Section 11.3 pnpv OUl I, 7

Do not remove NATIONAL BUREAU OF STANDARDS REPORT

NBS Project

3Q11“60“0002

1951 NBS Report

13U

COMBINING TOLERANCES

by

Ei1 Fo Xlcg

U. S. DEPARTMENT

NATIONAL BUREAU

O F

OF

COMMERCE

STANDARDS

This report is issuec Approved for public release by the

production m any foj Director of the National Institute of writing is obtained ,

J ^ , Washington 25. D. c. Standards and Technology (NIST) on October 9, 2015

.ion, reprinting, or re- ted unless permission in

al Bureau of Standards,
FOREWORD

This report was prepared in the St; tistieal Engineering Laboratory (Section 11 o3 g National Bureau of Standards)

in response to a request from Hr 0 Robert S 0 Hoff of the

Projector Fuze Development Sec: Ion (Section 13o6 a National

Bureau of Standards

)

0

Jo H e Curtiss

Chiefs National Applied

Mathematics Laboratories

Ho Vo /istln

.acting Director

National Bureau of Standards
COMBINING TOLERANCES

by

E 0 P 0 King

Statement of the Problem

Knowing the tolerance limita on the various component

parts of on electrical circuit, we wish to determine tolerance

limits on the performance of this circuit® More precisely

knowing the means and standard deviations of the distributions

of the components, how can we obtain limits which include a

certain percent of the distribution of performance? In view

of the particular applications to be made, we make the follow**

ing assumptions:

Xo Performance is a linear function of the

components ©

2c The components ar© selected at random from

their respectiv© distributions

©

Summary

The statements that can b© made about the expected varla«

bility in performance depend on the amount of information avails

able concerning the distribution of performance® The appropriate

statements are explained, according as the form of this district

bution is (1) unspecified, (2) unimodal and symmetric, and

(3) normal 0

An important approximation is given when the number of

components is large©
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Fop simplicity, let us assume that

romance., depends on t^Po components

specify the functional relation as

the measure of per°

C~, and C^

0 Let us

a

Let the !cnc^: means

(nominal .. values) of and Cp be H, and Mof respectively p and their standard deviations and ap

,

respective lyo . The

mean performance, P

(

-, , is then given by

P « a C + a C 11 2 2

where a.

} and a

p

are known constants c

F

0 " a

lMl * &

2M2

and the standard deviation of performance, o , by

P — yai Oj t a2

0 2 *

the latter relation only holding when C-j and Cp are selected

at random from their respective distributions

«

Knowing P0 and o , we now wish to know what percent of

.performances will fall within fixed limits*, The sharpness of

the results depends on how much additional knoalledg© tfe

possess concerning the performance distributions
' it

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.

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«3«

Case 1; Performance Distribution unspec ified

In this case we know only ?

(J and a,.o Using a form of

the ,T ?chebycheff Inequality” {!) # re can make the following

statement

©

For any given constant K> at least 100(l«^g=) Percen *

the performance distribution is included in the inter-ffal

P 4* K o o 0 •» p

For example 9 v?hen R»3, *r e knov that the interval of 3

standard deviations on either side. of the' mean' includes at least

100(1—i-) percent , or 8B 0 9 percent of the distribution 0 3

Case ?.% Performance Distribution Unimodal and Symmetric

T-his case would arise, for example, if both components

Were drawn at random from tha same unimodal symmetric distri^

bution 0 Using the n Gauss Inequality'' (2) the sharper state**

ment can now be made that at least IQOll^^g) percent of the

performance distribution is included in the interval k a ? 0 - p

• hen K«3, we can now say that the interval of 3 standard

deviations on either side of the mean includes more than 95

percent of the distribution <>
- I

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: it*.

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;> 1
Cas e 3: Performance Distribut i on Normal

This case only arises when both components are drav;n at

random from normal distributions® rte can no\\r make exact

statements based on the Table of the Formal Integral (3)o

Denoting the normal frequency function by f(x) we have the

fraction of the performance distribution included in the interval

The latter integral is tabulated in (2) as a function of Ko

When K**3, we find that the interval of 3 standard a viations on either side of the mean includes 99 «7 percent of the

distribution

Although the above outline deals exclusively with tw©

components, the same argument applies to any finite number * For example 9 if

P_ + K o is

0 ® p

K gd

ftlx) dx 1 « 2 jP f(x) dx

P * « a2c

2 ^ a 3^3

vre have

the rest following exactly as before 0
-.

rrouuMijx *s 2-^

o tm»b**S* 1 lotutrxain!. arid dmid >iJj W

*

.
Approximation when. the Lumber of Components is Large

as the number of components (each selected at random)

increases 9 the distribution of performance approaches a normal

distribution - under general conditions regardless of the type

of dis tribution of the components * Thus v;e can use Case 3

above to give an approximate answer, the larger the number of

components the better the approximation® A rough working rule

is to use Case 3 when there are four or more components 0

Illustrative Example

Let U3 consider a circuit whoso performance can be c-x« pressed in terms of the four capacitors , CPf and Cj

by the equation

P *C*j ^ Cg ^ Cj

(In this case * ap » a

j

» a;, ® 1)

