Statistical quality control

In short

Statistical quality control (SQC) is the use of statistics to judge whether a manufacturing process is behaving consistently and whether its output meets specification. Its three arms are process control (control charts), process capability analysis (Cp, Cpk) and acceptance sampling (lot-by-lot inspection plans). It rests on Shewhart's distinction between chance (common) causes that are inherent in the process and assignable (special) causes that can be found and removed. It is examined in GATE PI and ME (industrial engineering section), in university Quality Control / Quality Assurance and Reliability papers, and is used daily by quality engineers at Maruti Suzuki, Bosch, Tata Motors, Sundram Fasteners, TVS, Bharat Forge, Cipla and Sun Pharma, and by every IATF 16949 or ISO 13485 supplier audited on SPC evidence.

Explained

Variation, causes and the basic logic

• Chance causes: many small, unavoidable sources (machine vibration, raw-material scatter, operator-to-operator jitter). They give a stable, predictable distribution — the process is then said to be in statistical control.
• Assignable causes: a tool wearing out, a wrong setting, a new material lot, a changed operator. They shift the mean or inflate the spread; a control chart signals them.
• In control ≠ capable. A process can be perfectly stable and still make scrap if its natural spread is wider than the tolerance. Control charts look at the process's own voice; capability compares that voice with the specification.
• Specification limits (USL, LSL) come from the designer and apply to individual parts. Control limits are computed from the process data and apply to a plotted statistic (usually a subgroup mean). Never draw specification limits on an X̄ chart.
• Why ±3σ: for a normal distribution 99.73% of values lie within ±3σ, so a false alarm occurs on about 0.27% of points (1 in 370). Tightening to ±2σ would raise false alarms to 1 in 22.
• Rational subgrouping: choose subgroups so that variation *within* a subgroup is only chance cause (consecutive parts, same setup) and any shift appears *between* subgroups.

Control charts for variables (measured data)

• Used when the characteristic is measurable — diameter in mm, weight in g, torque in N·m. Always run a pair: one chart for central tendency and one for spread. Read the R (or s) chart first: limits of the X̄ chart depend on R̄.
• X̄ – R chart: subgroup size n = 4 or 5 is standard (small n keeps within-subgroup variation to chance causes). Used for most machining and assembly characteristics.
• X̄ – s chart: preferred for larger subgroups (n > 10) or unequal n, since s uses all the data and R only the two extremes.
• Median – R chart: simpler shop-floor arithmetic, less efficient statistically.
• Individual – moving range (I-MR) chart: n = 1, used for slow processes, batch chemistry, destructive or costly tests, and single-piece flow.
• Constants A₂, D₃, D₄, d₂, c₄, A₃, B₃, B₄ are tabulated against n; they convert R̄ or s̄ into 3σ limits. d₂ links range to standard deviation: σ̂ = R̄/d₂.
• Procedure: collect 20–25 subgroups → compute X̄ and R for each → X̄̄ and R̄ → trial limits → delete subgroups with findable assignable causes and recompute → freeze as standard limits → plot live and act on signals → revise when the process genuinely changes.

Control charts for attributes (counted data)

• Used when the item is classified good/bad or defects are counted. Cheaper (go/no-go gauges) but needs much larger samples to detect the same shift.
• p chart — fraction defective, variable or constant sample size; limits recomputed for each n (or based on n̄ when n varies by less than about 25%). Binomial basis.
• np chart — number defective, constant sample size only.
• c chart — number of defects in one inspection unit of constant size (blemishes per painted bonnet, pinholes per 10 m² of sheet). Poisson basis.
• u chart — defects per unit when the inspection unit size varies (defects per m² of fabric when roll lengths differ).
• Defect versus defective: a defective *item* may carry several *defects*; p/np count items, c/u count defects.
• Limitation: since limits are computed as a mean ± 3√(variance), the LCL often comes out negative — it is then taken as zero, and the chart has no lower bound to detect genuine improvement.

