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TABLES FOR ONE-SIZED STATISTICAL TOLERUCE LIMITS

Z:::J.BY

(Z~GERALD J. LIEBRM

TECHNICAL REPORT NO. 34NOVEKBER I, 1957

PMEARE UNDER CONTIRACT N6onr -25126

GERALD J, LIKEKAN, PROJRT DIRECT'ORFOR

OFFCR OF NAVAL R ACH

REPOUTION IN WHOLE OR IN PART IS KI1TKD FORANY rURPOSE OF THE UNITED STATE

APPLIED MATHD(ATICS AND STATISTICS LABORATORYSTANFORD UNIVERSITYSTANFORD, CALIFORNIA

.- .*1, - .- \.

" " .

. A- ,. ,.q

TABLES FOR ONE-SIDED STATISTICAL TOLERANCE LIXITS

by

Gerald J. Lieberman

'A

1. INTROUCTION.

The quality of manufactured product is often specified by setting

a range, the bounds of which are called tolerance i ts. These limits

have the property that a certain percentage, 00(]-a) % , of the product

may be expected to fall within these limits. If the quality of the items

is normally distributed with known mean, 14 , and known standard deviation,

a , two-sided tolerance limits are formed by adding to and subtracting from

p" the quantity K a' , where K2 is the normal deviate correspondingU/22to a/2 . An upper one-sided tolerance limit Is given by p + K or and alover one-sided tolerance limit is given by p- K a . Unfortunately, In

apractice, the values p and a, are rarely known and estimates of these

A values, - and a , respectively, are used. However, where before it

could be stated that 95% of a manufactured product lies below 1 + .64 a,

the upper one-sided tolerance limit, this statement cannot be extended to

the limit i + 1.645 a . The quantities i and s arp random variables,

and hence the limit depends upon the particular outcome of the sample.

Different samples will lead to different limits. How close thebe 11mit

are to u + 1.645 a depends upon how good the estimates are.

It is evidnt then that the fraction of the ;tems lnc'u,&ed bet.--,r

A,, A~ -

- ~; -2

",a. t -K., , i + K a.,s I for two-sided tolerance limits,

b. - o, + K ] for an upper one-sided tolerance limit,

c. Ix- Kas , co] for a lower one-sided tolerance limit

will not always contain a specified proportion I- a of the manufactured

items. However, it is possible to determine constants K such that in a

large series of samples from a normal distribution, a fixed proportion 7

of the int( rvals

a. [-Ks, + Ks ] for two-sided tolerance limits,

b. [- ;0, i + Ks] for an upper one-sided tolerance limit, and

-c. - Ks, co] for a lower one-sided tolerance limit

C will include 100(1-a)% or more of the distribution. The finite limits

of these intevals are known as statistical tolerance limits. Thus, an

upper one-sided statistical tolerance limt is given by x + Ks and has

the property that the probability is equal to a preassigned value 7 that

the interval [- co, i + Ks] includes at least a specified proportion

1 - a of the distribution. Note that in most practical situations y is

usually a large number close to 1 . Statistical tolerance intervals

should not be confused with confidence intervals for a parameter of the

distribution. Confidence limits for the mean of a normal distribution are

such that in a given fraction, say .95 , of the samples from which they

are computed, the interval bounded by the limits will include the true mean

of the distribution. For confidence interval estimation, .95 is a!lso

called the confidence coefficient.

Many extensive tables of factol-i K for two-sided o~lera~n:e limitshave been computed. A. R. Bovker presents such a table in [1) and very

recently A. Lieberman issued a set of tables [2) which are an extension

of Bowker's tables in that they can be used when s is computed from saple

si~es other than n . Resnikoff (33 and JMitra (14) computed values of

K when a Is estimated by the average range of subgroups. Tables of

factors K for one-sioe. tolerance limits have never been conputed anid It

is the purpose of this paper to present such tables.

2.* TABfL OF ONE-SIDED T01"ACE LIM.

Factors K such that i+ Ku and -Ks are an upper one-sided

tolerance limit and a lower one-sided tolerance limit, respectively, for

a normal universe are given in Table 1 for n a3,2d,...,2530,354,245,50;

for Y a 0.75, 0.90, 0.95, 0.99; and for Q a 0.25, 0.1000.05,0.01..0.001

iis the sample mean a (x1 + x', x.. x)/n ) and a is the sample

st~zdard deviation

(( a!__( )2 + (x2- ) + .+(Xn )2 /(n-l).For n greater than 50, the values of K can be obtained using the

asymptotic normality property of x+ Ks . With tLhis approximation, K

is given by

K a

a

where

K'~

a 1

K2

.... .-. .. . . . . . . . .1* - b

. . . 3C M V. '1 ° -.. 4. .

-4--

and K and K (which equals -K, that 2 ) are thea -7 7

normal deviates corresponding to a and 7 respectively, i.e.

o-1 2 00 1 2

fe nds a dz -,

7

3. .XAMPLZ.

