1)1.40130

1.9

2

2.2

2.5

3

(-5)1.51616

(-5)2. 50009

(-5)5.94824

(-4)1.69299

(-4)6. 15093

(-4)1.71191

(-4)2.50094

(-4)4.82772

(-3)1.06949

(-3)2.86273

(-4)9.32711

(-3)1.25235

(-3)2.09010

(-3)3.89034

(-3)8.42686

(-3)1.93773

(-3)2.50943

(-3)3.93528

(-3)6.79984

(-2)1. 34581

(-3)5. 10853

(-3)6.30946

(-3)9. 11927

(-2)1.42956

(-2)2.51427

(-2)1. 06881

(-2)1.27415

(-2)1.73273

(-2)2.52587

(-2)4.06714

(-2)2.25808

(-2)2.59963

(-2)3. 32930

(-2)4.51730

(-2)6.66469

(-2)6.30700

(-2)6.93951

(-2)8.21875

(-1)1. 01506

(-1)1.33387

(-1)1.49702

(-1)1.59104

[-1)1. 77389

(-1)2.03537

(-1)2.43864

3.5 4

4.5

5

6

(-3)1.56292

(-3)3. 17233

(-3)5.53902

(-3)8.69860

(-2)1. 73247

(-3)5.85331

(-2)1.01018

(-2)1.55567

(-2)2.21047

(-2)3.79164

(-2)1.48281

(-2)2.28812

(-2)3.23129

(-2)4.28494

(-2)6.62792

(-2)2.22142

(-2)3.26821

(-2)4.44782

(-2)5. 72643

(-2)8.47300

(-2)3.81666

(-2)5.27450

(-2)6.83739

(-2)8.46634

(-1)1. 18117

(-2)5.79813

(-2)7.64404

(-2)9.55126

(-1)1. 14820

(-1)1.53161

(-2)8.92487

(-1)1. 12235

(-1)1.35136

(-1)1.57658

(-1)2.00909

(-1)1.64255

(-1)1.93764

(-1)2.21779

(-1)2.48281

(-1)2.96917

(-1)2.80480

(-1)3. 13810

(-1)3.44246

(-1)3.72132

(-1)4.21407

7

8

9

10

12

(-2)2.86702

(-2)4.21841

(-2)5. 73173

(-2)7. 35904

(-1)1.08064

(-2)5.64188

(-2)7.66545

(-2)9. 78857

(-1)1. 19570

(-1)1.62860

(-2)9. 15932

(-1)1. 17704

(-1)1.43893

(-1)1.69700

(-1)2. 19136

(-1)1. 13415

(-1)1.42270

(-1)1. 70658

(-1)1.98202

(-1)2.50026

(-1)1.51532

(-1)1.84052

(-1)2. 15230

(-1)2.44863

(-1)2.99295

(-1)1.90187

(-1)2.25322

(-1)2.58349

(-1)2.89241

(-1)3.44941

(-1)2.41274

(-1)2. 78602

(-1)3.12986

(-1)3.44623

(-1)4.00580

(-1)3.40219

(-1)3. 78848

(-1)4. 13430

(-1)4.44514

(-1)4.98008

(-1)4.63554

(-1)5.00000

(-1)5. 31822

(-1)5.59845

(-1)6.06915

15

20

24

30

40

(-1)1.60881

(-1)2.43730

(-1)3.02417

(-1)3. 77637

(-1)4. 75026

(-1)2.24667

(-1)3. 14862

(-1)3. 75412

(-1)4.50103

(-1)5.43063

(-1)2.86288

(-1)3.79364

(-1)4.39472

(-1)5. 11592

(-1)5.98817

(-1)3. 18907

(-1)4. 12241

(-1)4.71486

(-1)5.41702

(-1)6.25549

(-1)3.69550

(-1)4.61868

(-1)5. 19109

(-1)5.85823

(-1)6.64111

(-1)4. 15209

(-1)5.05350

(-1)5.60216

(-1)6.23321

(-1)6.96362

(-1)4.69509

(-1)5. 55737

(-1)6.07207

(-1)6.65587

(-1)7.32186

(-1)5.61693

(-1)6.38515

(-1)6.83089

(-1)7.32633

(-1)7.87966

(-1)6.61269

(-1)7.24724

(-1)7.60607

(-1)7.99762

(-1)8.42657

60

120

CO

(-1)6.02580

(-1)7.71869

1.00000

(-1)6.59931

(-1)8.08543

1.00000

(-1)7.05256

(-1)8.36449

1.00000

(-1)7.26513

(-1)8.49237

1.00000

(-1)7.56684

(-1)8.67082

1.00000

(-1)7.81497

(-1)8.81501

1.00000

(-1)8.08640

(-1)8.97024

1.00000

(-1)8.50092

(-1)9.20252

1.00000

(-1)8.89851

(-1)9.42023

1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

IT
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Z/i = 9vz p = 0.0001 p =0.001 p = 0.005 p = 0.01 p =0.025 p = 0.05 p=0.l p = 0.25 p=0.5

1

I.I

1.2

1.3

(-9)1.84483

(-8)1.00607

(-8)4. 14712

(-7)1.37817

(-7)1.84483

(-7)6.61929

(-6)1.92493

(-6)4. 76178

(-6)4.61212

(-5)1.23503

(-5)2.81443

(-5)5.66443

(-5)1.84491

(-5)4.35547

(-5)8.93645

(-4)1.64581

(-4)1. 15333

(-4)2.30540

(-4)4. 11817

(-4)6.74627

(-4)4.61704

(-4)8. 14039

(-3)1. 31001

(-3)1.96506

(-3)1.85282

(-3)2.88404

(-3)4. 18525

(-3)5. 75422

(-2)1. 18529

(-2)1.57062

(-2)1.99474

(-2)2.45153

(-2)5.20147

(-2)6. 13222

(-2)7. 06719

(-2)8. 00079

1.4

1.5

1.6

1.7

1.8

(-7)3. 86634

(-7)9.47121

(-6)2.07790

(-6)4. 16274

(-6)7. 73067

(-5)1.03724

(-5)2.04059

(-5)3.69535

(-5)6.25026

(-5)9.98624

(-4)1.03393

(-4)1. 74522

(-4)2. 76420

(-4)4. 15437

(-4)5.97632

(-4)2.78414

(-4)4.40001

(-4)6.57930

(-4)9.39917

(-3)1.29261

(-3)1.03241

(-3)1.49609

(-3)2.07396

(-3)2. 77178

(-3)3.59305

(-3)2.78910

(-3)3. 78722

(-3)4.96049

(-3)6.30691

(-3)7.82213

(-3)7. 58211

(-3)9.65579

(-2)1. 19597

(-2)1.44771

(-2)1.71909

(-2)2.93557

(-2)3.44221

(-2)3.96744

(-2)4.50783

(-2)5.06046

(-2)8.92905

(-2)9.84917

(-1)1.07592

(-1)1. 16576

(-1)1.25436

1.9

2

2.2

2.5

3

(-5)1.34683

(-5)2.22231

(-5)5.29402

(-4)1.50956

(-4)5.50048

(-4)1.52073

(-4)2.22309

(-4)4.29684

(-4)9.53657

(-3)2. 56030

(-4)8.28585

(-3)1. 11328

(-3)1.86043

(-3)3.46951

(-3)7.53879

(-3)1.72150

(-3)2.23092

(-3)3.50320

(-3)6.06524

(-2)1.20430

(-3)4.53929

(-3)5.61038

(-3)8. 12033

(-2)1.27564

(-2)2.25128

(-3)9.50014

(-2)1. 13338

(-2)1.54362

(-2)2.25525

(-2)3.64473

(-2)2.00844

(-2)2.31415

(-2)2.96859

(-2)4.03775

(-2)5.98087

(-2)5.62285

(-2)6. 19287

(-2)7. 34877

(-2)9. 10200

(-1)1.20148

(-1)1.34164

(-1)1.42756

(-1)1.59525

(-1)1.83633

(-1)2.21120

3.5

4

4.5

5

6

(-3)1.40150

(-3)2. 85216

(-3)4.99247

(-3)7. 85915

(-2)1.57239

(-3)5.24991

(-3)9.08532

(-2)1.40283

(-2)1.99840

(-2)3.44444

(-2)1.33055

(-2)2. 05918

(-2)2.91619

(-2)3.87763

(-2)6.02873

(-2)1.99407

(-2)2.94264

(-2)4.01648

(-2)5. 18565

(-2)7. 71356

(-2)3.42886

(-2)4. 75386

(-2)6. 18158

(-2)7.67701

(-1)1. 07699

(-2)5.21438

(-2)6.89786

(-2)8.64694

(-1)1.04271

(-1)1. 39889

(-2)8.03964

(-1)1.01469

(-1)1.22592

(-1)1.43486

(-1)1. 83938

(-1)1.48578

(-1)1. 75960

(-1)2. 02142

(-1)2.27071

(-1)2. 73232

(-1)2.55480

(-1)2.87028

(-1)3. 16065

(-1)3.42864

(-1)3.906.77

7

8

9

10

12

(-2)2.61307

(-2)3.85979

(-2)5.26357

(-2)6. 78088

(-1)1.00183

(-2)5. 14823

(-2)7.02378

(-2)9.00363

(-1)1. 10373

(-1)1.51290

(-2)8.37076

(-1)1.08039

(-1)1. 32607

(-1)1.56966

(-1)2.04015

(-1)1.03752

(-1)1.30728

(-1)1.57452

(-1)1.83548

(-1)2.33066

(-1)1.38864

(-1)1.69437

(-1)1.98966

(-1)2.27218

(-1)2.79573

(-1)1. 74607

(-1)2. 07829

(-1)2.39298

(-1)2.68937

(-1)3.22864

(-1)2.22066

(-1)2. 57639

(-1)2.90669

(-1)3.21277

(-1)3. 75920

(-1)3. 14777

(-1)3.52191

(-1)3.85966

(-1)4. 16554

(-1)4.69697

(-1)4.32046

(-1)4.68178

(-1)5.00000

(-1)5.28237

(-1)5. 76128

15

20

24

30

40

(-1)1.50306

(-1)2. 30005

(-1)2.87158

(-1)3.61168

(-1)4.58152

(-1)2. 10370

(-1)2.97827

(-1)3.57294

(-1)4.31423

(-1)5.24808

(-1)2.68686

(-1)3.59661

(-1)4. 19185

(-1)4.91366

(-1)5.79728

(-1)2.99682

(-1)3.91312

(-1)4.50249

(-1)5.20846

(-1)6.06170

(-1)3.47992

(-1)4.39281

(-1)4.96644

(-1)5.64212

(-1)6.44460

(-1)3.91761

(-1)4.81518

(-1)5. 36886

(-1)6.01247

(-1)6. 76633

(-1)4.44104

(-1)5. 30734

(-1)5.83143

(-1)6.43216

(-1)7. 12554

(-1)5. 33800

(-1)6. 12337

(-1)6. 58518

(-1)7. 10377

(-1)7.68958

(-1)6.32157

(-1)6. 98579

(-1)7. 36631

(-1)7. 78565

(-1)8.25006

60

120

CO

(-1)5.86991

(-1)7. 60888

1.00000

(-1)6.43876

(-1)7.97803

1.00000

(-1)6.89066

(-1)8.26022

1.00000

(-1)7. 10344

(-1)8. 39001

1.00000

(-1)7.40652

(-1)8.57170

1.00000

(-1)7.65688

(-1)8. 71910

1.00000

(-1)7.93208

(-1)8.87848

1.00000

(-1)8. 35570

(-1)9. 11872

1.00000

(-1)8. 76720

(-1)9.34655

1.00000

The numbers in parentheses indicate the power of ten

the number following is to be multiplied, e.g., (-1)1.2345

by which

6 = 0.123456.

