Shewhart control chart constants: A2, A3, B3, B4, D3, D4, d2, c4

Updated: 2026-08-04 · ISKRIT

Control limits on a Shewhart chart are not computed from a normal distribution directly. They come from tabulated factors that convert an average range or an average standard deviation into a three sigma distance for the given subgroup size.

The table

What multiplies what

A2 multiplies the average range on an averages chart, A3 multiplies the average standard deviation for the same purpose. B3 and B4 bound a standard deviation chart, D3 and D4 bound a range chart. d2 converts an average range into an estimate of the process standard deviation, and c4 does the same correction for an average standard deviation.

So the limits on an averages and range chart pair are: centre line plus or minus A2 times the mean range for the averages, and D3 and D4 times the mean range for the ranges themselves.

The zeros are not rounding

B3 is zero up to n = 5 and D3 is zero up to n = 6 because the standard does not define a lower control limit for those subgroup sizes at all. With so few observations the lower tail of the range distribution carries no useful signal, so the chart has an upper limit only. A calculation that prints a negative lower limit for n = 4 has extrapolated a formula past the point where the table stops.

Individuals charts

When there is one measurement per occasion, variation is estimated from the moving range of consecutive pairs, so n = 2 and d2 = 1.128. The limit factor for the individuals chart is tabulated as 2.66, which is three divided by 1.128 rounded. The moving range chart itself uses D4 = 3.267 with no lower limit.

Questions

Where do the numbers come from? They are expectations of the range and standard deviation of samples from a normal distribution, so they are constants of the sampling distribution, not of your process.

What if my subgroup is larger than 10? Use the standard deviation chart with A3, B3 and B4 rather than the range chart, since the range loses efficiency as n grows.

The same table drives the limits in the control charts and internal quality control tool.