Walter Shewhart’s average and range chart had been around for more than a decade when, in 1941, W. J. Jennett came up with the chart for individual values and moving ranges (an XmR chart). The XmR is built on the same foundation as the average and range chart, but it involves two unique ideas. First, it works directly with the original data rather than averages; second, it uses the successive differences to compute the limits. These unique aspects of an XmR chart have caused more than one statistician to ask, “Can it really be that simple?”

When confronted with the simplicity created by the two unique aspects of the XmR chart, various statisticians have come up with three erroneous caveats regarding the chart.¹ The first of these is, “The XmR chart requires normally distributed data.” The second is, “The XmR chart will result in too many false alarms.” And the third is, “The XmR chart will miss signals of a process change.” The contradictory nature of the last two reveals the lack of any real theoretical basis for any of these caveats. In this article, I’ll use an empirical example to illustrate the falsehood of each of them.

Normally distributed data

One source for the misconception that the XmR chart requires “normally distributed data” is a faulty understanding of why the average and range charts work. Since the central limit theorem assures us that averages will have a distribution that tends toward normality as the subgroup size increases, many have erroneously thought that the central limit theorem is what makes the average chart work. However, this naive view fails to account for how the range chart works. Ranges have skewed distributions. While the central limit theorem is true, it doesn’t apply to ranges in the same way that it applies to averages. So the central limit theorem doesn’t explain why an average and range chart works. Therefore, it can’t be used to argue that an XmR needs normally distributed data.

A second source for this erroneous idea that the XmR chart requires normality is the notion that you need a fixed probability of a false alarm, as was explained in my May column.² This idea comes from a failure to understand what Shewhart was doing with his generic, fixed-width limits. As he argued on page 277 of his 1931 book Economic Control of Quality of Manufactured Product,³ rather than seeking limits that would always result in a fixed probability of a false alarm, he was seeking generic limits that would merely keep the probability of a false alarm economically small. Based on experience, he found that three-sigma limits centered on the average would do this. Thus, arguing that we need normality to get a fixed probability of a false alarm completely misses the point behind Shewhart’s charts. Without the need to maintain a fixed probability of a false alarm, there’s no need to have “normally distributed data.”

A third source for this hazardous idea that the XmR chart requires normality is a failure to understand what standard deviations tell us. Regardless of what probability model you may use, you will always have approximately 95% within two standard deviations of the average, and approximately 99–100% within three standard deviations of the average. ^{4, 5, 6}

To examine the erroneous idea that the XmR chart requires normality, I modified my bead board to produce a skewed histogram. (This was done by blocking the pin block so that no bead could go below 8, as shown in Figure 1.)

Figure 1: My modified bead board with funnel at 9.5

With the funnel centered at 9.5, I ran 1,041 beads through this modified quincunx. The histogram is shown in Figure 2 along with the proportion at each value. The average is 9.604, the standard deviation statistic is 1.428, and this process is definitely not producing “normally distributed data.”

Figure 2: A “non-normal” process

So we have a physical system that produces data that have a skewed histogram. This histogram ranges from about one standard deviation below the average to about three standard deviations above the average. We’ll use this modified bead board to see how well the XmR chart works when the original data are non-normal.

False alarms

Another erroneous idea about an XmR chart is that you will have too many false alarms. This idea is usually tied up with the erroneous notion that a control chart is supposed to have exactly three false alarms per 1,000 points. Of course, a false alarm cannot happen unless the process is being operated predictably. So to examine the false alarm rate we’ll collect data from our modified bead board without changing the funnel position, thus simulating predictable operation.

Independent of the data in Figure 2, I collected 10 sets of 25 consecutive values from the bead board while it was configured as shown in Figure 1. Each set was used to create an X chart with its own set of limits. While limits may be computed using fewer data, we generally like to have at least 20–25 values in our XmR chart baseline when possible. So here we are using typically small amounts of data to compute our limits. (In statistical jargon each set of limits is based on 15 degrees of freedom, which is half of what is commonly recommended for good estimates.) These 10 X charts are shown in Figures 3 and 4.

Figure 3: X charts for observations 1–125

Figure 4: X charts for observations 126–250

As should be expected when using only 25 values, the limits vary. Nevertheless, the story told by these 10 charts remains the same. This process was operated predictably. With a process that produces skewed data, and with limits based on only 25 data, we have one false alarm (point 78) out of 2,500 points. This is equivalent to four false alarms per thousand. So our symmetric, three-sigma limits result in a reasonably small risk of a false alarm even when our predictably operated process has a skewed histogram.

Predicting process outcomes

How well do these limits, based on 15 degrees of freedom, do in predicting how the process will work? Figure 5 shows the histogram of Figure 2 with the 10 sets of limits from Figures 3 and 4 added. (Remember, the 250 values of the charts are not part of the 1,041 data of Figure 2, so here we’re extrapolating from the observed data of the charts to the process outcomes shown in the histogram.)

Figure 5: Computed limits bracket 99.4% of process outcomes

Three of the 10 upper limits round off to a value of 13, while the remainder round off to either 14 or 15. So here we find at least 99.4% of the 1,041 values falling on or within the computed limits of Figures 3 and 4! While the boundary value places a limit for the process outcomes on the low side, the computed limits cover virtually all of the process outcomes on the high side.

This happens because, regardless of the shape of the histogram, symmetric, three-sigma limits will capture virtually all of the routine variation when a process is operated predictably. Moreover, symmetric, three-sigma limits are robust enough that we don’t need to wait until we have hundreds of data before computing the limits.

Therefore, it’s fallacious to think that an XmR chart will have appreciably more false alarms than will be found on an average chart. And it’s also a fallacy that the data have to be normally distributed before the XmR chart will work properly.

