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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.%--
-
-'.. -
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Technical Reports Distribution List
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Attn: Quality Control Office, MCQComander Wright-Patterson Air
Force BaseAir Materiel Force, European Area Ohio 10Attn: Quality
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York 1
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GeneralAttn: Quality Control Office Chemical Corps Materiel
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4 BranchSCommander
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346 BroadwayClark Air Force Base New York 13, New YorkAPO 74 Attn:
Dr. J. LadermanSan Francisco, California I
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Commanding Officer Commanding OfficerOffice of Naval Research S.
C. Supply Agency
Branch Office 225 South 18th Street1000 Geary Street
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SIGSU-H6d 2
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Laboratories U. S. Naval Underwater OrdnanceWashington 25, D. C.
Station
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Air ForceWashington 25, D. C. 2 Washington 25, D. . 3
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AgencyComm ding Officer 84 Sands Street
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Office of Naval Research
- Attn: Qu~lity Surety Division I Washington 25, D. C. 2Quality
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Sperry Gyroscope C:.Commanding Officer Great Neck, New York
* I Rock Island Arsenal Attn: Mr. Herbert Jaffe, CP-25 LRock
Island, Illinois
Attn: R&D Division
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Eq. Mobile Air Materiel Area office of Operations AnalySis
Materiel Quality Division (MOSQA) DCS/Operationgs
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of Technical Services
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Defense (S&L, Washington 25, D. C.
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52HN.A. ^ 3' • N .. t;tiattcal Laboratory,\ %-."1512 11 S. N,- ,
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Atthr: Div. of Rezc?1-ch Berkeley 4, Calif'Olnia
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TchcaLiryAOffice, Assistant Secretary of
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Office of Naval Research, Code 200 Technical Operation, Tnc.
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F.- Monrre, Vlrgin ia.
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Dr. Adam Abruzzi Mr. Carlton M. BeyerDept. of Economics of
Engineering Office of Ou-.ded MissilesStevens Inst. of Technology
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E)
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of MathematicsUniversity of Nebraska University of KansasLincoln 8,
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Sandia CorporationProf. J. N. Berrettoni Albuquerque, New
Mexi'oWestern Reserve UniversityCleveland, Ohio I Dr Louts
COuurt
Division .'N-t.nnal u.reau o: Stnl.arWasnington 2'., D.c'
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Mrs. J. Henley Crosland Mr. Donald FlischelDirector of Libraries
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Prof. Meyer Dvass c/o Office of Naval ResearchDept. of
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Prof. P rank M. Oryna, Jr. Mr. P t. LindreniQuality Control
Consultant Joy Man,I'etu, g CompanySignal Corps ftgneering Lab. -
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Walter V. Box" Chief, Statistics BranchVought Aircraft, Inc. Allied
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Research Project Prof. a. W. Maclrath65 Administration Bld. Dept.
of canical angneeringUniversity of California net of M in
nesota~University of MinnesotaLos Angeles 24, California i
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ngineer Department of MathematicsChrysler Division-Chrysler Corp.
Washington State College12200 E. Jefferson
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Mathematical Institute N.A.D. BangorFaculty of Science
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Kyusyu University Dr. R. B. MurphyFukuoka, Japan c/o Bell
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Prof. E. G. Olds J:io-'. Seymour ShermanDept. of Mathemwtics
Morc Sehool of Electrical Eng.Carnegie Institute of Technology
Unersity of Pennsylvania
Pittsburgh, Pa. 1 Philadelphia , P.
Mr. Cyril Peckham Mr. Walter Shewart
Project Globe Bell Telephone Labs., Inc.
University of Dayton Mrray Hill, ew Jersey
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Supervisor
Chief Statistician Boeing Airplane Company
Aeronautical Research Laborary Seattle i, Washington
Wright Air Development Center
Wright-Patterson Air Force Base Dr. Rosedith SitgreavesOhio 1
Teachers College
Columbia UniversityDr. Harry G. Romig New York 27, N.Y.351 Alma
Real
Drive
Pacific Palisades, California 1 Dr. Milton SobelBell Telephone
Labs.
Dr. Alan J. Rowe 555 Union Blvd.General Electric Co.
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Production Control Service, 20. 2Aea
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Columbia UniversityProf. Herman Rubin New York 27, N.Y.Dept. of
MthematicsUniversity of Oregon Dr. M. D. SpringerEugene, Oregon
Head, Statistical Analysis BranchPU.S. Naval Ordnance PlantProf.
Norman Rudy Indianapolis, Indiana,1
Statistics Department
Sacramento State College Mr. Selig StarrSacramento, California 1
Research & Development Dept.Pro- U.S. Naval Powder FactoryProf.
h- nry Schieff Indianhead, Md.
Statistical Laboratory
University of California Mr. Arthur SteinBerkeley 4, California
1 Cornell Aeronautical Laborator4 Inc.
P. 0. Box 235
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General Dry Batteries, Inc.
13000 Athens Avenue Mr. Dan TeichroewCleveland 7, Ohio National
Cash Register Co
Product Development Dept.14l) E, E! Segundo Blvl.
Hawtorn~, " ' £
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Eli Xizabeth VaU4±I&flD.?.A.'on?325 7th StreetDrM.A
iobyBremerton~, V&Shlyton De~pt. of )ktheatics
cesareo, vilUeo York U4vrstInsetitute ds *tetc& Y s~ltc New
York 53, N. Y.AY. J. iferrtr. e isi .R .~i
M. Vnir' Che NPt. 2220.'1Tw, Project RAN.MW~weerin fttlsicsLb ~
Ctn'"Vo0a0t Al re taft,chemicla Corps ftgt~aeg Agency Da. - ?-BX
5WArW cbdj Cefat., aiam alad*~
P. Stts.ca A"Itloval copies for ~jcKin~ oz MfCndLd ~e, wid
reserve ror futurK~ ga on C tArio, OCad& I r qure
Pro0f W - Allen Walliscomittee on Statistics
t~z'res~y of (2hIcASChica3 P, Ilinois
Prof - A. Wuto?ebatsebe xSohchui.Lerutett, ()rawy
* ~&resu Of iSUM Co4. 20D*rmU~t of the jhyVasbgW-Oft 25, D.
C. 2
Riysicia loa rahg, %ftm'*an's 8 1uI£boMLoy aIBelmr, New Jrey
Dr. Irvin4 Vjejebell ? -epbooe Labs.lortb Ardoyer, Is.
yx. Silas William, Jr.StrA#ndazs BranchProcurement
DivisionDCS/Logisticz c U. S. AqWshigton 25, 1b. C.
M~r. j. W. YoungQilaty Control
7.?rhAMriclAn Aviation, Icr~t-cntioa1Airport
knge)E 45 Cal