Reading a chart: out-of-control rules

• Any point outside a control limit (the primary Shewhart rule).
• Run rule: seven (some texts nine) consecutive points on one side of the centre line — a mean shift.
• Trend: six or seven consecutive points steadily rising or falling — tool wear, temperature drift, depleting bath concentration.
• Hugging the centre line: too little variation, usually from mixing two streams, wrong subgrouping or fabricated data.
• Cyclic pattern: shift changes, ambient temperature, operator rotation.
• Two of three points beyond 2σ, four of five beyond 1σ on the same side (Western Electric zone rules).
• Type I error (α): calling an in-control process out of control — cost of needless hunting. Type II error (β): failing to detect a real shift — cost of bad parts. ARL = 1/α when in control (≈370 for 3σ limits), and ARL = 1/(1−β) after a shift.
• CUSUM and EWMA charts accumulate information from past points and detect small sustained shifts faster than a Shewhart chart, which is better for large sudden shifts.

Process capability and performance

• Requires the process to be in control and approximately normal, and the measurement system to be adequate (gauge R&R, usually below 10% of tolerance).
• Natural tolerance = 6σ. Capability indices compare tolerance width with this spread.
• Cp uses spread only; Cpk also penalises off-centring; Cpm penalises deviation from a target. Cp = Cpk only when the process is exactly centred, and Cpk ≤ Cp always.
• Cp vs Pp: Cp/Cpk use short-term (within-subgroup) sigma from R̄/d₂; Pp/Ppk use the overall long-term sigma of all individual readings, so Ppk is usually lower. Automotive PPAP typically asks for Ppk on a short run and Cpk once stable.
• Industry benchmarks: Cpk ≥ 1.33 (4σ) is the common customer requirement in Indian auto supply; ≥ 1.67 for safety-critical and new processes; 1.00 means the tolerance is just filled and roughly 2,700 ppm fall outside.
• Six Sigma link: a 6σ process with the conventional 1.5σ long-term drift gives 3.4 defects per million opportunities; Cp = 2.0, Cpk = 1.5.
• Non-normal data: transform (Box-Cox), fit another distribution, or quote percentile-based capability — applying Cpk blindly to a skewed distribution (e.g. flatness, roundness, which are bounded at zero) overstates quality.

Acceptance sampling

• Judging a whole lot by inspecting a sample. Used on incoming purchased parts, outgoing finished lots and in-process stages where 100% inspection is uneconomic or destructive.
• Why sample: 100% inspection is costly, slow, and with inspector fatigue is itself only about 80–95% effective; sampling also pressures the supplier to control the process rather than rely on sorting.
• Types: single sampling (n, c), double sampling (second sample only if the first is inconclusive — lower average inspection), multiple and sequential sampling, and continuous sampling (CSP-1) for conveyor flow.
• Key quantities: AQL (quality level routinely accepted, producer's target), LTPD or RQL (quality level the consumer wants almost surely rejected), producer's risk α (good lot rejected, usually 5%), consumer's risk β (bad lot accepted, usually 10%).
• OC curve: plots probability of acceptance against incoming fraction defective. Steeper = more discriminating. Increasing n steepens it; increasing c shifts it right (more lenient). For a fixed sampling *fraction*, a larger lot with proportionally larger n is more discriminating — so lot size barely matters, sample size does.
• AOQ and AOQL: with rectifying inspection (rejected lots 100% inspected and defectives replaced), average outgoing quality rises then falls with incoming quality; its maximum, AOQL, is the worst average quality the plan can let through.
• Indian and international standards: IS 2500 (Part 1) is the Indian sampling-inspection standard for attributes, aligned with ISO 2859-1 / MIL-STD-105E — AQL-indexed plans with normal, tightened and reduced inspection switching rules; IS 2500 (Part 2) covers variables sampling, aligned with ISO 3951. Check the current part/edition on the BIS portal before quoting it in a document, as parts are periodically revised.
• Limitation: sampling never improves a lot; it only sorts lots. It always carries both risks and gives no information about *why* the process produced defectives — which is why SPC on the process is preferred to inspection of the output.

