Statistical quality control

In short

Statistical quality control (SQC) is the use of probability and sampling to judge whether a manufacturing process is behaving consistently and whether a lot of product is fit to accept. Its three branches are process control (control charts), process capability analysis (Cp, Cpk) and acceptance sampling (OC curves, AQL/LTPD plans). It matters because 100% inspection is costly, fatiguing and still imperfect, whereas a small sample taken at the right time tells you *when the process changed*, not merely that a part is bad. It is examined in GATE (ME and PI), university Industrial Engineering papers and almost every quality/production interview at Indian employers — Maruti Suzuki, Tata Motors, Bosch, TVS, Ashok Leyland, Hindustan Unilever, Sundaram Fasteners, L&T, BHEL, HAL and their Tier-1 suppliers, who must show capability evidence under IATF 16949/PPAP.

Explained

Variation, and the logic on which SQC rests

• Every process varies. Chance (common) causes are many small, inherent, unavoidable influences — tool wear in the normal band, slight material and temperature variation. A process affected only by these is in statistical control: stable and predictable.
• Assignable (special) causes are few, large and identifiable — a wrong setting, a blunt tool, a new operator, a changed material batch. They make output unpredictable; SQC exists to detect them quickly and cheaply.
• "In control" is not the same as "meeting specification". A stable process can be stably producing scrap; conversely, a capable process can be out of control. Control limits come from the process (its own variation); specification limits come from design/customer. Never plot specification limits on an X̄ chart.
• Quality characteristic types decide the tool: variables are measured on a continuous scale (mm, kg, N/mm², °C); attributes are counted as go/no-go (defective / not defective) or as number of defects.
• Defect vs defective: a defect is one nonconformity; a defective (nonconforming unit) is an item carrying one or more defects. One casting may have 4 blowholes — 4 defects, 1 defective.
• Rational subgrouping: pick samples so that variation *within* a subgroup is only chance cause (consecutive pieces, same setting), and any process shift shows up *between* subgroups. Typical practice: n = 4 or 5, 20–25 subgroups to set up the chart.
• The three-sigma (Shewhart) limits are an economic compromise: for a normal variable, 99.73% of points fall inside, so false alarms are about 0.27% (1 in 370 points), while real shifts of about 1.5σ or more are caught within a few subgroups.

Control charts for variables

• Plotted in pairs — one chart for central tendency, one for spread. Interpret the range/sigma chart first: control limits of the X̄ chart are computed from R̄, so they are meaningless if the spread itself is unstable.
• X̄ – R chart (n = 2 to about 9, the normal shop-floor choice): centre lines X̄̄ and R̄; limits use factors A₂, D₃, D₄.
• X̄ – s chart (n ≥ 10, or unequal n, or computerised gauging): uses s̄ with A₃, B₃, B₄ — statistically more efficient than the range.
• X – MR chart (individuals and moving range): for slow processes, destructive or expensive tests, batch chemistry, one reading per shift. Moving range of two successive values; σ̂ = M̄R̄/1.128. No averaging, so it is insensitive to small shifts.
• Median and mid-range charts: simpler arithmetic for manual shop use; slightly less efficient.
• Specialised charts: CUSUM (cumulative sum of deviations from target) and EWMA (exponentially weighted moving average) detect *small sustained* shifts far faster than Shewhart charts; pre-control (green/yellow/red zones at the quarter-tolerance points) is a crude but quick set-up tool; short-run / target (DNOM) charts plot deviation from nominal so one chart serves many part numbers.
• Process σ from charts: σ̂ = R̄/d₂ or s̄/c₄. This "within-subgroup" σ is the short-term spread used for Cp/Cpk.

Control charts for attributes

• p chart — fraction nonconforming; sample size may vary (limits then recomputed per sample, or an average n̄ used if sizes vary within ±25%). Most general attribute chart.
• np chart — number nonconforming; only for constant n. Easier for operators (plot the raw count).
• c chart — number of defects per constant inspection unit (one car body, one 10 m weld, one roll of cloth). Based on the Poisson distribution.
• u chart — defects per unit when the inspection unit size varies (defects per m² of plate, per 100 m of wire).
• Demerit / U chart: defects weighted by seriousness (critical, major, minor) — used in appliance and vehicle final audits.
• Attribute charts need much larger samples (often 50–400) than variables charts because a go/no-go result carries less information. Rule of thumb for a usable p chart: n large enough that np̄ ≥ about 5, so the normal approximation holds; a positive LCL needs more — np̄ > 9(1 − p̄), i.e. roughly np̄ ≥ 9.
• If the computed LCL is negative it is taken as zero (a count cannot be negative). A point below a positive LCL is good news and must still be investigated — it often means inspection was not done.

