Statistical quality control and control charts

Why it matters

No machine makes two identical parts. Statistical quality control (SQC) tells you whether the variation you see is the normal "noise" of a stable process or a signal that something has changed — a worn tool, a new batch of material — so you adjust only when you should. Capability indices then tell you whether a stable process can actually hold the drawing tolerance.

Key ideas

• Two kinds of variation.
  • Common (chance, random) causes — many small, always-present effects; a process with only these is in statistical control and predictable.
  • Assignable (special) causes — identifiable events such as tool wear, a wrong setting or a bad material lot. Control charts are designed to detect these.
• Control chart logic. Plot a sample statistic over time with a centre line (CL) and control limits at ±3 standard errors of that statistic. For a normal statistic, 99.73% of points fall inside when the process is in control, so a false alarm happens about 3 times in 1,000 samples. A point outside the limits, or a non-random pattern (a run of about 7–8 points on one side of CL, a steady trend, cycles), signals an assignable cause.
• Control limits are not specification limits. Control limits come from the process data and describe what the process does; specification (tolerance) limits come from the design and describe what the customer needs. A process can be in control and still make defective parts.
• Rational subgroups. Take small samples (often n = 4–5) of consecutive parts so variation within a sample is only common-cause; differences between samples then reveal shifts. Typically 20–25 subgroups are used to set trial limits.
• Charts for variables (measured data).
  • X̄ chart — monitors the process mean (centring).
  • R chart — monitors spread using the range; used for small n. An S chart uses the sample standard deviation for larger n.
  • Always read the R chart first: X̄ limits depend on R̄, so they are meaningless if spread is out of control.
  • By the central limit theorem, sample means are close to normal even if individual values are not, and their standard deviation is σ/√n.
• Charts for attributes (counted data).
  • p chart — fraction defective; works with constant or varying sample size.
  • np chart — number of defectives; constant sample size only.
  • c chart — number of defects per unit (one unit or a constant inspection area), based on the Poisson distribution.
  • u chart — defects per unit when the inspected amount varies.
  • Note the difference: a defective is a rejected item; a defect is a single non-conformity, and one item may have several.
• Process capability. For a stable process with normal output, the natural spread is 6σ. Cp compares the tolerance width with 6σ (potential capability, ignores centring). Cpk uses the distance from the mean to the nearer specification limit (actual capability). Cpk ≤ Cp, with equality only when the process is centred. Cp < 1 means defects even when centred; Cp ≈ 1.33 or more is a common industrial minimum (customer requirements vary).
• Acceptance sampling (link). Accepting or rejecting lots from samples, described by an operating-characteristic (OC) curve, producer's risk α and consumer's risk β, is the inspection side of SQC; control charts are the process side.

Formulas

• Standard error of the mean: σ_x̄ = σ / √n
• X̄ chart (from R̄): CL = X̿, UCL = X̿ + A₂·R̄, LCL = X̿ − A₂·R̄
• R chart: CL = R̄, UCL = D₄·R̄, LCL = D₃·R̄
• Estimate of process σ: σ̂ = R̄ / d₂
  • X̿ = grand mean of subgroup means (mm); R̄ = mean subgroup range (mm); n = subgroup size; A₂, D₃, D₄, d₂ = factors from the SQC constants table for that n (for n = 5: A₂ = 0.577, D₃ = 0, D₄ = 2.114, d₂ = 2.326).
• X̄ chart with known σ: UCL, LCL = μ ± 3σ/√n
• p chart: CL = p̄, UCL, LCL = p̄ ± 3·√(p̄(1 − p̄)/n)
• np chart: CL = n·p̄, UCL, LCL = n·p̄ ± 3·√(n·p̄(1 − p̄))
• c chart: CL = c̄, UCL, LCL = c̄ ± 3·√c̄
• u chart: CL = ū, UCL, LCL = ū ± 3·√(ū/n)
  • A negative LCL is set to zero for all attribute charts.
• Capability: Cp = (USL − LSL) / (6σ)
• Cpk = min[(USL − μ) / (3σ), (μ − LSL) / (3σ)]
  • USL, LSL = upper and lower specification limits; μ = process mean; σ = process standard deviation (all in the same unit, e.g. mm).

