Practical Statistitcs for The Textile Industry Part I

9. Analysis'of Diskrcto and Ranking Data

last row of LhlS Lable dnd suggest Lhat Lhcre class 01 problems ar'ises ,.,.hen data are classi.
may be differences in absentee rate among Lhe fied accordilig to two characteristics; the
factories. We therefore ask whether these data are then said to be arranged in a
di ffe"cl1ces MC signi fical1t.
~_o_n_t.i_ll.9_C.!l.Cy._' ~_a_~~c~.
Ihe null IlypoLl""", is i.IlclL Lhe cibcL'lILeIeaLe
is the same for all foetor-ies. 10 get an es- Suppose lhat, in addition to counting the
timate of this common or overall absentee numbers of stoppages on eac~) of the five
rate, we combine the data for all factories IIlJchine~.in Example 9.1, Vie also recorded the
and find type of ~toppage, of Vlhich there are three
niibelled A, B, and C). Each stoppage then has
total number of absences 174 two char~cteristics associated with it, namely
tota 1 number of employees TOlD" the machine on which it occurred and the type
of stoppoge. The complete data are shown in
If the null hypothesis is true, this rate Table 9.S.
would be expected to apply to all factories.
Thus the expected number of absences in Fac- The colulnn totals at the foot of the table are
tory A, for example, would be equal to the the totai numbers of stoppages on each machine
overall dbsentee rate x the number of em- and are identical with the numbers already
ployees in Factory A ~ 0.1626 x 212 ~ 34.5. presented in Table 9.1.
Proceeding in this way for each factory, the
expected frequencies fe shown in Table 9.4 are The extra information concerning the type of
found.
stoppage allows us to ask a different question
The details of the calculation of x~ are -Iso
from tha~ posed in Example 9.1. There we asked
shown in this table, the result being x~ =
whether the total stoppage rate on each ma-
10.469. The degrees 0& freedom are 3, Slnce
only three of the devidtions are independent chine cO'Jld reasonably be the same. The data
because the fourth must be such that their sum
is zero. Comparison of x~ with Table A3 shows of TablE 9.5 allows us to ask whether the
that the significance level c of the
experiment is such that pattern )f stoppages is the same on each ma-

This is ,ather a low probability, indicating ~i.e., whether the proportions of A, B,
that only rarely would deviations from expec-
tat i(,n such as those in Table 9.4 be found if and C types of stoppage could be identical on
the null hypothesis is true. The data there-
each machine. At first sight, the data cer-
fore throw doubt on the truth of the null hy-
pothesis, and we conclLje that there are real tainly seem to suggest that this is not so.
differenCes in absenteeism among the facto-
ries. In fac" the calculations of Table 9.4 Table 9.0 shows the percentage frequency of
suggest that Factory C has an absentee rate
significantly greater than average, while each type of stoppage on each machine and from
Factory 0 has a better-than-average record.
The next stage of the investigation would be this it san be seen, for example, that the
to discover why this should be SJ. proportion of Type A stoppages on machine 1
(51.8t) is ne ..rly twice that on machine 3
(26.3%) and that the proportion of Type B
stoppages on machine 5 is much greater than
that on any other machine. Many other examples
of this kind can be found in the data, and

this prompt~ ~s to ask whether the five ma-
chines are behaving in the same way.

Another way of putting this questi)n is to ask
whether the pattern of type of stoppage is
indepen:ent of machine, and we take this as
ournuTnypothesi s:

Ho: the type of stoppage pattern is
independent of machine number.

In Examples 9.1 and 9.2, the data were clas- Type of Machine No. Row
sified according to a single criterion, stoppi'ge 234 5 totals
'machine number' in the case of Example 9.1
and 'factdry' in Example 9.2. An important A ------------- S 58

14 14 10 12

B 7 3 8 5 12 35

C 6 19 20 9 2 56

Columr I 27 36 38 r 22 149
tota l.~--.l- o~

Factory fa fe fo-fe (fo-fel2/fe
A 41
B 72 Table 9.6 Percentage Frequencies of Stoppages on a
C 27
0 34 34.5 6 .5. 1.225 Set of Fi~~ Machines
73.3 -1.3 0.023
Totals 17~ 17.7 9.3 4.886 ---- l
48.5 -14.5 4.335
Type of i·1achnie No. I

stoppage 4 36.4

----------------

A 51.8 38.9 26.3 46.2

174.0 0.0 10.469 ~~ ~8.3 21.1 19.2 54.5
C 22.2 52.8 52.6 34.6 9.1

IISSLllll';IIhH;j'; hypoI.lH";i, La be VII(', .11'(,111 To ,.0Il1p,lrtOhis \Vith the crilicill villues in
I"hlc 113, vie [1 c('(1to blOl'1the associilted
(l11c\tl,lt0 ('\.\II'("tf'<\ 1I'('(j1l1'JlC1C:. cnn'co;ptl'HlillCj nUIII',eorf degrees of freedom. It \·,1i1 be re-
called that the degrees of freedom are equal
/,0 Lilt' ObSCI"'/cd frequencies in ft1blc 9.J. :hc lo the nUlllbrrof independent. deviat.ions con-

.-IlleLhodnf analysis can t.h,'nbe IIsed to Lo,;t lrihuting to xb. N"ow:-tlie-deviations (fo-fe)

~!il!'Ul('r' the (\h~;r'l'vcd Cl'('qtll'llcil'S d'iffpr' \~iq- sho\Vn in the fourth column of Table 9.7 are
subject to a nUlllberof constr"ints. One )f
nifiCilnlly f"1111t1hc I'XPl'ctl'J·ol1es. three is that :.he sum of the deviations .'0"
r,l(I lIIachine1lI11',"h.i1d. to zero, heciluse the
Consider first the II-type stoppages. If the SUIll:o.f the expected and observed frequencies
for each machine are the same. Thus there are
rate of type A is dentical for all machin~s. onl; t\Vo independent deviations for each ma-
chi ,e-:--5'imilarly,the sum of the deviations
our nest cstimat',; fOl' this I'att'is provided by for each type of stoppage must be zero, since
we ~ade the sums of the expected and observed
(()lllhinill'l .\11 f\ lypc' ',I.Cl!lp,llj('(; 10<)('1.11(',', ir'- frequencies equal. Hence, for each type of
stoppage, there are four independent devia-
rcspectlve of Lh" ma(hine all ,·,hichthev oc- tions. Overall, therefore, there are 2 x 4 = 8
independent del iat ions, and thi s is the number
curred. Thus, overall A-type stoppage ~ate of degrees of freedom. Again, this is an ex-
amp~e of a general rule, which is
to t_L".f1y_n~b~__eo_t:.._~,~_~tp£.,.:?_toppa_g~_5s8
nLml,lerof degrees of freedom in a contingency
total nUlllhrrof sloPPilges 149 tab,e

Now, on machine 1 a total of 27 stoppages was We can now carry out the test of Ho. From
observed; hence, if this machine was behaving Table A3, we find that th significance level
identically to the others; we would expect a of the experiment is
thot, on average, a proportion 58/149 of these
27 stoppages would be of type A. Hence the ex- The chance of getting frequencies like those
pected number of type-A stoppages on machine 1 sho •.n in Table 9.5 is therefore very small if
the null hypothesis is true. We are thus lea
is tclreJect Ho and to conclUde that the pattern
of stoppages was not the same for all ma-
and this corresponds to the observed frequency chines.
of 14 shown in Table 9.5. Expected frequencies
to correspond with all the other observed The analysis does not stop there, of course,
frequencies can be found in the same way. In becnuse, having shown that there are real difo
fact, Equation (9.3) exemplifies a general ferr,nces among the machines, one would \~ant to
rule. which is that know more about them. Table 9.7 is useful for
thL purpo'se, since it shows that about half
expected frequency the total x6 value is contributed by tre en-
tries for machine 5. Further examinati,n sho\Vs
row total x column total that this machine had consi.1erably more type-B
grand total stoppages and fevler type-C stoppages than ex-
pecLed.
Thus, to find the expected frequency for
type-B stopoages on machine 4, \V~ have The contributions to x6 for the other machines

Table 9.7 shows the expected frequencies cal- are smaller chan that of machine 5, and this
culated in this way, together with the calcu- suggests that the latter might be an odd ma-
lation of x~, which proceeds exactly as in chine. To confirm this, an analysis similar to
previous examples. We find that x~ = 27.025. that above, but using onll the results for
!I1~c"ines 1 - 4, ./as carried out. This yielded
Table 9.7 Calculation of x'0 for Example 9.3 xiJ = 10.828 \·lith6 degrees of freedom, .,hich
is "ot significant at the 5/. level and sug-
Class fa fe fn-fe (fO-fe~ gest.s that machines 1 - 4 could well havl' been
behGving similarly, thus confirming that ma-
I,A 14 10.51 3.49 1.159 chi ne, 5 vias the odd one among the group "f
I,B 0.069 fiv? The next step in the investigation would.
1,C 6,34 0.66 1.697 be to try to discover \Vhy, but that is a tech-
n01(gical problem rather than a statisti"al
2,A 6 10.15 -4.15 0.000 one.
2,8 3,524
2.C 14 14.01 -0.01 2.211 Th" method of analysis desciibed above is val-
id ~rovided that the expected frequencies are
3,A 8.46 -5. ~.6 i .551 large enough to justify approximating the
0.097 Poi<son distribution by the norlllaldistribu-
"J,D" 1 ? 13.53 5.47 2.291

3,C 10 14.79 -4.79 0.349
0.202
4,A 8 8.93 -0.93 0.061
a ,B 20 I·~.28 5.72
4,C 0 037
12 10.12 1.88 9.023
5,A 4.754
5,0 5 5.11 -1.11
5,C
9 3.77 -0.77

3 3.56 -0.56
1 ? 5. 17 6.83
2 8,27 -6.27

Totals 1·19 149.00 0.00 27.025=xb

tion. As stated in Section 9.3, a rule often \·,Lih one degree 01' fn,edolli.Comparing this
quoted is that dl I the expected frequencies w'rh the values in Table A3, we find tha' the
should be ~r'~"Le•. UIJn S. /I(J\'ieverf,or con- slgnificance level 0 is greater than O.OS
ting.cncy tables, i L II",;hl:', ~hol'lfL1II"L, if (,X). Hence we conclude that the data (0 not
fel'Ierth"n one ill five of the l:xpecLed value', r'ovide suff'cient evidence to reject 'he
is less than S, a minimu", expected value of 1 h/pothesis that the two lines produce, on
is allowable. average, the same proportion of defectives.

The general contingency table has r rows and c ~.6 Subjective Tests: Ranks
columns and is therefore often referred to as
an r x c table. A specia"1 case is the 2 x 2 Tnere are several interesting properties of
table; in such a table, a~ allowance can be textile materials that are not capable of
made for ,.mall frequencies, knO\m as precLe mea3uremenL Examples are handle,
the contil.uity correction. A convenient comfort, and attractiveness of design. HO\'I-
notatiOi1Tor the general 2 x 2 table is shO\'In ever, in such cases, it is often possible to
in Table 9.8. A sample of size n has been rtrrange individuals in order subjectively and
classified according to whether its then to assign numbers or ranks to the indi-
members satisfy, or do not satisfy, two cri- ,ljduals, 1 to the individuaTCOnsidered best,
teria A and 8. On applying the method de- ~ to the next best, and so on. The resulting
scribed in the last section, it can be shown .,umerical data can then be used in an attempt
that to draw meaningful conclusions about the elu-
sive properties under consideration, and in
2 n(ad-bc)2 ~he next few sections we shall consider sev-
Xo = (a+b)(c+d)(a+c)(b+d) eral methods of analysis that have been found
useful for this purpose.
.with one degree of freedom. However, it is
usually desirable to apply the continuity In many investigations, we are interested in
correction (and essential to do so if any of the degree of ~ssociation between two sub-
the expected frequencies is s~all), which jectively assessed variables, and the rank
modifies the above equation to correlation coefficien 1as been devisecrto
measure the strength of su~h an association.
n( I ad-bc I-O.5n)2
Seven fabrics were first ranked by an expe-
(a+b)(c+d)(a+c)(b+d)
rienced technologist according to handle. They
In a factory, two prodJction lines are pro-
ducing the same article. Random samples from were then made into identical dresses and worn
each line were examined and the numbers of
defectives counted, with the results shown in by a lady, who ranked them according to com-
Table g.g. The object of the experiment was
to discover whether the two lines differed in fort. The ranks are shown in Table 9.10; we
the proportion of defectives they produced,
i.e., whether the pattern of defectives/non- are interested in knowing whether handle and
defec~ives was independent of which line was
used. comfort are associated. r , ••

Category A Row In general, suppose n individuals have been
A not A totals ranked according to two properties X and Y,
and let the two ranks assigned to the ith
ab a+b article be xi and y j. Thus, in Tahle 9.10,:
cd c+d x I = 7, y, = 7; X 2 = 3, y 2 = 2; and so on. The
rank correlation coefficient Rs d,per.ds on the
a+c b+d n differences di = xi-Yo in ranks a;signed to
each of the individuals and is calculated from
the equation

"Category 8 6rdi'
1-----
8
not 8 (n-1 )n(n+1)

Column
totals

Table 9.9 Numbers of Defectives Produced by Handl e a~d Comfort Ranki ngs of~;:.J'
Two Production Lines
D F --G1i!I
I Line 1 Line 2 !Row total s Fabric
177
Defective I 14 Handle rank 2 4 6!
Non-defective 366 Comfort rank 3 \i
I r4
o~

180 380 Rank J~.•...
differences
l

The value of Rs can range from +1, when the _~_a!·II?I_e_~.?_(~.t.il1..lI.-e_d1.

t\;O sets of r'allkinqsore the ''''"C, to -1, \-Ihell IIIthis example, Il = 7, since seven fabrics
"ere ranked, and we found Rs = 0.86. Comparing
the two sets of rankings are as different as this with the critical values in Table 9.11,
possible. i.e" when the article with the we s~e that Rs is certainly significant at the
i,ighestJ.,rank has the lowest Y rank, the arti- 5% l2vel and almost approaches the 1% critical
cle with the second-highest xrank has the value. The null hypothesis of no association
second-lm;est Y ,'ank, and so on It is, of is tnerefore rejected, and "e can be quite
course, rare to find these extreme values in con:ident that the conclusion reached at the
an actual experimellt, However, the closer the end of Section 9.7 Vias justified. ***
exp~rimental value is to ±1, the more closely
are the two variables associated, while a When the numblr of items ranked is greater
value close to zero suggests that there is than 10, an approximate test is to calculate
little associGtion between the two variables,
Rs (~)}
The rank differences di are shown at the foot l-R~
of Table 9.10 (note that the sum of these dif-
ferences is always zero). From this we find and then to compare the value obtained with

the values in Table A4, with (n-2) degrees of

freedom. For example, if n = 16 and Rs = 0.59,

then

tto 0.59 (~1)-0.592 = 2.73

so that, by using Equation (9.7), with 14 degrees of freedom. Comparison with
the values of Table A4 shows that this value
6x8 corresponds to a single-tail area of less than
6x7x8 0.01; this must be doubled because we are car-
rying out a two-tail test, so that the sign'f-
This lS quite close to the extreme value +1, icance level is <0.02. The null hypothesis
"hich suggests that the 'handle' of cloths .woule therefore be rejected.
and the comfort of garments made from them
are indeed associated; the bettpr the handle, In sr;rneinvestigations, the observers may find
the greater is the comfort, at ,east for the it difficult to express a preference between
cloths useD in the experiment. ._** two or more individuals. This situation will
occur either because there really is no dif-
The experiment actually carried out in an in- ference between the individuals or because the
vestigation can al"ays be thought of as one of observer is unable to detect a small differc
a succession of sil.lilarexperiments that could enee. In s~ch case~, the articles are said to
be carried out, and it is a matter of common be ti2d. It is then usual to allot to each
experience that repeat experiments tend to tieOilndividual the mean of the ranks it would
yield slightly different results. Experiments have -eceived if the-rnaividuals were not
involving ranks are no exception; consequent- tied. For example, suppose four objects A, B,
ly, "r must test whether the high value of Rs C, 0 are being ranked. The observer considers
ob~ained in Example 9.5 really does indicate a .A definitely the best and C definitely the
re,il association bet"een handle and comfort, worst but is unable to distinguish between 8
or "hether it could have arisen purely by and D. If these individuals had not been tied,
chance. they would have heen given rank~ 2 and 3.
SinCE they are tied, we allot each of them the
The null hypothesi s '1 n the general case is rank

. The Ho: the:e is no association between the two and the result of the ranking procedure would
,n the variables-;- therefore 'Je

to "hile the alternative hypothesis is
\ fraln
H,: there is an associatior between the tVIO
variables . Examp~e 9.6

Note that the lest is t"o-tailed, since the In an experiment to investigate the effective-
association, if the null hypothesis is re- ness cf a softening agent, samples of a cer-
jected, could be either positive or negative. tain fabric were treated with increasing
amounts of the softener. The treated cloths
When the number of articles being ranked is were then ranked for '~Qftness' by two ob-
between 5 and 10 (the most common case), the serve"s. In order to avoid as far as possible
critical values that Rs must exceed to be any bias in the results, the letters identi-
significant are shov," in Table 9.11. fying the cloths were assigned to the cloth,
at random so t~at the observers had no idea of
n 5 6 7 8 9 10

5% Rs 0.90 0.83 0.75 0.71 0.68 0.64
1% Rs - 0.94 0.89 0.86 0.82 0.78

9. I\ne lys is of Oi skrcte clnd 1~i:lIIki nCj Duta

the amount of softener each cloth had re-

ceived. The rcsul ts of tile runking arc g1Vcn

in Table 9.12. In this experiment "Ie are in- There wtre ten entries.in a dpcign competi-
tion, vlhich '"as to be judged b) -our jUdges.
terested in ~~ether or not the two observers Each jU~ge ranked the entries in order of
prcfcn:,,((~,with thc n;sull.s sh()~m in Table
are in agreement in their assessment of soft- 'J. U. In till, CXJlllplc,'tiC "Iould "IJnt to klloVi

ness, and this is rcflected ill the dcg"ccc. of .*.Vihether or not there Vias a general consensus

ossocio.tion orl)c("l~l:1\ Ull'-ll' ~,(.'L~, l·dilks. of opinion among the judges as a whole as a
preliminary to finding the prize winners.
Hie rank di ffen;'!lces al-(:' sl1GI,;'n dt the foot of
Table 9.12, from ,·,hierV,ie find that [d 2 " When mor~ than two rankings are involved, it
20.00. Since n 0 8, Equation (9.7) gives is nece'sary to adopt a different approach
from th~t of rank correlation. The reason for
1 ---- 6 x 20 this is that, while two rankings can disagree
7x8x9 complet~ly, more than two cannot. For example,
if threE rankings are denoted by X, Y, and Z,
Comparing this with the critical values in and if X and Z disagree completely and Y and Z
Table 9.11, ~Ie see that Rs is significant at disagree completely, then X and Y must agree
the 5% level. W~ therefore conclude that there completely. In other words, there is no SUC1
is some measure of agreement between the two thing as complete disagreement in more than
observers. This suggests that the cloths could tvlO rani.ings.
be given an overall ranking based on the sums
of the ranks given by the two observers. The A measure of the degree' of agreement among r
rank sums in this case are shown below. rankings of the same n objects that overcom'S
this difficulty ·is the coefficient of con-
An examination of these sums reveals that the cordance W, ~Ihic" is deTlned as follo\'ls.Sup-
cloths fall into Llo groups. Cloths C, D, E, pose the sum of the ranks given to the ith
and G have low rank sums, ranging from 4.5 to object i,,"lfj and that the mean of these rank
6, while cloths A, 8, F, and H have high rank sums is ~. In fact, it can be shown that
sums, between 11 <lnd 16. This suggests that
there may be an optimum, or threshold, amount The coefficient of concordance is then given
of softener for the fabric used in the ex-
periment. To use more than thi s amount \'Iould by
be uneconomic, and it'would therefore be of
interest to compare the rank sums with the 125
known levels of softener applied to the cloths r2n{n-1) (n+l)
to see whether this suggestion is valid. *••
The v3lues of W range from 0 eo +1. If all
We hove so tar been concerned with measuring rankings pre the same, i.e., there is complete
the association betVieen two sets of rankings. agreemen~, then W = 1; if the rankings differ
in many experimental stums, hO\vever, more very much among themselves, then W will be
than two rankings may be involved. For exam- close to zero.
ple, in the investigation of Example 9.6,
there is no reason why three or more observers
should not be used - in fact, generally speak-
ing, the more observers the better in such an
investigation. Another example is given below.