Missed signals

Another erroneous idea about an XmR chart is that it will miss signals of a process change. The likely source of this idea is a naive comparison between the width of the limits on an average chart and the width of the limits on the X chart. While it may seem that we would detect more signals with the tighter limits of an average chart, the fact that the two sets of limits apply to two different random variables makes this direct comparison of the width of the limits bogus.

A rigorous comparison requires tables of the power function for process behavior charts. ^{7, 8} From these tables we find, for example, a theoretical probability of 0.991 that an average chart based on subgroups of size five will detect a two-sigma shift within two subgroups. These same 10 data, when placed on an XmR chart using detection rules 1 and 2, will detect a two-sigma shift with a theoretical probability of 0.973. These probabilities are both virtual certainties, and will be indistinguishable in practice.

Of course, these power functions are based on the assumption of a normal distribution for the original data. So what happens with our skewed data?

Figures 3 and 4 show how the XmR chart works with skewed data when the process is operated predictably. To see how the XmR chart reacts to a process change, we’ll need to move the funnel on the modified bead board and collect additional data. Since we’re only interested in detecting signals of changes that are large enough to be of economic importance, we shall center the funnel over 12.5, which will result in a two-sigma process shift. Observations 251–350 were collected following this two-sigma shift.

These were taken in the order they occurred, divided into groups of 10, and these groups were then added to the 10 charts of Figures 3 and 4. The idea is to see how well the limits computed from the skewed data do in detecting the shift represented by observations 251–350. The resulting charts are seen in Figures 6 and 7.

Figure 6: Charts 1–5 with a two-sigma shift

Figure 7: Charts 6–10 with a two-sigma shift

All 10 charts detect the two-sigma shift. Eight of these charts contain multiple signals within the 10 added observations. Chart 6 has only one point out (point 307), but this point is part of a run of 10 successive values above the central line. And Chart 8 has a single point out (point 329) that admittedly looks similar to the false alarm at point 78.

So how do we separate false alarms from signals of process change? In practice, we don’t need to. False alarms are rare (one out of 250 points here) while signals are not rare (23 outside the limits in 100 values here). If we treat all 24 values that fall above the upper limits as potential signals and look for an assignable cause, then we’ll be right 23 times out of 24, or 96% of the time.

This is why we have a license to hunt whenever we get a point outside the limits. These points are so much more likely to represent a signal than to be a false alarm that we’re not concerned with false alarms.

As signals of economic importance go, the two-sigma shift used here is rather on the small side. So, this example shows that with skewed data, and with limits based on only 25 values, the XmR chart is able to reliably detect signals that are large enough to be of economic importance without an excessive number of false alarms.

Rational sampling

Only two things matter when constructing an XmR chart. One, the XmR chart is intended for use with sequences of values that are logically comparable. You have to organize things so that you’re not comparing apples to oranges. And two, the moving ranges must logically represent the routine variation of the measure being plotted on the X chart. Both of these requirements can only be met by a rational process that incorporates the context for the data and the purpose of the chart. This means that you should always be able to explain what the values on the X chart represent and should avoid sample frequencies that are too high or too low to capture the routine variation. Since both of these requirements depend upon judgment, they’re grouped under the heading of rational sampling.

Summary

XmR charts let you plot a point every time you get a value, which is of utmost importance when your data come along one value at a time. Moreover, every point has to sink or swim on its own. This means that nothing can hide on an XmR chart, which makes it the easiest process behavior chart to use.

Since XmR charts work with skewed data, you don’t need to be concerned about the “normality” of your data. The symmetric, three-sigma limits will still filter out virtually all of the routine variation.

When your skewed process is operated predictably, the XmR chart will filter out the noise of the routine variation without too many false alarms. When false alarms do occur, they’ll tend to look like that in Figure 3—an isolated point that falls just outside the limits.

When your skewed process is operated unpredictably, the problem isn’t false alarms but missed signals. As seen in Figures 6 and 7, changes large enough to be of economic importance (generally shifts of two sigma or greater) are easily detected by an XmR chart.

Once we eliminate the obstacles created by the hazardous ideas about an XmR chart, it becomes the easiest chart to create and use. These hazardous ideas have no foundation in theory or practice. They’re merely superstitious nonsense based on misunderstandings. So, yes, the analysis of your data really can be as simple as an XmR chart, even though the purveyors of complexity may wish it wasn’t so.

Donald J. Wheeler’s complete “Understanding SPC” seminar may be streamed for free; for details, see spcpress.com.

References

1. Wheeler, Donald J. “A History of the Chart for Individual Values.” Quality Digest, June, 2024.

2. Wheeler, Donald J. “Some Hazardous Ideas.” Quality Digest, May, 2026.

3. Shewhart, Walter. Economic Control of Quality of Manufactured Product. D. Van Nostrand, 1931; republished by ASQ, 1980. Reprinted by Martino Fine Books, 2015.

4. Wheeler, Donald J. “What You Need to Know About Lognormal Models.” Quality Digest, Jan., 2024.

5. Wheeler, Donald J. “What You Need to Know About Gamma Probability Models.” Quality Digest, Feb., 2024.

6. Wheeler, Donald J. “What You Need to Know About Weibull Distributions.” Quality Digest, March, 2024.

7. Wheeler, Donald J.; Stauffer, Rip. “When Should We Use Extra Detection Rules?” Quality Digest, Oct., 2017.

8. Wheeler, Donald J. “Detecting a Shift in Process Average: Tables of the Power Function for X-bar Charts.” Journal of Quality Technology, Oct. 1983, vol. 15, no. 4, pp. 155–169.