The most common way to measure variability, at least for data coming from a normal distribution, is standard deviation. Another less common approach is to use mean range. Standard deviation is mathematically simple but operationally a little complicated. Mean range, on the other hand, is complicated to analyze mathematically but operationally very simple.

## ASQ/ANSI Z1.9

The ASQ/ANSI Z1.9 standard, Sampling Procedures and Table for Inspection by Variables for Percent Nonconforming, gives several options for measuring variability, and one of these is the mean range method. Specifically, several samples of five items each are drawn, and the average of the ranges is the variability estimate. The ANSI Z1.9 standard grew out of, and is still very similar to, the US military standard MIL-STD-414 from 1957. The ANSI standard, last updated in 2018, is not that different from the military standard from six decades earlier.

The average mean is obviously simple to carry out: take five samples, subtract the smallest value from the largest, and write that down. Repeat this a few times and average the numbers you wrote down. No squares, no square roots, easy to carry out manually. This was obviously a benefit in 1957, but not as much now that computers are ubiquitous. The more important advantage today is that the mean range can be more robust for heavy-tailed data. More on that here.

## Probability distribution

The distribution of the range of a sample is not simple to write down, even when the samples come from a normal distribution. There are nice asymptotic formulas for the range as the number of samples goes to infinity, but five is a bit far from infinity [1].

This is a problem that was thoroughly studied decades ago. The random variable obtained by averaging *k* ranges from samples of *n* elements each is denoted

or sometimes without the subscripts.

## Approximate distribution

There are several useful approximations for the distribution of this statistic. Pearson [2] evaluated several proposed approximations and found that the best one for *n* < 10 (as it is in our case, with *n* = 5) to be

Here σ is the standard deviation of the population being sampled, and the values of *c* and ν vary with *n* and *k*. For example, when *n* = 5 and *k* = 4, the value of ν is 14.7 and the value of *c* is 2.37. The value of *c* doesn’t change much as *k* gets larger, though the value of ν does [3].

Note that the approximating distribution is chi, not the more familiar chi square. (I won’t forget a bug I had to chase down years ago that was the result of seeing χ in a paper and reading it as χ². Double, triple, quadruple check, everything looks right. Oh wait …)

For particular values of *n* and *k* you could use the approximate distribution to find how to scale mean range to a comparable standard deviation, and to assess the relative efficiency of the two methods of measuring variation. The ANSI/ASQ Z1.9 standard gives tables for acceptance based on mean range, but doesn’t go into statistical details.

## Related posts

[1] Of course every finite number is far from infinity. But the *behavior* at some numbers is quite close to the *behavior* at infinity. Asymptotic estimates are not just an academic exercise. They can give very useful approximations for finite inputs—that’s why people study them—sometimes even for inputs as small as five.

[2] E. S. Pearson (1952) Comparison of two approximations to the distribution of the range in small samples from normal populations. *Biometrika* 39, 130–6.

[3] H. A. David (1970). Order Statistics. John Wiley and Sons.