TESTING THE ASSUMPTIONS OF TESTING THE ASSUMPTIONS OF VARIABLES CONTROL CHARTS AND AN VARIABLES CONTROL CHARTS AND AN

APPLICATION ON FOOD INDUSTRYAPPLICATION ON FOOD INDUSTRY

Berna YAZICI

Department of Statistics, Anadolu University Eskisehir,Turkey

E-mail:bbaloglu@anadolu.edu.tr

ABSTRACTABSTRACT

In this study the statistical assumptions to construct

variables quality control charts have been held. Testing those

assumptions are mentioned and the solutions for the researcher,

in case of violation of the assumptions are also explained.

INTRODUCTIONINTRODUCTION

Quality is generally desirable characteristics of a product or

service should have. The customers have many options to select a

product or service. So companies need improve the quality of they

produce to survive. Quality improvement is the reduction of

variability in processes and products. Variability is described by

statistical methods. Control charts are one of the tools that is used

to detect whether the process is under control or not. But those

charts may cause misunderstandings if the researchers make

decisions in case of violation some assumptions. These

assumptions are uncorrelated measurements, normality,

homoscedasticity and homogeneity of means.

INTRODUCTIONINTRODUCTION

For application, 144 measurements are taken from a factory

that produces wafers. The thickness of wafers is in question for

statistical process control studies in this company. For the study

16 samples, with 9 measurements each, are used. All samples are

taken by the same worker half an hour periodically. Each of the

assumptions mentioned above has been tested on this data set.

Recommendations in case of violation of each assumption are

mentioned.

1. UNCORRELATED MEASUREMENTS ASSUMPTION

All the samples selected randomly are independent of the one immediately preceding and the one immediately following, briefly independency of the measurements.

• In Eq. L is the amount of lag.

XXX

XXLXnXLXXLXXrnL 2

)...2211(

• In runs test, duration of completed runs (d) is important. The expected number of completed runs of deviations is and the expected number of completed runs of all durations is

1d2

1dnf̂

2

3n)f̂(E

FOR THE DATA SET OF WAFER THICKNESS

• Lower point: –0.462 upper point is 0.328. The result is not between the confidence interval limits

54.0r161

• Runs test  

= 45.71 > =2.167 2calc

2table

We reject the null hypothesis of uncorrelated measurements

1. UNCORRELATED MEASUREMENTS ASSUMPTION

If the assumption is violated

• In this case researcher may fit an ARIMA model and apply standard control charts to residuals instead of the raw data. Residuals will give uncorrelated results.

• Exponentially weighted moving average (EWMA) control charts can be used by moving centerline, with control limits based on prediction error variance.

• To decide whether or not an autocorrelated process may be considered in control, one must investigate the reasons for the autocorrelation. After that, it will be easier to eliminate the autocorrelation by using an engineering controller. In this case, the reason of the autocorrelation can be determined and uncorrelated measurements can be held.

• One way to remove autocorrelation is taking the samples in larger sampling intervals if the process structure is suitable. In this way, Shewhart control charts become appropriate.

2. NORMALITY ASSUMPTION

The distribution of means will be normal if the population is normally distributed.

2 test of goodness of fit

• Shapiro-Wilk’s W test for normality

• where bi is calculated as follows mi representing

the expected values of the order statistics from a unit normal distribution

2

i

2n

1iii

)XX(

)Xb(

W

n

1i

2i

ii

m

mb

• Graphical methods that the researcher can apply using computer packages for testing the normality, such as Q-Q Plot, Lilliefors Test

FOR THE DATA SET OF WAFER THICKNESS

 

= 0.959 < =1.635

2calc

2table

• Shapiro-Wilk’s W test for normality Wcal = 0.9694 critical value for =0.05 and 16 from table is 0.887. Wcal > W(16; 0.05)

We cannot reject the hypothesis of normality.