18
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Z/, = 10vz p = 0.0001 p=o.ooi p = 0.005 p = 0.0l p =0.025 p = 0.05 p=0.l p = 0.25 P=0.5

1 (-9)1.65120 (-7)1.65120 [-6)4. 12804 (-5)1.65127 (-4)1.03228 -4)4.13254 -3)1.65851 [-2)1.06155 -2)4.66872

I.I (-9)9.01004 (-7)5.92801 [-5)1. 10605 (-5)3.90062 (-4)2.06466 -4)7.29057 -3)2.58325 [-2)1.40777 [-2)5.51009

1.2 (-8)3.71617 (-6)1.72490 [-5)2.52197 (-5)8.00786 (-4)3.69031 [-3)1. 17396 [-3)3.75116 -2)1.78935 [-2)6.35702

1.3 (-7)1.23567 (-6)4.26940 [-5)5.07872 [-4)1.47564 (-4)6.04890 [-3)1.76205 -3)5. 16077 -2)2.20088 [-2)7.20444

1.4 (-7)3.46850 (-6)9.30513 (-5)9.27548 (-4)2.49770 (-4)9.26225 (-3)2.50247 [-3)6.80461 [-2)2.63757 [-^2)8.04873

1.5 (-7)8.50136 (-5)1.83164 (-4)1.56652 (-4)3.94954 (-3)1.34299 [-3)3.40007 (-3)8.67137 [-2)3.09530 [-2)8.88730

1.6 (-6)1.86615 (-5)3.31877 (-4)2.48254 (-4)5.90902 (-3)1.86281 (-3)4.45612 (-2)1.07475 [-2)3.57049 (-2)9.71829

1.7 (-6)3.74055 (-5)5.61636 (-4)3.73311 (-4)8.44629 (-3)2.49102 (-3)5.66908 (-2)1.30184 [-2)4.06010 [-1)1.05404

1.8 (-6)6.95033 (-5)8.97826 (-4)5.37322 (-3)1. 16221 (-3)3.23095 (-3)7.03534 [-2)1.54692 (-2)4.56153 (-1)1. 13526

1.9 (-5)1.21152 (-4)1.36795 (-4)7.45370 (-3)1.54868 (-3)4.08417 (-3)8.54978 [-2)1.80850 (-2)5.07255 [-1)1.21543

2 (-5)2.00008 (-4)2.00080 (-3)1.00201 (-3)2.00805 (-3)5.05076 (-2)1.02062 [-2)2.08516 (-2)5.59125 (-1)1.29449

2.2 (-5)4.76949 (-4)3.87119 (-3)1.67625 (-3)3. 15665 (-3)7.31870 (-2)1.39174 (-2)2.67842 (-2)6.64534 (-1)1.44923

2.5 (-4)1.36202 (-4)8.60483 (-3)3. 13092 (-3)5.47401 (-2)1. 15167 (-2)2.03706 (-2)3.65032 [-2)8.24983 (-1)1.67269

3 (-4)4.97467 (-3)2.31577 (-3)6.82038 (-2)1.08977 (-2)2.03819 (-2)3.30196 (-2)5.42455 (-1)1.09303 (-1)2.02249

3.5 (-3)1.27039 (-3)4.75962 (-2)1.20674 (-2)1.80907 (-2)3. 11282 (-2)4.73775 (-2)7.31465 [-1)1.35638 [-1)2.34564

4 (-3)2.59093 (-3)8.25549 (-2)1.87207 (-2)2.67632 (-2)4.32719 (-2)6.28499 (-2)9.25953 (-1)1.61163 (-1)2.64450

4.5 (-3)4.54463 (-2)1.27750 (-2)2.65741 (-2)3.66183 (-2)5.64124 (-2)7.89997 (-1)1. 12191 [-1)1.85712 (-1)2.92141

5 (-3)7. 16847 (-2)1.82371 (-2)3.54153 (-2)4.73888 (-2)7.02331 (-2)9.55098 (-1)1.31669 (-1)2.09218 (-1)3.17856

6 (-2)1.43965 (-2)3. 15608 (-2)5.52993 (-2)7.08038 (-2)9.89883 (-1)1.28756 (-1)1.69638 (-1)2.53074 (-1)3.64116

7 (-2)2.40098 (-2)4.73509 (-2)7.70897 (-2)9.56276 (-1)1.28179 (-1)1.61420 (-1)2.05730 (-1)2.92911 (-1)4.04541

8 (-2)3.55831 (-2)6.48295 (-2)9.98672 (-1)1.20950 (-1)1.57013 (-1)1.92903 (-1)2.39662 [-1)3.29083 (-1)4.40155

9 (-2)4.86756 (-2)8.33773 (-1)1.22999 (-1)1.46186 (-1)1.85038 (-1)2.22922 [-1)2.71386 (-1)3.61978 (-1)4.71763

10 (-2)6.28902 (-1)1.02523 (-1)1.46056 (-1)1.70965 (-1)2. 12009 (-1)2.51368 (-1)3.00969 (-1)3.91964 (-1)5.00000

12 (-2)9.34092 (-1)1.41309 (-1)1.90916 (-1)2. 18338 (-1)2.62378 (-1)3.03537 (-1)3.54216 (-1)4.44507 (-1)5.48306

15 (-1)1.41098 (-1)1.97869 (-1)2.53227 (-1)2.82755 (-1)3.28933 (-1)3.70946 (-1)4.21429 (-1)5.08634 (-1)6.05494

20 (-1)2. 17852 (-1)2.82676 (-1)3.42056 (-1)3.72565 (-1)4.18965 (-1)4.59995 (-1)5.08035 (-1)5.88324 (-1)6.74249

24 (-1)2.73505 (-1)3.41011 (-1)4.00870 (-1)4.31029 (-1)4.76229 (-1)5. 15604 (-1)5.61078 (-1)6.35760 (-1)7. 14114

30 (-1)3.46257 (-1)4. 14434 (-1)4.72890 (-1)5.01748 (-1)5.44347 (-1)5.80880 (-1)6.22473 (-1)6.89542 (-1)7.58457

40 (-1)4.42659 (-1)5.07976 (-1)5.62054 (-1)5.88188 (-1)6.26158 (-1)6.58193 (-1)6.94121 (-1)7.50951 (-1)8.08076

60 (-1)5.72440 (-1)6.28834 (-1)6.73841 (-1)6.95108 (-1)7.25497 (-1)7.50695 (-1)7.78512 (-1)8.21627 (-1)8.63967

120 (-1)7.50435 (-1)7.87548 (-1)8.16038 (-1)8.29183 (-1)8.47637 (-1)8.62660 (-1)8.78968 (-1)9.03703 (-1)9.27398

CO 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0. 123456.

19
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

p = 0.0001 p=0.00l p = 0.005 p = 0.01 p =0.025 p = 0.05 p=0.l p = 0.25 p=0.5

-9)1.36463

-9)7.45283

-8)3.07656

•7)1.02386

7)1.36463

•7)4.90347

6)1.42802

6)3.53759

6)3.41160

6)9. 14893

5)2.08791

5)4.20821

5)1

•5)3

5)6

4)1

36469

22649

62963

22272

5)8.53133

4)1.70786

4)3.05525

4)5.01233

4)3.41545

4)6.03093

4)9.72004

3)1.46024

3)1.37087

3)2.13727

3)3. 10655

3)4.27809

3)8.78142

2)1.16591

2)1.48370

2)1.82714

2)3.87462

2)4.58028

2)5.29278

2)6.00789

-7)2.87640

-7)7.05599

-6)1.55015

-6)3.10969

-6)5.78278

6)7.71667

5)1.52023

5)2.75680

5)4.66916

5)7.47010

5)7.69214

4)1.30020

4)2.06221

4)3. 10360

4)4.47080

4)2

•4)3

•4)4

4)7

4)9

07136

27816

90868

02229

67069

4)7.68172

3)1.11479

3)1.54762

3)2.07133

3)2.68893

3)2.07573

3)2.82283

3)3.70296

3)4.71523

3)5.85699

3)5.64632

3)7.20246

3)8.93584

2)1.08348

2)1.28875

2)2.19234

2)2.57594

2)2.97504

2)3.38714

2)3.81011

2)6.72254

2)7.43451

2)8. 14217

2)8.84433

2)9.54010

-5)1.00880

-5)1.66674

-5)3.98075

-4)1.13936

-4)4. 17656

4)1.13908

4)1.66736

•4)3.23109

4)7.19854

-3)1.94452

• 4)6.20691

4)8.35075

• 3)1.39923

-3)2.61971

•3)5.72903

3)1,

3)1

3)2

3)4

•3)9

28972

67365

3)3.40196

3)4.21074

(-3)6. 11206

3)9.64291

15690|(-2)1. 71391

58111

3)7.12432

3)8.51244

2)1. 16292

2)1.70686

2)2.77943

•2)1.50819

-2)1.74068

2)2.24046

2)3.06271

2)4.57405

2)4.24212

2)4.68157

2)5.57757

2)6.94883

2)9.25923

1)1.02289

1)1.09101

1)1.22490

1)1.41958

1)1.72751

-3)1.07030

-3)2. 19016

-3)3.85411

-3)6.09838

-2)1.23211

3)4.01103

3)6.98151

• 2)1.08405

• 2)1.55272

2)2.70434

2)1.01752

2)1.58443

2)2.25733

2)3.01907

2)4.74640

•2)1

• 2)2

2)3

2)4

•2)6

52612](

:>i (

11289U

04337 (

08396 (

• 2)2.62865

•2)3.66926

•2)4.80279

2)6.00279

2)8.52334

(- 2)4.00599

2)5.33755

2)6.73755

2)8.17896

1)1. 11113

2)6. 19773

2)7.88234

2)9.59337

1)1. 13074

1)1.46855

1)1. 15527

1)1.37974(

1)1.59766

1)1.80817

1)2.20577

1)2.01550

1)2.28490

1)2.53718

1)2.77379

1)3.20519

-2)2.06652

-2)3.07906

-2)4.23330

-2)5.49568

-2)8.23433

-2)4.08192

2)5.62059

2)7.26744

2)8.98132

•1)1.24929

-2)6.65925

2)8.67877

1)1.07490

1)1.28311

1)1.69312

-2)8.

