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SPC and Capability: Method

Open the SPC and Capability tool

Control Charts, Run Rules, Capability and Gauge R&R

Common and special causes

Every process varies. Shewhart (1931) separated that variation into two kinds: common causes, the many small influences always present that make the output vary in a stable, predictable pattern, and special (assignable) causes, identifiable events that shift the mean or inflate the spread for a while. A control chart is a running test of whether only common causes are acting. While they are, the process is in statistical control and its future output can be predicted; when a special cause appears, the chart signals it and the cause can be found and removed. Reacting to common-cause variation as if it were special (tampering) makes a process worse, and ignoring a special cause lets a defect stream continue; the chart tells the two apart.

Rational subgroups

Measurements are taken in small subgroups chosen so that only common causes can act within a subgroup (consecutive pieces from one machine, one shift, one lot) while a special cause would act between subgroups. The within-subgroup dispersion (the range R or the standard deviation s) then estimates the common-cause sigma, and the chart of subgroup means tests whether the means wander more than that sigma allows. Subgroups of four or five are the usual compromise between sensitivity and sampling cost; Shewhart's rule of thumb asks for about 25 subgroups before limits are trusted. When measurements come one at a time (batches, laboratory results) the individuals chart uses the moving range of consecutive values as its subgroup.

Three-sigma limits and the constants

For a plotted statistic w with mean μw and standard deviation σw the centre line is μw and the limits are μw ± 3σw. Three sigma is Shewhart's economic choice: for a normal statistic the false-alarm probability per point is 0.27 % (one in 370 points), small enough that a signal is worth investigating. The unknown σ is estimated from the subgroup dispersion through the constants of the standard table (ASTM E2587, Montgomery Appendix VI):

  • d2 = E[R]/σ and d3 = sd(R)/σ for the range of n normal values, so σ̂ = R̄/d2 and sd(R) = d3 R̄/d2. The tool computes them by numerical integration of the range distribution, E[R] = ∫ [1 − Φ(x)n − (1 − Φ(x))n] dx, and the corresponding double integral for E[R²], and rounds them as the tables print them.
  • c4 = E[s]/σ = √(2/(n − 1)) Γ(n/2)/Γ((n − 1)/2), so σ̂ = s̄/c4; sd(s) = σ √(1 − c4²).
  • A2 = 3/(d2√n) and A3 = 3/(c4√n) give the X̄ limits x̿ ± A2 R̄ or x̿ ± A3 s̄; D3, D4 = 1 ∓ 3 d3/d2 give the R limits; B3, B4 = 1 ∓ 3 √(1 − c4²)/c4 the s limits. A negative lower factor is set to zero (n ≤ 6 for R, n ≤ 5 for s).
  • Individuals: UCL/LCL = x̄ ± 3 MR̄/d2(2) = x̄ ± 2.66 MR̄; the moving range chart has UCL = D4(2) MR̄ = 3.267 MR̄.

The range is used for subgroups up to about 10 (its efficiency relative to s is 0.955 at n = 5 and 0.85 at n = 10); the s chart is preferred beyond that.

Phase I and Phase II

Phase I is the retrospective study: limits are computed from the data at hand (trial limits), the chart is examined, subgroups with an assignable cause are removed and the limits recomputed, until the remaining data show control. Those limits are then frozen and used in Phase II to monitor new subgroups one at a time. The tool's revision option makes one such pass, dropping either the points beyond the limits on either chart (the standard practice, Montgomery §6.2.3) or every flagged subgroup, and lists what was removed; the analyst must still confirm that each removed subgroup had an assignable cause, because removing points that merely look odd biases the limits inward. The dispersion chart is read first: if the R or s chart is out of control the sigma estimate, and with it the X̄ limits, is meaningless.

Run rules and their cost

A single point beyond 3σ is Shewhart's rule. The Western Electric handbook (1956) added patterns that are about as improbable under control: two of three consecutive points beyond 2σ on the same side, four of five beyond 1σ on the same side, and eight in a row on one side of the centre line. Nelson (1984) formalised eight tests (one beyond 3σ; nine in a row on one side; six in a row steadily rising or falling; fourteen alternating; two of three beyond 2σ; four of five beyond 1σ; fifteen in a row within 1σ, stratification; eight in a row beyond 1σ on either side, mixture). The zones are defined on the chart's own sigma, (UCL − CL)/3, and the tool flags the point that completes a pattern. The rules buy sensitivity to small shifts at the price of false alarms: the plain 3σ chart alarms every 370 in-control points on average, the four Western Electric rules together about every 92 (Champ and Woodall 1987). Use them when a small sustained shift matters and the cost of a false search is modest.

Attribute charts

When the measurement is a count, the binomial and Poisson distributions supply the sigma. The p chart plots the fraction nonconforming of each sample about p̄ = Σ defectives / Σ inspected with limits p̄ ± 3 √(p̄(1 − p̄)/ni), so a varying sample size gives each point its own limits; the np chart plots the count itself for a constant n with limits np̄ ± 3 √(np̄(1 − p̄)). The c chart plots the number of nonconformities in a fixed inspection unit with limits c̄ ± 3 √c̄; the u chart plots nonconformities per unit for a varying number of units with ū ± 3 √(ū/ni). A lower limit below zero is set to zero. The run rules apply to these charts in the same way, using each point's own sigma.