Reading a chart: decision rules, errors and speed of detection

• Out-of-control signals (Western Electric / Nelson rules): (1) any point outside the 3σ limits; (2) 2 of 3 consecutive points beyond 2σ on the same side; (3) 4 of 5 beyond 1σ on the same side; (4) 8 or 9 consecutive points on one side of the centre line (a run); (5) 6 points steadily increasing or decreasing (trend — tool wear, bath depletion); (6) 14 points alternating up and down; (7) 15 points hugging the centre line (stratification — often mixed streams or a wrongly computed limit); (8) 8 points with none within 1σ (mixture — two machines on one chart).
• Errors: Type I (α, producer's risk) — signalling when the process is fine, 0.0027 per point at 3σ. Type II (β, consumer's risk) — missing a real shift.
• ARL (average run length) = 1/P(signal). In control, ARL₀ = 1/0.0027 ≈ 370 subgroups. Widening limits to 3.5σ cuts false alarms but slows detection; narrowing to 2σ does the opposite. The OC curve of a control chart plots β against the size of the shift.
• Larger n makes the X̄ chart more sensitive (limits tighten as 1/√n); more frequent sampling detects shifts sooner. Both cost money — this is the economic design of control charts.
• Charts require normality (X̄ charts are robust by the central limit theorem even if individuals are not), independence (autocorrelated data gives false alarms) and a stable measurement system.

Process capability and capability indices

• Capability compares the voice of the process (6σ spread) with the voice of the customer (tolerance). Valid only after the process is shown to be in control; otherwise σ has no single meaning.
• Cp measures potential (spread only); Cpk measures actual (spread *and* centring). Cpk ≤ Cp always; they are equal only when the mean sits exactly at the middle of the tolerance.
• Cpu / Cpl are the one-sided indices, used when only one limit exists (e.g. minimum tensile strength, maximum runout).
• Cpm (Taguchi) penalises departure from the *target*, not just from the tolerance midpoint. Cr = 1/Cp is the capability ratio (expressed as a percentage of tolerance consumed).
• Pp / Ppk use the total (long-term) standard deviation of all individual readings and describe *performance* over time including drift; Cp/Cpk use short-term within-subgroup σ. Pp ≤ Cp in practice; the gap measures instability.
• Accepted benchmarks (customer-specific, not statutory): Cpk ≥ 1.33 for ongoing production, ≥ 1.67 for new or safety-critical characteristics; automotive PPAP studies commonly require Ppk ≥ 1.67 on a 300-piece run. A Cpk of 1.00 still leaks about 2,700 ppm.
• Machine capability (Cm/Cmk) is a short, single-setting study of the machine alone; process capability includes material, operator, tooling and environment.
• For non-normal data (flatness, roundness, waiting time — bounded at zero), use percentile-based indices or transform the data; applying the normal ppm table directly overstates quality badly.

Acceptance sampling

• Used on incoming lots, outgoing lots and between operations when inspection is destructive, costly or the lot is already made. It sorts lots, it does not improve quality — prevention through process control is always superior.
• Single sampling plan (N, n, c): draw n, accept if defectives d ≤ c, else reject. Double sampling: a second sample only for the indecisive middle zone — lower average inspection. Multiple/sequential (Wald SPRT): smallest average sample size, most complex administration.
• OC curve: probability of acceptance Pa versus incoming fraction defective p. The ideal curve is a vertical step at the limiting quality; real curves are S-shaped. Type A OC curve applies to an isolated lot (hypergeometric); Type B to a continuing stream (binomial, or Poisson when n ≥ 10 and p ≤ 0.1).
• Four defining points: AQL (acceptable quality level — the worst *average* quality considered satisfactory; Pa ≈ 1 − α, typically 0.95), producer's risk α, LTPD / LQ (lot tolerance percent defective, the quality that must be rejected; Pa = β, usually 0.10), consumer's risk β. Plans are built to pass through two of these points.
• Rectifying inspection (rejected lots are 100% inspected and defectives replaced) gives AOQ = outgoing quality after this screening; its maximum over all p is the AOQL. ATI is the average total inspected per lot.
• Normal–tightened–reduced switching rules (ISO 2859-1) adjust inspection severity on the basis of recent history — the real driver of economy in an AQL system.
• Sampling by variables (ISO 3951 / IS 2500 Part 2): measure n pieces, accept if (X̄ − LSL)/s ≥ k (or USL − X̄ ≥ k·s). Needs a known distribution shape, but a far smaller sample than attributes for the same protection; cannot be used for go/no-go gauging.
• Continuous sampling plans (CSP-1 etc.) for conveyor production: 100% inspect until i consecutive conforming units, then sample a fraction f.