1)1.

1)1.

1)1.

•1)1.

27139(

05262

27955

50443

93977

(-

1)1. 11129

1)1.36996

1)1.62400

1)1.87086

1)2.33794

1)1.40291

1)1.68750

1)1.96184

1)2.22441

1)2.71250

-1)1.79407

1)2.10396

1)2.39702

1)2.67318

1)3. 17719

(-

1)2.57241

1)2.90986

1)3.22049

1)3.50681

1)4.01580

(-

(-

1)3.58837

1)3.93085

1)4.23872

1)4.51694

1)5.00000

-1)1.25807

-1)1.97231

-1)2.50026

-1)3.20203

•1)4. 15076

1)1.77002

1)2.56822

1)3.12846

1)3.84579

1)4.77842

1)2.27283

1)3.11843

1)3.69009

1)4.40239

1)5.30242

1)2.

1)3.

1)3.

1)4.

1)5.

54259

40290

97487

67894

55726

1)2.96681

1)3.83804

1)4.40417

1)5.08954

1)5.92963

1)3.35542

1)4.22556

1)4.78082

1)5.44418

1)6.24595

1)5.45942

1)7.30870

1.00000

1)6.01303

1)7.68276

1.00000

1)6.45839

1)7.97198

1.0000C

1)6.

1)8.

1

67012

10616

00000

(-1)6. 97429(-l)7. 22815

1)8. 29544(-l)8. 45045

1.0000C 1.00000

1)3.82606

1)4.68292

1)5.21929

1)5.85110

1)6.60341

1)4.64974

1)5.45736

1)5.94873

1)6.51560

1)7. 17576

1)5.58378

1)6. 30330

1)6.72962

1)7.21195

1)7.76209

1)7.51041

1)8.61980

1.00000

1)7.95294

1)8.87937

1.00000

1)8.39537

1)9. 13215

1.00000

J

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

20
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Z/| = 15 vz p = 0.0001 p=0.00l p = 0.005 p = 0.0l p =0.025 p = 0.05 p=0.l p = 0.25 p=0.5

1 (-9)1.08267 (-7)1.08267 (-6)2.70670 (-5)1.08272 (-5)6.76865 (-4)2.70984 (-3)1.08778 (-3)6.97342 (-2)3.08671

I.I (-9)5.91806 (-7)3.89369 (-6)7.26488 (-5)2.56206 (-4)1.35618 (-4)4.78927 (-3)1.69751 (-3)9.26941 (-2)3.65473

1.2 (-8)2.44510 (-6)1. 13492 (-5)1.65937 (-5)5.26894 (-4)2.42824 (-4)7.72579 (-3)2.46973 (-2)1. 18100 (-2)4.23003

1.3 (-8)8. 14409 (-6)2.81390 (-5)3.34734 (-5)9.72595 (-4)3.98715 (-3)1. 16169 (-3)3.40442 (-2)1.45612 (-2)4.80921

1.4 (-7)2.28990 (-6)6.14323 (-5)6. 12375 (-4)1.64904 (-4)6. 11589 (-3)1.65284 (-3)4.49764 (-2)1. 74928 (-2)5. 38980

1.5 (-7)5.62195 (-5)1.21126 (-4)1.03597 (-4)2.61201 (-4)8.88325 (-3)2.24977 (-3)5.74290 (-2)2.05788 (-2)5.96998

1.6 (-6)1.23612 (-5)2. 19834 (-4)1.64449 (-4)3.91448 (-3)1.23430 (-3)2.95394 (-3)7. 13214 (-2)2.37963 (-2)6.54840

1.7 (-6)2.48177 (-5)3. 72636 (-4)2.47698 (-4)5.60471 (-3)1.65343 (-3)3.76491 (-3)8.65653 (-2)2.71260 (-2)7. 12405

1.8 (-6)4.61884 (-5)5.96658 (-4)3.57109 (-4)7. 72494 (-3)2.14829 (-3)4.68088 (-2)1.03070 (-2)3.05510 (-2)7.69619

1.9 (-6)8.06401 (-5)9. 10548 (-4)4.96189 (-3)1.03109 (-3)2.72034 (-3)5.69901 (-2)1.20743 (-2)3.40569 (-2)8.26423

2 (-5)1.33339 (-4)1.33391 (-4)6.68116 (-3)1.33915 (-3)3.37002 (-3)6.81577 (-2)1.39499 (-2)3.76313 (-2)8.82775

2.2 (-5)3. 18962 (-4)2.58901 (-3)1. 12130 (-3)2. 11210 (-3)4.90030 (-3)9. 32879 (-2)1. 79921 (-2)4.49434 (-2)9.94002

2.5 (-5)9. 15029 (-4)5.78156 (-3)2. 10444 (-3)3.68075 (-3)7. 75157 (-2)1.37309 (-2)2.46713 (-2)5.61965 (-1)1. 15688

3 (-4)3.36676 (-3)1.56772 (-3)4.62057 (-3)7.38765 (-2)1.38382 (-2)2.24647 (-2)3. 70349 (-2)7.53244 (-1)1.41732

3.5 (-4)8.65884 (-3)3.24587 (-3)8.23894 (-2)1.23631 (-2)2. 13170 (-2)3.25296 (-2)5.04345 (-2)9.45166 (-1)1.66405

4 (-3)1. 77807 (-3)5.67040 (-2)1.28794 (-2)1.84355 (-2)2.98846 (-2)4. 35412 (-2)6.44585 (-1)1. 13497 (-1)1.89768

4.5 (-.3)3. 13959 (-3)8.83637 (-2)1.84197 (-2)2.54211 (-2)3.92830 (-2)5.52082 (-2)7.88244 (-1)1. 32108 (-1)2. 11898

5 (-3)4.98428 (-2)1.27012 (-2)2.47286 (-2)3.31492 (-2)4.93019 (-2)6. 73116 (-2)9.33355 (-1)1.50258 (-1)2. 32879

6 (-2)1.01349 (-2)2.22733 (-2)3.91620 (-2)5.02576 (-2)7.05633 (-2)9.22072 (-1)1.22278 (-1)1.85002 (-1)2.71704

7 (-2)1.71029 (-2)3.38401 (-2)5.53287 (-2)6.88198 (-2)9.26946 (-1)1. 17329 (-1)1.50591 (-1)2. 17579 (-1)3.06824

8 (-2)2.56325 (-2)4.68880 (-2)7.25860 (-2)8.81767 (-1)1. 15078 (-1)1.42156 (-1)1. 77923 (-1)2.48022 (-1)3.38731

9 (-2)3.54388 (-2)6.09876 (-2)9.04642 (-1)1.07873 (-1)1.37320 (-1)1.66385 (-1)2.04108 (-1)2.76440 (-1)3.67843

10 (-2)4.62528 (-2)7.57966 (-1)1.08625 (-1)1.27596 (-1)1.59172 (-1)1.89842 [-1)2.29081 (-1)3.02969 (-1)3.94506 i

12 (-2)6.99926 (-1)1.06536 (-1)1.44892 (-1)1.66333 (-1)2.01146 (-2)2.34134 [-1)2.75401 [-1)3.50939 [-1)4.41622

15 (-1)1.08354 (-1)1.53020 (-1)1.97247 (-1)2.21132 (-1)2.58927 (-1)2.93820 [-1)3.36449 [-1)4.12155 (-1)5.00000

20 (-1)1.72976 (-1)2.26177 (-1)2.75750 (-1)3.01565 (-1)3.41318 (-1)3. 77009 (-1)4.19516 (-1)4.92564 (-1)5. 74210

24 (-1)2.21877 (-1)2.78808 (-1)3.30197 (-1)3.56448 (-1)3.96292 (-1)4.31537 (-1)4.72942 (-1)5.42848 (-1)6. 19406

30 (-1)2.88245 (-1)3.47660 (-1)3.99542 (-1)4.25516 (-1)4.64344 (-1)4.98156 (-1)5.37306 (-1)6.02174 (-1)6.71684

40 (-1)3.80293 (-1)4.39540 (-1)4.89499 (-1)5. 13979 (-1)5.49990 (-1)5.80830 (-1)6. 15990 (-1)6.73084 (-1)7.32847

60 (-1)5. 11372 (-1)5.65123 (-1)6.08785 (-1)6.29693 (-1)6.59924 (-1)6.85350 (-1)7. 13856 (-1)7.59150 (-1)8.05369

120 (-1)7.04241 (-1)7.41889 (-1)7.71254 (-1)7.84968 (-1)8.04425 (-1)8.20471 (-1)8.38135 (-1)8.65539 (-1)8.92730

00 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

21
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

V\= 20

I'* p = 0.0001 p=0.00l p = 0.005 p = 0.0l p=0.025 p=0.05 p=0 p = 0.25 p=0.5

I

I.I

1.2

1.3

(-10)8.

-9)4.

-8)1.

-8)6.

05272

40556

82176

07301

• 8)8.05272

7)2.89857

7)8.45588

• 6)2.09831

6)2.01320

6)5.40818

5)1.23634

5)2.49611

-6)8.05311

5)1.90727

5)3.92572

5)7.25268

5)5.03447

-4)1.00959

4)1.80925

4)2.97335

4)2.01562

4)3.56548

4)5.75678

•4)8.66396

(-4)8.09189

3)1.26395

(-3)1.84069

(-3)2.53977

3)5. 19143

•3)6.90868

3)8.81261

2)1.08786

2)2.30514

2)2.73370

2)3. 16908

2)3.60874

1.4

1.5

1.6

1.7

1.8

-7)1.

7)4.

7)9.

-6)1.

6)3.

70900

19929

24084

85681

45854

6)4.58484

•6)9.04750

5)1.64341

5)2. 78800

5)4.46775

-5)4.57033

5)7. 73823

4)1.22939

-4)1.85329

4)2.67412

4)1.23074

4)1.95109

4)2.92647

•4)4. 19363

•4)5.78493

4)4.56478

4)6.63606

•4)9.22863

3)1.23732

3)1.60906

3)1.23381

3)1.68094

3)2.20909

3)2.81817

•3)3.50706

(-3)3.35862

(-3)4.29277

(-3)5.33657

(-3)6.48375

(-3)7. 72781

•2)1.30846

•2)1.54118

2)1. 78435

2)2.03654

2)2.29654

(- 2)4.05083

2)4.49397

2)4.93712

2)5.37951

2)5.82052

1.9

2

2.2

2.5

3

6)6.

•5)1.

5)2.

5)6.

4)2.

04316

00005

39603

88983

54476

5)6.82370

4)1.00045

4)1.94491

4)4.35357

3)1. 18514

-4)3.71866

4)5.01129

-4)8.42428

3)1.58496

3)3.49426

-4)7.72793

-3)1.00453

-3)1.58702

-3)2. 77270

-3)5.58872

•3)2.03931

3)2.52858

•3)3.68332

•3)5.84216

2)1.04768

•3)4.27388

3)5. 11620

3)7.01588

2)1. 03563

2)1. 70257

(-3)9.06225

(-2)1.04807

(-2)1.35460

(-2)1.86333

(-2)2.81189

•2)2.56328

-2)2.83583

2)3.39532

2)4.26129

2)5. 74670

2)6.25971

2)6.69670

2)7.56308

2)8.84106

1)1.09077

3.5

4

4.5

5

6

4)6.

