Shewhart X-bar and R chart

The Shewhart X-bar and R chart is a pair of variables control charts for monitoring a process mean and process variability. The X-bar chart plots subgroup means with limits based on the average range, and the R chart plots subgroup ranges with limits based on factors D3 and D4. It is intended for small subgroup sizes, typically up to about 10.

Source checked8/8 values · 2026-10-25Methodology

Specification

1 source

Kind	methodology

Formula

UCL_X̄ = x̄ + A2(n) R̄; CL_X̄ = x̄; LCL_X̄ = x̄ − A2(n) R̄; UCL_R = D4(n) R̄; CL_R = R̄; LCL_R = D3(n) R̄

Symbol	Name
x̄	average of subgroup means
R̄	average subgroup range
A2	X-bar chart factor depending on subgroup size
D3	R chart lower-limit factor depending on subgroup size
D4	R chart upper-limit factor depending on subgroup size
n	subgroup size

Study notes

The R chart controls process variability because the sample range is related to the process standard deviation.
The document states the range approach is satisfactory for sample sizes up to around 10; for larger sample sizes subgroup standard deviations are preferable.
A typical sample size is 4 or 5.
For n=2 to 10, the factor table gives A2 = 1.880 to 0.308, D3 = 0 to 0.223, and D4 = 3.267 to 1.777.
For an X̄ chart with 3-sigma limits, the document gives false-alarm probability p = 0.0027.
The document defines A2 = 3/(d2√n), D3 = 1 − 3d3/d2, and D4 = 1 + 3d3/d2.