Cloth A 8 CDE F G H
7 6
Observer 1 8 6.5 3.5 1 3.5 5 2 5
Observer 2 1 4 2.5 6.5 2.5
I8 0.5 1I
Rank I0 2.5 -3 1 -1.5 '0.5
differences

Table 9. '13 Rankings of Entries in a Design Competition

Entry A BCD F G IIIH

JUdge 1 ~ 92 4 10 8 157
Judge 2 2 10 9
Judge 3 8 76 39 4 345
Judge 4 8 56 36 4
7 98 2 10 7
2 10 5 I

Rank sums Ri 29 6 30 22 12 35 25 8 29 24

Ri-R 7 -16 8 J -10 ~3 3 -14 2

In this case, r - 4 and n In general, the numbers of degrees of freedom
from Equation (9.81 that are n)t integers, and rough interpolaticn in
Table A5 is necessary. In the present example,
The penultimate row of Table 9.13 gives the the nrarest tabular entry is that for k, ~ 8,
rank sums Ri fa" each entr'y, ,·,hIie the last k, ~ ~O, and we find that our value of F
row gives the deviations Ri-~, obtained in greatly exceeds the 1% value for these degrees
this case by subtracting 22 from each rank of frpedom. Hence the significance level ~ is
sum. Note once again th~t th~ sum of these very ,nuch less than 0.01 (1%). The null hy-
deviations must be zero. pothecis of complete independence of the rank-
According to Equation 19.10), S is the sum of ings '5 therefore rejected, and we conclLde
squares of these devidtions, i.e. thatchere' is a real measure of agreement
among the judges. A final, overall, rank'ng Df
12 x 896 the entries, according to the rank sums, would
4'xl0x9xl1 therefore be justified and WOuld be

which sugges~s a certain amount of agreement The line under A and I indicates that they had
among the judges. *** identical rank sums. ***

,9.11 Test of Significance for W 9.12 (,efficient of Concorddnce when Tied
It is, however, pos~ible that a value of W as 'R3ri"KS are Allovled
high as 0.679 could arise purely by chance,
even when the judges cannot really distinguish When the observers are allowed to return ties
between the entries and thus assign ranks at in th~ir rankings, the calculation of W has to
random. (Wr ~uppose for the moment that ties be chonged slightly. The modification consists
ar~ not allowed.) A test of significance is in subtracting rE(t'-t) from the denominator
necessary to settle this point. of Equation (9.9), where t is the number of
A test of the null hypothesis ranks that are tied, and the summation extE'nds
over c,l1 ties. The procedure is illustrate( 'in
Ho. the rankings are independent the next example.
uses the F distribution. A slight correction
to the value of W has to be made, the adjusted Four observers ranked six fabrics in order of
value being 'streckiness' and gave the results shown in
Table 9.14. The mean rank total in this case,
12($-1 ) for which r ~ 4 and n ~ 6, is, from Equation
r'n(n-1 )(0+1 )+24 (g. 8),

F ~ (r-l )\01' and tre deviations from this are shown in the
1-W' last row of the table. Hence

is then found and referred to a table of the Observ~r 1 There is one tie involving two
variance ratio (Table A5) with k, and k, de- ranks. Hence t ~ 2 and t'-t ~
grees of freedom where 2'-2 ~ 6.

Example 9.7 (continued) ~ AB C D EF
Equation (9.11) 9ives
Fabric
w' ~ 12x (896-1) ,
Observer 1 2.5 4 2.5 5 61
42x10x9xll+24
the F value is then 9iven by Equation (9.12) Observer 2 5 3 1 4 6 2
and is
Obsf>rver 3 4.5 2 2 2 4.5 6
F ~ (4-1) X 0.677
1 - 0.677 Obse~ver 4 1 5 2 3':5 3.5 6

Ri 13 14 7.5 14.5 20 15
Ri-R -1 0 -6.5 0.5 6 1

______ ---J)~[ ~_

lliel'" Me LI'IU :,cL:, ut Li"d I'dllk", 1. The numbers of thin places in equal length
of similar yarn spun on four differen~ s
une illvo!viny L\IO rJnh (L ;/) ,pinning machines ,Iere counted, l'lithtile
and one involving three ranks
(t ; 3), giving, respectively, , i) 11 O~'1i li~1 r'l,::,\i 1 L:,.

There is one tie of two ranks 00 these ddta suggest that there were sig-
It; 2), so that ';iflcant differences among the machines?

Adding all the values of t'-t found above, we 2. Red, green, and blue fibres are nominally
obtain ~ixed in the proportions 2:3:5 to form a
certain blend. A sample of 1000 fibres
~hosen at random from a large batch of the
nixture was examined and the number of fi-
~res of each colour counted.

12S
r'n{n-l )(n+l )-rl:{t'-t)

12 x 80.5 966 Did the proportions differ significantly
4' x 6 x 5 x 7 - 4 x 42 3192 from the nominal?

which suggests there is not a great deal of ~\'n experiment ,Ias carried ut to compare
agreement among the observers. 0,

The test of significance proceeds as before; ~he effect of four shrink-resist treatments
thus on wool fibres. A number of fibres from
each treated batch was examined, and the
fibres were classified according to the
amount of damage they had sustained. The
results are shown below.

12{S-1 ) i Degree of Treatment
r'n(n-1) (n+l )-r[{t3-t)+2·' damage BC
I
A 60 71 D
12 x 79,5 954 I 34 20
4' x 6 x 5 x 7 - 4 x 42 + 24 3216 53 69 52
I, None 30 32
Mild I 17 16

I Severe

Fo = (4-1 ) x 0.2',7 1. 27 Do these results suggest that some treat-
ments are significantly more or less likely
1-0.297 to damage fibres than others?

with

k1 6-1-2/4 4.5 4. A company owns three factories, each making

degrees of freedom. Reference to Table A5 the same product on similar machines. Over
shows that the ob ;erved value of F (= 1.2:) "s a period of time, the machines in each fac
,considerably smaller than the corresponding tory were randomly observed, and the number
tabu1af values. Hence the significance level of occasions on which they were runniny or
is greater than 0.05 (5%), and the null stopped (together with the cause of stop-
hypothesis, that the rankings are independent, page) was noted. The data are shown I,elow.
cannot be reject~d. Possible explanations for
this are I State of machi nes at instant of observation
IFactory Running Breakdown Operator Short of
(i) tnere really were no differences Re- ,
among the fabrics and the observers
were therefore assigning ranks at absent ra\~ materi aI load!
random;
AI 1278 29 78 46
(ii) what, perhaps. amounts to the same
thing is that there are differences, ~ J,-_~_:_~~ ~_~ :_~ ~__~ -~
but that they are so-small that the
observers 'are not capable of de- Was the pattern of m$chine behaviour
tecting them; similar at eaco factory?

(iii) the observers differ in their in- 5. An investigation was carried out to compare
terpretation of what constitutes the absentee rates of male and female em-
'streaki ness'. ployees in a certain company. The numbers
of employees \~ho had been absent frOin \~ork
at least once during a period of time were
counted, with the following results.

C,lculate the cr.efficient of concordance
and test whether there was significant
agreement among the judges.

Do these data suggest that 'the absentee 10. Four groups of people, technologists,
rates for males and females were signif- sales directors, retail buyers, and memblrs
icantly different" of the public, were asked to rank six gar-
ments in order of 'saleabili':y'. The rinks
6. ["/0 people "ere dsked to rank eight sweat- 9i;/"n by the four c)roups were as folloYI';,
ers in order of attractiveness. Their rank-
ings are given below. Gi:1 rment AB C 0 EF

Technologists 6 54 13 2
Sales directors 62 1354
Re ai 1 buyers 3 12 4 5 6
ruhlic 3 12 6 4 5

248 11. Five observers ranked nine cloths in order
of 'surface smoothness'.
38
Cl c,th A B C 0 E F G H
Calculate the rark correlation coefficient
and test whether the t,\;/jOudges were in
significant agreement.

7. A training officer ranked a group of new Db:;. 1 6, 6, 1 5 2 3, 8' 3, 9
employees in order of potential by-using a Obs. 2
preliminary test. After six months at work, Obs. 3 8 4, 6 4~ 1 2 7 3 9
the same employees were ranked again by
their department head, The two rankings are Obs. 4 5 6 2 4 22 9 7 8
given belol'l. Obs. 5
4 7 11 5 3 11 B 6 9

4, 8 3 4, 2 1 6, 6, 9

Was there significant agreement among the
judJes?

Employee No. 12 3 4 5 6

Preliminary rank 1 2 3 4 5 6

6 months' rank 1 4 11 3 5 18

Employee No. 78 9 10 11 12

Preliminary rank 7 8 9 10 11 12

6 months' rank 17 ---8--- 2 16 10 12

Emp 1oyee No. 13 14 15 16 17 18

Preliminary rank 13 14 15 16 17 18
6 months' rank 13 15 9 6 7 14

Do these data suggest that the preliminary
test was" good predictor of employee po-
tential?

8.' Seven fabrics had their flexural rigidity
(a measure of fabric bending stiffness)
measured and were then ranked according to
the results of these tests. The fabrics
were also assessed for 'handl e' by an ex-
perienced technologist. The 'rigidity'
rankings and the 'handle' rankings are
given beloYI.

,ion

:e- Do these duta tend to confirm that there is
an as soc iat ion bet"leen flexural rigidity
oadir
and fabric handle?

69

61 9. Four judges ranked five cloths in order of
52 preference, with the following resul'ts.

Cloth 1\ B C 0

,ludge 1 3 5 14 2
Judge 2 2 5 134
Judge 3 34 2 5 1
Judge 4 4 5 132

I\CCl~I)LJIICC SiJIlI1J Ii IllJ

Quality contrnl is not a ·le.,act vity; it is cation the tetter, since, if they do not meet
probably as old as lIl<lnufactureitself. When il specificaticn, there could be complaints from
C 'aftsman makes an individual article, he customers (among llther undesirable conse-
checks as he goes along to make sure that what quences). T~us, if a yarn h~s a specified mean
he is making will be fit for its eventual end- linear density of 36±! tex, a production batch
use. Quality control here is a continuous pro- haVing a mean linear density within this range
cess and is total. When mass-production tech- has good "udlity )1' confo'mance; if the mean
niques a,oe used, however, it is usua,ly not linear dens·ty is outside tolerance, the qual-
possible, either physically or economically, ity of confnrmance is poor.
to check every aspect of every article that is
produced. At best only a representative few Quality control, as we shall understand it, is
articles (i.e., a sample) can be checked, and almost entirely concerned with quality of con-
a decision must then be made, based on the formance, i.e., it is a means of ensuring that
information giv~n by the ~ample, as to whether as many articles as possible meet their spec-
the process does, or does not, need correc- ification.
tion. Because this decision is based on sam-
pling, it is natural that statistical methods 10.2 Sampl'ng Inspection
should be used, and so the science of statis-
tical quality control has been devrlope(f()ver We have seen that quality control is a means
The last 50 orwjiars. of check in.' the qual ity of conformance of
manufactur;d articles. The act of checking
Before going on to consider this topic in more involves i~specting or testing the articles.
detail, it will be as well to discuss first There are places 1n the-rextlTe industry where
what we shall mean by quality. The dictionary traditionaily every article produced is in-
defines quality as 'the-Gegree of excellence spected. This lscalled 100% inspection, and
possessed by an article', and one of the there may be many good reasons why such a high
meani ngs of control is' a means of checking'. level of inspection i~ sometimes necessary. In
Thus quality control literally is 'a means of most circumstances, however, 100% inspection
checking the degree of excellence of an arti- cannot be justified economically, the cost of
cle'. While this is interesting, it does not inspection being greater than the cost of al-
really help very much for we now have to ex- lowing a few defective articles to go on to
plain what we mean by excellence in this con- the next 'cage of manufacture, o~ to the even-
text. tual customer. There are also instances in
.,hich 1007. inspection is not feasible on other
At least two kinds of excellence can be dis- grounds. An example is when the inspection is
tinguished. Perhaps the most famous exall~~leof destructi~e; testing yarn for linear density
'high quality' is the Rolls-Royce car. Indeed, or for tenacity falls into this category. In
the name Rolls-Royce is often used colloquial- most praccical situations, therefore, it is
ly as a synonym for excellence; one might, for most appr:priate to inspect samples of the
example, hear someone say of a superb piece of producti or,.
cloth that it is 'the Rolls-Royce of suit-
ings', meaning that it could hardly be im- Where sho"ld such inspection be carried out?
proved on. Why is this? The Rolls-Royce serves To deal with this question, it is convenient
the same basic purpose as any other car, name- to imagine a production process s consisting
ly, to transport ~eople from one place to of three ~arts, namely, the inpuL, the con-
another. But the Rolls-Royce is made from version s:age, and the output, as shown in
thicker metal, is given more coats of paint, Fig. 10.1.
has more comfortable seats, and so on, than
almost any other make of car. In other words, The Input
its specification is different from that of
mostOThe" cars-:-This kind of quality is some- This is tile raw material of the process, l'ihich
times called Quality of Design. Of course, would idl'ally be inspected to make sure that
generally speaklng, the better the quality of it is wo,th processing. For example, if yar)
design, the more expensive the article will be of inferiJr strength is supplied, there will
to produce, and it '.-1I1i therefore sell at a inevitab:y be too many loom stoppages; yarn of
bigher price. At what level the quality of de- incorrec, linear density will lead to fabric
sign is pitched has nothing to do with sam- of the wrong weight; and so on.
pling or statistics - it is a conscious deci-
sion by the manufacturer to enter the market This is the point at which the input is con·
at a certain level. The manufacturer of high- verted into something useful, e.g., fibre i
quality suitings is entering a different mar- turned i~to yarn, cr yarn i{1to fabric. It i,;
ket from the manufacturer of jeans, and the usually necessary to }nspect the conversion
specification of his cloth will therefore be
different. [1;J---..-1 1~~~~:rs ionJ---- •• output]

The other kind of excellence is called Quality

or-an--of Conformance. Once the specification

artlcle has been decided on, the individual
articles now have to be produced. Obviously,
the more closely they agree with the specifi-

process itsc If to l'1l<;urtchat. it 1 S \',or'k'ing not its mean lin~ar density was correct, a
r~rrectlv. The objective usually lS to check
whether the conversion stage is working within variables-ins~ scheme would be appro-
tolerances and to detect as quickly as pos-
sible any drift away from the SICcification. priate.

This could he inspected to minimize the risk ~hichever kind e '-heme is used, the outcome
of send ing 011 poor -qu"l itv prodliCts to the of i~s appJicati, is a decision either to
accept or to rejl~t the batch. What subs~-
next p~'ocess 0" Lo t..lJe customer'. In l!lurly quently happens when a batch is accepted is,
of course, obvious. But what options are
cases, if the input and the conv~rsion stage available when a batch is rejected? In gen-
h3ve been properly inspected, the level of eral, one of three courses of action can be
inspection at the output stage can be very adop'ed.

10~1. (a) rhe batch may simply be returned to the
producer.
The action to be taken as a result of inspec-
tion is usually different in the three parts (b) rhe batch may be accepted, but at a lower
of the process. For example, the input to a ~rice that would be negotiated by the
process often arrives in batches - a consign- producer and the consumer.
ment of yarn from a spinner to a knitter, or a
delivery of garments to a retailer, for ex- (c) In some circumstances, the rejected batch
ample. In such cases, the objective of the in- would be 100% inspected and any defectives
spection will be to decide whether or not to found replaced by non-defectives. This is
accept the batch as a whole. This is called called rectifying inspection.
acceptance sampling. Slmllar considerations
may apply at the output stage, for example Which of these is adopted in any particular
when a batch of finished goods is being sent case would depend on factors such as the
to a ~ustomer. economic situation, contractual obligations,
whether the consumer has machines standing
As explained earlier, the object of inspection idle waiting for raw material, and so on, and
at the conversion stage is the detection of woula be a matter for individual judgement uy
any need for remedial adjustmen. to the pro- the parties involved.
cess. This is known as process control, the
principal statistical tool belngtli'el:",ntrol We shall begin our detailed discussion of
chart. acce"tance-sampling schemes by considering
sampling by attributes, but many of the ideas
The remainder of this chapter will be devoted developed can be carried over when we consider
to acceptance sampling; process control is the sampling by variables.
subject of Chapter 11.
T~e simplLst type of attributes-sampling in·
10.3 Acceptance Sampling: Attributes and spection scheme is called single sampling. One
Vanables sample of size n is selected at random from
the ~atch, and, if the number of defectives in
We shall suppose that articles are supplied by the sample is not greater than a certain
a producer to a consumer in lots or batches. number c, the batch is accepted; otherwise it
Scme examples are: is rejected. For example, a sample of ~ize
n = ~OO might be chosen and the batch accepted
(i) a needle maker (producer) sends batches if the number of defectives in the sample ~'as
,of needl~s to a knitwear manufacturer four or fewer, i.e. c = 4. Thus, once the
(consumer) ; valll~s of nand c are chosen, the sampling
plan is defined and can be put into operation.
(ii) the kn itwear Inanufacturer (producer) However, these values are best not chosen
supplies lots of garments to a chain haphazardly but should be calculated on a ra-
store (consumer); tion,)l basis.

(iii) a spinning mi 11 (producer) supplies yarn To d~velop this basis, we begin by noting that
to a weaving mill (con,Jmer). the 'quality' of a batch may be measured by
the proportion of defectives it contains. Thus
Each batch will be examined, either by the a batch with only 2% defectives in it is of a
producer before despatch or by the consumer on better quality than one containing 5% defec-
receipt, to determine whether or not the batch tives. In generJl, suppose a batch contains a
is acceptable. Since it is usually uneconomic proportion p of defectives. Since ~e examine
to inspect the batch 100~, a sampling scheme only a sample chosen at random from the batch,
'is adopted, and such schemes can be divided rath~r than inspect the batch 100%, there is a
into two classes depending on the type of in- chance of wrongly sentencing the batch. There
spection to be adopted. are, in fact, two kinds of error that can be
Sometime~ the articles inspected are merely made:
classified as defective or non-defective, in
which case the sa~~ling scheme is said to be (i) a batch with an acceptable proportion of
sampling by attributes. It is probable that defectives could be rejected;
tli'eTt1Spectlon ln exa~lple (i) above vlOuld be or
of this type. the sample needles being sub-
jected to some type of go/no-go teet. (ii) a batch with an unacceptable proportion
of defectives could be accepted.
In the other class of sampling scheme, a "-
property of the articles inspected is actually
measured, and tile scheme is then sampling by Obviously, we should ncii want to make these
variables. !f the yarn in example (111) above mistakes too often, and so we must design the
were to be inspected to determine whether or plan (i.e., choose nand c' so that two con-
ditions are satisfied. Thes" )re:

'---------------------~---- f
'---___________ )~

(a) if pis "",\II, 111,'1',I'', " 11i'111I"',d,,di,III y 1

or ,)",'1'1111'I1II,'1",t.,II;,'lId It.I.IJlII",JI~i".e., I',,(p)h " IUlicLion of p.
11115 luw.tioll i, called the operating char-
acterist\c (DC) of the samplIng plan.

