Statistical quality control and control charts

Why it matters

A car contains thousands of toleranced dimensions — piston-pin diameters, brake-disc thickness, bore sizes, torque on wheel nuts — and suppliers must prove that their processes hold them consistently, not just that the last batch passed inspection. Control charts show, while production is running, whether a process is stable or has been disturbed by a special cause, and capability indices (Cp, Cpk) show whether a stable process is good enough for the specification. Both are standard topics in GATE and are demanded under the automotive quality system (IATF 16949, PPAP).

Key ideas

Variation. Every process varies.

• Common (chance) causes are many small, random sources built into the process (small vibrations, slight material variation). They give a stable, predictable distribution. Reducing them needs a change to the process (management action).
• Special (assignable) causes are identifiable events — a worn tool, a new batch of material, a wrong offset. They make the process unstable and should be found and removed at the machine (operator action). A process with only common-cause variation is in statistical control.

Control charts (Shewhart). Samples (subgroups) are taken at intervals and a statistic is plotted against a centre line (CL) and control limits usually at ±3 standard deviations of that statistic. With normal data, about 99.73 % of points fall inside ±3σ when the process is in control, so a point outside is strong evidence of a special cause. Other signals: a run of 7–8 points on one side of the CL, a steady trend of 6–7 points rising or falling, cycles, or points hugging the limits (Western Electric / Nelson run rules).

Control limits are not specification limits. Control limits come from the process (what it does); specification limits come from the design (what it must do). A process can be in control but still produce out-of-specification parts if it is not capable.

Charts for variables (measured data).

• X̄ chart — subgroup means; detects shifts in the mean.
• R chart — subgroup ranges; detects changes in spread. Use with small subgroups (n ≤ about 10).
• X̄–s chart — standard deviation for larger subgroups.
• Always check the R chart first: X̄ limits depend on R̄, so they are meaningless if spread is out of control.

Charts for attributes (counted data).

• p chart — fraction defective (sample size may vary); np chart — number defective (constant n).
• c chart — number of defects per unit of constant size (paint defects per car body, porosity per casting); u chart — defects per unit when unit size varies. p and np assume a binomial model; c and u assume a Poisson model.

Process capability. For a stable process, estimate σ from R̄/d₂ (or from s). Cp compares the tolerance width with the process spread (6σ) and ignores centring; Cpk uses the distance from the mean to the nearer specification limit, so it penalises an off-centre process. Cp = Cpk only when the process is centred. Typical automotive targets: Cpk ≥ 1.33 for ongoing production and ≥ 1.67 for new or critical characteristics (check your customer's requirement).

Acceptance sampling (link). When incoming lots are inspected by sampling (n items, accept if defectives ≤ c), the operating characteristic (OC) curve gives the probability of accepting a lot at each quality level; the producer's risk α is rejecting a good lot (at AQL) and the consumer's risk β is accepting a bad lot (at LTPD).

Control-chart constants (from standard tables, n = subgroup size).

n	A₂	D₃	D₄	d₂
2	1.880	0	3.267	1.128
3	1.023	0	2.574	1.693
4	0.729	0	2.282	2.059
5	0.577	0	2.114	2.326
6	0.483	0	2.004	2.534

Formulas

X̿ = Σ X̄_i / k, R̄ = Σ R_i / k

• Grand mean and average range over k subgroups (same unit as the measurement, e.g. mm).

X̄ chart: CL = X̿, UCL = X̿ + A₂·R̄, LCL = X̿ − A₂·R̄

R chart: CL = R̄, UCL = D₄·R̄, LCL = D₃·R̄

X̄ chart with known σ: X̿ ± 3·σ/√n

• σ = process standard deviation of individual values.

p chart: CL = p̄, limits = p̄ ± 3·√(p̄(1 − p̄)/n)

• p̄ = total defectives / total inspected; LCL taken as 0 if negative.

c chart: CL = c̄, limits = c̄ ± 3·√c̄

• c̄ = average defects per unit.

σ̂ = R̄ / d₂

Cp = (USL − LSL) / (6σ), Cpk = min(USL − μ, μ − LSL) / (3σ)

• USL, LSL = specification limits; μ = process mean.

Worked examples

Example 1 (standard → GATE level) — X̄–R chart and capability. Given: piston-pin diameter, 20 subgroups of n = 5; X̿ = 20.010 mm, R̄ = 0.020 mm; specification 20.00 ± 0.03 mm.

1. X̄ chart: UCL = 20.010 + 0.577 × 0.020 = 20.0215 mm; LCL = 20.010 − 0.0115 = 19.9985 mm
2. R chart: UCL = 2.114 × 0.020 = 0.0423 mm; LCL = 0 × 0.020 = 0
3. σ̂ = R̄ / d₂ = 0.020 / 2.326 = 0.00860 mm
4. Cp = 0.060 / (6 × 0.00860) = 1.16
5. Cpk = min(20.030 − 20.010, 20.010 − 19.970) / (3 × 0.00860) = 0.020 / 0.0258 = 0.78

Answer: X̄ limits 19.9985–20.0215 mm, R limits 0–0.0423 mm; Cp ≈ 1.16 but Cpk ≈ 0.78 — the spread is acceptable but the process is off-centre and must be re-centred at 20.000 mm.

Example 2 (GATE level) — p chart. Given: 25 samples of 200 brake pads each; 150 defectives in total.

1. p̄ = 150 / (25 × 200) = 0.030
2. σ_p = √(0.03 × 0.97 / 200) = 0.01206
3. UCL = 0.030 + 3 × 0.01206 = 0.0662; LCL = 0.030 − 0.0362 < 0 → 0

Answer: CL 0.030, UCL 0.066, LCL 0 — a sample with 14 defectives (p = 0.07) would signal a special cause.

Example 3 — c chart. Given: average of 4 paint defects per car body. UCL = 4 + 3√4 = 10, LCL = 4 − 6 < 0 → 0. Answer: a body with 11 or more defects signals an out-of-control paint process.

Common mistakes

• Treating control limits as tolerance limits, or computing control limits from specification limits.
• Using A₂ for the wrong subgroup size, or using the number of subgroups k instead of the subgroup size n to pick constants.
• Assuming R̄ equals σ; σ̂ = R̄/d₂.
• Reporting a negative LCL for an R, p or c chart; it is set to zero.
• Using Cp alone for an off-centre process — Cpk is the one that reflects actual defects.
• Using a p chart for defects per unit (should be c or u) or a c chart for fraction defective (should be p).

For GATE ME

Expect calculations of X̄ and R chart limits with given constants, p and c chart limits, Cp and Cpk, and conceptual questions on chance versus assignable causes, which chart suits which data, and OC-curve risks. Practise reading the constants table correctly and checking for negative lower limits.

Quick check

1. n = 4, X̿ = 50.0, R̄ = 2.0. Find the X̄ chart UCL.
2. USL = 10.6, LSL = 9.4, σ = 0.1, mean 10.0. Find Cp and Cpk.
3. Which chart for the number of scratches per bumper?
4. Is a point inside the control limits but outside the specification a control problem or a capability problem?
5. c̄ = 9. What are the c-chart limits?

Answers: 1. 50.0 + 0.729 × 2.0 = 51.46. 2. Cp = 1.2/0.6 = 2.0; Cpk = 0.6/0.3 = 2.0. 3. c chart. 4. Capability. 5. UCL = 18, LCL = 0.