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Chapter 7Is the Six Sigma statistical modeltechnically sound?
When I use a word, Humpty Dumpty said, in a rather scornful tone
it means just what I choose it to mean neither more nor less.
The question is, said Alice, whether you can make words mean
so many different things. The question is, said
Humpty Dumpty, which is to
be master thats allThrough the looking glass, Lewis Carroll
Overview
Purists challenge the statistical component of the Six Sigma model on the
grounds that it is, at worst, fundamentally flawed and, at best, based on very
tenuous grounds. There are a number of reasons for this. The critique and
response to the critique given in this chapter are prompted by four constructive
and positive considerations:
to place potential users in a better position to deal with technical queries on
the standard Six Sigma statistical model as they arise; to point out frailties in the conventional or original Six Sigma statistical
model and to propose more appropriate metrics where required by a partic-
ular situation;
to indicate the probable marketing reasons for the model that has given the
initiative its name; and
to put this matter of model frailty into proper perspective in relation to the
merits of the overall Six Sigma business initiative.
There are five very controversial aspects of the Six Sigma statistical modelthat are criticized, cause confusion, and hence prompt discussion and
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Is the Six Sigma statistical model technically sound? 209
resolution:
1 the different use of the universally accepted term denoting standard deviation,
the Greek term sigma;
2 the method of the linking of 6 Sigma to a claimed world-class performance
standard of 3.4 defects per million opportunities;
3 the claim that 3.4 defects per million opportunities represent world-class
performance;4 the use of the term defects when non-conformities or undesirable events is
intended;
5 the multiple meaning of the term opportunity.
Let us deal head-on with the most significant criticisms of the Six Sigma
model. The critique and response in the body of this chapter are dealt with in as
non-statistical a manner as possible. A more statistical approach, to justify the
conclusions drawn is given in the Chapter 8. These criticisms are dealt with either
by explaining and justifying the rationale, advising in interpretation and appli-
cation, and/or proposing alternative approaches in particular circumstances.
Sigma versus sigma
Criticism
The originators of the Six Sigma initiative use a unit of measurement, a Sigma, to
measure performance, the higher the value of Sigma the better the performance.
For example, a 6 Sigma process is rated better than a 5 Sigma one. This meaning
contrasts with the standard statistical meaning and usage of the term, sigma. The
symbol, sigma, a Greek letter symbolizing a statistical measure of variation, termed
the standard deviation is in universal use. The lower the value of the standard devi-
ation, sigma, the less the variation present and the better the performance of a
characteristic. The opposing meanings inevitably cause confusion. This confusioncan be compounded as the internationally accepted standard sigma provides the
derivation of the Sigma metric used in Six Sigma projects and also the relationship
between non-conformities per million opportunities and the Six Sigma, Sigma.
Throughout this book, to distinguish between the two uses of the term the
Six Sigma Sigma is represented by an upper case initial letter thus, Sigma.
The standardized conventional use of the term is represented wholly in lower
case italic, thus sigma.
Response to criticism
An advantage of the Sigma measure is its simplicity and practicality. This appeals
to all those who do not wish to get too embroiled in statistical niceties but
just want a simple readily understandable scale of performance measurement.
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The fact that 6 Sigma denotes something better than 5 Sigma and that
5 Sigma is better than 4 Sigma, and so on makes good practical sense to them
and they are quite prepared to run with it.
Others, who are already actively engaged in the application of statistical
process control, will readily understand the use of Sigmas in a different way to
sigmasbut may feel uncomfortable and uneasy with both the frailty of the under-
lying concept and the outcome. This is because an alternative soundly basedapproach is available with the use of the internationally defined measured data
process capability indices such as Cp. A Cp of 3 indicates less variation than a Cp
of 2, and so on. Such measures require the pre-establishment of both process
stability and a knowledge of the underlying pattern of variation.
Linkage of Sigma value to defect rate
Criticism
The value of Sigma is linked to a defects per million opportunities value in a
pseudo-statistical manner in two ways. Two gross assumptions are made.
1 The pattern of variation of the characteristic is symmetrical and bell shaped.
This pattern is termed normal or Gaussian distribution by statisticians.
2 A drift in one, unspecified, direction of the value of a characteristic amounting
to 1.5 standard deviations.
Response to criticism 1
It should be noted that the use of the term normal here is not that of usual but
rather just a statistical name given to a particular distribution. In the technical
sense it is quite appropriate to state, quite categorically, that: in diagnostic
activities leading to process improvement and in the estimation of process per-
formance normality of data should never be assumed. An erroneous assumptionof normality can, and undoubtably will, lead to false trails and take one up the
garden path in Six Sigma project-improvement activities.
Having said this, we need to recognize that a lot of people throughout the
world do and have being doing so, very successfully in the control of processes,
for a great many years. Walter Shewhart (Shewhart, W. A., 1931) introduced sta-
tistical quality control charts for both measured and attribute data. Standard
action control limits on such charts are set, quite simply, at 3 sigma (standard
deviations) of the measure plotted. This is so for both measured and attribute
characteristics. The assumption is that the data are normal when it is patently
not for range and standard deviation charts and frequently not for measured
individuals and attribute ones. Such procedures are, in general, used throughout
the world, and are specified by the International Standards Organization (ISO)
and the British Standards Institution in various standards. Examples are
210 Six Sigma: Continual Improvement for Businesses
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