(b) if p is large, tllere is a low probability

of accepting the batch. This argument sug- A 'lraph of the DC can he dr,wII, illidJ Lypical

gests that, in order to proceed with the lie UJl'Vl' ", "III1Wiiln ri~. IO.? Iwo p01JlL, Ue

calculatioll nf II JIi,IC, \'1(' 1I11"tc.,,"~,idl'I' lOlllllllt\.1.Iull DC cUI·ves. If the butch contaIns

the p"ull,dJiIIIy ,II ,JlI""L "I<J " 1I,'LchIl\I' no d2fectives (p ~ 0), it \'Iillalvlays be ac-

vJr'iOll~ V•.tllll'~ ul p. cepted, i.e., if p ~ 0, then Pa~l. More-

The calculatIon of this probability is fairly over, if ~very member of the batch is defec-
straightforward for single sampling. If a
batch contains a proportion p of defectives, tive (p 0 1), the batch vii11 a hlays be re-
then, when a single article is chosen at ran-
dom, the probability that it will be defective jected, i.e., Vii11 never be accepted; thus, ';f
is p, by the definition of probability (I;
4.2). Moreover, if the batch size is large p ~ 1, th2n Pa(l) ~ O. T~ese points are shown
compared with the sample size n, this proba-
bility will be virtually constant for each as A and II in Fig. 10.2.
member of the sample. Thus the probability of
finding exactly r defectives in a sample of We can now make more precise the eJnditions
size n is given by the binominal distribution (a) and (I,) that we Vlant the sampling plan to
(I; 5.4, Equation 5.8), i.e., if x is the satisfy by defining two critical values of p.
number of defectives in the sample
This is the quality level the consumer expects
NO\'Ithe batch "i 11 be accepted if r~c, i.e. , the producer to achieve. In fact, it should be
if x ~ 0 or 1 or 2 or ,.. or c. Therefore the chosen as ~he maximum value of p that Vlould be
probability that ,he batch "ill be accepted considered acceptable as a process average.
is, by using the addition rule of probability, Thus, ideelly, the producer should try to sub-
mit lots of quality better than Pl; however,
c if he consIstently maIntaIns an average qualic
ty, over meny batches, equal to Pl, his per-
n: formance is just acceptable. Again ideally,
the consumlr (and, indeed, the producer) would
~ r~(n-r): like to be certain of accepting such accept-
able batches. But this is not possible with a
r=O sampling scheme. The be~t that can be done is
to ensure that there is a high prorability,
This shows that, once nand c have been se- say 1-0, of accepting a batch of quality Pl.
lected, the probability of accepting a batch This is the precise version of condition (a)
depends only on the proportion of defectives
above. The values of a and P1 are usually
chosen by il<jreement betvleen the producer and
the consumer (a typir.al value for a is 0.05),
and they define another point on the DC curve:

Pa (p) = probabi 1ity of
accepting a batch

of quality p

1 -0 -= -=-1. 0 A ]Producer' s
- -~

I Consum,~r' 5
ri sk ~ B
I
I p ~ proportion of defec-
I tives in a batch
I
I
I

-1---

I

I

I

This point is shown as P on Fig. 10.2. Note \,ith 2(c+1) degrees of fr('('dollIln. fact., it.

that, sinCo' 1-, i",1111' prnh"hi Iity Dr (','" hI' ~hil\'II! Ih,lI 1"\,1,111111\', (IO,?) (,III ",

,\((1'1\l111q.1 Il'llll! (II (PHIl! qll.llllv 111.1111' '1\ li I {I~, i 111.11 i'r! (.11 lly

pl'll!1,1111111 Y III j"l',ll'{ I 11l'l '.111.11 ,1 lldlrli i" q.

lIencl'" ie,("il1lI'd Llh:~~Jduc~~~s Risk or the

sampl ing plan.

It l\ltl~t 1H' lli',llly lllldc'l"'.lc)(){! \'J!I,lt Iii;', (!I·r

Itlll 11111 (11 I11111 !I",. It I', 111.11. 11 IIII' pIll

dlnt'l (111l',\',t,')d!y ·,tllll1l\l~; Ih\Lche~, cutlLlini!ltj

iIproportion 01 of defectives, then a propor-

tion I-a of such batches will be accepted, and fren which,on dividing Equation (10.4' by
Equation (10.3), we obtain
a p-oporlion ~ will be rejected.
Thi; is an equation involving conly, "'hich
The Lot-tolerance Proportion Defective (LTPD): can be solved by using Table A3 when tl,e
values of 0, G, PI, and P2 are given. he
~'::.P..1l. pro"edure is b~s 1.exp 1ained by means of an
exaill1pe.
This is the quality level that is definitely
unacceptable to the consumer. Ideally, we Design a sampling plan for which the AQL is
would w,sh--ro be certain of rejecting a batch 2%, the LTPD is 5%, and the producer's and
of this poor quality, but again this is im- consumer's risks are both 5%. Translating
possible to achieve with a sampling plan. these conditions into the above notation, we
There will be a smilll chance G of acceptirg have
such a bad batch, and this is called the
Consumer's Risk. Once more, the values of P2
and G (wh,ch are the equivalent of condition
(b))can be selected by agreement between the
producer and the COilsumer (a typical value of
is 0.1), and once they have been chosen a
fourth point on the DC curve is fixed:

We have seen that the producer is expected to Xz c., h 0.02
keep his quality level p below the AQL of PI, z(
and if he does so'the vast majority of his o. 95
batches will be accepted. However, as p
increases beyond PI towards p2, the form of To solve this, we look down the columns headed
the OC curve shows, that the batches are more 0.05 and 0.95 in a x'table until we find a
and more 1ike ly to be rejected. The DC curve pair of entries whose ratio is about 2.5.
therefore tells us how the sampling scheme Doing this, we find that, for 26 degrees of
,,;11 dea~ with batches of varying quality. freecom,

Furthermore, it can be shown that, when the
four points A, P, C, B of the OC curve are
known (and, once the va Iues of PI, P2, 0 , and
B ar~ chosen, they are known), this is suffi-

cient to determine tne values of nand c. In

fact, nand c are the solutions of the equa-
tions

which, by using Equation (10.1), can be x2 38.89
wri tten 7.6,0.05 15.38
Z

X
26,0 *9 5

c

~ n: p,r(l_p,)n-r 1-0 , which is very close to what is required. Note
(10.2) that, since 2(c+1) is an even number, only
r=O r: (n-r): even values of the degrees of freedom k can be
8 considered. Equating 2(c+l) and the degrees of
c freedom found by the above procedure we obtain

L n: p,r(l_pz)n-r

r=O r: (n-r):

These rather formidable looking equations To find the value of n, we ret"urn to Equations
cannot often be solved exactly, since nand c
must necessarily be integers. Consequencly, a (10.3' and (10.4). The first of these gives
number of approximate methods of solution have
been devised, of \;I.ichperhaps the most 15.38 0 2n x 0.02
convenient is based o~ the x'distribution

)\lPi'G~;e >Ie novi cllJIIUC tll" 1 11'0 I ,'Gill ~'1, tu b',;
and keep the producer's risk at 10~, We now
hove

These two values of n would be equal if we had 0,06
been able to find a pair of x' values whose 0,02
ratio was exactly 2,5, However, they are quite
close, and it is sufficient to take their mean 23.69
and choose: 7.79

The required sampling plan therefore requires

us to take a sample of size 387 and to reject

the batch if there are more than 12 defectives

in the sample. ***

The choice of risks, AQL and LTPO in Example The mean value of n is found to be 196, on
10.1, has resulted in a very large sample using Equations (10.3) and (10.4).
size, and this would probably be very expen-
sive to operate in practice. We would there- Increasing the LTPD has resulted in a signif-
fore be led to investigate possible ways of icant reduction in the sample size required,
reducing the amount of inspection required. and this much more economical scheme might
There are basically two ways of doing this, well be adopted. Essentially what is being
namely, by increasing the risks we are pre- attempted in choosing a sampling plan is to
pared to take of wrongly sentencing batches, balance the cost of sampling against the cost
or by changing the AQL and the LTPO. The of maki':g wrong decisions. This is very diffi-
following examples illustrate the effect of cult to do, and the final choice of sampling
plan is often made subjectively by someone
such changes. with ar. intimate knowledge of the processes
involved and of current operating condition~.
Suppose we change the producer's risk from 5%
to 10%. We then have

10.6 The Military Standard 105D Sampling
--scr;eme

0.05 One aid sometimes found useful in making de-
0.02 cisions concerning rival sampling plans is a
compari,on of their operating characteristics.
Looking down the colum~s headed 0.90 and 0.05 The OC curves for our examples are shown in .-
in the x' table, we find that, for 20 degrees Fig. 10.3. They illustrate the fact that, in
of freed)m, general, changing the producer's risk alone
has little effect on the probability of ac-
x2 ceptin~ poor lots. Altering the L,PO, on the
20) 0 .05 other hand, generally has only a minor effect
on the chance of accepting good batches.

31.41 In nor~al circumstances, it is the good
12.44
batche~ that are most important, since on the

whole "hey will determine the eventual quality

reaching the consumer. If the producer's pro-

cess average is good, the vast majority of

batches wi 11 be accepted by the plan, and the

average qua11ty reaching the consumer will be

high. If the process average is poor, most of

the batches will be rejected and ~jll not be

receivra by the CGnsumer. In an ideal situa-

tion, we would want to accept all 'good'

Also, the mean value of n calculated from batches and reject a11 'bad' ones. It vii11 be
Equations (10.3) and (10.4) is r. = 313.
recall~d that the A~Pl) was defined in
Doubling the producer's risk has therefore
resulted in a reduction of the sample size Sectio~ 10.4 as the maximum value of p that is
required, though the reduction is not very
great. We therefore investigate the effect of consid~~ed acceptabl~process average.
increasing the tTPD.
Thus, ideally, if pcpl we would want to be

certair. of accepting the batch, whereas if

P>Pl w, would want to be certain of rejecting

it. The ideal OC curve would then be as shown

in Fig. 10.4. This idealized situation is not Fi g. 1 J. 4
realizable in practice (except by usin~ per- Pol (p)
fectly efficient 100~ inspcction). and prac-
tical OC CIlr'Vt'S c,I', llt' «'<j"rd,'d d'; ,lpprox ill'''- t
tions to it. !lowcver', the discllssion high-
lights the fundamental importance of the ,~QL
in designing sampling plar;.

This fact has been made the basis of perhaps
the most famolls and comprehensive collection
of sampling plans, nJmely, the Military Stan-
dard 1050 sampling-inspection scheme, which is
the frJit of a study by a joint American-Brit-
ish-Canadian working party. Th~s set of plans
is indexed according to AQL, with values rang-
ing from {lQL - 0.1'::' (P1 0 0.001) to {lQL - 10'10
'(Pl 0 0.101. ln addition to this, the plans
are also determined tj a feature called the
Inspection Level. In our calculations of sam-
pling plan~ in Section 10.4, no mention (,as
~ade of batch size, except that it was assumed
to be large in comparison with the sample
size. Thus a very large batch and a relatively

\
\
\

\

\

\

\ n = 196, c = 6

V
\

\

\

\

\

"'"
"" .•....•.
--.•••....•..
I -...,
0.06 0.07 0.08

rl --------------------------,.i .,

-,----------_._.- ------- ---------

10. QUality (untr'ol: l\ccept<1r1ce SJlllplillg

small one would be inspected Lo Llw same (:x- dcfecLives reaching him"lould be small. SUCh
tent. It can be argued that this is vlrong, an objective can be uchleved by ~
since the losses involved in incorrectly sen- ~.ection.
tencing a 'Iery large batch vlOU].. be greater
than those entailed in wrongly sentencing a The simplest rectifying-inspection plans
smaller one. TI,is difficulty is overCt ,e to utilize a single sample of size n and an ac-
some extent in Hi 1. Std. 1050 by defining ceptance number c, ,as ln the previous schemes
three levels of inipection, which effectively HO\vev~r, an essent 1 a1 feat lire nO\'1is the re-
determine the relationship between batch size placement of any defectives found by non-
and sample size. This relaticnship replaces defectives. The procedure is therefore as
the LTPO-Consumer's-risk Doint on the DC curve fol ]()\·IS.
as a condition to 'le met ,n calculating the
sampling plans. (1) Inspect a sample of size n.

There are three inspection levels. Level II is (2) If the number of defectives found is .$c,
the 'normal' level, used in most applications. replace the defectives by non-defectives
Level I involves smaller sample sizes and and accept the batch.
wculd be used, for example, when the items
bEing inspected are inexpensive, or when the (3) If the number of defectives found is >c
inspection is destructive. Level III gives inspect the batch 100%, and replace de-'
greater sample sizes than'normal and would be fcctives by non-defectives.
used for complex, expensive articles when the
consequences of recLiving poor quality coule It will be seen that rectifying inspection
be serious. play< an active role in determining the
qual'ty level that eventually reaches the
For a specified AQL and inspection level, Mil. custo,ner.
Std. 1050 tabulates pairs of values (n, c)
that are to be used so long as the producer Ave::.:,gOeutgoi ng Qual ity
submits lots of AQL quality. Another refine-
ment of the scheme is that it suggests It i' possible, in filCt, to calculate \vhat the
'tightened' inspection when Lhere is evidence mean value of this quality level will be, and
that the prooucer's process average has de- this is called the average outgoing quality
teriorated. There is also a provision for (AOQ). Su~pose the artlcles are submlEted In
'reduced' inspection when there is strong batches of size N and a sing1e~sampling plan
evidence that the producer's process average (n, cl is used. ~et Pa(P) be the probability
is particularly good. The standard further of accepting a batch of quality p as a result
provides the OC curves for its single-sampling of inspecting only the'sample. (The value of
plans. Pa(p) is given by the DC of the scheme.) Then
l-Pa(p) is the probability that the batch will
In addition to single~sampling plans of the be 10)% inspected.
type we have been considering, Mil. Std. 105D
gives details of double-sampling plans (in Now if a batch is 100% inspected and all de-
vlhich two samples .nay be taken in certain fect ill'S found are rep1 aced by non-defecti ves,
circumstances) and of multiple-sampling plans then, provided that the inspection itself has
(in which more than two samples might be in- been perfect, the final rectified batch will
volved). However, such plans are complicated contain no defectives. On the other hand, if a
to operate and to administer, and for most batch is accepted on the basis of the sample
textile appllcations single sampli~g should resul t, the remaining (N-n) items in the batch
suffice. will not be inspected and will still con:ain a
proportion p of defectives. The actual number
2-0.7 Rectifying Inspection of defectives remaining in the batch wi~
therefore be (N-n)p. We can 'hus draw up the
The sampling-inspection plans we have so far follO\ving table, which is, ill effect, a prob-
discussed deal with individual lots or batches abilicy distribution.
and are desijned primarily to achleve two ob-
jectives: x = numb('r of PrIx)
defE:ctives in
(a) to give protection against the possibility outgoing batch 1-Pa(p)
of an individual bad batch passing through Pal pI
to the next stage of manufacture, or to 0
the customer; and
(N-n)p
(b) to encourage the producer to consistently
submit good lots by rejecting most batches It wl1l be recalled (I; 5.2, Equation (5.1))
if the rrocess average goes beyond the that the mean of a probability distribution is
AQL. giver by

Such plans do nothing to affect the quality [ xiPr(x=xi)
level of the batches; they merely provide a i
rule for deciding whether or not to accept a
batch. Using this result on th~,above distribution,
we f'nd that
There are situations, however, in which the
sampling can be made to playa less passive
role. For example, suppose a manufacturer
regularly supplies batches of the same product
to an important customer. The customer might
not be too concerned if an odd poor batch
slipped through the inspection net, provided
he could be assurec that, in the long run,
over many batches, the average proportlon of

After applying the inspection plan, the out- that the batch should be 100% inspect,'d. ~Jhen
going batch still contains il tot"l of N items, this happens, all defectives are rf'l<dced, "nd
so that the "veril~e proportion of defectives CI~"in I.lIeoutgo'in~ (1I,,,lityis 'loud. Ihus,
',ince 1.11(o'ut(loi,uJ 'IU"lity is good ror bol.l
left ill I ill' h,lll'h (I.',; ': ·llI." AOI)) i '. gllod ~nd poor incoming qUillities, one "ou1d
expect there to be a maximum value for the
1I1<·.,n."ulll,b0"f"I'de f~(,c,t2.v.r,';_J!!~0,.t:.0! 1100. This maximum is called the "verage out-
tutal numbel' of i temJ in batch going ~uillity limit (1I00L), ilnd the custom('r
car he ilssured tll"t. in the long run, tile
(N-n )I,P;I (,,) ;"'eril~" level of defecl.'ives in the batches
reilch;ng him will never be greater than this.
N In most applications, it Vlill be less, since
it "ould be very unusual for a producer con-
This equation shows that the AOO depends on sistently to supply lots of quality equal to
the incomi ng qua 1ity p. Thus, if the producer that at Vlhich the 1I00L occurs. It is important
consistently s'llmils lots of quality p, the to rea'ize that individual batches passing on
OUf(jOlr1g-li-.JCcll"CiS11 have a proport ion of de- to the consumer may have a proportion of de-
fectives given by equation (10.7). fectives greater than the AOOL, but these are
balanced by the many batches that eventually
The form of the relatiooship betweeo ADO and p contain a 10Vier proportion of defectives.
is illustrated in Fig. 10.5, Vlhere the ADO
curve for the plan 10.8 Tile Dodge-Romig Tables

has been dra"n. It is typical of "hat is gen- The de~igo of sampling plans for rectifying
erally found, and an essential feature is that inspectioo, i.e., the calculation of nand c,
AOO reaches a maximum value. In this example, can be a lengthy process. However, much of the
the maximum AOO is about 0.0248 (2.48%) and is labour can be avoided by using a set of plans
achieved when p = 0.034 (3.4%). The reason Vlhy originally devised by H.F.Dodge and H.G.Romig*
this maximum occurs is easy to understand. If during the 1930s. They provide tViO sets of
the incoming quality is good (i.e., the value plans.
of p is 10")' the vast majority of such
batches "ill be accepted on the sample alone, (I) Here the plans are indexed according to
and the outgoing quality "ill therefore be specified values of the AOOL and are de-
good. On the other hand, if the incoming qual- signed to givi the long-term protection
ity is poor (i.e., the value of p is high), discussed in the last section.
the sample Vlill almost inevitably ,indicftte

AOOL ~ 0.0248 (2.48%)

n = 150, c = 6, N = 5000.