Related techniques and Indian practice

• Seven QC tools: check sheet, histogram, Pareto chart, cause-and-effect (Ishikawa) diagram, scatter diagram, stratification/flow chart, and the control chart itself.
• Measurement system analysis (MSA): bias, linearity, stability, repeatability (equipment) and reproducibility (appraiser). A chart drawn with an unfit gauge measures the gauge.
• Documented requirement: IATF 16949 and ISO 13485 suppliers must show control plans, SPC charts and capability studies; PPAP (production part approval process) bundles them with MSA and FMEA.
• BIS operates the ISI mark scheme under the BIS Act, 2016, with factory-level sampling and testing; mandatory certification applies to notified products such as cement, helmets, pressure cookers and many electrical goods.
• Who built it: Shewhart (control chart, 1924, Bell Labs), Dodge and Romig (sampling plans), Deming and Juran (management of quality), Taguchi (loss function, robust design), Ishikawa (QC circles).

Key points

• Chance cause - inherent random variation; leave it alone, improvement needs a process change.
• Assignable cause - a findable, removable disturbance; the control chart's purpose is to flag it.
• Control limits - computed from process data (±3σ of the plotted statistic); specification limits come from design and are never plotted on an X̄ chart.
• X̄ – R chart - variables data, n = 4–5; read the R chart first because X̄ limits use R̄.
• I-MR chart - n = 1, for slow, batch or destructive measurement.
• p and np charts - fraction/number defective, binomial; c and u charts - defects per unit, Poisson.
• σ̂ = R̄/d₂ - short-term standard deviation estimated from the mean range.
• Cp = (USL−LSL)/6σ - potential capability, ignores centring; Cpk - actual, penalises off-centre.
• Cpk ≥ 1.33 - the usual Indian automotive customer requirement; ≥1.67 for new or safety-critical processes.
• Cp vs Pp - short-term (within-subgroup) sigma versus long-term overall sigma.
• Type I error α - false alarm, ≈0.27% with 3σ limits, ARL ≈ 370 points.
• Type II error β - a real shift missed; reduced by larger n or CUSUM/EWMA charts.
• OC curve - P(accept) against lot fraction defective; larger n steepens it, larger c makes it lenient.
• AQL / LTPD - quality routinely accepted / quality that must be rejected; α producer's risk, β consumer's risk.
• AOQL - the worst average outgoing quality a rectifying plan can deliver.
• IS 2500 Part 1 - Indian attribute sampling standard, aligned to ISO 2859-1, with normal/tightened/reduced switching.
• Seven QC tools - check sheet, histogram, Pareto, Ishikawa, scatter, stratification, control chart.
• In control is not capable - stability and conformance are two different questions.

Formulas and examples

Constants used below (standard SQC tables): n = 5 → A₂ = 0.577, D₃ = 0, D₄ = 2.114, d₂ = 2.326, A₃ = 1.427, c₄ = 0.9400.

Example 1 — X̄ – R chart limits. A shaft diameter is measured in 25 subgroups of n = 5. ΣX̄ = 1,250.0 mm over 25 subgroups and ΣR = 1.00 mm.

• X̄̄ = 1,250.0/25 = 50.00 mm; R̄ = 1.00/25 = 0.040 mm.
• R chart: UCL = 2.114 × 0.040 = 0.0846 mm, CL = 0.040 mm, LCL = 0.
• X̄ chart: A₂R̄ = 0.577 × 0.040 = 0.02308 → UCL = 50.0231 mm, LCL = 49.9769 mm.
• σ̂ = R̄/d₂ = 0.040/2.326 = 0.01720 mm; natural tolerance 6σ̂ = 0.1032 mm.

Example 2 — capability of the same process. Specification 50.00 ± 0.05 mm (USL 50.05, LSL 49.95), process mean 50.01 mm, σ̂ = 0.0172 mm.

