Sigma (the lower-case Greek letter σ) is used to represent standard deviation
(a measure of variation) of a population (lower-case 's', is an estimate, based
on a sample). The term "six sigma process" comes from the notion that if one has
six standard deviations between the mean of a process and the nearest
specification limit, there will be practically no items that fail to meet the
specifications. This is the basis of the Process Capability Study, often used by
quality professionals.
The term "Six Sigma" has its roots in this tool, rather
than in simple process standard deviation, which is also measured in sigmas.
Criticism of the tool itself, and the way that the term was derived from the
tool, often sparks criticism of Six Sigma.
The widely accepted definition of a six sigma process is one that produces
3.4 defective parts per million opportunities (DPMO). A process that is normally
distributed will have 3.4 parts per million beyond a point that is 4.5 standard
deviations above or below the mean (one-sided Capability Study). This implies
that 3.4 DPMO corresponds to 4.5 sigmas, not six as the process name would
imply. This can be confirmed by running on QuikSigma or Minitab a Capability
Study on data with a mean of 0, a standard deviation of 1, and an upper
specification limit of 4.5. The 1.5 sigmas added to the name Six Sigma are
arbitrary and they are called "1.5 sigma shift" (SBTI Black Belt material, ca
1998). Dr. Donald Wheeler dismisses the 1.5 sigma shift as "goofy".
In a Capability Study, sigma refers to the number of standard deviations
between the process mean and the nearest specification limit, rather than the
standard deviation of the process, which is also measured in "sigmas". As
process standard deviation goes up, or the mean of the process moves away from
the center of the tolerance, the Process Capability sigma number goes down,
because fewer standard deviations will then fit between the mean and the nearest
specification limit. The notion that, in the long term, processes usually do not
perform as well as they do in the short term is correct. That requires that
Process Capability sigma based on long term data is less than or equal to an
estimate based on short term sigma. However, the original use of the 1.5 sigma
shift is as shown above, and implicitly assumes the opposite.
As sample size increases, the error in the estimate of standard deviation
converges much more slowly than the estimate of the mean. Even with a few dozen
samples, the estimate of standard deviation often drags an alarming amount of
uncertainty into the Capability Study calculations. It follows that estimates of
defect rates can be very greatly influenced by uncertainty in the estimate of
standard deviation, and that the defective parts per million estimates produced
by Capability Studies often ought not to be taken too literally.
Estimates for the number of defective parts per million produced also depends
on knowing something about the shape of the distribution from which the samples
are drawn. Unfortunately, there are no means for proving that data belong to any
particular distribution. One can only assume normality, based on finding no
evidence to the contrary. Estimating defective parts per million down into the
100s or 10s of units based on such an assumption is wishful thinking, since
actual defects are often deviations from normality, which have been assumed not
to exist.
