Nov 7 2025
The Lowdown on the Range Chart
To use in-process measurements for quality control 100 years ago, Walter Shewhart proposed the \bar{X}-\sigma charts. It entailed arranging workpieces in rational subgroups, summarizing measurements by subgroup into means and standard deviations, charting both, and checking new values against control limits.
When Harold Dodge tried to implement the \bar{X}-\sigma charts at Western Electric, the engineers balked at calculating sample standard deviations with paper, pencils, and slide rules. To gain acceptance, Dodge let them use sample ranges and plot them in R charts instead. While easier to understand and to use daily, sample ranges are mathematically more complex and more sensitive to extreme values than standard deviations.
The SPC literature glosses over the motivation for the R chart and its math. It provides recipes for using these charts, but no explanation. We shouldn’t ask manufacturing professionals to use a tool without explaining its purpose and its theory. This is what this post is trying to remedy.
Like all control charts, the R chart uses limits calculated for the Gaussian distribution. As no simple formula is available for the R chart, setting control limits for it requires numerical approximations that must have consumed months for human computers in 1924. Today, you can replicate them instantaneously with software. These calculations reveal that the \pm 3\sigma limits in the books for the range chart do not actually encompass the 99.73% of the distribution that they do in \bar{X} charts.
The R chart was an ingenious workaround to technical and human constraints of the 1920s that no longer exist. Today, rather than blindly applying these tools, we must draw inspiration from their inventors and develop solutions to meet the process capability challenges we are actually facing.
The people of the Honda plant in Anna, OH, claim to make the best engines in the world. On the floor, there is neither a single control chart nor any engineer trained in SPC.
Apr 3 2026
Deviating Standard Deviations
This basic concept deserves revisiting. The following is from a blog post from 2022 hosted by a supplier of statistical software intended to explain the meaning of some notations in plain, simple terms:
The author calls two different things by the same name. If the standard deviation of each variable is 1, how could its expected value be anything else? The confusion within this nonsensical statement is the same we make when we equate the temperature of a soup with a thermometer reading. In our mental model of a bowl of soup, it has a temperature that exists regardless of our ability to measure it, and the thermometer reading is only an estimate of it.
For the purposes of eating soup, confusing the two is harmless, unless the thermometer, poorly calibrated, always gives you an answer that is 15°F off. This is the situation we have with the most commonly used estimator of the standard deviation of a random variable from a small sample. It is biased, and c_4(n) is a correction factor applicable when the random variable is Gaussian.
To describe c_4(N) accurately, we need to dig into probability theory. It is, in fact, the expected value of the estimator S=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}\left ( X_i -\bar{X} \right )} of the standard deviation from a sample of N independent Gaussian variables \left ( X_1, \dots, X_N \right ) with unit standard deviation, \sigma = 1. This is an accurate statement, but every term in it needs an explanation.
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By Michel Baudin • Technology 0 • Tags: Control Charts, Probability, Quality, Six Sigma, SPC, Standard Deviation, statistics