Control Chart Limits (X̄–R, X̄–s, I–MR)
Paste subgroup measurements and get Shewhart control limits: X̄–R or X̄–s charts for subgroups of 2–10, I–MR for single values, with out-of-limit flags.
Centre lines, UCL and LCL for the mean and spread charts, the process sigma estimate, a per-subgroup table with the flags, and notes on trial limits and the order in which to read the charts.
Example: Five subgroups of 5 with X̿ 10.10 and R̄ 0.48: X̄ limits 10.10 ± 0.577 × 0.48 = 9.823 … 10.377, R chart UCL 2.114 × 0.48 = 1.015, σ̂ = 0.48 ÷ 2.326 = 0.206.
Three-sigma limits
from your subgroups.
How X̄, R and s limits are built, the factors used, and what a flag means.
The limits
For subgroups of n values: the grand mean X̿ (mean of the subgroup means) is the X̄ centre line, and the limits are X̿ ± A2·R̄ with the range method or X̿ ± A3·s̄ with the standard-deviation method. The R chart has centre R̄ and limits D3·R̄ … D4·R̄; the s chart B3·s̄ … B4·s̄. The process sigma estimate is R̄ ÷ d2 or s̄ ÷ c4. One value per line is read as an individuals chart: X̄ ± 2.66·MR̄ and an MR upper limit of 3.267·MR̄.
The factors
A2, D3, D4, d2, A3, B3, B4 and c4 for n = 2 to 10 are the standard Shewhart control chart factors, tabulated in ISO 7870-2 and the ASTM manual on presentation of data and control chart analysis; they follow from the sampling distribution of the range and standard deviation of n normal values, and set the limits at ±3σ̂ of the subgroup statistic. Subgroups above 10 are refused rather than approximated.
Reading it
A subgroup outside the limits is flagged. Read the spread chart first: the X̄ limits are built from R̄ (or s̄), so they mean nothing until the spread is in control. Fewer than 20 subgroups give trial limits; recompute after an out-of-control subgroup has been explained and dropped. Only the ±3σ rule is tested — run and trend rules are not — and control limits are not specification limits. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Shewhart chart factors A2, D3, D4, d2, A3, B3, B4, c4 for n = 2–10 as tabulated in ISO 7870-2 and ASTM MNL7 (presentation of data and control chart analysis); I–MR limits X̄ ± 2.66·MR̄, 3.267·MR̄
- Control limits ±3σ̂ from the within-subgroup spread; the run and trend rules (Western Electric / Nelson) are not applied
Last reviewed 20 September 2026. How results are checked: How we verify.