3)1.

3)2.

3)3.

3)7.

56918

35389

39916

82223

82558

•3)2.46324

3)4.31963

•3)6. 75690

3)9.74863

2)1.72215

3)6.25613

3)9.81967

2)1.41008

2)1.90064

2)3.03374

•3)9.39237

2)1.40651

2)1.94764

2)2.55033

•2)3.89821

•2)1.62122

2)2.28312

2)3.01461

2)3.80023

2)5.48606

-2)2.47734

-2)3.33192

-2)4.24479

-2)5. 19953

-2)7. 18703

(-2)3.84938

-2)4.94518

-2)6.07795

-2)7.23244

-2)9.56529

•2)7.25395

2)8. 76100

1)1.02545

1)1. 17262

1)1.45852

(-

1)1.28925

1)1.47963

1)1.66222

1)1.83736

1)2. 16686

7

8

9

10

12

2)1.

•2)2.

2)2.

2)3.

2)5.

32941

00532

78984

66313

60672

•2)2.63522

2)3.67658

2)4.81406

2)6.02138

2)8.56555

2)4.31888

2)5.70760

2)7. 16351

•2)8.65948

•1)1. 16961

•2)5. 38011

•2)6.94554

2)8.55840

1)1.01929

1)1.34576

•2)7.26633

•2)9.09204

1)1.09305

1)1.27598

•1)1.63364

-2)9.22382

-1)1. 12666

-1)1.32882

-1)1.52718

-1)1.90865

1)1. 18856

1)1.41611

1)1.63736

(-1)1.85134

1)2.25591

1)1.73162

1)1.99127

1)2.23758

1)2.47100

1)2.90170

1)2.47110

1)2. 75276

1)3.01421

1)3.25751

1)3.69670

15

20

24

30

40

2)8.

1)1.

1)1.

1)2.

1)3.

81504

43852

87255

47770

34605

1)1.25033

1)1.89037

1)2. 36535

1)3.00434

1)3.88728

-1)1.61897

-1)2. 31600

-1)2.81533

-1)3.46978

-1)4.34932

-1)1.81955

-1)2.53953

-1)3.04723

-1)3. 70493

-1)4.57784

-1)2. 13925

-1)2.88643

-1)3.40206

-1)4.05936

-1)4.91677

-1)2.43702 (-1)2.80446

-1)3.20087

-1)3. 71901

-1)4.37107

-1)5.20988

60

120

00

1)4

1)6

1

63816

65337

00000

1)5. 14884

1)7. 03036

1.00000

1)5.56875

1)7.32770

1.00000

-1)5. 77167

-1)7.46771

1.00000

-1)6.06738

-1)7.66781

1.00000

-1)6.31847

-1)7.83425

1.00000

(-1)3.57930

1)4. 09535

(-1)4. 73588

(-1)5.54762

1)3.46790

1)4.24087

1)4. 74178

1)5.35080

1)6. 10536

1)4.25790

1)5.00000

1)5.46857

1)6.02708

1)6.70415

(-1)6.60286

(-1)8.01917

1.00000

1)7.06202

1)8. 31026

1.00000

1)7.54199

1)8.60551

1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

22
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Z/, = 24vz p = 0.0001 p = 0.00l p = 0.005 p = 0.0l p =0.025 p=0.05 p=0.l p = 0.25 p=0.5

1 (-10)6.68273 (-8)6.68273 (-6)1.67070 (-6)6.68306 (-5)4. 17799 (-4)1.67273 (-4)6. 71570 (-3)4.31014 (-2)1.91679

I.I (-9)3.65762 (-7)2.40647 (-6)4.49002 (-5)1.58347 (-5)8.38199 (-4)2.96025 (-3)1.04948 (-3)5. 73917 (-2)2.27495

1.2 (-8)1. 51311 (-7)7.02329 (-5)1.02688 (-5)3.26064 (-4)1.50275 (-4)4. 78170 (-3)1.52908 (-3)7.32509 (-2)2.63937

1.3 (-8)5.04625 (-6)1. 74355 (-5)2.07410 (-5)6.02651 (-4)2.47071 (-4)7. 19968 (-3)2. 11083 (-3)9.04772 (-2)3.00794

1.4 (-7)1.42066 (-6)3.81128 (-5)3. 79923 (-4)1.02310 (-4)3. 79476 (-3)1.02575 (-3)2.79275 (-2)1.08891 (-2)3.37912

1.5 (-7)3.49224 (-6)7. 52415 (-5)6.43536 (-4)1.62261 (-4)5. 51903 (-3)1.39811 (-3)3. 57128 (-2)1.28337 (-2)3. 75175

1.6 (-7)7.68812 (-5)1. 36727 (-4)1.02283 (-4)2.43480 (-4)7.67856 (-3)1.83824 (-3)4.44187 (-2)1.48677 (-2)4. 12497

1.7 (-6)1.54545 (-5)2. 32050 (-4)1.54254 (-4)3.49054 (-3)1.02995 (-3)2.34616 (-3)5.39947 (-2)1.69797 (-2)4.49812

1.8 (-6)2.87978 (-5)3. 72012 (-4)2.22667 (-4)4.81709 (-3)1. 33997 (-3)2.92104 (-3)6.43879 (-2)1.91594 (-2)4.87069

1.9 (-6)5.03393 (-5)5.68415 (-4)3.09772 (-4)6.43775 (-3)1.69903 (-3)3.56140 (-3)7. 55455 (-2)2. 13982 (-2)5.24227

2 (-6)8. 33372 (-5)8.33716 (-4)4. 17625 (-4)8. 37177 (-3)2. 10759 (-3)4.26532 (-3)8. 74161 (•2)2. 36884 (-2)5.61257

2.2 (-5)1.99830 (-4)1.62209 (-4)7.02637 (-3)1. 32375 (-3)3.07284 (-3)5.85468 (-2)1. 13101 (-2)2.83978 (-2)6. 34838

2.5 (-5)5. 75300 (-4)3.63533 (-3)1. 32360 (-3)2.31571 (-3)4.88052 (-3)8.65484 (-2)1.55827 (-2)3.57082 (-2)7.43782

3 (-4)2. 12902 (-4)9.91592 (-3)2.92417 (-3)4.67770 (-3)8. 77246 (-2)1.42637 (-2)2. 35788 (-2)4.83067 (-2)9.20988

3.5 (-4)5.50648 (-3)2.06506 (-3)5.24642 (-3)7.87848 (-2)1. 36066 (-2)2.08063 (-2)3.23658 (-2)6. 11648 (-1)1.09238

4 (-3)1. 13701 (-3)3.62852 (-3)8.25219 (-2)1. 18239 (-2)1.92067 (-2)2.80534 (-2)4. 16910 (-2)7.40950 (-1)1.25791

4.5 (-3)2.01859 (-3)5.68701 (-2)1. 18749 (-2)1.64089 (-2)2.54196 (-2)3.58280 (-2)5. 13768 (-2)8.69814 (-1)1.41770

5 (-3)3.22184 (-3)8.22111 (-2)1.60398 (-2)2. 15336 (-2)3.21186 (-2)4.39941 (-2)6. 12951 (-2)9.97493 (-1)1.57195

6 (-3)6.62033 (-2)1.45795 (-2)2.57091 (-2)3.30569 (-2)4.65793 (-2)6. 11029 (-2)8. 14772 (-1)1.24750 (-1)1.86474

7 (-2)1. 12867 (-2)2.23948 (-2)3.67494 (-2)4.58164 (-2)6. 19689 (-2)7.87828 (-1)1.01733 (-1)1.48875 (-1)2. 13815

8 (-2)1.70846 (-2)3. 13614 (-2)4.87595 (-2)5.93902 (-2)7.78715 (-2)9.66583 (-1)1.21769 (-1)1. 72032 (-1)2.39393

9 (-2)2. 38495 (-2)4. 12139 (-2)6. 14334 (-2)7. 34718 (-2)9.40039 (-1)1. 14487 (-1)1.41412 (-1)1.94198 (-1)2.63369

10 (-2)3. 14190 (-2)5. 17324 (-2)7.45397 (-2)8. 78377 (-1)1. 10170 (-1)1.32111 (-1)1.60559 (-1)2. 15383 (-1)2.85886

12 (-2)4.83961 (-2)7.40878 (-1)1.01390 (-1)1. 16809 (-1)1.42097 (-1)1.66363 (-1)1.97161 (-1)2.54934 (-1)3.27038

15 (-2)7.67648 (-1)1.09153 (-1)1.41696 (-1)1.59478 (-1)1.87934 (-1)2. 14568 (-1)2.47618 (-1)3.07845 (-1)3.80594

20 (-1)1.26899 (-1)1.67247 (-1)2.05489 (-1)2.25671 (-1)2.57131 (-1)2.85800 (-1)3.20513 (-1)3.81786 (-1)4.53143

24 (-1)1.66661 (-1)2. 11176 (-1)2.52096 (-1)2.73293 (-1)3.05878 (-1)3.35148 (-1)3.70116 (-1)4.30776 (-1)5.00000

30 (-1)2.23034 (-1)2.71314 (-1)3.14290 (-1)3.36119 (-1)3.69180 (-1)3.98424 (-1)4.32865 (-1)4.91497 (-1)5.56947

40 (-1)3.05703 (-1)3.56293 (-1)3.99804 (-1)4.21443 (-1)4.53696 (-1)4.81751 (-1)5. 14280 (-1)5.68530 (-1)6.27635

60 (-1)4.32344 (-1)4.81347 (-1)5.21943 (-1)5.41669 (-1)5.70555 (-1)5.95222 (-1)6.23330 (-1)6.69145 (-1)7. 17715

120 (-1)6. 37971 (-1)6.75503 (-1)7.05314 (-1)7. 19423 (-1)7.39675 (-1)7.56609 (-1)7.75527 (-1)8.05567 (-1)8.36428

CO 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

23
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

^, = 30

^2 p = 0.0001 p = 0.001 p = 0.005 p = 0.01 p =0.025 p

= 0.05 p=0.l p = 0.25 p=0.5

1 ;-10)5. 32397 (-8)5.32397 (-6)1.33101 (-6)5.32423 (-5)3.32851 (-4)1.33265 (-4)5.35060 (-3)3.43531 (-2)1.53008

I.I (-9)2.91517 (-7)1.91799 (-6)3.57862 (-5)1.26205 (-5)6.68063 (-4)2.35943 (-4)8.36542 (-3)4.57689 (-2)1.81742

1.2 (-8)1.20649 (-7)5.60004 (-6)8. 18785 (-5)2.5998^ (-4)1. 19824 (-4)3.81288 (-3)1.21941 (-3)5.84505 (-2)2. 11023