II I

0.08 0.09 0.11

L

--+--

10. Quality Control: Acceptance Sampling

(II) These plans are classified according to

specified values of the LTPD and are in-

tCl1ded to ~ivc pl'otcetlOn against ilHli-

vidu.i1 h,,,1 ",I.". --.. -

III (II). tI,C COII:,UIII".".·1'1" clJl'Te,pul1ldl1V tu II I.ileP"oCCS savel iJgc is P Pa, thcn lite
the LTPD is fixed al 10~. The producer's risk mean number of iLews inspected in a batch of
is not considered in eIther set of plans, and average quality is
the AQL-prod~cer's-risk condition is replaced
by a requi remcnt that the ilverage total amount ane thi, is the quantity minimized by Dodge
of il1sp"ctilll1pCt' '",tcilsholiid be Illinimized. and Romig in producing their plans.

This ilverilgetotill inspection is easily found. [Xi_'lIple10.4
Suppose a batch of quality p has been submit-
ted. If the biltch is accepted on the sample Find the appropriate rectifying sampling plan
result illone, only n items vlill be inspected. fa' batches of size 5000 if the process aver-'
and tile prGJability tilat this occurs is age is 1.9% and the AOQL required is 2.5%.
Palp). On the otiler hand, if the batch is re-
jected, it Vii11 be inspected 100%, i.e., all TaLle 10,1 is an extract from the Dodge-Romig
table. This refers to the required AOQL and
N items in the biltch ('Ii 11 be inspected, and the neces~ary sampling plan is obtained by
the probability of this occurrence is fi~ding the column appropriate to the stated
1-Pa{p). The probability distribution of x = process average (i .e., the column head·,d
number of items inspected per batch is there- 1.51-2.00 in this case) and the row ap"ropri-
fore as shown below.

~umber of items PrIx)

inspected per batch Pa(p)
1-Pa(p)
n

N

ate to the batch size (that labelled 4001-5000 Batches ~f YPI~ are supplied to a weaving
for this example). At the intersection of this mill. 1" c)r'"'' II.i,II.LIi('100ms will opcr,lLe
rOI'1 and column. \oIPfind saLisfacLor:, . it is desirable Lhat the
minimum stre, ',th of the yarn supplied should
(Pt is the value of p. as a percentage, such De great·'r tlian 300 gf. He require to devise a
that batches of quality Pl have J qO~ chancp mcans of assuring thc wcavcr that this con-
of being rejpcted by thp pl.ln). diLion c': 5,ltisfipd.

Ihc ,lve,·a'.!L'u<L.1i IW'I"'LlIOII lor tI,i.,pl.ln r.1I1 Tlie fir:,L point to be el1lpliilsizeids thilt the
be found f(om [qualHJn (IO.ti) ,·,henPa(PiJ) is requirement concerns the minin, 'yarn strength
known. It can be shown that Pa(Pa) • 0.975 50 since, in general, this determines the fre-
that quency of loom stoppages. Any assurance scheme
adopted would naturally be based on an assess-
ii 150+(500l,-150)( 1-0.975) = 271. ment of yarn strength, say, by using a stan-
dard single-thread-strength test, which mea-
It is instructive to see ho\olthis amount of sures the breaking strength of a comparatively
effort (and cost) is reduced if \oleare willing short length of yarn. Such tests are destruc-
to relax the AOQL a little. If an AOQL of 3% tive, S( that we are inevitably led to con-
is chosen, rather than 2.5%, the Dodge-Romig sider the possibility of devising an accep-
tables give tance-sampling plan. Such a plan has two com-
ponents. namely, a sample size n and a rule
and it can be s~o\olnthat, for this plan, for declrling whether or not to accept the
Pa(Pa) ~ 0.99. Therefore the average total batch.
inspectlon lS now
At first sight, it might be suggested that a
Thus a considerable reduction in inspection suitable decision rule is to require that all
costs (about 36%) has been brought about by a n tests in the sample should give results
slight reduction in the AOQL required. Balanc- greater than 300 gf. However, such a rule is
ing costs of inspection against the savings unreali,tic for at least two reasons:
achieved by the plan is one of the arts of
choosing a sampling plan, and no completely (a) it ignores the inevitable experimental
satisfactory objective method for doing so has error in a strength test, which may be
been developed. As before, a subjective consid~rable; and
judgement by one completely familiar with the
circumstances is needed. (b) to be fair, the spinner must be allowed a
reasonable tolerance, since otherwise hi"
10.9 Acceptance Sampling by Variables: prices may well increase considerably as
Assurance about a Mlnlmum (or Maximum) he strives to meet very s~ringent require-
Value ment,;.

One of the features of sampling inspection by To meet the second of these objections, sup-
3ttributes i~ ·that it usually involves rela- pose the consumer (i.e., the weaver) agrees
tively large samples, as we have seen in the that a batch of yarn \,i11 be acceptable if not
above examples. The reason for this is that more than 1% of its length has a strength le~s
each item inspected is merely classified as than 300 gf. Thus we have defined an accept-
'defective' or 'non-defective', and no in- able quality level, and it is importa~
formation is given about how defective an item underst~nd what It implies. It is that, if all
might be. In other words, an item that just the yarn in the batch were subjected to stan-
fails a test is treated in exactly the same dard strength tests, then 1% or less of such
way as one that fails the test disastrously. tests would give a result less than 300 gf.
This situation ~~n be remedied if the property
of the items being considered can be measured Because we shall be using a sampling plan,
against a continuous scale. Acceptanc~---- however, there will be a producer's risk
sampling schemes based on such measurements associated with this AQL, i.e., there is a
are said to be using acceptance sampling by chance :hat a batch of AQL will be rejected.
variables. Suppose this is fixed at 10%.

There are two main applications for this kind On the other hand, the weaver will want to
of acceptance scheme. They are: have a righ probability of rejecting a batch
with an excessive proportion of low-strength
(i) plans for giving assur3nce about a yarn in it, since such a batch would lead to
minimum (or maximum) requirement; too manj loom stoppages. To avoid this, we can
therefo:e define a lot-tolerance percent
(ii) plans for giving assurance about the mean defective of, say, 5%; 1.e., lf a batch has 5%
value of a batch. of~'rength weaker than 300 gf, the consumer
would W2nt to reject it. The chance of wrongly
In this section, we shall De concerned with accepti1g such a poor batch is the consumer's
the first of these. rlsk of the plan, which we fix at, say, 5%.

The ab0,e discussion has set the scene for a

general ization of the problem. Let x denote

the variable being measured (the result of a

standard yarn-strength test in the above ex-

ample), and let x have a normal distribution

with un~nown mean v. He s~up~9se that the

standard deviation 0 is known from past ex-

perience of testing similar lots. He wish to

define a sampling plan, i.e., to calculate a

sample 5ize n and a decision rule, that is

subject to the conditions imposed by

__I_.._~[

,

Acceptance Sampllng

(i) an AQL (Pl) Vlith its asso- ( 10.9) Note that, as ~ approaches the specified
ciated PR(a), ( 10. 10) minimum L, the proportion of unsatisfactory
test, increases, as one Vlould expect. Sub--
(ii) an l.TrD (PZ) I"i th its as so- trocling one of the last lVio equations fron
ci ,'leu CH(ll), Jild lhe lither Lo eliminJtc t, "Ie finel

We begin by noting a relation between the mean We r.ave so far said little about the decision
" of a batch and the proportion of values in rul~ that will be adopted once the n tesls
the b,1tch lying bclO\; the minimum L. The sit- making up the sample have been performed It
uation is shol'lIiln'Fig. 10.6. Since x has a is r.atural to base this on the mean x of the
rJrmal d'stribution, the standard normal vcri- sample. ,ie therefore define iJ 'critical' value
ate corresponding to a tail area p can be iL, Vlhich has to be determined and which is
found from Table A2. Let this value be up' suc;; that if X~Xt "e shall accept the bat.ch;
Then, as in (I; 5.6), otherViise Vie shall reject it.

~ To calculate the values of n and XL, Vie must
con!ider the AQL and LTPO conditions, Condi-
the minus sign being introduced because Vie are tions (10.9) and (10.10). Fig. 10.8(a) sho"s
dealing Vlith " left-hand tail area. This equa- the AQL condition, i.e., if a batch of AQL
tion can be rearranged to give (!' = "1) is submitted, there is to be a
probability a of rejecting it. Thus 'Ie require

Now, Vie have defined two special values of p, We "herefore consider the distribution of
namely, the AQL (p = p~) and the LTPD sample averages (the sampling distribution of
(p = P2). Let the means corresponding to these
be ~1 and "2, as shol'lIiln Fig. 10.7. Then, for xthe .nean (I; 6.5), \1hich wi 11 be normal, with
the batch of AQL (Fig. 10.7(a)), Equation
(10.11) gives a standard deviation 0 =0/1 n. If u" is the
standard normal variate corresponding to a
and, for the batch of UPO (Fig. 10.7(b)), Vie tai' area ", the diagram leads to
obtain
-ua XL-~l
i.e. , o/In

~l-xL = uaa/ In ( 10.13)

If a poor batch of LTPO is submitted, as shown

in Fig. 10.8(b), the LTPO condition, Condition
(10.10), requires there to be a chance B of

accepting such a batch. Hence, from the
'iJTa'S r am ,

n=o , (._u-a+u8)' The aJove account has assumed that x is re-
quired to be greater than a lower li•'.;, L. A
\JJI-IJ7. similar scheme can be devised if. required
to be less than an upper limit U.
To find ;\ we divide Equation (10.13) by
Equation (10.14) to ohtain 10.10 Acceptance Sampling by Variables:
Assurance about the Mean Value
Hence the required values of n and xL can be
calculated. We now consider a rather different problem. uf
\·,hichthe following is a typical example.

A spinner supplies yarn of nominal linear

density equal to 45 tex. His cortract with a

knitwear manufacturer states that a delively

of thp yarn will be acceptable if its mean

lineiH density lie~; within the range 45!1-:-5

tex. Ubviously the knitter cannot test all the

yarn to check this, and a sampling-inspection

plan "s therefore to be devised to assure the.

knitter that a delivery of yarn is acceptable.

The C01sumer's and producer's risks are to be

5}; a"d 10%. respectively. XH

This is an example of a general problem, which
can be stated as follows. A producer supplies
batches of articlos whose mean critical
'dimension' is nominally equal to D. (This
'dimension' can be any measurable property,
such as a length, a weight, a linear density,
etc.). The dimension can vary among individual
items \Iith a standard devi ation 0', and the
producer and consumer have agreed an allowable
tolerance ±T, i.e., if ~ is the actual mean
dimension of a batch, the batch WTTTlDe ac-
ceptab~e to the consumer if

In addition,we shall assume that past ex- If a sampling plan is to be used to provide
perience gives if = 15 gf. assurance that Condition (10. '7) is satisfied
by individual batches, certair, :isks will, as
The appropriatl ~ values can be found from usual, be associated with it. It is found con-
Table A2 and are venient to define the risks in the following
~Iay.
n = (,.28'6 + 1.6449) 2 = 18.44
(1) Producer's risk (o) =
2.3263 - 1.6449 ' the probability of rejecting
a I·erfect delivery, i.e.,
which would be rounded up to n = 19, since n one for which ~ = D.
must be an integer. Furthermore, Equation
(10.16) yields (2) Consumer's risk (8) =
the probability of accepting
a just-defective delivery, i.e.,
on~ for which ~ = D±T.

Let the required sample size be n and the
sample ~ean be R. It is natural to base the
acceptance rule of the sampling plan on i ~nd
a convenient form is:

300+ 15x (2.3263X 1.6449+ 1.6449 x 1.2816)
1.2816 + 1.6449

The required sampling therefore calls for 19 We therefore have to calculate the sample size
strength tests to be carried out and their nand tre sample tolerance t in such a way
mean value i found. If i~330.4, the batch of
yarn is accepted; if ,i<330.4, it is rejected. that the producer's- and consumer'\~risk con-
ditions are satisfied.

i Fig. 10.9(a) shows the sampling distribuLion Eouations (10.21) and (10.22) can be solved
I s.diultaneously for nand t, to give
of x from a batch I-Ihosemean" is exact ly
i' a~d we shall suppose that the standard devi-
cqual to the nomillal valuc O. This distribu- ation of standard count tests within a deliv-
I tion tCllds Lo be nO"'1IilIl-IlthsLandard devi- ery is a = 1.2 tex. This estimate has been
cbtained from ~ast experience of testing sim-
j ation "x = a/In, as shm-In. Such a perfect ilar deliveries of yarn. The u values corre-
sponding to <>/2 and B/2 are, from Table A2,
i batch \'I1i1 be erroneous ly rejected if the mean
of the sample aCLually mcasured falls outside F.quation (10.23) gives
1' I' thc rangc D+t. I\ccor'dillgto the PR condition, n = 1.22(1.6449+ 1.9600)2/' .52
\ the p,'obal)iIity of thi s Ilclppening is to be u, ~hich should be rounded up to n 9, since n
Ii; Si'nce the disLribution is sy'"l1leLricalilbout LlUst be an integer. Moreover, Equat',on (10.24)
iLs mca,l, the LoLal probilbil ty" is split gives
j Ii betlveen the two tai Is of the distribution, as
! i; shol·ln.Hence, if u<>/2' is the stan(~rd normal 1 .5 x 1.6449
variate corresponding to a tail area of <>/2, 1.6449+ " .9600
I Ii we have (consi(2ring the right-hand tail): The appropriate plan is therefore to do nine
II stanuard count tests and accept the delivery
I; (O+t)-O if their mean lies in the range 45±0.68 tex,
i.e., betwe:n 44.32 tex and 45.68 tex. Other-
a/In wise the delivery is rejected.

1, F'g. 10,9(b) shows the sampling distributions
of i from deliveries whosp means are just
"1!~: equa 1 to the all ol-/edtolerances O:!:T.Such poor
deliveries will be wron91y accepted if the
II, observed sample mean falls inside the range
O:!:t.According to the CR condltlon (10.19),
I I' the probability that this will occur is re·
I! quire1 to be e. Again, this total probability
r is split into two equal parts, one part for a
I. delivery on the upper tolerance and the other
for a delivery on the lower tolerance, as
!I, shown in the diagram. Thus, from the right-
hand tail of the distrib~tion centred on O-T,
\ve fi nd

(O-t)-(O-T)
u s/ 2 =

a/In

where uB/Z is the standard normal variate
corresponding to a tail area B/2.

II

I

1

i

I

I x=a/'Ifl S/2
~
J /

I

i
I

],

~ i
0- T O-t 0

/In a"Cl'pt,1IlCe-Silililnp9l plcln fo,' ottJ"ibutes where g ond p ~re the producer's and
is to be <cL up wiLh on dcccplohle qualiLy consultler's risks and pI and p2 are the AQL
lev,,1 of;' ,I lid " 111L-tolel',1IlcpCeru'nL and the LTPD. respectively. Show also that
de'ectivp of b", Ir the pr<.>ddcer's ond the balch should be accepted if i<iU,
consultlPr's risks ,1r,"hoth 5"" cdlcuL1tc tile
'.altlplsei?!.'and t:Je allol'lablenultlbCIo'f de- \;here
fectives,
_ Up 1 un + up2 U(l~ •
If the occ"ptable quality level is chonged xU = U 0 ( ------
to 1.S~, hO'd ;(, lrh' \tlmplill(J plan changed'~
Ua + Us
2. Articles are delivered in batches of size
10 000, and the process average is usually 'arn is received from a spinner and is to
2.2% defectives, A rectifyin9 inspection have a CV, as measured on an irregularity
scheme is being installed with an average ,ester, not greater than 20%. A sampling-
outgoing-quality limit of 2.5%. Use the inspection scheme in which n bobbins are
Dodge-Romig table in the text to find n tested for CV is being devised. If the AQL
and c. is 1.5%, the LTPD is 4%, and the produc-
er's and consumer's risks are to be 5%,
If the process average changed to 1.8%, calculate n and xU.
what would the plan become?

If the batch size were then increased to
25 000, how would the sampling plan be
affected?

3. /I certain fabric has to meet a specifica-
tion that its tensile strength, as measured
by an agreed test ing procedure. shou 1d not
be less than 50N. Previous E :perience of
such te~ts has shown that the coefficient
of variation of the fabric-strengtr tests
is 10%.

The specification also states that an
acceptable quality level is 1% and that the
lot-tolerance percent defective on each
batch of fabric \.Ji 11 b, 4% ..If the pro-
ducer's risk is 10% and the consumer's risk
is 5%, calculate an appropriate sampling
pI an.

If the producer's-risk is changed to 5%,
whut is the effect on the plan?

4. Deliveries of a certain type of garment are
made to a retailer, and a batch is accept-
ab 1e if the mean, lass of the garments in It
lies in the range 450-460 g. Past experi-
ence hcs shown trat the standard deviation
of the masses of garments within a delivery
is 9 g.

An acceptance sa~pling scheme is being set
up viith producer's and co; sumer' s ri sks
equal to 57. Calcul1te the sample size and
the accepta1ce tolerances.

If the producer'S risk is changed to 10%
and the consumer's risk to 6%, calculate
the new sampling plan.

5. Show that, if it is required that a mea-
sured property shoulj be less than an upper
limit U, then an appropriate sampling plan
is to take a sample of size

n (Uc< + uB ~ 2

Upl - Up2)

~[-------

They dll so by noting that, if only random

sourcec of variation are operating and if

Acceptance sampling, considered in the last observAtions of the process are made sequen-
chapter, is concerned "lith the sentencing
(i .e., acceptance or rejection) of batches of tially in time, these observations will exhi-
finished or semi-finished articles. The ulti-
mate decision regar~ing each batch submitted bit no systematic variations such as cycles
for inspection depends on the quality level of or trends, or runs above or below a mean '
the batch, e.g., the proportion of defectives
it contains, or whether the mean value of a level. In other words, the data will behave
particular property lies within specific tol-
erances, and so on. This qual ity level in turn random~y. However, such data, though random in
depends on the efficiency of the manufacturing
process that produced the articles in the ~\ITll, in fact, tend to conform tQ known
batch; consequently, much of the effort of a
I'lell-Ol-ganizedqual ity-control department I'lill statistical patterns. For example, the numbers
be devoted to checking the processEs in the
factory, i.e., to process control, of defectives found in random samples taken

Essentially, process control is concerned with from a stable production line would tend to
ensuring as for as possible that a manufactur-
ing process maintains a specific stable level, have a binomial distribution; the means of
and the function can be split into three main
areas. These are samples would tend to have a normal distribu-

tion; and so on. Thus, once some knowledge

about the process behaviour in stable condi-

tions is available, the extent of the variat-

ion expected from random caue - alone can be
calcul1ted and allO\'ledfor. If the process is

then inspected regularly, the variation it

exhibits at these inspections can be compared

with t~e allowable random variation. If the

observed variation conforms to the expected

random variation, the process is said to be in

contrr •. If, hO\;ever, the observed variation-

(i) the detection of significant changes in O'O'eSf:5tso conform, the process is said to be
qual ity level;
out oTl:ontrol; it would then be concluded
(ii) investigating the causes of any changes
detected; and ~~one assignable cause was operat-

(iii) removing the causes by whatever means ing, "nd efforts would be made to discover
are necessary.
what i~ was and hence to remove it.

