Question 1

How are UCL and LCL calculated for the X-bar control chart?

Accepted Answer

Control limits for the X-bar chart are: UCL = X̄ + A2×R̄ and LCL = X̄ − A2×R̄. The constant A2 depends on sample size n (e.g., A2 = 0.577 for n=5). For the R chart: UCL_R = D4×R̄ and LCL_R = D3×R̄ (D3 = 0 when n ≤ 6). These are based on the within-subgroup sigma estimate using the average range R̄.

Question 2

What are the Western Electric rules for control charts?

Accepted Answer

The Western Electric rules are 8 criteria for detecting non-random patterns on control charts. Key rules include: ① 1 point beyond 3σ (outside UCL/LCL); ② 2 of 3 consecutive points beyond 2σ on the same side; ③ 4 of 5 consecutive points beyond 1σ on the same side; ④ 8 consecutive points on the same side of the centerline; ⑤ 6 consecutive points trending monotonically up or down. These patterns are statistically unlikely under random variation and signal process anomalies.

Question 3

What is the difference between Cp and Cpk?

Accepted Answer

Cp (Process Capability Index) = (USL−LSL)/(6σ) indicates how small the process variation is relative to the specification width. Cpk accounts for the offset between the process center and the specification center: Cpk = min((USL−μ)/3σ, (μ−LSL)/3σ). Even if Cp is high, Cpk will be low if the process mean is off-center. In manufacturing, Cpk ≥ 1.33 is typically the acceptance criterion.

Question 4

When should I use a p chart versus a c chart?

Accepted Answer

The p chart monitors the proportion of defective items (defectives/sample size) and can handle variable sample sizes. The np chart monitors the actual count of defectives with a constant sample size. The c chart monitors the number of defects per inspection unit of fixed size. The u chart monitors defects per unit when sample sizes vary. Choose the chart type based on whether you are counting defective items or individual defects, and whether your sample size is constant.

Control Chart (X-bar·R·p·c) SPC Calculator

Control Limit Formulas