1.3 (-8)4.02534 (-6)1.39081 (-5)1.65449 (-5)4.80731 (-4)1.97091 (-4)5.74353 (-3)1.68414 (-3)7.22393 (-2)2.40682

1.4 (-7)1. 13372 (-6)3.04150 (-5)3.03189 (-5)8. 16469 (-4)3.02842 (-4)8. 18656 (-3)2.22931 (-3)8.69941 (-2)2. 70598

1.5 (-7)2.78806 (-6)6.00696 (-5)5. 13774 (-4)1.29544 (-4)4.40640 (-3)1. 11635 (-3)2.85219 (-2)1.02592 (-2)3.00678

1.6 (-7)6. 14041 (-5)1.09202 (-5)8. 16928 (-4)1.94470 (-4)6.13325 (-3)1.46845 (-3)3.54927 (-2)1. 18926 (-2)3.30851

1.7 (-6)1.23484 (-5)1.85412 (-4)1.23254 (-4)2.78911 (-4)8.23031 (-3)1.87507 (-3)4.31662 (-2)1. 35905 (-2)3.61066

1.8 (-6)2.30194 (-5)2.97367 (-4)1.77992 (-4)3.85071 (-3)1.07125 (-3)2.33561 (-3)5. 15016 (-2)1.53447 (-2)3.91280

1.9 (-6)4.02550 (-5)4.54549 (-4)2.47725 (-4)5.14844 (-3)1.35890 (-3)2.84898 (-3)6.04577 (-2)1. 71486 (-2)4.21461

2 (-6)6.66698 (-5)6.66978 (-4)3.34114 (-4)6.69798 (-3)1.68643 (-3)3.41371 (-3)6.99942 (-2)1.89961 (-2)4.51584

2.2 (-5)1.59994 (-4)1.29874 (-4)5.62604 (-3)1.06000 (-3)2.46101 (-3)4.69027 (-3)9.06565 (-2)2.28016 (-2)5. 11578

2.5 (-5)4.61167 (-4)2.91421 (-3)1.06115 (-3)1.85672 (-3)3.91415 (-3)6.94374 (-2)1.25106 (-2)2.87265 (-2)6.00742

3 (-4)1.71002 (-4)7.96504 (-3)2.34930 (-3)3.75876 (-3)7.05193 (-2)1. 14724 (-2)1.89821 (-2)3.89859 (-2)7.46641

3.5 (-4)4.43140 (-3)1.66212 (-3)4.22404 (-3)6.34481 (-2)1.09640 (-2)1.67772 (-2)2.61280 (-2)4.95191 (-2)8.88782

4 (-4)9. 16784 (-3)2.92642 (-3)6.65840 (-3)9.54358 (-2)1.55136 (-2)2.26788 (-2)3.37488 (-2)6.01741 (-1)1.02703

4.5 (-3)1.63073 (-3)4.59589 (-3)9.60220 (-2)1.32742 (-2)2.05814 (-2)2.90380 (-2)4. 17033 (-2)7.08550 (-1)1. 16140

5 (-3)2.60773 (-3)6.65723 (-2)1.29982 (-2)1.74593 (-2)2.60680 (-2)3.57471 (-2)4.98890 (-2)8. 14985 (-1)1.29197

6 (-3)5.37867 (-2)1. 18538 (-2)2.09243 (-2)2.69232 (-2)3.79851 (-2)4.98982 (-2)6.66676 (-1)1.02514 (-1)1.54218

7 (-3)9.20412 (-2)1.82810 (-2)3.00384 (-2)3.74812 (-2)5.07718 (-2)6.46507 (-2)8. 36683 (-1)1.23012 (-1)1.77864

8 (-2)1.39835 (-2)2.57019 (-2)4.00235 (-2)4.87972 (-2)6.40920 (-2).7. 96954 (-1)1.00641 (-1)1.42891 (-1)2.00238

9 (-2)1.95914 (-2)3.39077 (-2)5.06347 (-2)6.06234 (-2)7.77119 (-2)9.48271 (-1)1. 17430 (-1)1.62106 (-1)2.21435

10 (-2)2.59013 (-2)4.27234 (-2)6; 16840 (-2)7.27760 (-2)9. 14658 (-1)1.09906 (-1)1.33936 (-1)1.80642 (-1)2.41543

12 (-2)4.01728 (-2)6. 16348 (-2)8.45501 (-2)9.75422 (-1)1. 18932 (-1)1. 39554 (-1)1.65872 (-1)2. 15700 (-1)2. 78805

15 (-2)6.43439 (-2)9. 17400 (-1)1. 19424 (-1)1.34622 (-1)1.59046 (-1)1.82031 (-1)2. 10724 (-1)2.63534 (-1)3.28316

20 (-1)1.07924 (-1)1.42704 (-1)1. 75898 (-1)1.93509 (-1)2.21097 (-1)2.46389 (-1)2. 77213 (-1)3. 32206 (-1)3.97292

24 (-1)1.43198 (-1)1.82089 (-1)2. 18108 (-1)2. 36874 (-1)2.65871 (-1)2.92082 (-1)3.23611 (-1)3.78905 (-1)4.43053

30 (-1)1.94205 (-1)2.37130 (-1)2. 75651 (-1)2.95337 (-1)3.25315 (-1)3.52005 (-1)3.83659 (-1)4. 38152 (-1)5.00000

40 (-1)2.70996 (-1)3. 17050 (-1)3.57001 (-1)3.76998 (-1)4.06969 (-1)4.33215 (-1)4.63863 (-1)5. 15552 (-1)5. 72804 60 (-1)3.92975 (-1)4.39083 (-1)4.77619 (-1)4.96465 (-1)5.24219 (-1)5.48074 (-1)5.75450 (-1)6.20561 (-1)6.69156

120 (-1)6.01685 (-1)6.38761 (-1)6.68455 (-1)6.82593 (-1)7.02995 (-1)7.20157 (-1)7.39456 (-1)7. 70408 (-1)8.02676

00 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0.123456.

2k
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Z/| = 40vz p = 0.0001 p=0.00l p = 0.005 p = 0.0l p =0.025 p=0.05 p=0.l p = 0.25 p=0.5

1 (-10)3.97638 (-8)3.97638 (-7)9.94108 (-6)3.97658 (-5)2.48602 (-5)9.95349 (-4)3.99654 (-3)2.56689 (-2)1. 14503

I.I (-9)2.17822 (-7)1.43312 (-6)2.67394 (-6)9.43007 (-5)4.99181 (-4)1.76302 (-4)6.25130 (-3)3.42184 (-2)1.36113

1.2 (-9)9.01866 (-7)4. 18611 (-6)6. 12055 (-5)1.94346 (-5)8.95716 (-4)2.85033 (-4)9.11665 (-3)4.37251 (-2)1.58168

1.3 (-8)3.01027 (-6)1.04009 (-5)1.23728 (-5)3.59507 (-4)1.47394 (-4)4.29549 (-3)1.25972 (-3)5.40722 (-2)1.80543

1.4 (-8)8.48183 (-6)2.27547 (-5)2.26829 (-5)6. 10842 (-4)2.26578 (-4)6.12535 (-3)1.66831 (-3)6.51558 (-2)2.03146

1.5 (-7)2.08673 (-6)4.49594 (-5)3.84539 (-5)9.69594 (-4)3.29818 (-4)8.35656 (-3)2. 13550 (-3)7.68852 (-2)2.25908

1.6 (-7)4.59773 (-6)8. 17671 (-5)6.11694 (-4)1.45616 (-4)4.59272 (-3)1.09973 (-3)2.65876 (-3)8.91818 (-2)2.48776

1.7 (-7)9.24991 (-5)1.38888 (-5)9.23280 (-4)2.08933 (-4)6.16576 (-3)1.40490 (-3)3.23524 (-2)1.01978 (-2)2.71712

1.8 (-6)1.72504 (-5)2.22844 (-4)1.33388 (-4)2.88581 (-4)8.02884 (-3)1.75078 (-3)3.86195 (-2)1. 15214 (-2)2.94682

1.9 (-6)3.01789 (-5)3.40774 (-4)1.85723 (-4)3.86000 (-3)1.01893 (-3)2.13662 (-3)4.53591 (-2)1.28840 (-2)3. 17664

2 (-6)5.00024 (-5)5.00238 (-4)2.50596 (-4)5.02391 (-3)1.26509 (-3)2.56138 (-3)5.25417 (-2)1.42811 (-2)3.40637

2.2 (-5)1.20093 (-5)9.74866 (-4)4.22326 (-4)7.95750 (-3)1.84783 (-3)3.52263 (-3)6.81249 (-2)1.71642 (-2)3.86497

2.5 (-5)3.46578 (-4)2.19017 (-4)7.97578 (-3)1.39568 (-3)2.94299 (-3)5.22286 (-3)9.41659 (-2)2. 16663 (-2)4.54918

3 (-4)1.28769 (-4)5.99837 (-3)1.76957 (-3)2.83170 (-3)5.31481 (-3)8.65111 (-2)1.43274 (-2)2.94999 (-2)5.67558

3.5 (-4)3.34360 (-3)1.25430 (-3)3.18861 (-3)4.79076 (-3)8.28324 (-2)1.26842 (-2)1.97766 (-2)3.75916 (-2)6.78119

4 (-4)6.93100 (-3)2.21295 (-3)5.03734 (-3)7.22263 (-2)1. 17495 (-2)1.71910 (-2)2.56173 (-2)4.58268 (-2)7.86438

4.5 (-3)1.23528 (-3)3.48262 (-3)7.28065 (-2)1.00694 (-2)1.56262 (-2)2.20695 (-2)3.17448 (-2)5.41321 (-2)8.92470

5 (-3)1.97924 (-3)5.05520 (-3)9.87773 (-2)1.32750 (-2)1.98411 (-2)2.72403 (-2)3.80828 (-2)6.24579 (-2)9.96223

6 (-3)4.09837 (-3)9.03911 (-2)1.59730 (-2)2.05670 (-2)2.90559 (-2)3.82235 (-2)5. 11741 (-2)7.90426 (-1)1. 19704

7 (-3)7.04073 (-2)1.39989 (-2)2.30341 (-2)2.87666 (-2)3.90287 (-2)4.97805 (-2)6.45734 (-2)9.54047 (-1)1.38929

8 (-2)1.07385 (-2)1.97641 (-2)3.08283 (-2)3.76250 (-2)4.95076 (-2)6. 16755 (-2)7.80835 (-1)1. 11449 (-1)1.57343

9 (-2)1.51032 (-2)2.61825 (-2)3.91742 (-2)4.69567 (-2)6.03141 (-2)7.37479 (-2)9. 15767 (-1)1.27123 (-1)1.74994

10 (-2)2.00442 (-2)3.31251 (-2)4.79300 (-2)5.66215 (-2)7. 13186 (-2)8.58854 (-1)1.04967 (-1)1.42398 (-1)1.91924

12 (-2)3.13225 (-2)4.81711 (-2)6.62522 (-2)7.65466 (-2)9.35644 (-1)1. 10056 (-1)1. 31228 (-1)1.71711 (-1)2.23791

15 (-2)5.07123 (-2)7.25208 (-2)9.46972 (-1)1.06932 (-1)1.26689 (-1)1.45390 (-1)1.68889 (-1)2. 12612 (-1)2.67153

20 (-2)8.64797 (-1)1. 14772 (-1)1.41984 (-1)1.56507 (-1)1.79384 (-1)2.00496 (-1)2.2.6415 (-1)2.73207 (-1)3.29585

24 (-1)1.16124 (-1)1.48264 (-1)1.78286 (-1)1.94029 (-1)2. 18500 (-1)2.40775 (-1)2.67777 (-1)3. 15717 (-1)3.72365

30 (-1)1.60019 (-1)1.96252 (-1)2.29074 (-1)2.45965 (-1)2.71850 (-1)2.95068 (-1)3.22827 (-1)3.71223 (-1)4.27196

40 (-1)2.28288 (-1)2.68326 (-1)3.03413 (-1)3.21106 (-1)3.47802 (-1)3.71361 (-1)3.99102 (-1)4.46497 (-1)5.00000

60 (-1)3.41920 (-1)3.83772 (-1)4. 19127 (-1)4.36552 (-1)4.62385 (-1)4.84767 (-1)5. 10666 (-1)5.53895 (-1)6.01344

120 (-1)5.50749 (-1)5.86758 (-1)6. 15897 (-1)6.29875 (-1)6.50174 (-1)6.67378 (-1)6.86876 (-1)7. 18522 (-1)7.52092

CO 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0. 123456.