Of these, (ii) and (i'ii) are essentially tech- ~~e General Principle of Control Charts

nological operations; only someone with an At any instant of time, a process is either
intimate knowledge of the process, such as a 'in ccntrol' or it is 'out of control'. I~ a
foreman or departmental manager, can deal with "Iell-r,rganized factory, it should normally be
in contra I, and what is requi ..ed' is a means
them. However, detecting when a process has for detecting when there has been a signifi-
departed from a normal, staDTe, level is an- cant jeparture from the usual state of af-
fairs. This can be provided by a test of the
other matter. Since it is usually physically null hypothesis
and,economically impossible to check every
artlcle produced by a modern production pro- against the alternative hYP1thesis

cess, any decision about the current state of ~1: the process is out of control,
the process must necessarily be based on sam-
pling. Thus the,detection of changes in qual- and t1e general principles described in
lty level is a statistical problem, and one of (I; 8.2) can be employed to develop such a
the most important statistical tools that has test.
been developed for its solution is the control
chart, invented by,W.A. Shewhart in the-,g~ Suppose that a sample of size n has been se-
lecte~ from the process and that a paramet~r 0
The basis of all control charts is the obser- has been measured or calculated. For examp'e,0
might be:
vation that, ln any production process, some
variation is unavoidable. Shewhart divides the the nu~ber of defectives in a sample of
sources of variation into two groups, namely, garments;
random variation and variation due to
.'!.sslgnabl(causes. the number of defects (e.g., thick and
thin places, neps) in 1000 m of a
a) ?and~~iat~ is variation in quality cotton yarn;

produced by a multitude of causes, each one

of them slight and intermittent in action.

There is very 'little one can do about this

kind of variation, except drastically (and
expensively!) to modify the process.

b) Assignable variation, on the other hand, (iv) the standard deviation of such masses,
etc.
consists of the relatively large variations
over \vhich ';e have sc,me control. Examples If tl~2 null hypothesis is correct, the pattern
are differences among machines and/or oper- of variation of 0 will, as explained in the
atives, variations'in quality of raw mate- last section, be calculable. Suppose this dis-
rials, and so on. The effect of such causes tribution is as shovin in Fig. 11.1. If the
tends to be permanent, or at least long- process is in control, we should expect most

term, and it is this kind of variation that
control charts are designed to dEtect.

observed values of 0 to cluster around the by turn ing the di agram of Fi g. 11.1 anti-
mean of the distl"ibution, 1'0. Hm'lever, should
an observe~ (,v,1lue f,l11 in one of the I.ai1" clcckwi5e through gO"and extending the control
or the di';lrih,l!ion, h'f' ':hould (fnl h.Mill~ the limits 10 the ri~ht, as shmvn in Fig. 11.2.
"rgulllcntgivpn in (I; S.ill be inclined I.uhave Th" operdt ion pr'oduces a She\~lla,·tcontro~
doubts Jboul. the truth cf the lIul1 hypothesis chad. (In pl'acticc. the distribution otO is
and consider adopting the alternative, thus 'usli',1'lolyilitted, 50 that the chart looks muc'
co.lCluding that the process has gone out of lile an ordinary graph.)
control.
The practical application of this idea depends The results of regular inspect ion are plotted
on ho'''t'll('.tai Is' of the distl'ibution are de- on chis chart. So long as the plotted points
fined. The 1Il0stconvenient way in the present lil \Vithin the control lililits,the pro,:ess is
context is to choose a significance level d-., ass.lmed to be in control. A point fall ing out-
and to define the corresponding tails as hav- sid" ei ther control limit (like that 0:' sampl:
ing areas equal to 0/2, as sho\'Jn in Fig. 11.1. No.6 in Fig. 11.2) is an indication that the
This procedure defines two control limits, a proc~ss has gone out of control and that an
10l',erlimit LL. and an upper l,mlt DC, ana once inv2stigation to find the assignable cause re'
these have been found the following rules can sponsible is indicated.
be adopted.
A cnnventional choice for 0 in the above is
The distribution OC 8 when the process a = 0.002. If the distribution of e shuwn in
is in control Fig. 11.1 is normal, the position of the con-
It is envisaged that these rules will be used tra', 1imits iseasTly found. The standard nor-
on a routine basis as the process is inspected mal variate corresponding to a tail area 0/2 =
at regul ar interval s. It is, therefore, con- 0.001 is, from Table A2, uO.001 = 3.09. In this
venient to have a means of recording the re- case, therefore, the cont.-ol limits would be
sults of the inspections, and this is provided
set at

These limits are often referred to as ac:ion

limits. Another pair of limits, called-WJ~ng
T1mIfS, is often defined; these are set~

tha:~= 0.05, and are calculated as

~6 ± 1.96°6 (11.2)

When both sets of limits are used, the control

chart takes on the appearance shown in F·g.
11.~. Some quality controllers, particularly
in the USA, modify these limits slightly and
set them at

+ :ime
10 11 (sample no,)

11.1 Random and ~ssisnable Val-ialion They dl' so by noting that, if only random

sourcec of variation are operating and if

Acceptance sampling, considered in the last observntions of the process are made sequen-

chapter, is concerned with the sentencing tially in time, these observat10ns will exhi-
(i.e., acceptance or rejection) of batches of
finished or semi-finished articles. The ulti- bit no systematic variations such as cycles,

mate decision regarc'ing each batch submitted or trends, or runs above or below a mean
for inspection depends on the quality level of
the batch, e.g., the proportion of defectives level. In other words, the data will behave

it contains, or whether the mean value of a randomly. However, such data, though random in
particular property lies within specific tol-
erances, and so on. This qual ity level in turn ~illll, in fact, tend to conform to known
depends on the efficiency of the manufacturing
process that produced the articles in the statistical patterns. For example, the numbers
batch; consequently, much of the effort of a
of defectives found in random samples taken
well-organized qual ity-control department viiI1
be devoted to checking the processEs in the from a stable production line would tend to

factory, i.e., to process control. have a binomial distribution; the means of

samples would tend to have a normal distribu-

tion; and so on. Thus, once some kn0\1ledge

about the process behaviour in stable condi-

tions is available, the extent of the variat-

ion expected from random cau' - alone can be

calcu11ted and allo\'ledfor. It the process is

Essentially, process control is concerned \1ith then inspected regulal-ly, the variation it

ensuring as fur as possible that a manufactur- exhibits at these inspections can be compared
ing process maintains a specific stable level,
and the function can be split into three main with t1e allowable random variation. If the
areas. These are
observed variation conforms to the expected

random variation, the process is said to be in

contrr,. If, h0\1ever, the observed variation-

(i) the detection of significant changes in ~.ot so conform, the process is said to be
quality level;
out or-;:ontrol; it would then be concluded
(ii) investigating the causes of any changes
detected; and ~~one assignable cause was operat-

(iii) removing the causes by whatever means ing, ~nd efforts would be made to discover
are necessary.
what i~ was and hence to remove it.

Of these, (ii) and (i'ji) are essentially tech- ~~e General Principle of Control Charts

nological operations; only someone with an At any instant of time, a process is either
'in control' or it is 'out of control'. In a
intimate knowledge of the process, such as a well-Grganized factory, it should normally be
foreman or departmental manager, can deal with in control, and what is requi,'ed'is a means
for detecting when there has been a signifi-
them. However, detecting when a process has cant departure from the usual state of af-
departed from a norma l, staDTe, 1eve 1 is an- fairs. This can be provided by a test of the
other mattel-. Since it is usually physically nu 11 hypothes is
and economically impossible to check every
art1cle produced by a modern production pro- against the alternative hyp1thesis
~,: the process is out of control,
cess, any decision about the current state of
the process must necessarily be based on sam- and t1e general principles described in
pling. Thus the,detection of changes in qual- (I; 8.2) can be employed to develop such a
1ty level is a statistical problem, and one of te st.
the most important statistical tools that has
been developed for its solution is the control Suppose that a sample of size n has been se-
chart, invented by.W.A. Shewhart in the~~ lecte~ from the process and that a parameter 0
has been measured or calculated. For examp'e,0
The basis of all control charts is'the obser- might be:

vation that, in any production process, some the nUI1:berof defectives in a sampl,? of
variation is unavoidable. Shewhart divides the garments;
sources of variation into two groups, namely,
random variation and variation due to the number of defects (e.g., thick and
~s Sl gnab Ie causes. thin places, neps) in 1000 m of a
co tton yarn;
a) ~and~~iat~ is variation in quality
(iii) the mean mass-of 100 m lengths of yarn;
produced by a multitude of causes, each one
(iv) the standard deviation of such masses,
of them slight and intermittent in action. etc.

There is very 'little one can do about this If He null hypothesis is correct, the pattern
of variation of 0 will; as explained in the
kind of variation, except drastically (and last section, be calculable. Suppose this dis-
expensively') to modify the process. tribution is as shovm in Fig. 11.1. If the
process is in control, we should expect most
b) Assignable variation, on the other hand,

consists of the relatively large variations
over which \1e have sc,me control. Examples
are differences among machines and/or oper-
atives, variations in quality of raw mate-
rials, and so on. The effect of such causes
tends to be permanent, or at least long-

term, and it is this kind of variation that
control charts are designed to detect.

11.1 Random and ~ssisnable Val-ialion They dl' so by noting that, if only random

sourcec of variation are operating and if

Acceptance sampling, considered in the last observntions of the process are made sequen-
chapter, is concerned vlith the sentencing
(i.e., acceptance or rejection) of batches of tially in time, these observat10ns will exhl-
finished or semi-finished articles. The ulti-
mate decision regarc'ing each batch submitted bit no systematic variations such as cycles,
for inspection depends on the quality level of
the batch, e.g., the proportion of defectives or trends, or runs above or below a mean
it contains, or whether the mean value of a
particular property lies within specific tol- level. In other words, the data will behave
erances, and so on. This qual ity level in turn
depends on the efficiency of the manufacturing randomly. However, such data, though random in
process that produced the articles in the
batch; consequently, much of the effort of a ~iTlll, in fact, tend to conform to known
well-organized qual ity-control department viiI1
be devoted to checking the proceSSES in the statistical patterns. For example, the numbers
factory, i.e., to process control.
of defectives found in random samples taken
Essentially, process control is concerned \;ith
ensuring as fur as possible that a manufactur- from a stable production line would tend to
ing process maintains a specific stable level,
and the function can be split into three main have a binomial distribution; the means of
areas. These are
samples would tend to have a normal distribu-

tion; and so on. Thus, once some knO\;ledge

about lhe process behaviour in stable condi-

tions is available, the extent of the variat-

ion expected from random cau' - alone can be

calcu11ted and allovled for. It the process is

then inspected regula,-ly, the variation it

exhibits at these inspections can be compared

with t1e allowable random variation. If the

observed variation conforms to the expected

random variation, the process is said to be in

contre,. If, hO\;ever, the observed variation-

(i) the detection of significant changes in ~.ot so conform, the process is said to be
qual ity level;
out or-control; it would then be concluded
(ii) investigating the causes of any changes
detected; and that "~one assignable cause was operat-

(iii) removing the causes by whatever means ing, ~nd efforts would be made to discover
are necessary.
what i~ was and hence to remove it.

Of these, (ii) and (i'ii) are essentially tech- ~~e General Principle of Control Charts

nological operations; only someone with an At any instant of time, a process is either
intimate knowledge of the process, such as a 'in control' or it is 'out of control'. j~ a
foreman or departmental manager, can deal with well-Grganized factory, it should normally be
in control, and what is requi.'ed'is a means
them. However, detecting when a process has for detecting when there has been a signifi-
departed from a normal, staDTe, level is an- cant departure from the usual state of af-
other mattel-. Since it is usually physically fairs. This can be provided by a test of the
and economically impossible to check every nu 11 hypothes is
art1cle produced by a modern production pro-
cess, any decision about the current state of against the alternative hyp1thesis
the process must necessarily be based on sam-
pling. Thus the,detection of changes in qual- Y,: the process is out of control,
1ty level is a statistical problem, and one of
the most important statistical tools that has and t~e general principles described in
been developed for its solution is the control (I; 8.2) can be employed to develop such a
chart, invented by.W.A. Shewhart in the-,g~ te st.

The basis of all control charts is the obser- Suppose that a sample of size n has been se-
lecte~ from the process and that a parameter B
vation that, 1n any production process, some has been measured or calculated. For examp'e,B
variation is unavoidable. Shewhart divides the might be:
sources of Variation into two groups, namely,
random variation and variation due to the nUI1:berof defectives in a sample of
~s Slgnab Ie causes. garments;

a) ?and~~iat~ is variation in quality the number of defects (e.g., thick and
thin places, neps) in 1000 m of a
produced by a multitude of causes, each one co tton yarn;

of them slight and intermittent in action. (iii) the mean mass-of 100 m lengths of yarn;

There is very 'little one can do about this (iv) the standard deviation of such masses,
etc.
kind of variation, except drastically (and
expensively') to modify the process. If tl'.enull hypothesis is correct, the pattern
of variation of B will; as explained in the
b) Assignable variation, on the other hand, last section, be calculable. Suppose this dis-
tribution is as shovm in Fig. 11.1. If the
consists of the relatively large variations process is in control, we should expect most
over which \;e have sc·me control. Examples
are differences among machines and/or oper-

atives, variations in quality of raw mate-
rials, and so on. The effect of such causes
tends to be permanent, or at least long-

term, and it is this kind of variation that
control charts are designed to detect.

observed values of 0 to cluster around the by turning the diagram of Fig. 11.1 anti-
mean of the di st,'ibution. P". However, should
tin ()bsel'vc~ {'\ Vt111lC f'Jl1 in one of the tai ls clcckwi se through gOoand extending the control
limit.s 1:0 the righl:. as shO\vn in r··;g.11.2.
of the rli'i!,'ih,,1 ;n", "'c' ·:I'Ol,l(dfollo\"iIl0 lhe Thl, Opel'iltion produces a She\vhar-tcontro~
chart.. (In practice. the distribut'ion 010 is
argulI,~nLgiven in (I; G.ill be inclined Lu have 'us\iiillyomitted, so that. the chart looks muc:
doubts .1boul the trull, cf the null hypothesis lile an ordinary graph.)
and consider adopting the alternative, thus
cO,l(luding that the process has gone out of The results of regular inspect ion are plotted
contra 1. on shis chart. So long as the plotted points
The ~ractical application of this idea depends lie \"ithin the control lirnits, the pro';ess is
on hO'!11.11(' 'tai Is' of the distribution are de- ass·lmed to be in control. A point fall ing out-
fined. 'he rnost convenient way in the present sid, either control limit (like that oc sampl=
context is to choose a significance level do-, No.6 in Fig. 11.21 is an indication that the
and to define the corresponding tails as hav- proeass has gone out of control and that an
ing areas equal to 0/2, as sho\'lnin Fig. 11.1. inv2stigation to find the assignable cause re-
This procedure defines 1:\-/0 control limits, a sponsible is indicated.
lo\ver limit Ll. and an upper l,m,t 0[, ana once
these have been found the follO\'lingrules can A conventional choice for 0 in the above is
be adopted. o = 0.002. If the distribution of e shown in
Fig. 11.1 is normal, the position of the con-
The distribution o£ e \'Ihenthe process
is in control tro', 1imits iseasTly found. The standard nor-
It is envisaged that these rules will be used mal variate corresponding to a tail area 0/2 =
on a routine basis as the process is inspected 0.001 is, from Table A2, uO.001 = 3.09. In this
at regular intervals. It is, therefore, con- case, therefore, the cont~ol limits would be
venient to have a means of recording the re-
sults of the inspections, and this is provided set at

These limits are often referred to as ac:ion

limits. Another pair of limits, called-WJrnTng
TlffiTfS, is often defined; these are set--';o-----

tha:~ = 0.05, and are calculated as

~e!1.96oe (11.2)

When both sets of limits are used, the control

chart takes on the appearance shown in Fig.
11.~. Some quality controllers, particularly
in the USA, modify these limits slightly and
set them at

-+~ :ime
10 11 (sample no.)

j~l

-------,----,---,-----:---,----cc-c;:-:---------- --

11. Quality CantraJ: Central Charts

As explained in Section 11.2, the basic indi- outside the action limits and thus indi-
cation that a process has gone out of control cates a lack of control.
is given when a ,;ample point plots outside the
action limits. Experience in using control (d) Anuther indicaticn of possible trouble is
charts, however, leads to the evolution of a trend upwurds (or downwards) of the kind
other indications of 1ack of contre I, and some i l'ustrated in Fig. 11.4(d). When this
of these are illustrated in Fig. 11.4. ocrurs, it is prudent to check the process
for assignable causes before a point even-
(a) Fig. 11.4(a) illustrates the basic rule, tUJlly falls outside any-or-the limit
that a single point outside an action li- li.les (e) and (fl. Any non-random pattern,
mit is strong evidence that the process is such as those shown in Figures 11.4(e) and
out of contra I. (fi may indicate that the process is not
subject only to random sources of varia-
(b) A simi 1ar Iack of contra lis demonstrated ti0n, und it should be investigated.
if two consecutive points fall between the
same action and vlarning limits, as in Fiy. These examples indicate that a control chart,
11.4(b). The reason for this is that, if intelligently used, is a far more useful and
the process is in control, the probability versatile tool than its original design might
that a point will plot between a warning suggest.
ann an action limit is about 0.0214. Thus
the probability that two successive points Having discussed' control charts in general, we
wiil fall between the same limit lines is now go on to consider in more detail the indi-
(0.0214)2 or about 0.0005, if the samples vidual types of chart that are in common use.
are independent. This is a very small pro-
bability, and we should therefore conclude
that th@ process is out of clntrol To
·take account of this line of reasoning,
the following rule is often adopted

(i) If a poin_ falls between action and
warning limits, inspect another
sample immediately.

(iil If the second sa,"ple falls outside
the same warning limit, take actlon.

(i ii) If the second sample falls inside
the warning limit, assume t-he--
process is in control.

(c) A sequence of points sometimes occurs in
which all the points lie between the cen-
tral line and one· of the warning limits,
as in Fig. 11.4(r:). Such a sequence is
called a run. It can be shOl'm that, in
'probahiliVterrls, a run containing nine
points is equivalent to a single point

L-------~

In many situations, articles chosen from a
production line for routine inspection are
simply classified as 'defective' or 'non-
defective'. An example occurs in a making-up
room, when a garment may be counted defective
for ~any reasons - se~ms badly sewn, labels
stitched on upside down, buttons not aligned
with button holes, and so on. In such cases,
very careful definition of what counts as a
defective is necessary, and this is often one
of the most difficult tasks facing the quality
controller. Assuming, however, that this is
done, a measure of the quality of the current
prodlction.is the number of defectives in the
sample, i.e., in this case:

In order to apply the general theory of con-
trol charts, we need to know the distribution
of ewhen the process is in control, i.e.,
when it is subject only to random fluctua-
tions In this state, it will still produce a
certain number of defectives; suppose the pro-
portion of defectives produced under these-----.
condi':Tons is Pc. Then, when samples of size n
are cllosen at random, the numbers of defec-
tives they contain will vary from sample to
sampls according to a binomial distribution
(I; 5.4) with mean

If the sample size is large enough, the
Qinomial distribution can be approximated by
'the normal distribution (I; 5.8). The expres-
sions (11.1) and (11.2) can then be used to
calculate the control limit,; substituting for
~e and 00 they give

11. Quality Control: Control Charts

In order La cJlculJLe Lhe conL,'ul limlLs by TaDle 11.: Numbers of Defectives in Samples
of Size 180 taken at Random from
a Making-uD Room

using these equJLiolls, it is 1l1'ce"J,'y La hJve

an estimate of Pc, the proportion of defec- ----,

tives produced when the process is in control. INumber of Sample INumber Of,

If no pre"ious information is available, it Sample defectives ! number defectIVes
numbe,-
IIlusLbe obLJi'lcd by n1llllin~J p"eli",inJry ex- -~-----I--l-lJ i

1,-\\'JIl it'll (011'. i ',I' e 1/ I
Ill'I'illlt'IIl.. ',lllll'! Y ill '.f' Llll<1 tl
10 18
l'e,J~()"dIJ Ie IlUlIllwl·. /U',dy I 1'1 JIll I (I Ill. {J! rllil 12 19
12 20
dam samples ove,- J per'lod or L illl'"-li'd CUUIlLi"U 29 21
the number of 1efectlves In each. 25 22
13 23
Examp~~
9 24
Table 11.1 shows the data collected in such an 20 25
experiment, carried out in the making-up room 36 26
of a knitwear manufacturer; the sample size 24 27
used was n = 180. 1g 28
13 29
At the foot of the table, the calculation of
the control limits is shown, and this should 5 30
be self-explanatory. The corresponding control
chart ie, shown in Fig. 11.5, "here the data 11
have been plotted on the chart. It will be
seen that several points fall outside the ac- 0e {1Pc{I-Pc)}"i" {180x 0.D783x 0.921712
tion limits (sample numbers 6,11. and 16). 3.60
These should be investigated to discover
whether an assignable cause can be found to 14.09 ± 3.09 x 3.60
account for them. Of course, this is not al- 14.09±11.12 = (2.97, 25.21)
ways easy to do in retrospect, but the proce-
dure is helped if full records are kept of the
conditions pertaining at the times when the
~amples were taken (e.g., new batches of raw
material, new operatives starting work, new
• methods of working being introduced). If sa-
tisfactory explanations can be found, the
poi,nts outside the acLon limits indicate a
real lack of control at the times at which
these samples were taken. Consequently, they
should be deleted and the control limits re-
calculated.