Worked examples

Example 1 (standard): X̄ and R charts and capability Twenty subgroups of n = 5 shaft diameters give X̿ = 25.02 mm and R̄ = 0.05 mm. Specification 24.95 to 25.10 mm. For n = 5: A₂ = 0.577, D₃ = 0, D₄ = 2.114, d₂ = 2.326.

1. X̄ chart: UCL = 25.02 + 0.577 × 0.05 = 25.02 + 0.0289 = 25.049 mm; LCL = 25.02 − 0.0289 = 24.991 mm.
2. R chart: UCL = 2.114 × 0.05 = 0.106 mm; LCL = 0 × 0.05 = 0 mm.
3. σ̂ = R̄/d₂ = 0.05/2.326 = 0.0215 mm; 6σ̂ = 0.129 mm.
4. Cp = (25.10 − 24.95)/0.129 = 0.15/0.129 = 1.16.
5. Cpk = min[(25.10 − 25.02)/(3 × 0.0215), (25.02 − 24.95)/(3 × 0.0215)] = min[0.08/0.0645, 0.07/0.0645] = min[1.24, 1.09] = 1.09. Answer: X̄ limits 24.991–25.049 mm, R limits 0–0.106 mm; Cp = 1.16, Cpk = 1.09 — capable but slightly off-centre toward the LSL; below a 1.33 target.

Example 2 (GATE level): attribute charts (a) Samples of n = 200 bearings each give an average fraction defective p̄ = 0.04.

1. σ_p = √(0.04 × 0.96/200) = √0.000192 = 0.01386.
2. p chart: UCL = 0.04 + 3 × 0.01386 = 0.0816; LCL = 0.04 − 0.0416 < 0 → 0.
3. Equivalent np chart: CL = 200 × 0.04 = 8 defectives; UCL = 8 + 3√(200 × 0.04 × 0.96) = 8 + 3 × 2.771 = 16.31; LCL = 0.
4. A sample with 18 defectives (p = 0.09) is above both UCLs → look for an assignable cause. (b) Painted panels show on average c̄ = 9 defects per panel.
5. c chart: UCL = 9 + 3√9 = 18 defects; LCL = 9 − 9 = 0. Answer: p chart 0–0.0816 (CL 0.04); np chart 0–16.3 defectives (CL 8); c chart 0–18 defects per panel (CL 9).

Common mistakes

• Confusing control limits with specification limits, or putting tolerance limits on an X̄ chart.
• Using σ instead of σ/√n for the X̄ chart limits.
• Using the c chart for defectives or the np chart for defects.
• Leaving a negative LCL on an attribute chart instead of setting it to zero.
• Reporting Cp for an off-centre process and calling it capable; check Cpk.
• Computing capability from data that are not in statistical control.
• Picking A₂, D₄ or d₂ for the wrong subgroup size; take them from your SQC table for the actual n.

For GATE ME

Expect control-limit calculations for X̄–R, p, np and c charts (constants are given when needed), questions on which chart suits which data, Cp and Cpk numericals including the effect of shifting the mean, interpretation of patterns and out-of-control points, and the meaning of 3σ limits (99.73%). Acceptance-sampling basics (OC curve, α and β) can also appear. Practise quick square-root arithmetic and setting negative LCLs to zero.

Quick check

1. c̄ = 4 defects per unit. Find the UCL of the c chart.
2. USL = 50 mm, LSL = 30 mm, σ = 2 mm. Find Cp.
3. In Q2 the mean is 44 mm. Find Cpk.
4. Which chart would you use for the number of scratches on each car door?
5. What fraction of points falls inside 3σ limits for an in-control normal process?

Answers: 1. 4 + 3 × 2 = 10. 2. 20/12 = 1.67. 3. min(6/6, 14/6) = 1.0. 4. c chart. 5. 99.73%.