Understanding Standard Deviation and Control Charts

Jay Arthur
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Many people ask: "Why aren't my upper and lower control limits (UCL, LCL) calculated as:
µ + 3sigma (where µ is the mean and sigma is the standard deviation)?" If you are using a Levey Jennings chart, then that IS how your control limits are calculated.

However, if you are using another other control chart, you have to understand some key, underlying statistics: variation, standard deviation, sampling and populations.

Variance (stdev^{^2}) is the average of the square of the distance between each point in a total population (N) and the mean (µ).

If your data is spread over a wider range, you have a larger variance and standard deviation. If the data is centered around the average, you have a smaller variance and standard deviation.

Standard deviation (stdev or sigma) is the square root of the variance:
And it can be estimated using the average range (Rbar) between samples (Rbar/d₂) when the number of subgroups is 2-10, or using standard deviation Sbar/c₄ when n>10.
Rbar = Rave = ΣR_i/n

Sampling: Early users of SPC found that it cost too much to evaluate every item in the total population.. To reduce the cost of measuring everything, they had to find a way to evaluate a small sample and make inferences from it about the total population.

Understanding Control Chart Limits:

Ask yourself this question: "If a simple formula using the mean and standard deviation would work for any data, why are there so many different control charts?"

The short answer: to save money by measuring small samples, not the entire population.

Long answer: When using small samples or varying populations, the simple formula using the mean and standard deviation just doesn't work, because you don't know the average, µ, or sigma of the total population, only µ or sigma of your sample.

Why are there so many control charts? Because:
You have to estimate µ and sigma using the average and range of your samples.

In variable charts, the XmR uses a sample size of 1, XbarR (2-10) and XbarS (11-25). These small samples may be taken from lots of 1,000 or more.

In attribute charts, the c and np chart use small samples and "fixed" populations; the u and p charts have varying populations. So, you have to adjust the formulas to compensate for the varying samples and populations.

To reduce the cost of inspection at Western Electric in the 1930s, Dr. Walter S. Shewhart developed a set of formulas and constants to compensate for these variations in sample size and population. That's why they are sometimes called Shewhart Control Charts.

Reference: You can find these in any book on statistical process control (e.g., Introduction to Statistical Process Control, Montgomery, Wiley, 2001, pgs 207-265).

Other Resources

• Levey Jennings Chart
• Control Chart Formulas
• QI Macros Control Chart Wizard that selects the right control chart based on your data

Control Charts are just one of the tools included in the QI Macros for Excel SPC Software for Excel.

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