A variable control chart monitors a quality characteristic that is measured on a continuous scale — length, weight, temperature, hardness, voltage, etc. Because the data are quantitative, we track both the central tendency (mean) and the dispersion (range or standard deviation) of the process.
Tracking only the mean is insufficient — a process could have the right average but excessive variability, and vice versa. Hence variable charts are always used in pairs:
Samples (subgroups) of size \(n\) are taken at regular intervals. A rational subgroup is one in which the items within a subgroup are produced under essentially the same conditions (same operator, same machine setting, same short time window). This maximises the chance of detecting between-subgroup shifts.
For a normal population with standard deviation \(\sigma\), the sampling distributions of \(\bar{X}\), \(R\), and \(S\) are related to \(\sigma\) through well-known constants that depend only on the subgroup size \(n\):
| \(n\) | \(A_2\) | \(D_3\) | \(D_4\) | \(A_3\) | \(B_3\) | \(B_4\) | \(d_2\) | \(c_4\) |
|---|---|---|---|---|---|---|---|---|
| 2 | 1.880 | 0 | 3.267 | 2.659 | 0 | 3.267 | 1.128 | 0.7979 |
| 3 | 1.023 | 0 | 2.575 | 1.954 | 0 | 2.568 | 1.693 | 0.8862 |
| 4 | 0.729 | 0 | 2.282 | 1.628 | 0 | 2.266 | 2.059 | 0.9213 |
| 5 | 0.577 | 0 | 2.114 | 1.427 | 0 | 2.089 | 2.326 | 0.9400 |
| 6 | 0.483 | 0 | 2.004 | 1.287 | 0.030 | 1.970 | 2.534 | 0.9515 |
| 7 | 0.419 | 0.076 | 1.924 | 1.182 | 0.118 | 1.882 | 2.704 | 0.9594 |
| 8 | 0.373 | 0.136 | 1.864 | 1.099 | 0.185 | 1.815 | 2.847 | 0.9650 |
| 9 | 0.337 | 0.184 | 1.816 | 1.032 | 0.239 | 1.761 | 2.970 | 0.9693 |
| 10 | 0.308 | 0.223 | 1.777 | 0.975 | 0.284 | 1.716 | 3.078 | 0.9727 |
When no prior standard values of \(\mu\) and \(\sigma\) are given, we estimate them from the sample data (typically 20–25 subgroups of size \(n\)).
For subgroup \(j\) of size \(n\), compute:
where \(k\) is the number of subgroups.
Always construct the R chart first and verify it is in control before constructing the \(\bar{X}\) chart, because the \(\bar{X}\) chart limits depend on \(\bar{R}\).
20 subgroups of size \(n = 5\) from a steel-rod cutting process. Summary: \(\bar{\bar{X}} = 48.02\) mm, \(\bar{R} = 3.84\) mm.
For \(n = 5\): \(A_2 = 0.577\), \(D_3 = 0\), \(D_4 = 2.114\).
R chart: UCL = 2.114 × 3.84 = 8.12 mm; CL = 3.84; LCL = 0.
\(\bar{X}\) chart: UCL = 48.02 + 0.577 × 3.84 = 48.02 + 2.22 = 50.24 mm; CL = 48.02; LCL = 48.02 − 2.22 = 45.80 mm.
Process estimate: \(\hat{\sigma} = 3.84/2.326 = 1.651\) mm.
Continuing the same data, subgroup 14 has \(\bar{X}_{14} = 51.30\) mm and \(R_{14} = 9.50\) mm.
The range \(R_{14} = 9.50\) exceeds UCL\(_R\) = 8.12 → R chart flags an out-of-control signal. The mean \(\bar{X}_{14} = 51.30\) exceeds UCL\(_{\bar{X}}\) = 50.24 → \(\bar{X}\) chart also flags. Investigate: the cutting tool was replaced at subgroup 14, causing both a shift in mean and increased variability until the new tool settled in.
When the target (standard) values \(\mu_0\) and \(\sigma_0\) are specified by engineering design or historical process capability, the control limits are computed directly from these standards rather than from sample data.
Specified: \(\mu_0 = 50.00\) mm, \(\sigma_0 = 1.50\) mm, \(n = 5\).
\(d_2 = 2.326\), \(D_3 = 0\), \(D_4 = 2.114\).
\(\bar{X}\) chart: UCL = 50 + 3(1.50)/\(\sqrt{5}\) = 50 + 2.012 = 52.01; CL = 50; LCL = 50 − 2.012 = 47.99.
R chart: CL = 2.326 × 1.50 = 3.489; UCL = 2.114 × 3.489 = 7.375; LCL = 0.
Using the same standards, a subgroup of 5 measurements yields \(\bar{X} = 52.50\) mm. This exceeds UCL = 52.01 mm, so the process mean has shifted upward from the target \(\mu_0 = 50\). Investigation reveals a new operator (Man) set the machine at a higher offset.
For small subgroups (\(n \le 10\)), the range \(R\) is an efficient estimator of variability and is easy to compute. For larger subgroups (\(n > 10\)), the standard deviation \(S\) is preferred because it uses all the observations. The range uses only the two extremes, so as \(n\) grows it throws away more information and becomes a less efficient estimate of \(\sigma\).
Compute \(\bar{\bar{X}}\) and \(\bar{S} = \frac{1}{k}\sum_{j=1}^{k} S_j\).
Process \(\sigma\) estimate: \(\hat{\sigma} = \bar{S}/c_4\).
25 subgroups of size \(n = 8\). \(\bar{\bar{X}} = 62.5\), \(\bar{S} = 2.40\). For \(n = 8\): \(A_3 = 1.099\), \(B_3 = 0.185\), \(B_4 = 1.815\), \(c_4 = 0.9650\).
S chart: UCL = 1.815 × 2.40 = 4.356; CL = 2.40; LCL = 0.185 × 2.40 = 0.444.
\(\bar{X}\) chart: UCL = 62.5 + 1.099 × 2.40 = 62.5 + 2.638 = 65.138; CL = 62.5; LCL = 62.5 − 2.638 = 59.862.
\(\hat{\sigma} = 2.40/0.9650 = 2.487\).
Specified: \(\mu_0 = 200.0\) g, \(\sigma_0 = 3.0\) g, \(n = 9\). Constants: \(A_3 = 1.032\), \(B_3 = 0.239\), \(B_4 = 1.761\), \(c_4 = 0.9693\).
\(\bar{X}\) chart: UCL = 200 + 3(3)/3 = 203.0; CL = 200; LCL = 197.0.
S chart: CL = 0.9693 × 3.0 = 2.908; UCL = 1.761 × 2.908 = 5.121; LCL = 0.239 × 2.908 = 0.695.
Control charts show whether a process is in control (stable); capability indices show whether a stable process can actually meet the customer's specification limits (LSL, USL). They compare the specification width to the process spread \(6\sigma\).
Specifications LSL = 46, USL = 54; process \(\mu = 51,\ \sigma = 1\).
\(C_p = (54-46)/(6\cdot1) = 1.333\) (potentially capable), but \(C_{pk} = \min\!\big((54-51)/3,\ (51-46)/3\big) = \min(1.0,\ 1.667) = 1.0\). The process is off-centre (mean shifted toward the USL), so its actual capability (1.0) is well below its potential (1.33) — re-centring to \(\mu = 50\) would raise \(C_{pk}\) to 1.33.