fgfflOUB NATIONAL BUREAU OF STANDARDS REPORT

NBS Project

3737-60-0002 5 December 1952

NBS Report

2096

A TEST FOR STATISTICAL CONTROL APPLICABLE

TO A SHORT SERIES OF OBSERVATIONS

Churchill Eisenharfc and Edgar P 0 King

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

This report is issuf production in any fc writing is obtained

Washington 25, D. C.

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

ition, reprinting, or re- nted unless permission in nal Bureau of Standards,

A THAT FOP STATISTICAL CONTROL APP],TC.i -LE

TO A SITORT SERIES OF OhSKRVATTONS

hj

Churchill Kiaenhart and Edgar P« King

Introduction

In what follows w© shall call attention to»a useful test

for statistical control based on the ratio Cr m

Si \) ©f

IfS' ranges „ This teat has the attract!'/® features tnat it is

simple end it is applicable to a small amount of data*

Procedure

Given a sequence of 2n measurements £g g o©©* xgn

taken in time»order (sp.acc«»@rd©P* etc*} by a particular

measurement process* farm the sums and differences of successive

non<»©v©rl appi ng pairs from this sequence*? obtaining the two

columns s sum difference

*i " xi * *2

Sg » Xj xk

da

*“ k

3

*'

$ a a © o 0 © e $ a © © o

© © e a * o a <t * » * t> c

a n ” z

2n.l *

alR

d

n

3

*2n-l ~ *2n

Now compute the range of the sums (Hj)* the range of the

differences (Kg)* and form their rati© r * Ff^/Rgo If the

computed value of r is less than the critical value given in

Table I (attached) corresponding to ri* accept the hypothesis

that the measurement process is in a state of statistical

con troll otherwise* reject this hypothesis 0 Xf Xg* os©* sgin 3- s a sequence of independent measure**

manta ©n a process in statistical control* then the sums should

©xhi'fot the same variability as the differences 0 We would h@ne®

except the rati© ”rw to be "near*1 unity® On the other hand0

if some disturbance- . is present that tends to make x°s that

are close together in time (or in space., etc?) agree more
/ s

cF

'

-

:

/' ' -> f 3 -

r

: ?A

"

: 5 ( V
V 1

.«=> 2 05

closely than x®s that are nor© widely separated in tire© {or

spare .<? etc*), then there will be a corresponding tendency for

wnusual1 y large” values of r to resu.it©

Fix ample a

X 0 Suppose we apply this test to the following sequence taken

from a table of random numbers©

Xj « 19a * 39a « 64* » 92* * 3a x

& * 61* X

? • 14

X0 ?5 9 X^ a 90 ? x10 ® 55

Pot this sequence we compute the following sums and

differences?

sura difference.

58

156

64

89

145

®20

e»58

—61

35

We then obtain Rj * 156-58 » 98 «n^ Kg » 35 *° C®61) - 96®

and hence

p

: *s || * 1 «Q2

The critical value for r at the 5 percent l8?@l correspond®

ing to n®5 is found in Table I to be 2©5o Since l o 02 is

less than 2©5 we conclude that the sequence exhibis statistical

control©

Z 0 Tine following sequence was obtained from a table of random

normal numbers after "building in15 a line®? trend©

X

1

*9

*,46» ® 1©o6,5 x, ®

33 5o39* 5ol4®

® 7® 53* »io * 8o1^

3 o 69 @ x^ ® 4® 02

® 3®47* Xg « 6©65a

Applying- the same test in this ease yields*

sums differences

1©52 ct o 60

7o51 ** o53

lo©53 ©25

10*12 -3 ©18

15 ©9? - ©91

B| ^ l£ 0 97 « l 0 52 * lk<>k$

H2

*s-.«25— (»3«18) « 3«U3

r * s ko21

Since ij.o21 exceeds the critical vain© we correctly

conclude ' that a disturbance • of some sort is present* and

the figures themselves suggest an upward trendo

Concluding Remarks

A few remarks about the above procedure should b© made©

1) Th© critical values in Table I assume normality of th©

underlying di3tributlon 0

2) Systematic errors that tend to make th® x 0 s that ap©

near together in the sequence more- dissimilar then

z s s that are more widely separated will ordinarily moke

!5 pt3 unusually small 0 This situation was neglected in

th© above discussion because it seldom occurs 9

3) Th© test uses an even number of measurements© With

an odd number you should,* of course 9 neglect one©