2. NORMALITY ASSUMPTION

• The Camp-Meidell adjustment for normality can be made

• According to Tchebycheff inequality, no matter the

shape of the distribution

• If the population is not too skewed and unimodal larger sample sizes

suffice the normality assumption due to central limit theorem

• The development of X-bar, R and S chart mechanics is based on the

process metric being normally distributed. However, the chart itself is

robust to deviations from normality depending on the central limit theorem

2z25.2

100100

2z

100100

If the assumption is violated

3. HOMOGENEITY OF VARIANCES

The variances within each of the samples must be equal

• One way to test homogeneity of variances is Cochran’s g test

• 0 test

2

2

sofsum

estsarglg

m/12m

22

21

2T

0)s...ss(

s 1 test

m/12m

22

21

2m

22

21

1)s...ss(

m/)s...ss(

• Bartlett’s test

  

cM2

m

iisinpsmNM

1

2ln)1(2ln)(

mN1

1n1

)1m(31

1cm

1i imN

s)1n(

s

m

1i

2ii

2p

FOR THE DATA SET OF WAFER THICKNESS

0 = 1.727 > table = 1.41

1=1.31 >table=1.23

• Bartlett’s test  

= 59.846 > 2calc

2table

We reject the null hypothesis of equal variances.

3. HOMOGENEITY OF VARIANCES

• Taking new samples with equal number of observations may be the best

solution

If the assumption is violated

4. HOMOGENEITY OF MEANS

The control charts are constructed with the homogeny samples from a process

• 0 test

• ANOVA test Before constructing an ANOVA test, one must be sure whether there are extraordinary sample mean or not

m/12m

22

21

2T

0)s...ss(

s 

1jn

1i1ij

XX

XXr

2w

2B

s

sF

FOR THE DATA SET OF WAFER THICKNESS

• Critical value from Dixon’s table is 0.507>0.15 We cannot reject the H0 hypothesis and we conclude that there are not any extraordinary sample mean among 16 sample means.

• Fcal = 2.566>F0.05;15;128=2.11

We conclude that the means are not homogenous

4. HOMOGENEITY OF MEANS

• One can select new samples by equal time intervals

If the assumption is violated

The methods described here are summarized by a flow chart on next four slides

Test uncorrelatedmeasurements

assumption

Use a test depending on the circular

autocorrelation coefficientUse runs test

Fit ARIMA modelUse EWMA

control charts

Assumptionsatisfied

Research the reasonof autocorrelation

Take samplesin larger sample

intervals

Test the normalityassumption

No

Yes

Test the normalityassumption

Use 2 testUse

Shopiro-Wilk’s W test

Use Camp-Meidell

adjustment

UseTchebycheff

inequality

Assumptionsatisfied

Take larger samplesOnly use

X-bar charts

Test homogeneityof variancesassumption

No

Yes

Use Graphical methots

Test homogeneityof variancesassumption

Use

Cochran’s g testUse 0 test

Take new samples with equal numbers

of observations

Assumptionsatisfied

Test homogeneityof means

assumption

No

Yes

Use 1 testUse

Bartlett’s test

Test homogeneityof means

assumption

Use ANOVA

test

Take new samples by

equal time intervals

Assumptionsatisfied

Construct the control charts

No

Yes

CONCLUSIONS AND RECOMMENDATIONS

In statistical process control studies, variables control charts are one of the best guide for the researcher to detect the changes in the process. Before constructing those charts firstly some assumptions must be tested. The assumptions in question are uncorrelated measurements, normality, homogeneity of variances and homogeneity of means. To avoid the misunderstandings and wrong interpretations of these charts, one should test those assumptions and if the assumptions are satisfied, then the charts must be constructed.

5.REFERENCES

[1] Montgomery D., Introduction to Statistical Quality Control, Third Edition, John Wiley & Sons. Inc.,

1996.

[2] Farnum N. R., Modern Statistical Quality Control and Improvement, Duxbury Press, 1994.

[3] Cowden J. D., Statistical Methods in Quality Control, Prentice-Hall Inc., 1957.

[4] Şentürk S., Niceliksel Kalite Kontrol Grafiklerinin Varsayimlarinin Sinanmasi ve Bir Uygulama, Master

of Science Thesis, Graduate School of Natural and Applied Sciences, Statistics Program, Anadolu

University, 2002.

[5] Kolarik W. J., Creating Qulity Process Design for Results, McGraw-Hill, 1999.

[6] Kolarik W. J., Creating Quality Concepts, Systems, Strategies and Tools, McGraw-Hill, 1995.

[7] Sahai H. and Ageel M., The Analysis of Variance: Fixed Random and Mixed Models, Boston:

Birkhauser, 2000.

[8] Hansen L. B., Quality Control: Theory and Applications, Prentice-Hall Inc., 1963.

[9] DeVor R. E., Chang T. and Sutherland J. W., Statistical Quality Design and Control; Contemporary

Concepts and Methods, Prentice-Hall Inc., 1992.

[10] Summers D. C. S., Quality, Prentice-Hall Inc., 1997.