Control Chart Limits, Process Capability (Cp and Cpk), and Pareto Analysis
October 29, 2026
Quality tools turn raw measurements into decisions. A control chart shows whether a process is stable; capability indices compare the process spread to the specification limits; and a Pareto chart shows which problems matter most.
For an X-bar chart tracking the average of subgroups of size n, the control limits are center line ± 3σ ÷ √n, where σ is the process standard deviation. With a mean of 50.0, σ = 2.0, and n = 4: the standard error is 2 ÷ 2 = 1.0, so the limits are 47 to 53. With n = 9: limits of 50 ± 2, or 48 to 52.
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A point outside the control limits signals a special cause. Other common rules: 2 of 3 consecutive points beyond 2σ on the same side; 8 consecutive points on one side of the center line; a run of 6 steadily increasing or decreasing points. Control limits come from process variation and are not the same as specification limits, which come from customer requirements. Adjusting a stable process in reaction to random variation (tampering) makes things worse.
Process Capability: Cp and Cpk
Cp = (USL − LSL) ÷ (6σ) compares the spec width to the process width. With specs of 44 to 56 and σ = 2, Cp = 12 ÷ 12 = 1.0: the process just fits, with about 0.27% outside if centered. Cpk accounts for centering: Cpk = min(USL − mean, mean − LSL) ÷ (3σ). If the mean drifts to 51, Cpk = min(5, 7) ÷ 6 = 0.83, so the process is not capable: about 0.6% of output would fall beyond the upper limit. A Cpk of 1.33 is a common minimum for established processes, and 1.67 for critical features. Improving centering is often cheaper than reducing the spread.
| Mean | Cp | Cpk | Interpretation |
|---|---|---|---|
| 50 (centered) | 1.00 | 1.00 | Barely capable |
| 51 | 1.00 | 0.83 | Not capable; shift toward center |
| 50, σ = 1.5 | 1.33 | 1.33 | Capable |
| 50, σ = 1.2 | 1.67 | 1.67 | Highly capable |
Pareto Analysis
Count defects by cause: A 120, B 60, C 30, D 20, E 10, and F 10, total 250. Sorted, the cumulative shares are 48%, 72%, 84%, 92%, 96%, and 100%. Causes A and B (2 of 6) account for 72% of defects. Fixing them is the best use of effort, a version of the 80/20 principle. Reapply the analysis after improvements, as the ranking changes, and weigh the causes by cost or severity if their impacts differ.
Frequently Asked Questions
What are X-bar chart limits?
Center line ± 3σ ÷ √n.
What does a point outside the limits mean?
A likely special cause of variation.
What is Cpk?
A capability index that accounts for how well the process is centered.
What Cpk is generally considered capable?
At least 1.33.
What does a Pareto chart show?
Which causes contribute most to the problem.