Omitting the results for samples 6, 11 and 16
we find the following.

14.09 ± 1.96 x 3.60 21.15).
14.09± 7.06 = (l.03,

- - - - __ - - - - - - - - - - - - - - - '- - - -UAL

•0 ----- UAL
-,-~ - UWL
-- -- -- -- -- -- -- -- - -- -- -- --UWL
LP.L
•0 Q ____ ~-- time

o• "Q 0 , {sample no.1

• & l:L.- __ --L It
-_ --
-------LWL

• Ql 0 Ql

---------_-----------w

-+--1--+-+ ""I I'" I I' I
35
5 10 15 20 25 30

tht, action ilnd\Vilrninglimits ar'-' '~.09 and
!: 1.95, re spec tively. [f pos sib Ip . ..·E.''rv, it
is obviously much more convenir"' keep the
sarrple si.ze constant. A discus, of the size
of samples required \ViII be deferred until all
types of She\Vhart control charts have been

dea It \'ith.

{nPc{l-pc))2 1

[180x 0.0733x 0.9267)2 = 3.50 Anocher type of chart can be developed when
the parameter 0 is the number of fau1 ts or de-
13.19 ± 3.09 x 3.50 fects that occur in a unit of product. This
13.19 ± 10.82 might be, for example, the number of imper-
(2.37, 24.01) fections (neps, thick and thin places) in 1000
m of a yarn, or the number of defects (holes,
13.19± 1.95 x 3.50 streaky lines, etc.) in 50 m lengths of a
WOVL1 cloth, or the number of faults in a do-
13.19! 5.85 zen cut-and-sevi garments, and so on. When the
faults or defects occur at random (i.e., when
(6.33, 20.05) the process is in contron----wesnould expect
the number of them per unit production to vary
These limits have been marked on Fi9. 11.5~ acco~ding to a Poisson distribution (I; 5.5)
There is a point (sample No 7) lying outside h av ing mean
the revised action limits, but its deletion
would make little difference to the action and 1
warning limits. Thus the control chart is now
ready to be introduced to record the results and standard deviation 0e = ~C2 ,
of routi ne inspection. In the e rly s':ages of
its operat'on, however, the limits should be wher~ ~c is the mean number of defects per
revised again in the light of tht further in- unit under controlled conditions. If ~c is
formation acquired, any results outside the large enough, the normal distribution can be
action limits being deleted from the calcula- used as an approximation to the Poisson dis-
ti on. *** tribution (I; 5.9), and then Equations (11.1)
and (11.2) are used to calculate control 1i-
In all;he above, it has been assumed that the mits, which are
sample size remains constant and that the
parameter used to measure quality is the 1
number of defectives in a sample. In general,
this seems to be the. most convenient type of action limits = Vc ± 3.09~C2 ,
chart ':0 use, but some authorities recommend
the use of charts based on the proportion or 1
fra:tion of defectives in the sample. The ac-
tTOn and warning limits are then given (for warnirg limits =v c ± 1.96vc' .
constant sample size) by
Sa'1lple Thin Number of Neps
If, for some reason, the sample size has to places
vary from sample to samole, thE control limits _nur.lber :hick 55
must be recalculated fo. each sample. This .. 20 places 42
means that the chart cannot be prepared be- 20 36
forehand, and the appropriate limits must ad- 1 25 74 45
ditionally be marked on the chart when each 2 24 86 41
sample result is plotted. An alternative pro- 3 22 87 43
cedure is to plot the quantity 4 30 83 43
5 22 64 43
where Pi is the propcrtion of defectives in S 14 75 29
the ith sample and ni is its sample size. The 7 22 63 39
central line on the chart is then at zero, and 8 19 78 35
g 25 92 41
25 75 32
10 18 gO 40
11 24 77 46
12 25 87 28
13 20 94 54
1<,- 13 gO 45
1:' 23 88 47
18 77 30
1\; 13 106
84 814
17 422 73
18
19 1643
20

Totals

I Tab Ie 1"..2 Numbers of Imperfections in 1000 m Means (estimates of ~c) 21.1 82.2'" 40.7
Lengths of Yarn Action 1imits (6.9,35.5) (5· ?, 110.2) (21.0,60.4)
\'arning limits (12.1,30.1) (64.4,100.0) (28.2,53.2)

L-

If no prior inf'Jrmiltion about the value of Pc .:cmly chosen 1000 m lengths of a yarn. 1\ s
is available, a preliminary experiment mu<;t
again be run to provide ~"I estimate of it. I dte chart fa,' edch of these properties co~~a.

~x amp1e 1 I. 2 he run, or a ~;ingle chart could be kept for d

Table 11.2 shows the results of an experiment tne total number of faults. He shall acce t
in which the numbers of thin places, thick
places, and neps \'Iere counted in twenty ran- the "flrst of these alternatives. As an ex~ .

of the calculation of the control limit .. mp'2

sider the thin places. It is found that"'~ncon.

estimate of "c is 21.1. Using th's value in Us = Us
[Ql;;tions (11.5) and (11.6), \'Ie get
and
°0 = of/r,

Further, because the distribution of 0 tends

to be '2.'!..r:',al, control limits for the sample

averagcs Ire givcn by Equillions (11.1) and

(11.2). Ti,cy ilre

Cilrlrol limits for the other imperfections are To calcul ate these limits, the value of a must
calculated similarly and are given at the foot be known. ;f no prior information about the
of • ab I e 11. 2 . process is available, a preliminary experiment
must be run, which consists in selecting bet·
The three control chads are shovm in Fig. \'Ieen tVlenty and thirty small samples (usually
11.5, \,ith the data of Table 1'1.2 plotted on of size 3,4, or 5) at regular intervals.
the:;). There are no indications of any lack of
cor:".rol during the experimental period; hence Example 11.3
the control charts c"n be used to record rou·
tin" inspection results. As before, the limits The masses of samples of four garment blanks
wou'd be revised in the light of experience are shown in Table 11.3. These were taken from
gained. *** a process ~aking blanks whose nominal mass was
"s = 102 g. The mean and range of each sample
~ Control Charts for Averages and Ranges are shown in the table. ***

~ee the property being controlled is measured The sampll! ranges measure the variation exhi-
J1a'cst a continuous scale, for example, .the
linc,lr density of a yarn, the length of a gar- bited by the individuals within each sample.
En·. or the ma,~ per unit area of a cloth,
rather lhan being counted, the sample resul ts Since the members of each-sampre were produced
obtcined during routl'""iie1nspection can be used
~ calculate two parameters. namely,

within min'Jtes of each other, the variation

measured by the ranges is essentially short-

term variability, of the kind that would be

Iii ~the 'range of the sample. caused primarily by random sources. This is

the type of vari ati on that must be allowed for

The first of these is a measure of the general in computing the control limits, since it is

level of the process at the time the sa'11ple unavoidable,.

was taken. The second measures the variation Now, for ncrmally distributed variables, there
is a relatlon between the standard deviation ~
of individual items about this level. A pro-
Rand the mean range in samples of size n, of
~ss of this kind can go out of control either
the form
~couse it departs from a specified level

le.9.~ the linear density of the yarn being

spur. 1S no longer equal to the nominal volue)

~ because the variation between individuals

hH become greater than it usually is. In

extreme cases, both of these undesirable

',ents can occur simultaneously. In the rou- where an de;Jends on the sample size. (Values
of an are given in Table 11.4)
~ control of such processes, therefore, it

• usual to keep a chart for both the average

and the range of the samples.

11.7,1 Contrrl Charts for Averages

~~ose that articles are being produced to a

Specified dimension "s. Thus, for excm~le, the
hnear" density of a yarn may nominally be "S =

~ tex, a particu'ar style of garment should where An = 3.09 anlln. Similarly, Equation

bye length "s = 52 cm, or the mass per squarn (11.8) leads to

~mZIIetre of a certain cloth should be"s = 300
When the process is subject only to ran-

domvariation, suppose the individual items

prOduced vary with standard deviation o. Then,

~~ the process is producing articles of the

Ipecified level and is in contra', the aver-

;ges x of samples of size n \·/Ould tend to have

; nOrmal distribution with mean "S and stand-

Ir~ deviation a/In (I; 6.5). Hence if we set where Wn = '1.96 an/In. Values of· An and ~Jh are
given in Table 11.4.

~~I ,-----------

11. Quality Control: Co

Tubl e 11.3 -~'_lil.SSC-So-f -G-il-l'-li-I-cfl~JlllJn,s (

I - -~- ., J - - - ------ -- -----_.
SillllC~ n~:I~.~.~ 1
lJ 1\ ~ 10 11 1r".

- ----

103 108 103 111 9~ 104 107 105 106 99 99 100

I 105 102 100 100 99 i04 103 97 101 10" 99 94

I i05 102 103 108 102 100 102 105 99 99 103 95
99 J 0 1 100 103 i 01 104 109 102 9IT • 98 102 98

~ ~_. ------------

Sample average x 103 103,25 101. 5 105.5 98.5 103 105.25 102.25 9Q,75 100.75 96.75

Sample range R 6 7 3 i 1 10 4 7 8 46

~

Sample number 16 17 13 19 20 21 22 23 24 25

104 101 105 112 101 98 105 104 106 114
107 99 100 99 107 102
109 105 104 113 107 96 102 99 101 100
100 103 106 103 102 103
i05 99 102 102
103.25 102.75 107.75 i02.5 104 104.75
~ 106 96 102 100 64
-- 7 6 13 8
~.

Sample average x- 106 97.25 102.75 101.25
335
Sampl e range R 5

l:R = 161 ; R = l:R/25 = 161/25 = 6.44 102~ 0.750x 6.44 102±4.8 (97.2, 106.8)
102 ± 0.476 x 6.44 102d.l (98.9, 105.1)
Control 1 imits for aVt;rages chart: acti on 1imi ts
warn ing limits (0.10x 6.44, 2.57 x 6J4) (0.64, 16.6)
(0.29x 6.44, 1.93x 6.,14) (1.87, 12.4)
Control limits for ranges chart: acti on limits
Vlarning limits

Exalllple 11.3 (continued) control limits for a range chart. However, the
distribution has been worked out, and i~ can
The calculation of the control 1 imits for
averages is shown in Table 11.3. The average be shown that the control limits should be set
according to the following equations:
range R is calculated first, and then Equa-
where the values of C~ C~, On, O~ are as
tions (11. 9) and (11. 10) are use d, togeth~r shown in Table 11.5,
viith the va lues in Table 11.4 (for n = 4 in
this example) to find the contro 1 limits. *** The calculation of the control limits for the
rar,ge chart is shovm at the foot of Table 11.3
11.7.2 Control Chart for Ranges and is quite straightforward. ***

In th isease, e is put equal to the range R of
a sample. The distribution of sample ranges,
even from a normal population, is not itself
nonna l; consequently Equati ons ( 11.1) and
(11.2) c~nnot in genera 1 be u:~d to find the

Table 11.4 Val Les of An and \')n

n An Hn n Cn r' 0n O'
n
"'n

2 1.937 1.229 0.886 2 0.04 2.81 0.00 4.12

1.054 0.668 0.591 3 0.18 2.17 0.04 2.98
4 0.29 1.93 0.10 L.57
0.750 0.476 0.486
5 0.37 ,,, 1.81 0.16 2.34
0.594 0.377 0.430

0.498 0.31C 0.395 6 0.42 1.72 0.21 2.21
0.432 0.274 0.370
0.384 0.244 0.351 7 0.46 1.66 0.26 2 .11
0.347 0.220 0.337
0.317 0.202 0.325 8 0.50 1.62 0.29 ".04

9 0.52 1.58 0.32 1.9']
I

~O 0.54 1.56 0.35 -1-.9-3 ~i

Extracted from BS 600R (noVi Vlithdrawn) and Eytracted from BS 600R (now Vlithdrawn) and
BS 2564 by permission of the British Stand-
ards Institution and of the General Electric BS 2564 by permission of the British Stand-
Company p 1c. ads Institution and of the General Electric
Cc,mpany p1c.

limit). In such cases, it is usual to set the

centre line of the averages chart at the over-

all nlean X of 'he results of the preliminary

experiment. The control limi ts then become

The contt·o 1 charts ~or both averages and

ranges are shO\vn in Fig. 11.7, a"d the sample

"lie rages and ranges feom Table 11.3 have been

plotted on them. There are no points outside
the action limits on the range chart. This

suggests that the process was always in con-

tl-ol so far as short-term variability is For example, if in Example 11.3 the mean mass

concerned; thus the value of d las estimated us 0 102 g had not been specified, the over-

by R) used to calculate the limits on both all r'ean mass viOiTTd have been calculated:

charts was a reasonable one. No revision of

the control limi ts is needed, and the charts x 0 Sum of all measurements ~

are thus ready for use in routine control. If total number of articles measured

points had fallen outside the range action

limits, those samples would have been disre- 10215 0102.15 g
garded and the average range and control
160
limits recalculated.
The contraIl imits for averages would there-
There are several points outside the action fore be set at

limit< on the averages chart (sample nos. 12, 102. 15:!:0.750x6.44
102. 15:!:4.83
13, and 19), and a consecutive pair (sample

nos. 15 and 16) between the warning and action

limits, which suggests that at these times the

process had drifted off t,',rget ,nd \'Ias making

garments w~ose masses were significantly dif-

ferent from the nominal mass. Although the

reasons for this lack of control should be

investigated, no revision of the cantrall i-

mits is needed. **<

It hcs been assumed in all the above that a The li~its for the range chart would be as
specified objective value us exists. There are before.
instances when this may not be so. Properties
like yarn breaking strength and extension are
not always specified in this way (though there
may sometimes be a specified upper or lower

(!) Averag~s

- - -- -- - -- - ----- ---- - -r.;'- ---- - --- --- ---
f.:\ I!..)
,0
-- -- -- -- -- -- -- -- -- -- -- '6'- -- -- --- --- -- -- -- -.

;0

-------------\ff----------~------

~ (!)

, • Samp 1e no.
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

--- - -I~~-l-I- 11-'---'-'- -;--11-1-111-'- -. ~.

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

C.__

We have postponed until now the difficult It cal be seen that, in situation (al, the

questions of what sample size should be used probJlil ity of a sample result plotting out-

and how frequently a process should be in- slde the. actlon llini ts is 0.~02 (0.001 in each

spected during routine process control. The tJll (,f the dlstnbutlonl. A~te( a shift ha<

problem is basicillly an economic one Jnd taken ~lacc, this proDabll ity changes to "

consists essentially in bJlancing the costs of

running the control scheme against the penal-

ties incurred if poor qual ity is allowed to go

on to the next stage of manufacture, or to the

customer. If the consequences of poor qual i:y wherc B 1 and B2 are the ta i 1 areas shown. The
prob.:bility of a point not falling outside the
are not serious, or if the process is known to action limits when the smft has taken place
1s 1- B.
be extremely stable, it would obviously be
Now consider the probabil ity that the shift is
foolish to spend too much on complex quality- not detected until r samples have been lnspec-
ted, i.e., the rth sample is the flrst to plot
control procedures. But, if the process has a outsi,je the action limits after tl1'eS1l1ft h,s
occurred. This probability is
tendency to drift off target, or if the arti-
cles being produced are very expensive, then Pr(r samples before shift detected!
shift has occurred)
an increased expenditure on qUJlity control Pr(first sample inside AL and
second sample inslde AL ana--
could probably be justified.
------------------------and
A complete answer to these problems has not (r-1)th sample inside AL and
been given though several atte,.~ts to do so rth sample outside All
shift has 0ccurred)
have been made. However, one factor that plays
a part ln the decision is how quickly" con-

trol chart vlill detect a lack of control in a
process, and this can be investigated fairly

easi ly.

A useful concept in this connexion is the

average run lengt~ (ARL). Suppose a process,

prevlously In control at a mean level ~e,
suddenly goes out of control Jnd starts to

operate at a new mean level ~0' The average

run length is the mean number of samples that
will be inspected before a point plotted on

the control chart indicates a lack of control.

The situation is as in Fig. 11.8, which shows

the distribution of e (assumed to be normal):

(a) wh,"n the process is in control at level ~e, where we have assumed the samp les to be inde-
and pendent and have used the mul t. pl ication 'ule
of probabil ity ~I; 4.9).

(a) Before sh'ft (b) After shift
occurs occurs

Equation (11.13) i<;a gl'omctric probability lack of control, sucil as those described in
disU-ibution (I; ~l_3), wilicilhilS i\mcan SC'! . on 11.4, tend to further reduce the ARL,
wf:' " thus increases the efficiency of the
and this is the average run length. Note that c:'·..ts, but their precise effect on the ARL
this 'run' is not to be confused \'Iit.thhe (,. b~ difficult to determine. Larger shifts
'run' mcnlioncd in "'clion 11.4(c), \'Iilicwha:. (, >2) are detected quite quickly, on average,
i\ sequence of points lying belween the centre by th~ Shewhart chart, whatever rules ar~
line and tile \'Iarninglimit. adopt pd.