A mnufacturer of light bulbs would like to specify a single lower

limit above which he can be assured, with probability 0.95 , that at

least 99% of his production will lie. A sample of 30 bulbs is taken

and the sample mean and sample standard devir.tion are found to be 987.2

and 5 963 respectively. A value of K a 3.064 corresponding to a - 30

7 a 0.95 and a a 0.01 is obtained from Table 1 . The required lower

tolerance limit is given by i - Ks - 987.2 - (3.064)(5.963) - 968.9

4. C0TRT ION OF TRE TAM.U

For the case of a normal distribution with wkziown parameters P

and w , exact one-sided tolerance limits my be computed by the follow-

ing procedure. Let ta-1 , IS % be a non-central t statistic with

n-i degrees of freedom and non-etrality parameter n K . K isaadefined by

-2S e-Z dz -a

Determine t0 such that P tn K > t 1 - '• Let

K - to/ /'-. Then if i and s are the sample mean and the sample

standard deviation, respectively, bae. on a sample x ,x ,. .Vn .e

-2% X 2 _ _ix= - and s n

n n-i

i + Ks and x - Ks are upper one-sided tolerance limit and a lower

one-sided tolera-ice limit, respectively Thus, the probability is y that

. at least a proport!on l-a of the distribution will be less than x + Ks

(or greater than x .. Ks )

The value to may be obtained from tables of the distribution of the

non-central t sta'clstic. Such tables are those of Johnson & Welch [5]

and Resnikoff and Lieberman [6]. The latter tables are perhaps more

convenient for thiv application, and were used in the computation of Table 1.

,2

:i

!I

-p.

.1

---------.-- p~ . .

-S

.06-

TABLE 1.

Tolerance Factors for Normal DistributionsFactors K such that the probability is 7 that at least a proportion

"-a of the distribution will be less than x + Ks (or greater than x - Ks),where i and s are estimates of the mean and the standard deviation com-

puted from a sample of size n

7 - 0.75 I 7 - 0.90['-Cr-0.25 0.10 0.05 0.01 0.001 0.25 0.10 0.05 0.01 0.001

3 1.464 2.501 3.152 4.396 5.805 2.602 4.258 5.310 7.340 9.651

4 1.256 2.134 2.680 3.726 4.910 1.972 3.187 3.957 5.437 7.1285 1.152 1.961 2.463 3.421 4.507 1.698 2.742 3.400 4.666 6.112

6 1.087 1.860 2.336 3.243 4.273 1.540 2.494 3.O91 4.242 5.556

7 1.043 1.791 2.250 3.126 4.118 1.435 2.333 2.894 3.972 5.201

8 1.010 1 740 2,190 3.042 4.OO8 . 1.360 2.219 2.755 3.783 4.9559 0.984 1.702 2,141 2 977 3.924 1.302 2.133 2.649 3.641 4.772

1o 0.964 1.671 2.103 2,927 3.858 1.257 2.065 2.568 3.532 4.629

11 0.947 1.646 2.073 2.885 3.8c4 1 219 2.012 2.503 3.444 4.51512 0.933 1.624 2.048 2.851 3.760 1.188 1.966 2.448 3.371 4.42013 0.919 1.606 2.026 2.822 3.722 1..162 1.928 2.403 3.310 4.34114 0.909 1.591 2.007 2.796 3.690 1.139 1.895 2.363 3.257 4.274

" 15 0.899 1.577 1.991 2.776 3.661 1.119 1.866 2.329 3.212 4.215

16 0.891 1.566 1.977 2.756 3.637 1.101 1.842 2.299 3.172 4.164

17 0.883 1.554 1.964 2.739 3.6'r- 1.085 1.820 2.272 3.136 4.118V, 18 0.876 1,544 1.951 2.723 3.595 1.071 1.800 2.249 3.106 4.078

:19 0.870 1.536 1.942 2.710 3.577 1.058 1.781 2.228 3.078 4.O41i20 0865 1.526 1.933 2.697 3.561 I.o46 1.765 2.208 3.052 4.009

21 0.859 1.520 1.923 2.686 3.545 1.035 1.750 2.190 3.028 3-979022 .851, 1.514 1.916 2.675 3.532 1.025 1.736 2.174 3.007 3.952

23 0.849 1.508 1.907 2.665 3.520 1.016 1.724 2.159 2.987 3.927* 24 0.845 1.502 1.901 2.656 3.509 1.007 1.712 2.145 2.969 3.904

25 0.842 1.496 1.895 2.647 3.497 0.999 1.702 2.132 2.952 3.882

S3 0.825 1.475 1.869 2.613 3.441 0.966 1.657 2.c80 2.884 3.79435 0.812 1.458 1.849 2.588 3.421 0.942 1.623 2.041 2.833 3.73040 0.803 1.445 1.834 2.568 3.395 0.923 1.598 2.010 2.793 3.67945 0.795 1.435 1.821 2.552 3.375 0.908 1.577 1.986 2.762 3.638. 0 O 788 1.426 1.811 2,538 3.358 ,894 .56G 1.965 2 73c 3.604

iT-4

t --7.