Indian standards, measurement systems and practice

• IS 397 (Methods for statistical quality control during production) — control charts for variables and for attributes, and cumulative-sum charts, in its several parts; ISO 7870 series and ISO 7966 are the international equivalents. Check the current BIS revision before quoting clause numbers.
• IS 2500 (Sampling inspection procedures) — Part 1 attribute plans indexed by AQL (aligned with ISO 2859-1), Part 2 variables plans for percent nonconforming (ISO 3951), later parts cover skip-lot and isolated-lot (LQ) schemes. The old MIL-STD-105E was withdrawn and survives as ANSI/ASQ Z1.4; Dodge–Romig tables remain the classic LTPD- and AOQL-indexed plans.
• ISO 9001 (the 2026 edition, succeeding ISO 9001:2015 from 16 September 2026) requires monitoring and measurement of processes; IATF 16949 adds SPC and MSA as mandatory core tools for automotive suppliers. BIS operates the ISI mark scheme, in which conformity is demonstrated through a scheme of inspection and testing with retained control charts; NABL accredits the calibration and test laboratories that feed them.
• Measurement system analysis (MSA): a chart is only as good as the gauge. Study bias, linearity, stability, repeatability (equipment variation) and reproducibility (appraiser variation). Common acceptance: %GRR ≤ 10% of tolerance good, 10–30% conditional, > 30% unacceptable; number of distinct categories ≥ 5. Rule of ten — gauge resolution ≤ 1/10 of the tolerance.
• Seven basic QC tools supporting SQC on the shop floor: check sheet, histogram, Pareto diagram, cause-and-effect (Ishikawa) diagram, scatter diagram, stratification/flow chart, and the control chart itself. The chart detects; the Pareto and fishbone find the assignable cause.

Key points

• Chance cause – inherent, many, small; leave the process alone. Assignable cause – few, large, identifiable; find and remove it.
• Control limits ≠ specification limits – the first come from process data, the second from design; never mix them on one chart.
• 3σ limits – α = 0.27%, ARL₀ ≈ 370 points; an economic balance of false alarms against detection speed.
• Interpret R (or s) chart first – X̄ limits are derived from R̄ and are invalid if spread is unstable.
• Variables charts – X̄–R (n ≤ 9), X̄–s (n ≥ 10), X–MR (n = 1); small samples suffice because measurements carry more information.
• Attribute charts – p (fraction, variable n), np (count, fixed n), c (defects, fixed unit), u (defects per unit, variable size).
• Defect vs defective – c and u count defects (Poisson); p and np count nonconforming units (binomial).
• σ̂ = R̄/d₂ – the standard way to estimate process spread from a range chart.
• Negative LCL is reported as zero for R, p, np, c and u charts; a positive p-chart LCL needs about np̄ ≥ 9.
• Cp = tolerance / 6σ – potential capability; ignores where the mean sits.
• Cpk = min(Cpu, Cpl) – actual capability; Cpk < Cp means the process is off-centre.
• Cp/Cpk vs Pp/Ppk – short-term within-subgroup σ vs long-term total σ; the gap measures instability.
• Cpk benchmarks – 1.33 ongoing, 1.67 for new/safety-critical; Cpk = 1.00 still leaks ~2,700 ppm.
• OC curve – Pa versus incoming p; defined by AQL/α and LTPD/β; ideal curve is a vertical step.
• AQL – worst average quality accepted routinely; LTPD – quality that must be rejected with 90% certainty.
• AOQ and AOQL – outgoing quality under rectifying inspection, and its worst value over all p.
• Double/sequential sampling cuts average sample number but adds administrative complexity.
• MSA first – with %GRR above 30% of tolerance, the chart is plotting gauge noise, not the process.

Formulas and examples

Symbols: X̄̄ = grand average of subgroup means; R̄ = average range; s̄ = average subgroup standard deviation; n = subgroup size; k = number of subgroups; d = defectives in a sample; c̄ = average defects per inspection unit; ū = average defects per unit; N = lot size; c = acceptance number; Φ = standard normal CDF. Units follow the characteristic measured (mm, kg, N/mm²); ppm = parts per million.

Example 1 — X̄–R chart and capability (shaft diameter). 20 subgroups of n = 5; ΣX̄ = 2,000.0 mm, ΣR = 80.0 mm. Specification 100 ± 5 mm.

