Thanks for reading and replying!

Actually GMARCH and Jose...
Dr. Wheeler is a great guy, but his ongoing notion that normality is not needed for individuals charts is dead wrong (and demonstrably so). (Of course, x-bar type charts are less sensitive to departures from normality due to the central limit theorem.) Individuals charts are highly susceptible to producing alpha error (adjusting the process when it doesn't need it) and beta error (missing a process shift when it occurs) with departures from normality.

One can sympathize with the attitude, "Ennh, well assuming normality when it really isn't is close enough, and I will take the cost hit in unnecessary adjustments and missed signals in order to avoid having to do those hard [sic] normality tests and those difficult [sic] adjustments to the chart"... if one is in the 1970's and doing hand calculations. However, with modern software, these are easy to do, so why not do it the right way and save yourself money and aggravation?

Testing for normality is ALWAYS important when you get to measuring capability, regardless of chart type since those indices are calculated on the raw data, not the random sampling distribution of the averages. (Unless, of course, your customer cares about your ability to hit specs on average - NOT!) If you didn't take distribution shape into account, you would calculate your capability indices from formulae that assume normality from an estimate of the standard deviation that itself is generated from the theoretical distribution of a dispersion statistic assuming a normal distribution! That is crazy talk! The resulting "capability" will have nothing to do with your actual ability to meet specification. (You could use process performance metrics to quantify past performance which are kind of independent of shape and state control, but you can't calculate capability, which is a statement about future performance...)

And believe me, Deming and Shewhart knew full well the dependencies of the INDIVIDUAL charts and capability indices on the assumption of normality. (My old boss and now colleague Dr. Jeff Luftig worked for Deming at Ford and he is mentioned in Out of the Crisis if you want to look it up.)

Now on to Michael's comment...

As the other comments pointed out, SPC is the tool with which we determine if a process is in control - it would be odd to require a process to be in control before using the tool to determine if it is in control! However, it is true that if the chart indicates a process is out of control, you can't predict what the process is going to do in the future, nor can you use the estimated standard deviation nor can you calculate capability indices because you have no clue as to what it is going to do next. What you CAN and SHOULD do is use the control chart to identify and eliminate the sources of unexpected variation that is throwing it out of control. (Which is another reason why when using the individuals chart it is so important to check for normality - if the process is in control and non-normal, you will end up trying to chase down and trying to fix "differences" that are actually part of the random distribution.)

As above, the central limit theorem does protect you against non-normality of the population if you are using one of the x-bar type charts (the bigger the sample size, the more non-normal the population can be). But it is easy to adjust an individuals chart for non-normality, and it is just as valid once it is. Many processes in real life are totally in statistical control and non-normally distributed, so an individuals chart adjusted for whatever non-normality you have will still give you what you want: a heuristic for determining when to react and when not to. Plus, you can calculate capability for these non-normal but in control processes.

Michael, if you re-read the article, I think you will see why the aggregate data are non-normal - it is the sum of five (or three) normal distributions with different means (so it is probably platykurtic). The correct approach is to plot each "stream" on an individuals chart (and as you say, the streams are normally distributed so we can use the default calculations for the limits). In fact, even if the aggregate data happened to pass a normality test (beta error!) we STILL would not want to put the data on an x-bar and R chart because of the out of control condition on the means indicates something weird about the sources of variation. And don't forget that we also want to continue our investigations into why there is that difference across the width.