• Cp = (50.05 − 49.95)/(6 × 0.0172) = 0.10/0.1032 = 0.969.
• Cpu = (50.05 − 50.01)/(3 × 0.0172) = 0.04/0.0516 = 0.775; Cpl = (50.01 − 49.95)/0.0516 = 0.06/0.0516 = 1.163.
• Cpk = min = 0.775 — well below the 1.33 requirement, and the gap between Cp and Cpk shows the problem is centring, not spread.
• Fraction above USL = Φ(−3 × 0.775) = Φ(−2.326) ≈ 0.0100; below LSL = Φ(−3.488) ≈ 0.00024. Total ≈ 1.02% ≈ 10,200 ppm.
• Re-centre the mean to 50.00 mm: Cpk rises to Cp = 0.969, out-of-spec ≈ 2 × Φ(−2.907) ≈ 0.37% (3,700 ppm) — a free gain from an adjustment, with no new machine. To reach Cpk 1.33 the spread must also shrink to σ ≤ 0.10/(6 × 1.33) = 0.0125 mm.

Example 3 — p chart with varying sample size. 20 daily samples of about 400 castings each, giving 8,000 inspected in total with 240 defectives found.

• p̄ = 240/8,000 = 0.030.
• For a day with n = 400: √[0.03 × 0.97/400] = √(0.00007275) = 0.008529 → UCL = 0.030 + 3(0.008529) = 0.0556, LCL = 0.030 − 0.0256 = 0.0044.
• For a day with n = 200: √[0.0291/200] = √0.00014550 = 0.012062 → UCL = 0.0662, LCL = 0.030 − 0.0362 < 0 → 0.
• A day with 24 defectives in 400 gives p = 0.060 > 0.0556 → out of control; the same fraction in a sample of 200 (12 defectives) would not signal — showing why limits must be recomputed for each n.

Example 4 — c chart. Enamelled refrigerator doors inspected for paint blemishes; 30 doors give 120 blemishes in total.

• c̄ = 120/30 = 4.0 defects per door.
• UCL = 4 + 3√4 = 4 + 6 = 10; LCL = 4 − 6 < 0 → 0.
• A door with 11 blemishes signals an assignable cause (spray-gun fouling). Note the chart can never signal improvement, since LCL is zero.

Example 5 — single sampling plan and OC curve. Lot N = 2,000, plan n = 80, c = 2. Compute Pa at p = 1% and p = 4% (Poisson approximation).

• At p = 0.01: np = 0.8 → Pa = e^−0.8(1 + 0.8 + 0.32) = 0.4493 × 2.12 = 0.952. Producer's risk α = 1 − 0.952 = 4.8%, so AQL ≈ 1%.
• At p = 0.04: np = 3.2 → Pa = e^−3.2(1 + 3.2 + 5.12) = 0.04076 × 9.32 = 0.380. Consumer's risk at 4% is 38% — far above 10%, so 4% is not the LTPD.
• At p = 0.065: np = 5.2 → Pa = e^−5.2(1 + 5.2 + 13.52) = 0.005517 × 19.72 = 0.1088 ≈ 10% → LTPD ≈ 6.5%.
• AOQ at p = 0.01: 0.01 × 0.952 × (2,000 − 80)/2,000 = 0.00914 ≈ 0.91%.
• ATI at p = 0.04: 80 + (1 − 0.380)(2,000 − 80) = 80 + 0.620 × 1,920 = 1,270 pieces per lot — the hidden cost of buying poor quality.

Where it is used

• Automotive and auto components: control plans and Cpk studies on bore diameters, gear runout and fastener torque at Maruti Suzuki, Tata Motors, Bosch, Sundram Fasteners, Rane and Bharat Forge — mandated by IATF 16949 and submitted in PPAP to the OEM.
• Pharmaceuticals: fill-weight and tablet-hardness charts, content uniformity and acceptance sampling of incoming excipients at Cipla, Sun Pharma, Dr Reddy's, under Schedule M GMP, ICH Q8 (quality by design) and ICH Q10 (pharmaceutical quality system, continued process verification).
• Steel, cement and process plants: c and u charts for surface defects per m² of coil, X̄–R on cement fineness and compressive strength; BIS ISI-mark scheme sampling at Tata Steel, JSW, UltraTech.
• Textiles and garments: u charts for defects per 100 m² of fabric, AQL-based final random inspection (ISO 2859-1 / IS 2500 Part 1) on export shipments for buyers such as Zara and M&S.
• Electronics and EMS: defects per million opportunities on SMT lines, solder-paste thickness SPC, first-pass yield at Dixon, Foxconn India and Tata Electronics.
• Job roles: quality engineer, QA/QC inspector, process/manufacturing engineer, supplier quality assurance (SQA) engineer, Six Sigma Green/Black Belt, BIS and NABL-lab testing officers; placement tests at L&T, Ashok Leyland, TVS and Hero routinely ask control-chart limits and Cpk numerically.

