• Improvements on the traditional R and s charts for the normal distribution; calculation of exact quantiles of the sample range and sample standard deviation allow the user to set control limits with known risks (e.g. 0.00135 for Shewhart-equivalent risks)
• Distribution fitting for two and three parameter gamma and Weibull functions
• Distribution fitting of left-censored distributions, for applications in which there is a lower detection limit for the instrument involved.
  
• Multiple attribute control charts that use exact control limits for the binomial and Poisson distributions, thus rendering the traditional p, np, c and u charts (which rely on the normal approximation) obsolete. These are easily deployable to spreadsheets with no programming beyond the spreadsheet functions themselves.
  
• Range charts for the gamma and Weibull distributions
• Confidence limits for the nonconforming fraction (and thus the process performance index) for normal, Weibull, and tentatively gamma distributions
• Effect of gage reproducibility and repeatability on control charts and process performance indices
  
• A chapter on multivariate control charts
  

• Quantile quantile plots for normal and nonnormal distributions
• Chi square goodness of fit tests for normal and nonnormal distributions
• Distribution fitting for the gamma and Weibull distributions (including threshold parameters)
• Distribution fitting for left-censored Weibull and gamma distributions, for applications in which there are lower detection limits for a quality characteristic such as a trace impurity
  
• Cumulative distribution functions for the sample ranges of the normal, gamma, and Weibull distributions. This technically makes the traditional R chart (whose control limits rely on a normal approximation) obsolete; the user can select control limits with an exact Shewhart-equivalent risk of 0.00135 at each end, or anything else he or she chooses.
• Confidence intervals for the nonconforming fraction (tail area) of the normal, Weibull, and tentatively [1] gamma distribution, thus allowing the user to quote confidence limits on the process performance index.

• The range charts do NOT rely on normal approximations to the distribution of the actual range; they use the actual cumulative distribution function of the range.
  
• We can also calculate point estimates the process performance indices of these distributions, and confidence intervals for the performance index of a Weibull distribution.
• Little work has been done on confidence intervals for the survivor function of a gamma distribution, although we were able to reproduce the results from the only example of which we know.)

Shewhart control limits

Process capability indices CPU and CPL reflect the process' ability to meet the upper and lower specification limits (USL, LSL) respectively, and Cpk is their minimum. When Cp=2, we have a "Six Sigma" process, one that (if centered on the nominal) has six standard deviations between each specification limit and the mean. Such a process will have two parts per billion nonconforming (1 ppb at each end). Motorola assumes the possibility of a 1.5 sigma shift, which will result in 3.4 ppm nonconforming.

The Problem: What if it's not a bell curve?

There are two points that are out of control. The false alarm rate should be 0.135% at each control limit so, with 100 points, there's about a 27% chance of getting one false alarm. Two are far less likely. Furthermore, there is a problem with the way the points scatter around the center line.

alpha is the shape parameter gamma is the scale parameter delta is the threshold parameter (in reliability statistics, the guarantee time) The mean is alpha/gamma and the variance is alpha/gamma^2. Thus the mean is 2 and the standard deviation is the square root of 2. The long upper tail accounts for the two supposedly out of control points in the traditional Shewhart chart. The false alarm risk of the upper control limit is not 0.00135 but 0.014, more than 10 times what we expect.

F(x)= cumulative distribution Q(p)= quantile function. A guess must be provided for x1. rnd_gamma(z) generates a random number from the indicated gamma distribution. z is a dummy variable. Note that rnd(1) returns a random number from the uniform distribution [0,1], i.e. a random quantile.

The central limit theorem (CLT) says that, if we take a big enough sample, the sample averages will follow a normal distribution no matter what the individual measurements do. There are, however, a couple of problems with this:

1. It might not be convenient or possible to take a large sample. Some measurements, like impurity levels (in chemicals) or particle counts (in semiconductor processing equipment), yield only individual measurements. This gets into the subject of the rational subgroup. Five impurity measurements from one chemical batch does not constitute five independent representations of the process! It does not reflect the between-batch variation, it reflects only the within-batch variation (plus any gage repeatability variation).
2. Individual measurements, not averages, are in or out of specification. We cannot get accurate capability indices unless we use the underlying statistical distribution.
3. Transformations may behave sufficiently normal for SPC purposes but they do not, in our experience, yield accurate process performance indices.
• We have an example in which a Johnson transformation of a gamma distribution yields a beautiful normal probability plot, but the estimated nonconforming fraction is four times too large.
• We have another example (Levinson, Stensney, Webb, and Glahn, 2001) in which the square root transformation of a gamma distribution yields data that are apparently normal,  and the square root of the particle count data might work quite well on a control chart with normal control limits. The estimated nonconforming fraction is, however, off by an order of magnitude.
  

• Upper control limit: 0.99865 quantile, for a 0.00135 false alarm risk at the upper end
• Center line: 0.50 quantile or median. (The mean is the median of the normal distribution.) This allows us to use the Western Electric Zone C test for runs of eight consecutive points above or below the center line.
• Calculation of the appropriate quantiles for the Zone A and Zone B tests is also possible.
• Lower control limit: 0.00135 quantile, for a 0.00135 false alarm risk at the lower end.

This looks a lot better. The upper control limit, 8.9, is the 0.99865 quantile of this distribution. The median, 1.68, is a little below the mean.

The less people know about how laws and sausages are made, the better off they are. (Otto von Bismarck)

What does the capability index measure? It reflects the number of standard deviations between the specification limit and the mean. And that number (the standard normal deviate), in turn, should reflect the nonconforming fraction: the proportion of units that will be outside the specification limit. This calculation is quite straightforward and well-established for the normal distribution.

In this example, suppose for simplicity that the upper specification limit is 6.243 (the mean plus three standard deviations, the same as the UCL). This is not a capable process as the CPU is only 1. We have already seen that, if the distribution is normal, the nonconforming fraction above the USL will be 0.00135. We also saw that, for this gamma distribution, it's really 0.014, more than ten times as much! (Furthermore, since it's impossible to get less than zero from this distribution, there may well be no lower specification-- which is true for impurity levels and particle counts. We don't care how few impurities are in the product! In this case, the Cp and CPL indices don't exist, and Cpk=CPU.)

A capability index report that assumes a normal distribution can be off by several orders of magnitude (in terms of reflecting the nonconforming fraction) when the underlying distribution is nonnormal.

A nonconforming fraction of 0.0138 would result from a normal distribution whose USL was 2.195 standard normal deviates above its mean. This corresponds to a CPU of 0.732. This process has an equivalent Cpk of 0.732, which is even worse than the 1.00 reported.

Our papers on this subject:

• Levinson, W. "Watch Out for Nonnormal Distributions of Impurities," Chemical Engineering Progress, May 1997, pp. 70-76.
• Levinson, W. "Approximate Confidence Limits for Cpk and Confidence Limits for Non-Normal Process Capabilities," in Quality Engineering, 9(4), 635-640 (1997)
• Levinson, William and Polny, Angela. "SPC for Tool Particle Counts," Semiconductor International, June 1999.
• Levinson, "SPC for Real-World Processes: What to do when the Normality Assumption Doesn't Work." Presented at the ASQ's Annual Quality Conference (2000) in Indianapolis
• Levinson, W.A., Stensney, Frank, Webb, Raymond, and Glahn, Ronald. 2001. "SPC for Particle Counts," Semiconductor International, 10/01