TkA=L 1.

Tolerance Factors for Normal Distributions (continued).

7 - 0.95 7 - 0.99\i" 0.25 0.10 0.05 0.01 0.001 0.25 0.10 0.05 0.01 0.001

3 3.8C14 -. . 7.655 10.552 13.857C 4 2.619 4.163 5.145 7.o42 9.2155 2.149 3.4o7 4.202 5.741 7.501

6 1.895 3.006 3.707 5.062 6.612 2.849 4.4o8 5.409 7.334 9.5407 1.732 2.755 3.399 4.641 6.o61 2.490 3.856 4.730 6.411 8.3488 1.617 2.582 3.188 4.353 5.686 ;Z 252 3.496 4.287 5.811 7.566

1.532 2.454 3.031 4.143 5.414 2.085 3.242 3.971 5.389 7.0]4. 10 1.465 2.355 2.911 3.981 5.203 .954 3.048 3.739 5.075 6.60311 1.411 2.275 2.815 3.852 5.036 1.854 2.897 3.557 4.828 6.28412 1.366 2.210 2.736 3.747 4.900 1.771 2.773 3.41o 4.633 6.03213 1.329 2.155 2.670 3.659 4.787 1.702 2.677 3.290 4.472 5.82b14 1.296 2.108 2.t,14 3.585 4.690 1.645 2.592 3.189 4.336 5.65115 1.268 2.068 2.566 3.520 4.607 1.596 2.521 3.102 4.224 5.507

16 1.242 2.032 2.523 3.463 4..534 1.553 2.458 3.028 4.124 5.37417 1.220 2.001 2.486 3.415 4.471 1.514 2.405 2.962 4.038 5.26813 1.200 1.974 2.453 3.370 4.415 1.1481 2.357 2.906 3.961 5.16719 1.183 1.9149 2.423 3.331 4.364 1.450 2.315 2.855 3.893 5.078W20 1.167 1.926 2.396 3.295 4.319 1.424 2.275 2.807 3.832 5.003

21 1.152 1.905 2.371 3.262 4.276 1.397 2.241 2.768 3.776 4.932122 1.138 1.887 2.350 3.233 4.238 1.376 2.208 2.729 3.727 4.866. 23 1.126 1.869 2.329 3.206 4.204 1.355 2.179 2.693 3.680 4.80624 1.114 1.853 2.309 3.181 4.171 1.336 2.154 2.663 3.638 4.7550-4 25 1.103 1.838 2.292 3.158 4.143 1.319 2.129 2.632 3.6ol 4.706

3C 1.059 1.778 2.220 3.064 4.022 1.249 2.029 2.516 3.446 4.50835 1.025 1.732 2.166 2.99'. 3.934 1.195 1.957 2.431 3.334 4.36,.40 0.999 1.697 2.126 2.941 3.866 1.154 1.902 2.365 3.2'0 -.. 2 545 0.978 1.669 2.092 2.897 3.811 1.122 1.857 2.313 3.181 4.16850 0.961 1.646 2.065 2.863 3.766 1.096 1.821 2.296 3.124 !,.0 9 6

.'. "*,1

~-8-

BIBLIOGRAPHY

[i Bowker, Albert H., "Tolerance Limits for Normal Distributions,"

Techniques of Statistical A edited by Eisenhart, Hastay

. and Wallis, McGraw-Hill, New York, 1947, Chapter 2.

12) Lieberman, Alfred, "Table for the Determination of Two-sided Tolerance

A Limits for the Normal Distribution," Report No. 373-17(55), Bureau of

Ships, Navy Department, Washington 25, D.C., 20 August 1957.

(33 Resnikoff, George J., "Two-sided Tolerance Limits for Normal

- Distributions Using the Range," Technical Report No.33, Applied

- Mathematics and Statistics Laboratory, Stanford University, California,

April 25, 1957.

(41 Mitra, S. K., "Tables for Tolerance Limits for a Normal Pcpulation

* -Based on Sample Mean and Range or Mean Range," Journal of the

American Statistical Association March 1957.

(5] Johnson, N. L., and Welch, B. L., "Applications of the on-central

t- Distribution," Biometri Vol. 31, (1940), Pp. 362-389.

L61 Resnikoff, George J., and Lieberman, Gerald J., '"ables of the

Son-central t- Distribution," Stanford University Press, Stanford,

California, (1957), pp. 736.

Kr.oK.-

,F.2.2;

0.%--

-'.. -

STANFOR Ly~wRb.&aY

Technical Reports Distribution List

Contract N6onr-25126

Federal Telephone and Radio Co. Chief

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