25
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

Z/| = 60

p = 0.0001 p=0.00l p = 0.005 p = 0.01 p =0.025 p = 0.05 p=0.l p = 0.25 p=0.5

-10)2.

-9)1.

-9)5.

-8)2.

63990

44672

99249

00102

8)2.63990

8)9.51846

7)2.78148

7)6.91384

7)6.59984

6)1.77597

6)4.06683

6)8.22460

6)2.64004

6)6.26323

5)1.29135

5)2.38978

5)1.65046

5)3.31546

5)5.95172

5)9.79802

5)6.60819

4)1.17099

4)1.89400

4)2.85556

4)2.65347

4)4. 15240

4)6.05853

•4)8.37556

3)1.70489

3)2.27401

3)2.90748

3)3.59763

(- 3)7.61653

3)9.06116

2)1.05377

2)1.20380

8)5.

7)1

•7)3

7)6

6)1

64052

38828

06009

15898

14908

6)1.51322

6)2.99110

6)5.44215

•6)9.24779

•5)1.48441

5)1.50845

5)2.55832

5)4.07127

5)6.14768

5)8.88539

•5)4.06221

5)6.45071

5)9.69193

4)1.39121

4)1.92238

-4)1.50683

4)2.19436

4)3.05699

4)4. 10585

4)5.34887

-4)4.07385

4)5.56031

4)7.32078

• 4)9.35661

-3)1. 16657

-3)1. 10976

3)1.42124

3)1. 77037

3)2.15532

•3)2.57418

3)4.33769

3)5.12171

3)5.94454

3)6.80173

3)7.68945

(- 2)1.35559

2)1.50869

2)1.66275

2)1.81749

2)1.97272

•6)2.

6)3.

6)8.

5)2.

5)8,

oino

33349

01279

31525

61968

5)2.27091

•5)3.33495

5)6.50454

•4)1.46315

-4)4.01555

4)1.23768

4)1.67071

4)2.81801

4)5.32875

-3)1.18485

4)2.57244

4)3.34955

4)5.31005

• 4)9.32569

-3)1.89636

4)6.79123

4)8.43571

3)1.23328

3)1.96696

-3)3.56072

•3)1.42434

3)1. 70832

3)2.35172

•3)3.49204

-3)5.79912

3)3.02499

3)3.50586

3)4.55055

3)6.30040

3)9.61319

3)8.60436

3)9.54357

2)1. 14851

-2)1.45262

-2)1.98438

2)2. 12827

2)2.28400

2)2.59562

2)3.06236

2)3.83554

-4)2

-4)4

-4)8

3)1

3)2,

24269

65831

31907

33564

77700

-4)8.41438

• 3)1.48769

-3)2.34625

3)3.41307

3)6. 12961

-3)2.13974

-3)3.38798

3)4.90803

3)6.67426

•2)1.08437

3)3.21572

-3)4.85949

-3)6.79104

-3)8.97471

-2)1.39728

-3)5.56318

-3)7.91098

-2)1.05483

-2)1.34282

-2)1.97672

-3)8.52513

-2)1. 15853

-2)1.49135

-2)1.84582

-2)2.60430

-2)1.33076

-2)1.72880

-2)2. 14860

2)2.58513

2)3.49412

• 2)2.53708

2)3.10313

(-22)4.60028

-2)5.35521

-2)3. 67758|(-2)6. 09969

2)6.83350

2)8.26902

•2)4.25707

2)5.42216

•3)4.

3)7.

2)1.

•2)1.

•2)2.

79034

33644

03612

38078

17549

3)9.53506

2)1.35219

•2)1.79931

•2)2.28653

2)3.35439

•2)1.57118

•2)2. 11288

•2)2.69764

•2)3.31618

2)4.62651

2)1.96401

2)2.58150

•2)3.23758

2)3.92293

2)5.35407

-2)2.66907

2)3.40329

2)4. 16747

2)4.95285

2)6.56218

2)3.41032

2)4.24813

2)5. 10682

2)5.97855

2)7.73944

•2)4.43450

• 2)5.39280

•2)6.35999

2)7.32979

2)9.26041

2)6.58571

2)7.74031

•2)8.88144

1)1.00062

1)1.22001

2)9'. 66238

1)1. 10149

1)1.23280

•1)1.36033

1)1.60463

2)3

2)6

2)8

1)1

-1)1

56489

19623

43927

18587

7402 7

2)5.11488

•2)8.25829

1)1.08265

•1)1.46215

•1)2.05740

•2)6.70193

1)1.02589

1)1.30784

1)1.71518

1)2.33876

2)7.58234

1)1.13339

1)1.42681

1)1.84648

1)2.48193

2)9.01152

•1)1.30377

•1)1.61299

-1)2.04917

-1)2.69967

1)1.03730

1)1.46219

1)1.78383

1)2.23257

1)2.89362

1)1.20962

1)1.65828

1)1.99277

1)2.45390

1)3.12428

1)1.53408

1)2.01703

1)2.36895

•1)2.84554

1)3.52450

1)1.94631

1)2.45801

1)2.82285

1)3.30844

•1)3.98656

•1)2,

1)4,

72306

72805

00000

1)3.07478

•1)5.06303

1.0000C

1)3.37591

1)5.33781

1.0000C

1)3.52577

1)5.47092

1.00000

•1)3.74983

1)5.66582

1.00000

1)3.94584

1)5.83258

1.00000

1)4. 17502

1)6.02348

1.00000

1)4.56364

1)6.33805

1.00000

(- 1)5.00000

1)6.67906

1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0. 123456.

26
PERCENTAGE POINTS OF THE BETA DISTRIBUTION

VK = 120

^2 p = 0.0001 p=0.00l p = 0.005 p = O.Oi p =0.025 p=0.05 P=0.l p = 0.25 p=0.5

1 (-10)1.31446 (-8)1. 31446 (-7)3.28620 (-6)1.31453 (-6)8.21805 (-5)3.29041 (-4)1.32130 (-4)8.49264 (-3)3.79972

I.I (-10)7.20654 (-8)4. 74143 (-7)8.84663 (-6)3. 11991 (-5)1.65155 (-5)5.83323 (-4)2.06865 (-3)1. 13340 (-3)4.52397

1.2 (-9)2.98629 (-7)1.38612 (-6)2.02666 (-6)6.43530 (-5)2.96602 (-5)9.43901 (-4)3.01966 (-3)1.44997 (-3)5.26534

1.3 (-9)9.97606 (-7)3.44688 (-6)4. 10036 (-5)1.19143 (-5)4.88490 (-4)1.42374 (-4)4. 17649 (-3)1. 79521 (-3)6.01977

1.4 (-8)2.81324 (-7)7.54726 (-6)7.52350 (-5)2.02607 (-5)7.51566 (-4)2.03206 (-4)5.53653 (-3)2. 16581 (-3)6.78425

1.5 (-8)6.92702 (-6)1.49245 (-5)1.27651 (-5)3.21872 (-4)1.09497 (-4)2.77478 (-4)7.09398 (-3)2.55884 (-3)7.55651

1.6 (-7)1.52751 (-6)2. 71657 (-5)2.03228 (-5)4.83805 (-4)1.52608 (-4)3.65500 (-4)8.84110 (-3)2.97178 (-3)8.33483

1.7 (-7)3.07567 (-6)4.61816 (-5)3.07007 (-5)6.94765 (-4)2.05058 (-4)4.67359 (-3)1.07691 (-3)3.40245 (-3)9. 11788

1.8 (-7)5.74065 (-6)7.41591 (-5)4.43912 (-5)9.60440 (-4)2.67258 (-4)5.82972 (-3)1.28685 (-3)3.84897 (-3)9.90461

1.9 (-6)1.00514 (-5)1. 13499 (-5)6. 18605 (-4)1.28577 (-4)3.39479 (-4)7. 12128 (-3)1.51301 (-3)4.30970 (-2)1.06942

2 (-6)1.66675 (-5)1.66749 (-5)8. 35389 (-4)1.67492 (-4)4.21874 (-4)8.54523 (-3)1. 75447 (-3)4.78322 (-2)1. 14860

2.2 (-6)4.00971 (-5)3.25501 (-4)1.41027 (-4)2.65757 (-4)6. 17337 (-3)1. 17752 (-3)2.27975 (-3)5.76389 (-2)1. 30741

2.5 (-5)1. 16002 (-5)7. 33107 (-4)2.67022 (-4)4.67354 (-4)9.85990 (-3)1. 75114 (-3)3. 16166 (-3)7.30460 (-2)1.54623

3 (-5)4.32763 (-4)2.01622 (-4)5.95033 (-4)9.52521 (-3)1.78926 (-3)2.91567 (-3)4.83792 (-2)1.00122 (-2)1.94435

3.5 (-4)1. 12830 (-4)4.23391 (-3)1.07701 (-3)1.61902 (-3)2.80254 (-3)4.29784 (-3)6.71689 (-2)1.28445 (-2)2.34126

4 (-4)2.34844 (-4)7.50191 (-3)1.70925 (-3)2.45253 (-3)3.99560 (-3)5.85675 (-3)8.75210 (-2)1.57640 (-2)2.73615

4.5 (-4)4.20271 (-3)1. 18575 (-3)2.48198 (-3)3.43582 (-3)5.34170 (-3)7.56048 (-2)1.09103 (-2)1.87463 (-2)3. 12860

5 (-4)6.76169 (-3)1.72875 (-3)3.38330 (-3)4.55202 (-3)6.81835 (-3)9.38413 (-2)1. 31671 (-2)2. 17744 (-2)3.51841

