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NATIONAL BUREAU OF STANDARDS REPORT

I8k3

EFFECT OF TOTAL SAMPLE SIZE ON LEVEL OF

SIGNIFICANCE AT THE START OF A CONTROL CHART

fey

F» Proschan and I* R« Savage

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Aslinjjjpplp Director THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

3737-60=0002 7 August 1952 1814.3

EFFECT OF TOTAL SAMPLE SIZE ON LEVEL OF

SIGNIFICANCE AT THE START OF A CONTROL CHART

by

F« Press chan and X© Ho Savage

Statist leal Laboratory

Applied Mathexnaties Division

The publication, reprlntln

unless permission Is obta,r

25. D. C. Such permlssl

prepared If that agency

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

< Is prohibited

Is, Washington

en specifically

Its own use,
i L
FOREWORD

This report was prepared as part of a continuing program

of research on mathematical statistics and its applications

carried out at the National Bureau of Standards under the

general supervision of Dr„ Churchill Eisenhartj, Chief of the

Statistical Engineering laboratory . The Statistical Engineering

Laboratory is Section 11 0 3 of the National Applied Mathematics

Laboratories (Division 11, National Bureau of Standards) , and

is concerned with tfye development and application of modern

statistical methods in the physical sciences and ©ngineeringo

Jo Ho Curtiss

Chiefs National Applied

Mathematics Laboratories

A 0 Vo Astin

Director

National Bureau of Standards
'

•

'
EFFECT OF TOTAL SAMPLE SIZE ON LEVEL OF

SIGNIFICANCE AT THE START OF A CONTROL CHART

by

F 0 Pros chan and I„ R» Savage

Introduction s In starting a control chart to detect a shift

in a normal population mean, it is generally assumed that one

knows >u, the population mean, and 0 , the population standard

deviation, or that one has accumulated enough past data to

estimate them with negligible error a If these assumptions

are correct, the level of significance turns out to be 0 3

per cent (more accurately, *26 per cent) under the usual

practice of setting the control limits at m i 3n~ 0 However,

Suppose that it is important to start the control chart as

soon as possible so that only a relatively small amount of

past data is available,, How is the level of significance

affected by the number, m, and the size, n, of subgroups used to

set the limits?

Summary g For various sample sizes n, the largest number of

subgroups m for which the significance level a. exceeds 1 per

cent is given in Table I %

2 «=>

TABLE I

n m

10 4. 12 4

14 4

16 3

18 3

20 3

In each case above p a < 2 per cent 0 Deviation s Let

c=> 4» K

x± “ mean of i sample

x s mean of 1^ (i “

n - common sample siz ©

m * number of samples available at the start of the

control chart

R s average of the m sample ranges

expected value of the rang© in th© samples of n s randomly selected from a normal population of

unit standard deviation

u

“

cr ^ra+l ° x

Control limits are customarily set at/u, + 3o= 0 Suppose

u and a are unknown so that the control limits are set at

x + 3 ( 1 )

= 3

CD

The probability 9 of the next sample meanp x^^p falling

outside of the control limits is given by

« - Probo (fSm+1

CD

( 2 )

or

a « Probo ( > 3) m Probo (u » (3)

Values of 3 /

C=OT=7' are as follows

' m4i

3 fjSZ

“¥m+l

m 3 rw°

T m+1

1 2 0 1213 11 2.8723

2 2.1*95 12 2a8823

3 2.5981 13 2 08909

4 2.6833 14 2o8983

$ 2 o ?386 15 2 0 9047

6 2. 7775 16 2 0 9104

7 2 o 8062 17 2.9155

8 2 0848)4. 18 2 0 9200

9 2 0 BIj.60 19 2 0 9240

10 2o8604 20 ~2V92T?

Bounds for ©. may be determined by the use of the tables of

[1], which give the 90 * 95 & 98 & 99 0 99<>8p and 99 °9 percentiles

of u, By using Table 6 of [l]p which gives the 99 percentile

of Up for a given value of n the largest value ©f m was

determined such that a. » X per cento (See Table 1) Six point

Lagrangian interpolation was performed to determine the 99

percentile value of u for n - 2P m * 16 and four point Lagrangian

interpolation for the 99 percentile of u for n = 4s ® 88 7®

In addition p by using Table 5 of [1] it was established that a

was actually less than 2 percent for the combinations of m and

n given in Table I c

k

c=>

Using Sample Standard Deviation Instead of Range

With only a small amount of data available at the start

of a control chart for means s it might b© considered advisable

to estimate the population variance by pooling the sample

variances,, The control limits would then be established as

1 + 3 =vs

(k)

where

a « / Z af/m

Vial 1

2 and “ variance of 1^ sample 0 If this more efficient procedure for estimating the

population variance is followed*, will the level ©f significance

of the control chart for means come out closer to the desired

value s a 3 per cent? Or p putting it another way*, will there b®

any decrease in th© maximum number of subgroups 9 for which a 9 the level of signiffeane© 9 exceeds 1 per cent?

In general the answer is no 9 as th© following table shows 0 Table II gives the largest value of m for a given value of n

such that a > 1 per cent*, using (I4.) as the control limits,,

TABLE II

n m

18

20 3

tx> CO

In each case, a < 2 per cent.

Note that Table II differs from Table I in only 2 cases t n 3 2, li).. In both cases m differs in the two tables by 1

only* Hence, in general, control limits established by (4)

are not noticeably superior to those of (1) (using as the

criterion of superiority, the maximum number of subgroups for

which a > 1 per cent) 0

REFERENCE

E* Lord, The use of rar^ge in place of standard deviation in

the t«=test, Biometrika ,, Vol 0 34 » Parts 1, 2 (1947) 9 PP* 4l“67<*

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