Learn · updated 2026-10-21
The X̄-R chart (X-bar and range)
Use an X̄-R chart when you measure a few parts at a time, say 5 every hour. The X̄ chart watches the process average; the R chart watches the spread inside each subgroup. Read the R chart first: if the spread isn't stable, the X̄ limits mean nothing.
The formulas. X̿ is the mean of the subgroup means, R̄ the mean of the subgroup ranges, n the subgroup size.
X̄ chartUCL = X̿ + A2 × R̄LCL = X̿ − A2 × R̄R chartUCL = D4 × R̄LCL = D3 × R̄
For subgroups of 5 (NIST's table): A2 = 0.577, D3 = 0, D4 = 2.115. For 10 or more per subgroup, use the X̄-S chart instead.
Worked example: 8 subgroups of 5
| Subgroup | Values | Mean X̄ | Range R |
|---|---|---|---|
| 1 | 10.2 9.8 10.1 10.0 9.9 | 10.000 | 0.4 |
| 2 | 10.4 10.1 9.7 10.0 10.2 | 10.080 | 0.7 |
| 3 | 9.9 10.0 10.3 10.1 9.8 | 10.020 | 0.5 |
| 4 | 10.0 9.6 10.2 10.1 10.0 | 9.980 | 0.6 |
| 5 | 10.3 10.2 9.9 10.0 10.1 | 10.100 | 0.4 |
| 6 | 10.1 9.9 10.0 10.2 9.8 | 10.000 | 0.4 |
| 7 | 9.8 10.0 10.1 9.9 10.2 | 10.000 | 0.4 |
| 8 | 10.0 10.3 9.9 10.1 10.0 | 10.060 | 0.4 |
- Grand mean: X̿ = 10.030. Average range: R̄ = 0.475.
- X̄ limits: 10.030 ± 0.577 × 0.475 → UCL 10.304, LCL 9.756.
- R limits: 2.115 × 0.475 = 1.004; the lower limit is 0 because D3 = 0 for n = 5.
- Estimated process σ for capability: R̄ / d2 = 0.475 / 2.326 = 0.2042.
LimitLine computes the constants for any subgroup size from 2 to 25, and its values match NIST's printed table; see the validation page.
Picking subgroups
- Make them rational. Parts in one subgroup should be made close together under the same conditions, so differences between subgroups show real change.
- Don't mix sources. One part from each of five machines puts the machine-to-machine difference inside the range and widens the limits until nothing shows.
- Keep the size constant. The constants assume every subgroup has the same n.