The above has assumpd that. lack of cont.rol is The results of the last section can be used to
milk" some progrr~'; LtMMds solv'jng the problcm
indici\tCeionly ',.,i1('1l ilpoint plots o,Ii'.id"l.hr of \,I.h•lis the best ,ample size to use. lis
staled earlier, a complete solution would re-
ilclioll limit.", !I ~.!.tli"Tlilltj limit:. ,11"1' dl~:(l quire a knowl~dge of the costs involved in
letting defective material pass on, i.e., the
L1<;pf!. i.{1 .• oil t\'!(l <:.IICC(")SlVC po:nt.s lying cost of not detecting shifts when they occur
(the ARLlS a factor in det.ermining this), the
bclwecn the ,1CL i0.'-' Iimi Ls i\!'lleille\'Iarning costs of inspection itself (labour, materials,
space, etc.) and the probabi 1ity that shifts
limits arc takcn as iodicating lack of con- of various sizes \,ill, in fact, occur. The
probl~m has not yet been solved completely,
trol, the calculation of the average run and tile best that can be done here is to des~
length is more difficult. However, it can be cribe some suggestions that have been made in
the light of experience and on the basis of
shown that in this case the average run length what theoretical work has been done.

is A fairly obvious requirement is that, when a
shift occurs, there should be a good chance of
where y is the probability of a point falling detecting it fairly quickly. To make this more
between the action and warning limits. precise, consider the general process of Sec-
tion 11.2 vlorking at a mean level ~e with
When the distribution of 0 is normal, it is standard deviation 00' If a shift of magnitude
koe tokes place, suppose we want there to be a
quite easy to ca,culate y and ~ this ~ur- probcbility Pd of detecting the shift before
at most r samples have been inspected, i.e.,
pose, it is convenient to measure the size of
Pd = Pr(detecting a shift in the first
the shift ~0-~0in terms of 0e and to put
r samples/shift of magnitude koe)
The standard normal variate corresponding to 61
For s'mplicity, suppose the process is con-
in Figure 11.8 is then trolled by using only action limits, i.e., the
shift is detected only when a point falls out-
Thus, when k is known, Ul and hence 91 can be side the action limits. Then the probability
found. The values of62, '(1, and Y2 can be that cny given sample wi 11 not detect a shift
estimated in a similar way. Average run is 1-~, where p is given by--
lengths can then be calculated from Equations
(11.15) and (11.16), and the results of such a as in Section 11.8. A glance at Fig. 11.8 will
calculation are given in Table 11.6. show that, even for moderate shifts, 62 is
small compared with 61, so that we may put
A striking feature revealed by this table is
the effect of using warning limits in the way Now, if r successive independent samples are
described in Section 11.4(b). Small shifts, in taken ufter the shift has occurred, the proba-
the region 0.5 < k < 2.0, are detected, on aver- bility that none of them will detect the shift
age, about tJ.liceas quickly as when action 1i- is
mits alone are employed. However, it is also
evide~t that Shewhart charts are not particu- by the multiplication rule of probability (1;
larly good at detecting small shifts. The 4.9). fhis -isthe probability that the shift
adoption of additional rules for detecting will not be detected by the r samples. The
probaFTTity that it will be detected by one of
Table 11.6 Relation between Average Run the r samples is therefore
Length and 51ze of 5hlft
81 = 1- ( 1-Pd ) 1/r .
ARL vlhen:
k only AL used bot,! WL and AL used

0 500 238
99
0.5 200 26
1.0 55 8.9
1.5 18 4.1
2.5
2.0 7.3 1.8
1.4
2.5 3.6
1.2
3.0 2.2
3.5 1.5
4.0 1.2

l

-_____ -------'l~c=

When Pd and r have been specified, 01 can be We now ~ave all the information needed to cal-
calculated from this EUlation. When 81 is culate" from Equation (11.20), which gives
known, the correspondin,! sLandard normal vari-
ate can be hund from tables, and Equdtion 1.9;2 x 0.0733 x (1 - 0.0733)
(11.17) then gives k. Jut keo is tile size of (0.11 - 0.0733)
the shift required to be detected, and this
last condition provides an Equ~tion to solve \olethus'see that the original sample size of
for the sample size n. Some examples '.'1I1i ma~e 180 wns not too far away from what is requir-
the procedure c]ca,-. ed, ***

The making-up process of Example 11.1 was The data for Example 11.2 were obtained by
control Ied by means of a control chart for test ins 'unit' lengths of 1000 m of yarn, and
defectives based on ar estimated proportion of the average number of neps per unit was
defect ives es timat ed as

(This was the revised value after deleting Suppose we want to operate a control scheme
'out-of -control' data.) Suppose we want to that rds a 50% chance of detecting a shift to
design a chart that wi 11 have a 50% chance of
detecting a shift to before at most four samples have been in-
spect ,~d.
before at most five samples have been inspec-
ted. Suppo~e the required scheme involves testing
lengths of n units, i.e., lrf10n metres. Then 6
In 'this case, the basic distribution of vlOuld be the number of neps I- • n units and,
0(= number of defectives in a sample of size since the basic distribution is Poisson, we
n) is binomial, for ,.hich have

k2PC(l-PC) Since we require Pd =0.5 (50%) when r 4,
(p'c - PC)2 Equation (11.19) gives

For the stated conditions, we require that and the corresponding standard normal variate
Pd = 0.5 (50%) and r = 5.
Thus Equation (1-1.19) gives ~::

Table Al then shows that the corresponding
standard normal variate is

and, substituting the known values in Equation The tolerance ran~e is then 2T, and whether or
(11.21 ), ''Ieget not our control c art need~ modification de-
pends on the ratio
-2.0-92-x4-0.-7
.'hich's sometimes called the relative pre-
(50 - 40.7) 1 ci sion index.
When this ratio is small, the situation will
Hence lengths of yarn equal tc 2.06 'units', be as shown in Fig. 11.9(a). This shows the
or 2060 m('tl'C.'.~';hould he tC'sl.C'dH.'" tolerances ~s±T and the distribution of indi-
vidual articles when the process is in control.
An approach similar tJ ,he above can be In thi~; case, ther'e wi 11 be an unacceptably
adopted for control charts for averages, but high proportion of individuals outside toler-
these have been studied in considerable depth ances, as shown by the 1arge shaded areas in
by A.J. Duncan, and his general conclusions the ta'ls of the distribution. The lower the
can be surrmarized as follows. relati~e precision index, the greater will be
the proportion of out-of-tolerance work produ-
1. The conventional sample sizes are 4 or 5, ced. When this situation occurs, two remedies
and these are close to the optimum if it is are avoilable.
required to detect reasonably quickly (i) Reduce the process variability o. This
fairly large shifts, rif the order of 20e.
If it is necessary to detect smaller shifts would usually involve modifications to
of down to 00 say. then sample sizes of 15 tre process itself and could be very
or 20 are more economical. e>,pensive.

2. If the cost of not detecting a shift is distributi(n of
high, i.e., if ]tis expensive to let individual articles
faulty material go on to the next stage, it
is better to inspect small samples fre- distribution of
quently rather than large samples less fre- individual articles
quently,

3. [t may sometimes be better to use control
limits set at ~e±20e or at ~e±1.50e, rather
than the conventional limits at ~e±3.0goe.
This vlould be true, for example, if it is
possible to decide quickly and cheaply that
there is nothing wrong with a process when
a sample result happens by chance to fall
outside the limits. On the other hand, if
investigating trouble is very expensive, it
may be best to consider using wide limits
set at Pe±3.50e or even at ~e±40e.

4. If the cost of inspection itself is very
high, e,g" when it is destructive, then it
is most eccnomical to inspect small samples
infrequently, by lIsing narrow control li-
mits, say, atpe±1.50e.

11.10 Tolerances: fotodifiedControl Limits for
Averages

We have so far said nothing about any toleran-
ces that may be specified for a manufactured
article. Such tolel'ances are commonly used
when the property e being considered is a
measured ,continuous variable. For example, the
mass per running metre of a certain cloth may
be specified as 420±15 g/mt, the implication
being that, so long as individual metre
lengths of the cloth have masses between 405
and 435 glm', the c,loth is acceptable for fur-
ther processing. Thf' control charts developed
so far take no account of such tolerances,
since they are concerned entirely,with decid-
ing vlhether or not a process is 'in control'
in the sense defined in Section 11.1, i.e.,
whether it is subject only to random sources
of variati0n and is meeting a required average

val ue.

Let us suppose that our process is in control
in this sense and that the standal'd deviationO'
of the random sources of variation is known.
Suppose also that individual articles are
acceptable if they have a d'imension e lying
inside the tolerances

(ii) Check "iliether the tolerances have been ~lQ .....!...!10dified Control Limits fOI' Ave~~

set 1II0l'e narro\"ily than is necessary, It

is not unCOlllllon to find that tolerances The modification is based on the realization

are chosen fairly arbitrarily, and when that, in the situation of Fig. 11.9(b) the

this happens there is a tendency to set prosess average call be allmved to drift away
them 11I0,'12slt'ingently than is justified
fre,,11 the specified mean value Us before the

by the practical use for which the ar- process \'Ii 11 produce too many out-of-lolerance

ticle is intended. Fo;' example, it is pi,'ces. In f ac t, if we as sume that a propor-
needlessly strict to specify that the
length of a knitted sweater shall be ti~n of 1 in 1000 articles outside one of the

tolerance limits is acceptable and that the

within the range 52.5±0.5 cm, yet this distribution of individual measurements is

kind of specification does occur, no,.:lIal, the process average can drift until it

lils within 3.090 of tolerances, as shawn in

When neither of these remedies can be applied, Fir. 11.10(a), since 3.09 is the standard nor-
the best that can be done is to keep a control
mai variate corresponding to a tail area I)f
chart for averages of the conventional type,
Vihich ~Ii 11 keep the average level of the pro- 0.001. The process average can thus be allo"led

cess correct, and then sort out the defectives to drift between
by 100% inspection.

The other extreme, when the relative precision

index is high, is illustraLed in Fig. 11.9(bJ.

This shows that, if the process is maintained

'in control' at the specified mean level" s,

then virtually no out-of-tolerance articles

wi 11 be rroduced. In this case, it is not ne-

cessary to keep such a strict control on the ana these limits are shown on Fig. 11.10. Ho-
wever, Vie should not want the process m!an to
process, and a modified control chart can be

ado pted.

=--------3-T

0.05

(a) Distributions ,b) Distributions of means
of individuals
of samples of size n

(ii) Check ,'Jil(>ther Uw to Jeranc~s have lJeen ~_lQ ...2.!'lodified Control Limits for~~

set IIIOI'L'nal-ro\'lly than is necessary. It

is not unCOJJlllon to find that tolerances The modification is based on the realization

are chosen fai rly arbi trari ly, and when that, in the situation of rig. 11.9(b) the

this happens there is a tendency to set prosess average call be allm<ed to drift al<ay
them mOI-e stringently than is justified
fre,.11 the specified mean value "s before the

by the practical use for which the ar- p,-ocess wi 11 produce too many out-of-wlerance

t icle is intended. rm- example, it is pif'ces. In fact, if we assume that a propor-

needlessly strict to specify that the ti:;n of 1 in 1000 articles outside one of the

length of a knitted sweater shall be tolerance limits is acceptable and that the

within the range 5Z.5!0.5 cm, yet this distribution of individual measurements is

kind of specification does occur. nOI:l1al, the process average can drift until it

liLS viithin 3.090 of tolerances, as shawn in

When neither of these remedies can be applied, Fic. 11,10(a), since 3.09 is the standard nor-
the best that can be done is to keep a control
mai variate corresponding to a tail area of
chart for averages of the conventional type,
which wi 11 keep the average level of the pro- 0.001. The process average can thus be al iOI<ed

cess correct, and then sort out the defectives to drift between
by 100% inspection.

The other extreme, when the relative precision

index is high, is i llustraLed in Fig. 11.9(b).

This shows that, if the process is maintained

'in control' at the specified mean level "S,

then virtually no out-of-tolerance articles

\'Ii 11 be rroduced. In this case, it is not ne-

cessary to keep such a strict control on the ane these limits are shown on Fig. 11.10. Ho-
wever, we should not want the process m?an to
process, and a modified control chart can be

adopted.

=--------3-T

0.05

(a) Distributions ,b) Distributions of means
of individuals
of samples of size n

drift beyond these limits since the process ~s ± 3. 090 Iin ,
would then produce too mony defectives. Conse- \'Ihilethe modified limits are
quently, we require there to be ~ high proba-
bility of taking action if it did so. Now, it is Obviously sensihle to use whichever set
this action would be based on the means of of linrits is the wider.
samples of size n, as in an ordinary control
chart, and, when the process has drifted to ~U'
say, the distribution of sample overages would
be as shown in Fig. 11.10Ib). This distribu-
tion has a standard deviation olin (I; 6.5), and,
if we require 0 95' chonce of taking action

when the IJJaXlffiurl' allowable drifL h~s occurred
(and thus 0 5'.:chilnce of not taking action!,
the action 1imit \'lOuldbe---setat a di stance

1.6~n/ln inside Lhe drifL limits as shown in
Fiq. 11.1DTliT--(nH' valuc 1.6~ is thc stand-
ar~ normal variaLc corresponding to a tail
area of 5%). Thc upper action limit is
thereiore

. = ~s + T - (3.0~+ 1.64/1n)a T/R> (3.09+ 4.73lln)an = B~ •

after using Equation (11.22). Now, as stated Values of B~ are shown in Table 11.7.
in Section 11.7, there is a relation between
the st3ndard deviction 0 of a normal popula- Examr~

tion and the average range R in samples of A chart to control the mean linear density of
the yarn produced by a spinning process is
size n, of the form being set up. The nominal linear density of
the yarn is 36 tex, and a standard linear-
On combining Equations (11.24) and (11.25), density test is deemed acceptable if its value
the action limits are .given by lies within the tolerances 36±? tex. A preli-
minal'y experiment was carried out, which :011-
The values of A~ for various values of n are sist~d in takin~ twenty samples at regulae
given in Table 11.7. intervals, each sample consisting of suffi-
cient yarn to make five standard linear-
n A' Bn' density tests. The average range of these
n test; was found to be 0.68 tex. Calculate
appr0priate action limits for the chart.
2 3.765 5.701
From the data above, we obtair.
3 2.386 3.440 ~s = 36, T = 2, n = 5, . R = 0.68,

4 1.900 2.651 so th at

Fr£m Ta9le 11.7, we find that BS = 2.38. Hence

T/R ' B5' which indicates that modified con-
trol limits are appropriate. These are calcu-
lated by using Equation (11.26), which gives

action limits 36 ± (2 - 1.644 x 0.68l

where we have used the value A'5 = 1.644 taken
from Tab Ie 11.7 .

It is interesting to compare these limits with
conventional action limits, which would be set
at

5 1.644 2.238

6 1.485 1.983

7 1.373 1.805

8 1.288 1.672

9 1.226 1.573

10 1.173 1.490

~[-------

11. Quality Control: Control Charts

It \'I1i1 be seen tllat the modified limlts are Si-l ; d1+d,+d3+ ... +di-l
considerably \Iider than the conventional 1i- a"d

miLs, "Ilich <1llo\IS Lhe spin"i,.g mach'in" to

drift off torgeL before it begins La produce
u"acceptable yarn. ***

The basic rule for taking action when using so that the ith cumulative sum is easily cal-
Shewhart control charts is to do so when a c dated from the previous one. A Cusum chart
single point falls outside the action limits. i, a plot of the cumulative sums Si against
This simple procedure can be extended by pro- t·,me.
posing that action should also be taken "hen
t"o consecutive points lie between the actlon The effect of this procedure is as follows. If
and warning limits, and further rules based on He process is in control and making articles
trends, runs, and cycles in the last fe" plot- o~ the correct level, the deviations di will
ted poi nts were di scussed in Section 11.4. tend to cluster y'ound zero, and the values of
Fcl101·,ing on from these, it is a logical step the cumulative sums "ill tend to remain steady.
to attempt to devise procedures that take lnto .t, hOl'lever,the process shifts to produce ar-
account all previous sample data in arriving ticles "ith mean dimension greater than m, the
at a declSl0n to take action. Such a develop- aeviations di \Iill tend to be positive, ~nd
the cumulative sums Si \lil.ltherefore tend to
ment leads to the cumulative-sum control chart increase. The path of the points plotted on
or, more briefly, the cusum chart. Lhe ~usum chart will thus be sloping up\lards.
The reverse trend would occur if the process
Suppose a process is being controlled by means shifted to produce articles belo\l the refer-
c. a parameter 0 (the sample mean, the number ence level. Shifts in the process are there-
of defectives in a sample, etc.) and let the fore detected by changes in the slope of the
path of the points on the cusum chart, •.nd the
values of 0 in successive samples be main advantage of the cusum chart over the
Shevlhart chart is that it tends to detect re-
e e e i .1) iatively small shifts more quickly than the
2) ••• , She\lhart chart does. This point is illistrated
in the follo\ling example.
Suppose also that the process is to be con-
trolled about some reference valup m (for After the control limits for the garment mas-
example, a specified mean value ~ s)' The de- ses of Example 11.3 had been calculated, the
vi ations chart Vias used to control the process. The
next fifty sample averages are shown in Table
di = Gi-m 11.8 and are plotted on the control chart in
Fig. 11.11. This chart provides no evidence of
of the individual 0 values frolT,the reference any serious lack of control in the process.
can be found and the sum of these deviations
calculated, i.e.,

SI d1·,

S2 d1+d2
S3 d1+d2+d3

Table 11.8 M"an Masses of Samples of Four (1) (2) (3) I(4) (5 ) (6) (7) (8)
(;".;rmenBtlanks Xi xi-102
Sample
number 5ampl e

5i number Xi 'jo"~

1 100 -2 -·2 26 102 -2 I

2 lOOt -t ·-2i; 27 99~ -2~

3 101 -1, ··3l- 28 100~ -H -31 I
4 102~
5 103~ "H -2~ 29 991 -2~ -6
-1 30 101J, -f -6t-
6 100~ -H -2~ 31 100t -1-!;- -8

7 100~ -1~ -3t 32 102 0 -8
103!;- 1t
8 106t 4t ~ 33 99!;- -2~ -6t
-3
9 101 -1 1 34 99 -91
100t -1~ -121
-2 99f -14l-
99 -2t
10 100~ -1) -2 35 103~ -3 -16!
99~ 1) -191
11 101~ -i -2t 36 10n -18
1041- -2~
12 102 0, -21; 37 99!;- -~ -201
13 101~ 21- -2'
14 101) -I -3 38 101t ·16;-
102 -2~
-3~ 39 --l; -21l
10H
15 103~ H, -2 \- 40 102 0 -2H
16 102~ 102
17 101) -~" :f~ 41 lOOt 1 --w21t
103
-It 42 -2 i-22t

18 102 0 -1~ 43 0 -22i ,

19 101' -~ -2* 44 0 -23! I
_1J,...,.
20 102 0 -2t 45 ~
1
21 100 -2 -4* 46

22 102 0 -4* 47

23 103 ~ 11; -2~ 48
24 ~04t
2~, 1 '9
102-!-
25 1 T
;;

104

••

103

"

102

..c

101

"100

.0..•• ••••
•.I
..\ ••
•• e •• •• •• •••••• •• •• •• •• ••

-5 0 "

I •• •• ••

I

1 •• ••
-10
••

\

••

"-15

"
"

•-20 fl
•••••• • •

-25

,.

