Ranges vs. Standard Deviations: Which Way Should You Go?

A look at the mathematical difference between X-bar R and X-bar S

Recently, in one of the many online discussion groups about quality, Six Sigma, and lean, this question was posed: “Can X-bar R and X-bar S be used interchangeably based on samples size (n) if the subgroup size is greater than one and less than eight?” Answers varied, of course.

In some of these discussion groups, you get to see how far rule four of W. Edwards Deming’s funnel experiment has pushed some training programs off in one direction or another, especially when it comes to statistical process control (SPC). One set of answers that surprised me, though, came from a couple of consultants in France, who said, “Clearly not... the question is about a sample of 1 to 8. [The] response is definitely no. You can’t calculate a standard deviation with a sample of one or two. A sample higher than 8 is highly recommended.”

The point they were trying to make was that for subgroups of size eight or smaller, you could only use X-bar R charts.

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Comments

It was actually faster

One note: the first time I ran this model, it took a little less than 12 minutes. I added several outputs to that initial model, and it actually got faster. I ran it four times in all, and the last three runs took just over 6 minutes each.

The s chart is slightly more powerful

It is actually possible to calculate the chance of detecting a given change in process variation for the R chart and the s chart. The latter uses the chi square distribution, and the former is somewhat more complicated.

The powers of both tests are equal for a sample of 2, which is not surprising. The power of the s chart increases relative to that of the R chart for samples of 3 or more because the s statistic uses all the information, but the difference is not really much.

Efficiency

A great simulation, but this could have been done much faster and with actual theory to support. Well, maybe not faster since it requires calculus but rigorous.

In graduate statistics you learn about Efficiency as it relates to statistics. A standard deviation is an "efficient" statistic at any sample size (compared with other estimates). The range is "efficient" at n=2 and then begins to slowly degrade as n increases. However, the efficiency doesn't degrade significantly as related to standard deviation until n=9. Hence the rule about n=8.

I did this exercise as part of a graduate class 30 years ago so I'm rusty on the mechanics but it has stuck with me all these years.

Thanks for the comment

I'd be interested to know what you meant by "this could have been done much faster and with actual theory to support."

Efficiency is not really the prime consideration when using control charts...they are more about sensitivity. William Levinson's comment (along with an email from another friend) have made me think I might have expanded this simulation (which was about limits) to include known signal detection, especially when the underlying distributions are skewed. That wasn't the question I was trying to answer here, but that would provide a more comprehensive treatment of the difference betweent the two approaches.