• X̄̄ = 2,000.0/20 = 100.00 mm; R̄ = 80.0/20 = 4.00 mm.
• R chart: UCL = 2.114 × 4 = 8.46 mm, LCL = 0 × 4 = 0.
• X̄ chart: UCL = 100 + 0.577 × 4 = 102.31 mm, LCL = 100 − 2.308 = 97.69 mm.
• σ̂ = R̄/d₂ = 4/2.326 = 1.72 mm. 6σ̂ = 10.32 mm against a tolerance of 10 mm.
• Cp = 10/10.32 = 0.97 — the spread alone is already slightly too wide. Centred, z = 5/1.72 = 2.91, so scrap ≈ 2 × 0.0018 = 0.36% ≈ 3,600 ppm.
• If the mean drifts to 101.5 mm: Cpu = (105 − 101.5)/(3 × 1.72) = 3.5/5.16 = 0.68; Cpl = (101.5 − 95)/5.16 = 1.26; Cpk = 0.68. Out-of-spec: z_u = 3.5/1.72 = 2.04 → 2.07%; lower tail (z = 3.78) ≈ 0.008%. Total ≈ 2.1%, about 21,000 ppm — a 1.5 mm shift multiplies rejection nearly six-fold.

Example 2 — p and np charts (relay assembly). 25 samples of n = 200; total defectives 300.

• p̄ = 300/5,000 = 0.06. σ_p = √(0.06 × 0.94/200) = √0.000282 = 0.01679.
• p chart: UCL = 0.06 + 0.0504 = 0.1104, LCL = 0.06 − 0.0504 = 0.0096.
• np chart: CL = 12, σ = √(12 × 0.94) = √11.28 = 3.359 → UCL = 22.08, LCL = 1.92.
• A sample with 25 defectives gives p = 0.125 > 0.1104 → out of control; investigate that shift's assignable cause. A sample with 1 defective is below the LCL — almost always a missed or falsified inspection, not a miracle.

Example 3 — c and u charts (welded assemblies). 20 assemblies inspected, 80 defects found in all.

• c̄ = 80/20 = 4 defects per assembly. UCL = 4 + 3√4 = 10, LCL = 4 − 6 = −2 → 0.
• Same data re-expressed per metre of weld, each assembly carrying 5 m: ū = 4/5 = 0.8 defects/m. For a 5 m sample, UCL = 0.8 + 3√(0.8/5) = 0.8 + 3(0.4) = 2.0, LCL = 0.8 − 1.2 → 0. For a 20 m fabrication, UCL = 0.8 + 3√(0.8/20) = 0.8 + 0.6 = 1.4/m — larger samples give tighter limits.

Example 4 — single sampling plan, OC point, AOQ and ATI. N = 2,000, n = 50, c = 1.

• At incoming p = 0.02: np = 1.0 → Pa = e⁻¹(1 + 1) = 0.3679 × 2 = 0.736.
• AOQ = 0.736 × 0.02 × (1,950/2,000) = 0.01435 ≈ 1.44%.
• ATI = 50 + (1 − 0.736)(1,950) = 50 + 514.8 ≈ 565 pieces per lot on average.
• At p = 0.08 (candidate LTPD): np = 4 → Pa = e⁻⁴(1 + 4) = 0.0183 × 5 = 0.092, i.e. consumer's risk ≈ 9.2%.
• At p = 0.01 (candidate AQL): np = 0.5 → Pa = e⁻⁰·⁵(1.5) = 0.6065 × 1.5 = 0.910, so producer's risk α ≈ 9% — high; a larger n with a proportionally larger c would steepen the curve and protect both parties.

Example 5 — how fast will the chart catch a shift? Process σ = 0.60 mm, X̄ chart with n = 5. The mean shifts by 1.5σ = 0.90 mm.

• σ_X̄ = 0.60/√5 = 0.268 mm; limits sit 3σ_X̄ = 0.805 mm (= 1.342σ) from the old centre line.
• After the shift the subgroup mean averages 0.90 mm above the old centre: z = (0.805 − 0.900)/0.268 = −0.354 → P(point above UCL) = 0.638 (opposite tail negligible).
• ARL = 1/0.638 ≈ 1.6 subgroups — caught almost at once. Repeat for a 0.5σ shift: z = (1.342 − 0.5)/0.447 = 1.88 → P = 0.030, ARL ≈ 33 subgroups, which is why CUSUM or EWMA charts are preferred for small drifts.