\ -30 II .. Sample
number
\ 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

I

)~[------

It vii 11 be recalled that the specified garment and so on. When these sums are p lotted on a
cusum chart, as in Fig. 11.12, the points ex-
mass was ~s = 102 g. The devlations in this
hibit a marked downward trend, starting at
example are therefore given by about the tVlenty-sixth sa'!Iple. The over-all
where Xi is the mean of tlil' ith sample, and Velld of I.IlC plotted point s before: this \'Ias
their values are given in :olumns (3) and (7)
of Table 11.8. The cor~esoonding cumulative IIIJI'izontul, Vlhich lndicutl:d " process 'ill COII-
sums are in columns (4) and (8); they are trol. HO'tlv,er, the dOl'mwar-d trend in the se-
cond half of the chart suggests a shift in th0
(a) Process process below the required mean level, In
in control fact, the Jver-all mean of the first 25 sam-
ples vias '02 g, exactly or. target, \;hile the
(b) Shift above last 25 sa,nples had an over-all mean of 101 g,
objective
Thus a small shift in ave"age level \'/as indi-
(c) Shi ft be low cated qui ..e quick ly by the cusum char-t but was
object ive
not detecl'd by the Shewhart chart. ***

\·Jhereas the cusum chai't by it se If f,'equently
gives a good subjective indication that a pro-

cess change may have taken place, it is neces-
sary tc have an o~jective means of confirming
the suspic ion. One method that has been devel-
oped for this purpose uses a V mask. This is
placed on the cusum chart with the vertex of
the V a distance d ahead of the last plotted
point, as shovm in Fig. 11.13. If all the pre-
viously p'otted points fall between

the limbs of the V, as in Fig. 11.13(a), the
process is assumed to be in control. However,

one or mOI~ points outside the V mask, as in

Fig. 11. U(b) and (c), indicate a shift in the
mean leV€l of the process.

The-properties of the system depend on the

choice of d and ~, where 2~ is <"e angle bet-

ween the limbs of .the V mask. A , .1era 1 recom-

mendation given by Ewan*, which he states

gives bet:er discrimination than a Shewhart

chart, is as follows: choose

d = 10 horizontal plotting intervals

and

where vi :s the ratio of the vertical and hori-

zontal scale intervals on the cusum chart. A

more genEral discussion of the choice of a V

mask has been given by Wood\'lard and Goldsmith t.

We shall use Ewan's rules to prorluce an apprJ-

~riate V mask for this example. Ihe first rule

gives d ~ 10 immediately, and to find 0} we
Oxneed to knov'
(= a/In) and vi. Now, when

discussing Example 11.3, we found that the

Raverage range in samples of size 4 was =

6.44 g. An estimat~ of 0 is therefore

* W.D. E\;an. 'When and Hovi to IIse Cusum

Charts', Technometrics, 1963,5, 1.

t R.H. Woodward and P.L.Goldsmith.
'Cum~lative Sum Techniques' (iCI Monograph
No.3), Oliver and Boyd, London, 1964

when' the value of a4 has been token from 1. Over a period of time, daily samples of size
Table 11.4. Hence we have 200 dere chosen from a product,on line anl
the "umber of drfectives on each day was
Furthermore, in Fig. 11.12, equal intervals toun.ed. The results were as follows:
along the vertical and hOI'izontal scales
represent 5 and 4 units, respectively. Hence Number
\1; 5/4 1.25, and Equation (11.27) then
gives of De-

or t = 32°. This mask is sketched in Fig. fectives
11.12 and shows that the shift would first be
detected by sample no. 37, sample no. 26 then Number 5 8 2 15 5 2 3
lying outside the limbs of the V mask. In of De- 2
practice, the mask is usually marked on trans-
parent plastics material so that it can simply fective<;
be laid on the chart at any point. ***
Use these data to construct a control chart
11.11.2 Advantages ard Disadvantages of Cusum for the number of defectives in samples of
Ch ar ts size 200. Plot the data on the chart and
comment.
The principal advantages of cusum charts are
as follows. If you were setting up a chart for future use,
how would you modify the control limits?
(a) Shifts in the process are easily detected (Assume that assignable causes can be found
visually by a change in the slope of the for any points outside the limits on the
chart. original chart.)

(b) Because of this, it is fairly easy to es- 2. A menufacturer of woven cloth examines
timate when the shift occurred, and this 100 ~ lengths of cloth and counts the
may be useful in helping to find the cause number 01 faults in each. A sample of the
of the shift. results is given below.

(c) Small shi fts, of up to about 20";:,tend to Calculate control limits for a chart for the
be detected more quickly by the cusum number of defects per 100 m of cloth.
ch ar t.
3. SamVl es of size 4 were tak£:,lon each of 10
Against these advantages must be set some days from a machine making sweater blanks,
di sadvantages. whose lengths should be 57·cm. The fol:ow-

The charts are not as easy to understand ing length measurements were obtained.
as Shewhart charts.
Day 2 3 4 5 6 7 8 9 10
(ii) The Shewhart chart is usually more ef-
fective in detecting large shifts in the Lengths 57.7 57.2 57.7 57.9 56.1. 56.4 56.1 55.4 56.9 58.2
process average and trends in the pro- of 57.4 57.9 55.8· 56.1 55.6 57.7 57.4 56.6 55.6 58.4
ces s. 56.9 56.1 55.6 55.4 56.6 58.2 58.2 :;7.7 57.7 56.1
blanks 56.1 55.9 56.6 55.4 57.2 56.1 58.4 ~,6.9 56.4 5:/.2
(iii) If several charts are to be used, they (em)
must each have a separate V mask;
alternatively, if the same mask is to be
used for all charts, careful choice of
scales is nec8ssary.

Ewan, in the paper previously mentioned, sug-
gests that Shewhart charts should be used when
simplicity of operation is required and when
tests are ine(pensive. If testing is expensive
and if the type of process changes expected to
occur are sudden shifts in average level, then
cusum charts should be considered.

Use these data to calculate control limits for
a control chart for averages and ranges of

samples of size 4.

==--------'

(vi )-(ix) iJI'eoutside Lhe e.<perimenLe/"s Con,

trc I. The var i at ion they produce is of a ran-

I·lostexperiments arc CdlTicd out to inves- do~ nature; different pieces of fabric cut
tigate the effects of one or more treatments
on a respol;se variab~ Examples are: fro,))the same roll may behave quite different-

(a) the effect of different shrink-resist ly, Vlashing machines are notoriously variable
treatments on t.he area shrinkage of a
knitted fabric; fro n one run to another, anyone I·,hoha:;

(b) the effect of increasing the speed of a hanjled a piece of knitted fabric will know
spinning machine on the number of end-
breaks per hour. hov: difficult it is to measure, and it is well

The results of any experiment are affected by known that temperature and humidity af'ecl th2
variations dl'ising from Illanydifferent sour-
ces, For eXiJmple, in the case of experiment properties of textile materials. This Incon-
(a) above, possible factors influencing the
,'esult of an area-shrinkage test are: trc,ned randor; variation represents th: 'ex-

( i) th·' type of shrink-resist treatement, perimental error', and, in order to be able to

(ii) the type of fabric, interpret the results of an expe"iment proper-

(iii) the type of lYashlng machi ne, ly. it is necessary to have an estimat? of its

(iv) the length of the wash, magnitude. The vlay to do this is to carry out

(v) the temperature of the l'lash, more than ore test on each treatment being

(vii) which run of the washing machine is considered, and the number of such repeat ob-
used,
servations is called the number of replica-
(vi ii) errors in measuring fabric dimensions
before and after the wash, and tions. It is not possible to make a genem

and others could no doubt be written down af- 'sta~ement about hovi many replications are
tel' further thought; the list is long enough
already to emphasize the fact :hat even this needed in a given experiment. This will depend
simple experiment is not so uncomplicated as
it first appears. In this experiment, the pri- on the size of the difference betVleen two
mary interest is the effect of the shrink-re-
si st treatment. The other factors listed are treatments that is considered practically im-
subsidiary to this, and one of the arts of
good experimentation lies in ensuring that the portant and on the magnitude of the expErimen-
effects of interest can be separated from the
effects of the subsidiary factors. The activi- tal error. The discussion of (I; 8.2) is of
ty involved in planning an experiment so that
this is possible is called experimental 1esign relevance here. '
W8ich is a large topic needlng tar more space
than Vie have available to do it justice. HOIve- It is possible that some (or all) of the un-
ver, it is possible to remark briefly on some cortrolled sources of variation are functions
of the principles of good experimental design. of time, of place, and of the experimental
units themselves. As an example of this, con-
The planning should begin by making a list of sider experiment (b) to estimate the effect of
the factors likely to affect the experimental changing spinning-machine speed on the rate of
result" such as that given above for experi- occurrence of end-breaks. 5uppose five speeds,
ment (a), and czrefully distinguishing the labelled 51-55, are to be considered, where 5,
main factors from the subsidiary on~s. The is the slowest speed and 55 the highest. An
subsidiary factors can be divided into tViO ty- exreriment of this kind would have to be car-
pes. For example, factors (ii)-(v) are under ripd out over a considerable time period in
the control of the experimenter. He can choose orJer to get reliable estimates of end-break-
the type of fabric he Vlishes to use, the type ag~ rates. If the speed were increased in or-
of washing maChine, ard the length and tempe- der of ascending magnitude 51+5r53+54+55 '
rature of the wash, and he can standardize thQn 55 might be at a disadvantage compared
these. They can be thought of as the experi-
mental conditions, and they are repea~ with 5, ror-a number of reasons.
tram experlment to experiment. But factors
By the time the highest speed is ,
I'eached, certain parts of the machlne
could have deteriorated, and increased
breakage rate cou ld be caused by th',s,
rather than by increased speed.

Atmospheric condi tions cc·uld change
during the course of an experiment, and,
since such changes rarely occur randomly
but tend to change slowly over several
days, again breakage rates could be
affected by something other than m~chine
speed.

5ince the experiment would consume a
gr~at deal of fibre, this raw mate:ial
might change as the experiment pro-
ceeded, again affecting the final speed
used by comparison with the first.

F0rtunately, there is a relatively simple Vlay
of overcoming this difficulty, namely,. ~
hation. In the present experiment, this
simply 'I1eansusing the speeds 51 - 55 in a
ri.ndom order, rather than a systematic are.

The best method of choosing such a r?ndom It is oft~n a good idea to begin the analysis
D,-der is by means of a table of random num- of some data by constructing a pictorial re-
hors*, •.r•ich are used as follo'.'lsB_egin by presentation of it; this helps the researcher
i;bel1ing the speeds 0, 1, 2, 3, 4 to COlTes- to get a 'feel' for the data and sometimes en-
pond to Sl, S2, S3, S4, S5, respectively. The ables some tentative conclusions to be drawn.
~umbers in the random-number table Vlill usual- A suitable form of graph for this experiment
ly be found in pairs. Choosing J st0rting is Sh0l1n in Fig. I? 1. The individual data are
point "t ,'"ndom. \'11' find the ~equence ~ho\m JS spots Jnd the mc"n VJ 1ue for

86, 57, 34, 45, 67, 73, 43, 07, 48, etc. each fab ie, calculated in Table 12.1, is in-
dicated uy a short horizontal line. Th~ grand,
Divide each of these by the number of speeds or over-Jll, mean of the data is also shO\·m.
and find the rern"inde,-; thus VII'get The resulting picture suggests that there may
be di ffc"ences among the treatment means, and
The ,'"ncloronrdp,' of spl'pds corr~sponding to i1 procedure is required to test whether or not
this sequeflce lS the obsecvcd diffc,'enccs arc stiltistically
signi ficunt.
and the experiment should be arranged so that
the speed is varied in this order. If there had been only two makes of fabric
being co'npared, a t-test, as described in (I;
Randomiza~ion should be carried out whenever 8.41, w~,ld be appropriate. Since more than
practicable in order to eliminate as many un- two sample averages are being compared in this
wanted effects as possible. There are instan- experiment, a different approach is necessary.
ces when such a procedure is difficult, or Such an dlternative is provided by the analy-
even impossible_ For instance, ranc)mizing may sis of vilriance. ***
increase the S2tting-up time betVleen runs of a
production process, and this could be ver." ex- Graph of the data in Table 12.1
pensive. When this situation occurs, the expe- Make of fabric
rimenter should appreciate that he is running
risks by not randomizing and, if possible, 60
should take other steps to avoid the most ob-
vious pitfalls. These could include measuring a 50 •• A
second variable that v'ould ir dicate any un-
wanted trends in the experimental conditions CVl 40 t0 •e3l •
so that they could be taken account of in the .0_'0 c 30 -B
analysis of the experimental results. 20
+-'0 10 0
When the experimental design has been decided 0
on, a~d the experiment carried out, the final <au -10 •• ,I -• t~t X3l Grand
stage is the analysis of the results. Many of mean
the statistical techni~ues described in pre- "-<11 -.• 0 •
vious chapters have a role to play, when ap- +-' Vl
propriate, at this stage, and we shall see <11 •• ••
some of them in action ,n what follows. Howev- co
er, the remainder of this chapter wi 11 be pri- <110 D
mari ly devot~d to a. ver'y powerful and genera 1 0._
technique of analysis known as the analysis of
variance, often abbreviated to ANOVA. 4-"-

0<11
>

VlO
Vl
<11<11

xu·_E

w+-'

•

-20

The data shown in Table 12.1 are the results Table 12.1 Time in Seconds for Water to Pene-
'trate Fabric (Actual Time-lOa)

of an experiment designed to compare the ef-

fectiveness of four makes of showerproof fa-

bric. Five pieces of etoacha make were chosen at ~lake of fabric
random and subjected test in which the I II III

tlme for water to penet"ate the fabric was

measured. The data in Tabie 12.1 are the ori- IV

ginal times (in seconds) less 100; thus the 23
18
first piece of Fabric I had a penetration time 3 27 49 6
10 41 13 37
of 103s, \'ihichis recorded as 3 in Table 12.1. 7 36 46 6
-14 27 34
19 19 53 90

Totals Ti 25 150 195 18
Means x-. 5 30 39

1

* A convenient table may be found in: Grand total T = n. = 460 "
J.Murdoch and J.A.Barnes. 'Statistical 1
tabl es for Sc :ence, Engineel'ing, Management
and Business Studies', Macl'lillan, London. Grand mean x = T/4 x 5 = 460/20 = 23

~[--_.

~~~nde~_~~.U<ll.'.' Iile ',dm\' I Jill! uf equdt.lOII Cdll he I'lri tLcn dO\'IO
fo( every ub~erv<.lLioll~ ~i1u:...) for C,o(,;,HlljJI f':l

To dev"lop thL analysis of VarlallCC, VIE begin

by noting that anyone of the observations in

fable 12.1 can be considered to consist of

sever-,,] COlllpllllenls.I,lh i~ i Ilu"tr"l ~din

Fi g. 12. 'I, \-lIlh lile Ii nL ul)~,el'Vdll all 01 lhe

third fabric used as an example. Denotlng thlS

observation by X31, its numerical value in This con~eption can be g~neralizcd. Suppose

this experiment being 49, Vie have there are c treatments to be compared and that

n observations have been made on each

treatment. Denote :\I1Y observation by xi"

III Fig. 12.1, this value is represented by the where i is t.he treatment number and J ld the

distance OA, which can be divided into three replicat'on number. The scheme is shown ~n

Table 12.2. The treatment means xi are sample

parts, namely estimate~ of population treatment means ui'

xand the grand mean is an estimate of the

over-all population mean, l'. If we define the

population treatment effects by

DC represents the grand mean X,

CB represents the deviation of the fabric mean
from the X31

grand mean x3 -x, and

BA represents the deviation of observation

from ~he fabric mean = x31-x3.

Thus Equation (12.1) may be written and thi~ is the assumed model of the data.

Because the data ,are classified only according

X31 x+(x,-x)+(X31-X3) to the treatments, this particular form of

experin'ent is called a one-way classification.

The purpose of the expenment 1S to test

whether the treatment means are different,

i.e., tn test the null hypothesis

\1here t3 (= X3-X) me~sures the 'effect' of the

third treatment, i.e., hoVi it differs from the

over-all mean, while e31 (=X31-x3) may be re-

garded as the random error "ssociated with the

observation x31. For this particular observa-
tion, since x31 = 49, x = 23, )(3 = 39, we find

t3 = 39-23 = 16, e31= 49-39 = 10.

against the alternative hypothesis that at

least two of the treatment means are different.

Table 12.2 General Notation for a One-way c
ANOVA
xc1
Treatment xc2

1 2 ... i ...
x11 x21 ... xi 1 ..,
x12 x22 ... xi2 ...

~.: The Test of Significance

...x1j x .. We begin, as always, by $upposing that the
x2j '" lJ xcj null hypotnesis is true. If it is, then the
errcr variance 02 can be estimatea in two in-
dependent ways.

x1n x2n ... x.ln '" xen The first method is to consider the variation

~lithi,l each treatment. For example, the indi-

... T ... Vl<IUaT observations xi 1, xi2, ... , xin on the
1
Totals T1 T2 Tc ith tl'eatment vary about thei r mean xi, so
~leans x- 1
that an estimate of their variance is

x2 '" Xi ... X n

c 2 L - 2

Si = (xiJ·-Yi) /(n-1) ,

c J=1

Grand total T= L T and :his calculation can be carried out for
Grand mean i=1
x- = T/cn 1

each treatment, to yield c separate estimates

~~'10dl'~~_~_U~:.: Illl'',<illf'"i~ntiof equilLlolICJn I)(! "IrilLen dOl'in
lor l'VUj ubservdLiun; '.hU5, fot' "x""'lJll:,
To dev~lop the ~nalysis of varIance, w~ begin
by noting that anyone of the observatIons In This con~cption can be generalized. SUPlJose
rable 12.1 cun be considered to ConSIst of there are c treatments to be compared and that
sever',d cOIIIIJO'1l'n15.1.1ii"s i Jlw,L,'"l,d in n observations have been made on each
Fig. 12.1, ,·,illlIll'f i"sl obsl','vJLion of Lhl' treatment.. Denote .1ny observation by Xij'
third fabric used as an example. Denoting this "here i is the treatmcnt nu"lber dnd j is the
observation by x31, its numencal value in repl icati on number. The scheme is shovm ~n
this experiment being 49, we have Table 12.2. The treatment means Xi are sample
estimate,', of population treatment means ui'
In Fig, 12.1, this value is represented by the and the 9rand mean x is an estimate of the
distance DA, \~hich can be divided into three over-all population mean, 1'. If "e define the
parts, naineIy population treatment effects by

DC represents the grand mean X, and thi~ is the assumed model of the data.
Because the data.are classified only according
CS represents the deviation of the fabric mean to the treatments, this particuhr form of
from the experiwent is called a one-way classification.
The purpose of the experIment IS to test
SA represents the deviation of observation x31 whether the treatment means are different,
from ~he fabric mean = x31-x3. i.e., to test the null hypothesis

where t3 (= X3-X) me~sures the 'effect' of the against the alternative hypothesis that at
third treatment, i.e., how it differs from the least two of the treatment means are different.
over-all mean, vlhile e31 (=X31-X3) may be re-
garded as the random error cssociated with the ~.? The Test of Significance
observation x31. For this particular observa- We begin, as always, by supposing that the
tion, since x31 = 49, x = 23, )(3 = 39, we find null hypotnesis is true, If it is, then the
errer variance 02 can be estimatea in two in-
t) = 39-23 = 16, e)1= 49-39 = 10. dependent ways.
The f'irst method is to consieer the variation
Table 12.2 General Notation for a One-way withi,] each treatment. For example, the indi-
ANOVA VTCllJaT observations Xi 1, xi2, "., xin on the
Treatment - ith t,'eatment vary about thei I' mean xi, so
] that an estimate of their variance is
2 .., i '" c
n
Xj 1 x21 ,., Xi 1 '" xC]
xc2 2::j=l (xiJ,-yY/(n-1)
x12 x22 " , xi2 '"
and ~his calculation can be carried out for
xlj x2j '" xij '" xcj each treatment, to yield c separate estimates

x1n x2n ... x.In '" xcn
Tc
Totals T, T2 '" T .,. Xc
~leans - x-2 '" 1
Xl
-
x·I '"

c

Grand total T=L TI
Grand mean
i=1

x- = T/cn