Capability and performance

Control is about stability; capability compares a stable process with its specification. With the process mean μ and sigma σ and the limits LSL and USL, Cp = (USL − LSL)/(6σ) measures the room a centred process would have, CPU = (USL − μ)/(3σ) and CPL = (μ − LSL)/(3σ) the room on each side, and Cpk = min(CPU, CPL) = Cp(1 − k) the actual capability, where k = |m − μ|/((USL − LSL)/2) is the off-centring from the midpoint m. Cpm = (USL − LSL)/(6 √(σ² + (μ − T)²)) (Taguchi) penalises distance from a target T whether or not the specification is centred on it. Two sigmas are reported, following the AIAG and Minitab convention:

  • σwithin = R̄/d2, s̄/c4 or MR̄/d2, the common-cause sigma inside the subgroups: what the process could do if its mean held still. Cp and Cpk use it.
  • σoverall, the sample standard deviation of every measurement: what the process actually delivered over the study, including any drift between subgroups. Pp and Ppk are the same ratios with this sigma; they are often called performance indices.

When the process is in control the two sigmas agree and Cpk ≈ Ppk; a large gap is itself a sign that the process is not stable, and then the chart, not the index, is the thing to fix. A Cpk of 1.33 (a 4σ distance to the nearer limit) is the usual acceptance criterion for an existing process and 1.50 for a new one; 1.00 means the limit sits exactly 3σ from the mean.

Fraction out of specification, ppm and sigma levels

Under the normal model the expected fraction outside the specification is p = Φ((LSL − μ)/σ) + 1 − Φ((USL − μ)/σ), reported in parts per million for each sigma, beside the observed fraction counted in the data. A one-sided Cpk of 1.00 corresponds to Φ(−3) = 1,350 ppm, 1.33 to Φ(−4) = 32 ppm, and a centred Cp of 1.00 to 2,700 ppm on both sides. Z-bench = Φ−1(1 − p) converts the total fraction into one equivalent one-sided distance in sigma units. The Six Sigma convention quotes a sigma level as Z-bench plus 1.5: the long-term fraction (overall sigma) is assumed to come from a process whose mean drifts by up to 1.5σ, so a process delivering 3.4 ppm long term is called a 6σ process although Φ(−6) is 0.001 ppm. The tool prints Z-bench without the shift for both sigmas and the shifted sigma level for the overall sigma, and labels each, since the convention is often quoted without saying which is meant.

Gauge R&R by ANOVA

A capability figure is only as good as the measurement that produced it. In a crossed gauge study o operators each measure the same p parts r times. The two-factor analysis of variance splits the total sum of squares into part, operator, operator × part interaction and repeatability (the within-cell error), with degrees of freedom p − 1, o − 1, (o − 1)(p − 1) and op(r − 1). The variance components follow from the expected mean squares: σ²repeat = MSerror, σ²inter = (MSinter − MSerror)/r, σ²oper = (MSoper − MSinter)/(pr), σ²part = (MSpart − MSinter)/(or); a negative estimate is taken as zero. Following the AIAG manual the interaction is tested first and, when its p-value exceeds 0.25, pooled with the error, after which the operator and part components are computed against the pooled mean square. Reproducibility is operator plus interaction; gauge R&R is repeatability plus reproducibility; the total adds the part-to-part component. The report gives each component's %Contribution (σ² over the total σ²), %Study Variation (σ over the total σ, equivalently 6σ over 6σtotal) and, when a specification is given, %Tolerance (6σ over USL − LSL), and the number of distinct categories ndc = ⌊1.41 σpart/σGRR⌋, the number of non-overlapping groups the gauge can resolve among the parts. The AIAG acceptance guidelines on %Study Variation are: under 10 % acceptable, 10 to 30 % may be acceptable depending on the application and the cost of the gauge and of its improvement, over 30 % unacceptable; ndc should be 5 or more. Whether repeatability or reproducibility dominates says where to look: the instrument and fixture, or operator technique and the measurement instruction.

Worked example on the page

The default data are Montgomery's hard-bake photolithography example (7th ed., Example 6.1, Table 6.1): 25 subgroups of five flow-width measurements in microns. The book prints x̿ = 1.5056, R̄ = 0.32521, X̄ limits 1.69325 and 1.31795, R chart UCL 0.68749, σ̂ = R̄/d2 = 0.1398, Cp = 1.192 for the specification 1.50 ± 0.50 and a fraction out of specification of about 0.00035; the page reproduces them, and the test script checks the library against these values, the NIST/SEMATECH individuals and capability examples, the constants table and hand-worked gauge R&R sums.

Limits of the tool

  • Shewhart charts only: no CUSUM or EWMA charts, which detect small sustained shifts sooner than run rules do.
  • The data are assumed independent; autocorrelated measurements (continuous processes sampled closely) produce false signals and need a time-series model first.
  • Capability assumes a normal distribution. Skewed data need a transformation or a fitted distribution before ppm figures mean anything; the histogram on the page is the first check.
  • Variable subgroup sizes are supported for the p and u charts only; the X̄ charts need equal subgroups.
  • The Phase I revision makes a single pass and does not verify assignable causes.
  • The gauge study must be balanced and crossed (every operator measures every part the same number of times); nested designs, linearity, bias and stability studies are not covered.