# Shewhart X-bar and R chart

Operations · `X-bar and R chart` · verification: source checked

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.

## Specification

| Property | Value |
|---|---|
| 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
```

- `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.

## Sources

1. [6.3.2.1. Shewhart X-bar and R and S Control Charts](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm) (accessed 2026-10-25)

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