Suppose that the capacitors originally had capacitances

that followed a normal distribution with mean (nominal) value

l8o jjjif and standard deviation llj. upf s but later the supply

was screened and all capacitors beyond 1$ percent' of the

V. . nominal value were removed* If four capacitors are now drawn

at random and the circuit constructed, what tolerance limits

can be obtained on performance?
;oe

* c°

> a° * 4

- *

.

v.
Notice that the distribution of components in this case

is a ** truncated normal** distribution Its mean is

H«l80

To find its standard deviation, we note that the point of

truncation is (#15) (180) or 2? ppf from the nominal value of

180* This is §1, or lo93 times the at* dev* of the original

‘ r normal distribution* In terras of this factor, the desired

at* dev* is

/&* o

1.93

P f(x)dx

1 0 S3

x sli, dev 0 ox the original normal

fixate

x 0*3

The Table 01 Inooi»ipi<?te formal -/’unction?. 1 S’-

1*93 *

* -2 S' f

.£». -i. ^ A f V/Wu 3

-lo93

,706

The Table of the Normal 3!htagral gives

lo93

J

f (x)dx ** »9l-S-6 «lo93
I , 1 I ’i .001*1*1

J

'•

- I

U.«« i»

»• j m; laoMsna*- — * - **••*

- *

.
Therefor©

* 12 ol

It follows that the mean and standard deviation of performance

are

V’lth no additional information, we would have (following Case l)s

Tho interval J

3^1 c includes at least 90 percent of th©

distribution of performance©

The interval Pq J Op includes at least 95 percent of the

distribution of performance*

Converting these to percentage statements (noting that

Op is 3 ok percent of Pq):

At least 90 percent of the distribution falls within 10 ©5

cent of the nominal performance©

At least 95 percent of the distribution falls within 15 o3 per-*

&ent of the nominal performance*

If we knew only that the performance distribution was

unimodal and symmetric we would have (following Case 2 )

The interval P^ 4* 2*1 includes at least 90 percent of the

distribution©

The interval 3 o

r} Includes at least 90 percent of the

distribution o

P0 * 4(180)

720

P

* 240 2
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onoitfud. ^sJtb

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r 1 iJ T
At least 90 percent of the distribution falls ^ithin 7cl

cent of the nominal performance©

At least 9^ percent of the distribution falls within 10*2 per~

cent of the nominal performance

©

Fortunately we know d in addition, that the distribution

of performance is approximately norma?. 9 Hence i:@ can make the

following approximate statements has sd on a Table of th© Correal

Integral©

The interval Pq J 1 3 65 c0 includes 90 percent of the distribution

The interval iP_. + 1.96 o

77 95

' n n 11 n 0 p

The interval £1 ^ 2 Q $8 o

17 99 " i; 77 0 ® p

90 percent of th&• distribution falls within 5 06 percent of the

nominal performance © 95 percent of the distribution falls within 6 ©7 percent of the

nominal performance e 99 percent of the distribution falls within 8 C 3 percent of the

nominal performance.

ocac aoopoo^OfficsoQoe iseru>£jsa»es:&«suc:scM3 ar3Ssofilo

ftUsing this approximation,, the percent of thft distribution

included is in error by less than X©5 percent when there are

four components© For more details^ see (53

o
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•(y o

Thus, although the capacitances vary up to 15 percent of

8 0 8 percent of the nominal performance®

If no screening had been done, we would have

H»180

0^24 o In this case^ the mean and st e dev® of performance or©

P

0 - 720

Since the distribution ;jf performance is exactly normal p we

can make the following precise statements 2 90 percent of the distribution falls within 6 <4 percent of the

nominal performance e 95 percent of the distribution falls within 7o6 percent of the

nominal performance®

their nominal value, virtually all performances fall within

« 28 o

99 percent of the distribution falls within 10 percent of the

nominal performance 0
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131bir.g.;3?aphy

(1) So Vilks, Mathematical Statistics , Princeton University

Pre33, p, 30o

( 2 ) Ho Cramer. Mathematical Methods of Statistics, Princeton

University ITre^sTreTa^^

(35 HIsenhart, B&stsy, and ^'allis, Techniques of Statistical

Analysis » MeGrav^ITill , Table lo 5 » Vo 31° ^

{ Lt) K 0 Pearson. Tables for Statisticians and! Biometricians

Part I. Cambridge"

' "Onlvers i

t

’y"

(5) Birnbaum and Andrews, ”0n Sums of Symmetrically Truncated

Normal Random Variables”,,*•' . Armais -r--— ITT - of Mathematical mt - — Statistics --cr .

Lur'jt Volo 20, Koo 3# Sept o , l9kyTvVo
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