Where it is used

• Automotive and auto-component plants (Maruti Suzuki, Tata Motors, Bosch, Sundaram Fasteners, Rane, Endurance): X̄–R charts on bore, hardness and torque; Cpk/Ppk evidence in PPAP submissions and supplier audits under IATF 16949.
• Foundries, forging and machining job shops: c and u charts for blowholes and surface defects per casting; machine capability (Cmk) studies for CNC acceptance before commissioning.
• FMCG and pharma filling lines (HUL, Dabur, Cipla, Sun Pharma): X̄–R charts on fill weight against Legal Metrology packaged-commodity limits; attribute charts for seal and label defects; sampling plans for incoming packaging material.
• Textiles, steel and cement: u charts for defects per 100 m² of fabric or per plate; continuous sampling plans on rolling mills; BIS ISI-mark schemes of inspection and testing at steel and cement works.
• Incoming quality / vendor management: ISO 2859-1 (IS 2500 Part 1) AQL plans with normal–tightened–reduced switching for bought-out fasteners, bearings and electronics.
• Job roles: QA/QC engineer, SPC or SQC engineer, process/manufacturing engineer, supplier quality engineer, metrology and calibration engineer, Six Sigma Green/Black Belt projects, BIS and NABL-accredited laboratory roles.

Interview questions

• Q: What is the difference between chance and assignable causes? A: Chance causes are many small inherent influences that cannot be economically removed and define the natural process spread; assignable causes are few, large, identifiable and must be found and eliminated.
• Q: Why are control limits set at 3σ? A: It balances risks economically — only 0.27% false alarms (ARL ≈ 370) while still catching shifts of about 1.5σ within a couple of subgroups.
• Q: Can a process be in control yet produce rejects? A: Yes. Control limits describe the process's own variation; if that variation exceeds the tolerance, or the mean is off-centre, a perfectly stable process produces out-of-specification parts — that is a capability problem, not a control problem.
• Q: Which chart do you read first, X̄ or R, and why? A: The R chart, because the X̄ chart's limits are computed from R̄; if the spread is unstable those limits have no meaning.
• Q: Difference between Cp and Cpk? A: Cp compares tolerance with 6σ and ignores location; Cpk also penalises off-centring, so Cpk = Cp only when the mean sits at the tolerance midpoint, and Cpk is always ≤ Cp.
• Q: Difference between Cpk and Ppk? A: Cpk uses short-term within-subgroup σ (what the process can do); Ppk uses the long-term total σ of all individuals (what it actually delivered, including drift). A large gap signals an unstable process.
• Q: When would you use a c chart rather than a p chart? A: When you count defects on a constant inspection unit — blowholes per casting, solder faults per board — rather than classifying whole units as good or bad.
• Q: What does a negative LCL on a p chart mean? A: Nothing physical; it is reported as zero, since a fraction defective cannot be negative. It also means the sample is too small to detect improvement.
• Q: Define AQL and LTPD. A: AQL is the worst average quality level routinely accepted (Pa ≈ 0.95); LTPD is the poor quality level that must be rejected with about 90% probability (Pa = 0.10).
• Q: Why is acceptance sampling considered inferior to process control? A: It only sorts lots after the fact and accepts a defined level of defectives; it neither prevents nor diagnoses the cause. Control charts act on the process while it is running.
• Q: Eight consecutive points lie above the centre line but inside the limits — is the process in control? A: No. A run of eight on one side signals a mean shift; the probability of it occurring by chance is about (1/2)⁸ ≈ 0.4% on each side.
• Q: Your gauge has a %GRR of 35% of tolerance. What is the first action? A: Fix the measurement system before interpreting any chart or capability index — at that level most of the observed variation is measurement noise, and Cpk will be understated.

Common mistakes

• Drawing specification limits on an X̄ chart, or calling a process "capable" because its points lie within control limits.
• Computing Cp or Cpk on a process that has not first been shown to be in statistical control — σ then has no stable value.
• Using the individual σ where σ/√n belongs (or vice versa) in X̄ chart limits; and forgetting that for n = 5, A₂R̄ already contains the √n effect.
• Reporting a negative LCL instead of zero, or ignoring a point below a positive LCL that usually means inspection was skipped.
• Mixing up defects and defectives — using a c chart where a p chart is required, or taking c̄'s σ as √(c̄/n) instead of √c̄.
• Choosing the wrong constant from the table (A₂ for an s chart, D₄ for the X̄ chart) or reading the row for the wrong n.
• Treating AQL as "the quality level we aim at" rather than a sampling-plan parameter, and accepting repeated borderline lots instead of switching to tightened inspection.

Related topics

Lean manufacturing and Six Sigma · Work study and ergonomics · Production planning and control · Maintenance and reliability engineering · Metrology and inspection · Probability and statistics (distributions, hypothesis testing)