6 (-3)1.41185 (-3)3.11892 (-3)5.52397 (-3)7. 12350 (-2)1.00919 (-2)1.33168 (-2)1. 79064 (-2)2.79228 (-2)4.28964

7 (-3)2.44603 (-3)4.87440 (-3)8.04415 (-2)1.00651 (-2)1.37022 (-2)1. 75398 (-2)2.28657 (-2)3.41434 (-2)5.04941

8 (-3)3.76261 (-3)6.94535 (-2)1.08727 (-2)1.32995 (-2)1.75688 (-2)2. 19762 (-2)2.79777 (-2)4.03964 (-2)5. 79768

9 (-3)5.33758 (-3)9.28631 (-2)1.39532 (-2)1.67681 (-2)2. 16338 (-2)2.65720 (-2)3.31966 (-2)4.66558 (-2)6.53455

10 (-3)7. 14507 (-2)1. 18580 (-2)1.72409 (-2)2.04258 (-2)2.58538 (-2)3. 12877 (-2)3.84894 (-2)5.29036 (-2)7.26017

12 (-2)1. 13596 (-2)1.75652 (-2)2.43016 (-2)2.81735 (-2)3.46337 (-2)4.09669 (-2)4.92059 (-2)6.53169 (-2)8.67850

15 (-2)1.88693 (-2)2.71742 (-2)3.57431 (-2)4.05258 (-2)4.83349 (-2)5.58271 (-2)6.53861 (-2)8.36238 (-1)1.07270

20 (-2)3.35364 (-2)4.49174 (-2)5.60701 (-2)6.21091 (-2)7. 17491 (-2)8.07891 (-2)9.20850 (-1)1. 13065 (-1)1.39449

24 (-2)4.64714 (-2)5.99568 (-2)7.28247 (-2)7.96827 (-2)9.04999 (-1)1.00521 (-1)1. 12901 (-1)1. 35560 (-1)1.63572

30 (-2)6.69313 (-2)8.30695 (-2)9.80479 (-1)1.05898 (-1)1. 18124 (-1)1.29302 (-1)1.42946 (-1)1.67518 (-1)1.97324

40 (-1)1.01992 (-1)1.21495 (-1)1.39069 (-1)1.48114 (-1)1.62006 (-1)1. 74525 (-1)1.89600 (-1)2. 16261 (-1)2.47908

60 (-1)1.70038 (-1)1.93641 (-1)2. 14214 (-1)2.24586 (-1)2.40265 (-1)2.54160 (-1)2.70626 (-1)2.99131 (-1)3.32094

120 (-1)3.34705 (-1)3.61436 (-1)3.83802 (-1)3.94790 (-1)4. 11070 (-1)4.25191 (-1)4.41584 (-1)4.69179 (-1)5.00000

CO 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

The numbers in parentheses indicate the power of ten by which

the number following is to be multiplied, e.g., (-1)1.23456 = 0. 123456.

27
28
TABLE HI

VALUES OF THE COMPLETE BETA FUNCTION,

B(v

2 /2, v

x lZ) .

29
TABLE III. Values of the Complete Beta Function, B ( k2 /2, v,/2).

1 2 3 4 5 6

1 1

1 .2

1 3

3. 141 5927

2. 942 4685

2.774 5019

2.630 6299

2.000 0000

1.818 1818

1.666 6667

1.538 4615

1. 570 7963

1.401 1755

1.261 1372

1. 143 7521

1.333 3333

1. 173 0205

1.041 6667

(- 1) 9.324 0093

1. 178 0972

1.025 2504

(- 1) 9. 008 1231

(- 1) 7.979 6662

1.066 6667

(- 1) 9.200 1610

(- 1) 8.012 8205

(- 1) 7.036 9882

1 .4

1 5

1 6

1 .7 18

2.505 7956

2.396 2805

2.299 2878

2.212 6723

2. 134 7597

1.428 5714

1.333 3333

1. 250 0000

1. 176 4706

1.111 1111

1. 044 0815

(- 1) 9.585 1219

(- 1) 8.843 4147

(- 1) 8. 195 0825

(- 1) 7.624 1419

(- 1) 8.403 3613

(- 1) 7.619 0476

(- 1) 6. 944 4444

(- 1) 6. 359 3005

(- 1) 5. 847 9532

(- 1) 7. 118 7374

{- 1) 6.390 0813

(- 1) 5. 767 4444

(- 1) 5.230 9037

(- 1) 4. 765 0887

(- 1) 6.224 7121

(- 1) 5.541 1255

(- 1) 4. 960 3175

(- 1) 4.462 6670

(- 1) 4.033 0712

1 9

2

2.2

2.5

3

2.064 2239

2.000 0000

1. 387 1812

1. 743 0384

1.570 7963

1.052 6316

1.000 0000

(- 1) 9.090 9091

(- 1) 8.000 0000

(- 1) 6.666 6667

(- 1) 7. 118 0135

(- 1) 6.666 6667

(- 1) 5.897 4411

(- 1) 4.994 3953

(- 1) 3.926 9908

(- 1) 5. 398 1107

(- 1) 5.000 0000

(- 1) 4.329 0043

(- 1) 3.555 5556

(- 1) 2.666 6667

(- 1) 4. 357 9675

(- 1) 4.000 0000

(- 1) 3.402 3699

(- 1) 2. 724 2156

(- 1) 1.963 4954

(- 1) 3.659 7360

(- 1) 3. 333 3333

(- 1) 2.792 9060

(- 1) 2. 188 0342

(- 1) 1.523 8095

3.5

4

4.5

5

6

1.437 7683

1.333 3333

1.248 5988

1. 178 0972

1.066 6667

(- 1) 5. 714 2857

(- 1) 5.000 0000

(- 1) 4.444 4444

(- 1) 4.000 0000

(- 1) 3.333 3333

(- 1) 3. 195 0406

(- 1) 2.666 6667

(- 1) 2.270 1797

(- 1) 1.963 4954

(- 1) 1.523 8095

(- 1) 2.077 9221

(- 1) 1.666 6667

(- 1) 1.367 5214

(- 1) 1. 142 8571

(- 2) 8.333 3333

(- 1) 1.474 6341

(- 1) 1. 142 8571

(- 2) 9.080 7188

(- 2) 7. 363 1078

(- 2) 5.079 3651

(- 1) 1. 108 2251

(- 2) 8. 333 3333

(- 2) 6.435 3947

(- 2) 5.079 3651

(- 2) 3. 333 3333

7

8

9

10

12

(- 1) 9.817 4770

(- 1) 9.142 8571

(- 1) 8.590 2924

(- 1) 8. 126 9841

(- 1) 7.388 1674

(- 1) 2.857 1429

(- 1) 2.500 0000

(- 1) 2.222 2222

(- 1) 2.000 00001

(- 1) 1.666 6667

(- 1) 1.227 1846

(- 1) 1.015 8730

(- 2) 8.590 2924

k (- 2) 7.388 1674

(- 2) 5.683 2057

(- 2) 6.349 2063

(- 2) 5.000 0000

(- 2) 4.040 4040

(- 2) 3.333 3333

(- 2) 2.380 9524

(- 2) 3.681 5539

(- 2) 2.770 5628

(- 2) 2. 147 5731

(- 2) 1. 704 9617

(- 2) 1. 136 6411

(- 2) 2.308 8023

(- 2) 1.666 6667

(- 2) 1.243 2012

(- 3) 9.523 8095

(- 3) 5.952 3810

15

20

24

3040

(- 1) 6.580 7776

(- 1) 5.675 4639

(- 1) 5. 170 1948

(- 1) 4.614 7455

(- 1) 3.988 1731

(- 1) 1.333 3333

(- 1) 1.000 0000

(- 2) 8.333 3333

(- 2) 6.666 6667

(- 2) 5.000 0000

(- 2) 4. 112 9860

(- 2) 2.702 6018

(- 2) 2.068 0779

(- 2) 1.488 6276

(- 3) 9.727 2514

(- 2) 1.568 6275

(- 3) 9.090 9091

(- 3) 6.410 2564

(- 3) 4. 166 6667

(- 3) 2.380 9524

(- 3) 6.854 9767

(- 3) 3.525 1328

(- 3) 2.297 8644

(- 3) 1.353 2978

(- 4) 6. 786 4545

(- 3) 3.302 3736

(- 3) 1.515 1515

(- 4) 9. 157 5092

(- 4) 4.901 9608

(- 4) 2. 164 5022

60

120

(- 1) 3.249 5542

(- 1) 2.293 000 1

(- 2) 3.333 3333

(- 2) 1.666 6667

(- 3) 5.327 1380

(- 3) 1.895 0414

(- 3) 1.075 2688

(- 4) 2.732 2404

(- 4) 2.536 7324

(- 5) 4.622 0523

(- 5) 6.720 4301

(- 6) 8.813 6788

he numbers in parentheses indicate the power of ten by which the number following is to be multiplied, e.g., (-1)1. Z 3456 78 - 0. 12345678.

30
TABLE III. Values of the Complete Beta Function, B(v2 /2, i-j/2). v. 7 8 9 10 12 15

1

1 .