Interview questions

• Q: What is the difference between control limits and specification limits? A: Control limits are computed from the process's own variation and apply to the plotted statistic; specification limits come from the design and apply to individual parts. They are independent, and specification limits must never be drawn on an X̄ chart.
• Q: Can a process be in control and still produce rejects? A: Yes — stability only means the variation is predictable. If the 6σ natural spread exceeds the tolerance, or the mean is off-centre, a perfectly stable process produces out-of-spec parts.
• Q: Why are control limits set at ±3σ? A: It balances the two errors: 99.73% of a normal distribution lies inside, giving about one false alarm in 370 points, while still being tight enough to catch meaningful shifts.
• Q: Why read the R chart before the X̄ chart? A: The X̄ limits are computed from R̄, so if the spread is unstable, R̄ is not a valid estimate and the X̄ limits are meaningless.
• Q: When would you use an I-MR chart instead of X̄–R? A: When a natural subgroup does not exist — slow or batch processes, destructive or expensive tests, a single reading per shift.
• Q: Difference between a p chart and a c chart? A: A p chart plots the fraction of defective items on a binomial basis; a c chart plots the number of defects in a constant inspection unit on a Poisson basis — one item can hold several defects.
• Q: Cp is 1.8 but Cpk is 0.9. What do you do? A: The spread is fine and the mean is badly off-centre, so re-centre the process by adjusting the setting or offset — no machine or tooling investment is needed.
• Q: Why is Ppk usually lower than Cpk? A: Cpk uses short-term within-subgroup sigma; Ppk uses the overall long-term sigma, which also contains between-subgroup drift such as tool wear and material-lot changes.
• Q: What does the OC curve tell you, and what makes a plan more discriminating? A: It gives the probability of accepting a lot against its fraction defective; increasing the sample size steepens the curve, while increasing the acceptance number makes the plan more lenient.
• Q: Does doubling the lot size require doubling the sample? A: No — discrimination depends almost entirely on n and c, not on lot size, so a constant-percentage sampling rule is statistically wasteful on big lots and too weak on small ones.
• Q: When is CUSUM or EWMA better than a Shewhart chart? A: For detecting small sustained shifts (around 0.5σ–1σ), because they accumulate information from earlier points; the Shewhart chart is better for large sudden shifts.
• Q: Which Indian standard governs attribute sampling inspection? A: IS 2500 (Part 1), aligned with ISO 2859-1, giving AQL-indexed single, double and multiple plans with normal, tightened and reduced inspection switching rules.
• Q: What must you check before trusting any control chart? A: The measurement system — through a gauge R&R study — and rational subgrouping; an unfit gauge or wrongly formed subgroups make the chart report the measurement process, not the manufacturing process.

Common mistakes

• Plotting specification limits on an X̄ chart, or comparing a subgroup mean against the part tolerance — the mean's spread is σ/√n, far narrower than the individuals'.
• Computing Cp or Cpk on a process that is not yet in control, or on clearly non-normal data, and reporting the figure as capability.
• Using N (sample size) instead of n in p-chart limits, or forgetting to recompute limits when n changes day to day.
• Reporting a negative LCL instead of zero on p, np, c and u charts.
• Confusing "defect" with "defective" — counting defects on a p chart or defective items on a c chart.
• Taking a single point inside the limits as proof of control while ignoring runs, trends and zone patterns.
• Treating acceptance sampling as quality improvement: it sorts lots, it never makes a lot better, and it diagnoses nothing about the cause.
• Mixing the two sigmas — using R̄/d₂ where the long-term overall standard deviation is required (Pp/Ppk, ppm estimates).

Related topics

Lean manufacturing and Six Sigma · Reliability engineering · ISO 9001 quality management systems · Total productive maintenance · Design of experiments and Taguchi methods · Probability and statistics for engineers