1 .2

1.3

(- 1) 9.817 4770

(- 1) 8.403 6915

(- 1) 7.264 6154

(- 1) 6. 333 0684

(- 1) 9. 142 8571

(- 1) 7. 774 7839

(- 1) 6.677 3504

(- 1) 5.783 8259

(- 1) 8.590 2924

(- 1) 7.262 4495

(- 1) 6.201 5010

(- 1) 5.341 1420

(- 1) 8. 126 9841

(- 1) 6.834 9749

(- 1) 5.806 3917

(- 1) 4.975 3341

(- 1) 7.388 1674

(- 1) 6. 157 6350

(- 1) 5. 184 2783

(- 1) 4.402 9505

(- 1) 6. 580 7776

(- 1) 5.424 2015

(- 1) 4.516 7600

(- 1) 3. 794 3230

1 4

1.5

1 .6

1 .7

1.8

(- 1) 5.561 5136

(- 1) 4.915 4471

(- 1) 4.369 2760

(- 1) 3.903 6595

(- 1) 3.503 7417

(- 1) 5.047 0639

(- 1) 4.432 9004

(- 1) 3.916 0401

(- 1) 3.477 4029

(- 1) 3. 102 3624

(- 1) 4.634 5947

(- 1) 4.048 0153

(- 1) 3.556 3875

(- 1) 3. 140 8754

(- 1) 2. 787 0672

(- 1) 4.295 3735

(- 1) 3.732 9688

(- 1) 3.263 3668

(- 1) 2.867 9611

(- 1) 2.532 5408

(- 1) 3. 767 8715

(- 1) 3.246 0598

(- 1) 2.813 2472

(- 1) 2.451 2488

(- 1) 2. 146 2210

(- 1) 3.211 9797

(- 1) 2.737 4953

(- 1) 2. 347 2446

(- 1) 2.023 5989

(- 1) 1. 753 1963

1.9

2

2.2

2.5

3

(- 1) 3. 157 9474

(- 1) 2.857 1429

(- 1) 2.362 7569

(- 1) 1.816 1438

(- 1) 1.227 1846

(- 1) 2.779 5464

(- 1) 2.500 0000

(- 1) 2.043 5898

(- 1) 1.544 4947

(- 1) 1.015 8730

(- 1) 2.483 7789

(- 1) 2.222 2222

(- 1) 1.797 7498

(- 1) 1.338 2112

(- 2) 8.590 2924

(- 1) 2.246 0981

(- 1) 2.000 0000

(- 1) 1.602 8155

(- 1) 1. 176 7579

(- 2) 7.388 1674

(- 1) 1.887 4774

(- 1) 1.666 6667

(- 1) 1.313 7832

(- 2) 9.414 0631

(- 2) 5.683 2057

(- 1) 1.525 7697

(- 1) 1.333 3333

(- 1) 1.029 6088

(- 2) 7. 157 1463

(- 2) 4. 112 9860

3.5

4

4.5

5

6

(- 2) 8.674 3184

(- 2) 6.349 2063

(- 2) 4.779 3257

(- 2) 3.681 5539

(- 2) 2.308 8023

(- 2) 6.999 3165

(- 2) 5.000 0000

(- 2) 3.677 3684

(- 2) 2. 770 5628

(- 2) 1.666 6667

(- 2) 5.782 8790

(- 2) 4.040 4040

(- 2) 2.909 1548

(- 2) 2. 147 5731

(- 2) 1.243 2012

(- 2) 4.869 0897

(- 2) 3.333 3333

(- 2) 2.353 5158

(- 2) 1.704 9617

(- 3) 9.523 8095

(- 2) 3.606 7331

(- 2) 2.380 9524

(- 2) 1.623 1143

(- 2) 1. 136 6411

(- 3) 5.952 3810

(- 2) 2.488 6320

(- 2) 1.568 6275

(- 2) 1.022 4495

(- 3) 6.854 9767

(- 3) 3. 302 3736

7

8

9

10

12

(- 2) 1.533 9808

(- 2) 1.065 6011

(- 3) 7.669 9039

(- 3) 5.683 2057

(- 3) 3.343 0622

(- 2) 1.065 6011

(- 3) 7. 142 8571

(- 3) 4.972 8050

(- 3) 3.571 4286

(- 3) 1.984 1270

(- 3) 7.669 9039

(- 3) 4.972 8050

(- 3) 3.355 5830

(- 3) 2.340 1435

(- 3) 1.231 6545

(- 3) 5.683 2057

(- 3) 3.571 4286

(- 3) 2.340 1435

(- 3) 1.587 3016

(- 4) 7.936 5079

(- 3) 3. 343 0622

(- 3) 1.984 1270

{- 3) 1.231 6545

(- 4) 7.936 5079

(- 4) 3.607 5036

(- 3) 1. 713 7442

(- 4) 9.435 3531

(- 4) 5.452 8223

(- 4) 3.281 8619

(- 4) 1. 312 7448

15

20

24

3040

(- 3) 1.713 7442

(- 4) 7.050 2657

(- 4) 3.961 8351

{- 4) 1.933 2826

(- 5) 7.540 5050

(- 4) 9.435 3531

(- 4) 3.496 5035

(- 4) 1.831 5018

(- 5) 8.169 9346

(- 5) 2.823 2637

(- 4)5.452 8223

(- 4) 1.827 8467

(- 5) 8.946 0793

(- 5) 3.657 5617

(- 5) 1.123 0539

(- 4) 3.281 8619

(- 5) 9.990 -0100

{- 5) 4.578 7546

(- 5) 1.719 9862

(- 6) 4.705 4395

{- 4) 1.312 7448

(- 5) 3.330 0033

(- 5) 1.346 6925

(- 6) 4.299 9656

(- 7) 9.410 8790

(- 5) 4.016 5879

(- 6) 7.929 4794

(- 6) 2.694 1860

(- 7) 6.846 2583

(- 7) 1.091 2821 60

120

(- 5) 1.951 3326

(- 6) 1.848 8209

(- 6) 6.109 4819

(- 7) 4.196 9899

(- 6) 2.038 7057

(- 7) 1.019 0351

(- 7) 7. 187 6258

(- 8) 2.623 1187

(- 7) 1.026 8037

(- 9) 2.017 7836

(- 9) 7.336 7342

(-11) 5.835 1908

The numbers in parentheses indicate the power of ten by which the number following is to be multiplied, e.g. , (-1)1.2345678 = 0. 12345678.

31
TABLE III. Values of the Complete Beta Function, B (v2 /2, Vj/2). v,

•4\

20 24 30 40 60 120

1

1 .1

1.2

1.3

(- 1) 5.675 4639

(- 1) 4.611 4552

(- 1) 3.785 6711

(- 1) 3. 135 4443

(- 1) 5.170 1948

(- 1) 4.162 9025

(- 1) 3.386 6609

(- 1) 2.779 8172

(- 1) 4.614 7455

(- 1) 3.674 5389

(- 1) 2.956 3921

(- 1) 2.399 9825

(- 1) 3.988 1731

(- 1) 3.130 3503

(- 1) 2.482 7554

(- 1) 1.986 9167

(- 1) 3.249 5542

(- 1) 2.499 4730

(- 1) 1.942 7333

(- 1) 1.523 7039

(- 1) 2.293 0001

(- 1) 1.703 6761

(- 1) 1.279 1696

(- 2) 9.691 9209

1 .4

1.5

1 .6

1 .7

1.8

(- 1) 2.617 1063

(- 1) 2.199 4807

(- 1) 1.859 8479

(- 1) 1.581 3553

(- 1) 1.351 3095

(- 1) 2.299 5582

(- 1) 1.915 4310

(- 1) 1.605 3301

(- 1) 1.352 9260

(- 1) 1. 145 9722

(- 1) 1.963 6108

(- 1) 1.617 7606

(- 1) 1.341 1156

(- 1) 1.118 0142

(- 2) 9. 367 7636

(- 1) 1.602 6713

(- 1) 1.301 7825

(- 1) 1.064 0035

(- 2) 8.745 6953

(- 2) 7.225 5424

(- 1) 1.204 5502

(- 2) 9.589 5054

(- 2) 7.682 3820

(- 2) 6. 189 5613

(- 2) 5.012 6368

(- 2) 7.401 9559

(- 2) 5.693 0893

(- 2) 4.406 5202

(- 2) 3.430 2490

(- 2) 2.684 2051

1.9

2

2.2

2.5

3

(- 1) 1. 160 0200

(- 1) 1.000 0000

(- 2) 7.516 2178

(- 2) 5.019 9725

(- 2) 2. 702 6018

(- 2) 9.751 6061

(- 2) 8.333 3333

(- 2) 6. 155 7886

(- 2) 4.006 8715

(- 2) 2.068 0779

(- 2) 7.885 7611

(- 2) 6.666 6667

(- 2) 4.820 2434

(- 2) 3.039 1872

(- 2) 1.488 6276

(- 2) 5.997 6750

(- 2) 5.000 0000

(- 2) 3.515 8089

(- 2) 2. 126 5844

(- 3) 9.727 2514

(- 2) 4.078 7443

(- 2) 3.333 3333

(- 2) 2.252 7675

(- 2) 1.284 3290

(- 3) 5.327 1380

(- 2) 2. 110 4612

(- 2) 1.666 6667

(- 2) 1. 051 9074

(- 3) 5.413 8441

(- 3) 1.895 0414

3.5

4

4.5

5

6

(- 2) 1.534 5214

(- 3) 9.090 9091

(- 3) 5.577 7472

(- 3) 3.525 1328

(- 3) 1.515 1515

(- 2) 1. 126 7241

(- 3) 6.410 2564

(- 3) 3.780 0674

(- 3) 2.297 8644

(- 4) 9. 157 5092

(- 3) 7.703 6218

(- 3) 4.166 6667

(- 3) 2.337 8363

(- 3) 1.353 2978

(- 4) 4.901 9608

(- 3) 4.705 2778

(- 3) 2.380 9524

(- 3) 1.250 9320

(- 4) 6.786 4545

(- 4) 2.164 5022

(- 3) 2.338 9038

(- 3) 1.075 2688

(- 4) 5. 137 3159

(- 4) 2.536 7324

(- 5) 6. 720 4301

(- 4) 7.028 5053

(- 4) 2.732 2404

(- 4) 1. 104 8661

(- 5) 4.622 0523

(- 6) 8.813 6788

7

8

9

10

12

(- 4) 7. 050 2657

(- 4) 3.496 5035

(- 4) 1.827 8467

(- 5) 9.990 0100

(- 5) 3.330 0033

(- 4) 3.961 8351

(- 4) 1.831 5018

(- 5) 8.946 0793

(- 5) 4.578 7546

(- 5) 1.346 6925

(- 4) 1.933 2826

(- 5) 8. 169 9346

(- 5) 3.657 5617

(- 5) 1.719 9862

(- 6) 4.299 9656

(- 5) 7.540 5050

(- 5) 2. 823 2637

(- 5) 1. 123 0539

(- 6) 4.705 4395

(- 7) 9.410 8790

(- 5) 1.951 3326

(- 6) 6. 109 4819

(- 6) 2.038 7057

(- 7) 7. 187 6258

(- 7) 1.026 8037

(- 6) 1.848 8209

(- 7) 4. 196 9899

(- 7) 1.019 0351

(- 8) 2.623 1187

(- 9) 2.017 7836

15

20

24

3040

(- 6) 7.929 4794

(- 6) 1.082 5088

(- 7) 2.835 1422

(- 8) 5.098 7734

(- 9) 4.992 5087

(- 6) 2.694 1860

(- 7) 2.835 1422

(- 8) 6. 163 3525

(- 9) 8.628 6935

(-10) 5.905 1179

(- 7) 6.846 2583

(- 8) 5.098 7734

(- 9) 8.628 6935

(-10) 8.595 6334

(-11) 3.592 0169

(- 7) 1.091 2821

(- 9) 4.992 5087

(-10) 5.905 1179

(-11) 3.592 0169

(-13) 7.254 4446

(- 9) 7. 336 7342

(-10) 1.572 9567

(-11) 1.055 0320

(-13) 2.899 6650

(-15) 1.768 1885

(-11) 5.835 1908

(-13) 2.940 8958

(-15) 6.509 0249

(-17) 3.654 9504

(-20) 1.885 7342

60

120

(-10) 1.572 9567

(-13) 2.940 8958

(-11) 1.055 0320

(-15) 6.509 0249

(-13) 2.899 6650

(-17) 3.654 9504

(-15) 1.768 1885

(-20) 1.885 7342

(-19) 5.637 0780

(-26) 7.427 9529

(-26) 7.42.7 9529

(-37) 3.450 1231

The numbers in parentheses indicate the power of ten by which the number following is to be multiplied, e.g. , (- 1) 1. 2345678 = 0. 12345678.

32 GPO 841-614