Dimensional Capability: Method

Capability Indices, True Position Yield and the First-Article Study

What the study answers

A single-characteristic capability study asks whether one dimension of a stable process fits its tolerance. A drawing has dozens of characteristics, and a first-article or PPAP submission must show each of them. This tool runs the same study on every characteristic of a part at once, from one pasted box, and adds the two things a multi-characteristic study needs that a single one does not: the ranking that says which feature to work on first, and the assembly yield, the probability that a part meets all of its characteristics together. Each characteristic is treated by the normal model of the SPC and Capability tool; a true-position characteristic, whose deviation is a radius rather than a signed error, gets its own bivariate treatment.

Entering the characteristics

Block format: a header line #number name | nominal | lower tolerance | upper tolerance | type followed by the measurements on as many lines as needed (spaces, commas or tabs between values). The tolerance cells are deviations from the nominal. A bare lower value is read below the nominal, so -0.10, 0.10 and ±0.10 all give the same lower limit; a lower cell with a plus sign lies above the nominal, which is how a hole 25 +0.021 / +0.000 is written (25.000 | 0 | +0.021), and an upper cell with a minus sign lies below it, for a shaft 10 −0.010 / −0.030 (10 | -0.030 | -0.010). Types: bilateral (the default when both cells hold a value), max-only (flatness, runout, perpendicularity, roughness: the limit in the upper cell, nominal 0 or blank), min-only (a minimum wall or a minimum strength: the limit in the lower cell), position (one line per part with the x and y deviations of the feature from its true position, the zone diameter in the upper cell, an optional third number on each line being that part's MMC bonus) and position-3d (x y z, a spherical zone). A blank lower or upper cell with the bilateral type makes the characteristic one-sided.

Wide format: a first line part | name 1 | name 2 | …, then nominal | …, lower | …, upper | … and optionally type | …, then one line per part, its identifier first. The wide format aligns the parts, so the lot's own first-pass yield (parts with every characteristic in specification) is counted as well; position characteristics take the block format only.

Within and overall sigma

Two standard deviations are reported for each characteristic, following the AIAG and Minitab convention. The within (short-term) sigma comes from the dispersion inside rational subgroups of consecutive measurements: with a subgroup size of 1 it is the mean moving range of two consecutive values over d₂(2) = 1.128; with subgroups of m it is R̄/d₂(m) or s̄/c₄(m) by the select, the constants of the standard table (ASTM E2587, Montgomery Appendix VI; the tool computes them by integration of the range distribution and the gamma function and rounds them as the table prints them). The overall (long-term) sigma is the sample standard deviation of every measurement. When the process is stable the two agree; when they differ by more than about 15 % the process drifted during the lot and the capability figure is not yet the number to act on.

The indices

• C_p = (USL − LSL)/(6σ_w): the room a centred process would have. Its reciprocal, 6σ_w/(USL − LSL), is the percent of tolerance used.
• C_pu = (USL − μ)/(3σ_w), C_pl = (μ − LSL)/(3σ_w) and C_pk = min(C_pu, C_pl) = C_p(1 − k), with k = |midpoint − μ|/(half tolerance): the actual capability, allowing for the off-centring. A one-sided characteristic has only the index of its one limit.
• C_pm = (USL − LSL)/(6√(σ² + (μ − T)²)) (Chan, Cheng and Spiring 1988; Taguchi): penalises distance from the target T, here the nominal, whether or not the tolerance is symmetric about it. The tool evaluates it with the overall sigma, as Montgomery's Example 8.3 does.
• P_p and P_pk: the same ratios with the overall sigma, often called performance indices.
• Expected fraction out of specification, Φ((LSL − μ)/σ) + 1 − Φ((USL − μ)/σ), in parts per million for each sigma, beside the observed count. C_pk = 1.00 means 1,350 ppm on the near side, 1.33 means 32 ppm, 1.67 means 0.3 ppm.
• Mean offset, 100 (μ − nominal)/(USL − LSL): where the process sits inside its tolerance, positive above the nominal.

How sure is the Cpk?

A capability index from n measurements is an estimate. Bissell (1990) gave the approximation that is now standard (Montgomery 7th ed. eq. 8.15): the standard error of Ĉ_pk is about √(1/(9n) + Ĉ_pk²/(2(n − 1))), so the lower one-sided bound at confidence 1 − α is Ĉ_pk − z_α√(1/(9n) + Ĉ_pk²/(2(n − 1))). Kushler and Hurley (1992) compared this with the exact bounds and found it adequate for n of 30 and more. With 30 parts and Ĉ_pk = 1.33 the 95 % lower bound is about 1.03: a first-article lot of 30 cannot by itself demonstrate 1.33 with confidence, which is why submission criteria ask for 1.67 on small lots or for 100 or more pieces. The tool prints the bound for every characteristic.

Acceptance bands

The verdict compares C_pk with the entered target: 1.33 is the usual criterion for an existing process and 1.67 for a new process or a safety characteristic. The AIAG Production Part Approval Process (4th ed., §2.2.11.5) states the bands for an initial study: above 1.67 the process currently meets the acceptance criteria; 1.33 to 1.67 may be acceptable, contact the customer; below 1.33 does not currently meet the criteria. Both are printed. A characteristic with C_pk below 1.00 is producing nonconforming parts now; between 1.00 and the target it meets the drawing only while the mean stays exactly where it is.

True position

ASME Y14.5-2018 defines a position tolerance as a cylindrical zone of the stated diameter centred on the true position; the actual deviation of a feature is the diameter of the smallest such cylinder that contains its axis, 2√(x² + y²) for the measured x and y errors. That radius cannot be negative and is not normally distributed even when x and y are, so the ordinary Cpk formula on the radial values is wrong. The tool instead models x and y as independent normal variables with their own means and sigmas and computes the probability mass inside the zone circle of radius R = T/2:

• General case: Y = ∫_−R^R φ((x − μ_x)/σ_x)/σ_x [Φ((√(R² − x²) − μ_y)/σ_y) − Φ((−√(R² − x²) − μ_y)/σ_y)] dx, integrated by Simpson's rule on x = R sin t. This is the value used.
• Equal sigmas: (x² + y²)/σ² is non-central χ² with 2 degrees of freedom and non-centrality λ = (μ_x² + μ_y²)/σ², so Y = F_χ²(R²/σ²; 2, λ), computed as a Poisson mixture of central χ² terms and printed with the pooled sigma as a check.
• Centred process: Y = 1 − exp(−R²/(2σ²)), the Rayleigh distribution; the page prints it as the yield that recentring would reach.

Bothe (1997) converts the yield into an equivalent capability by treating the position as a one-sided characteristic with the same fraction out: C_pk,eq = Φ⁻¹(Y)/3, so that 99.865 % inside gives 1.00 and 99.9968 % gives 1.33. The within sigma gives the equivalent Cpk, the overall sigma the equivalent Ppk. When a maximum-material-condition modifier applies, each part's bonus (the departure of its feature size from MMC) is added to that part's zone for the conformance count; the statistical yield is computed on the fixed zone and is therefore conservative. A spherical zone (x y z) is handled by the 3-degree-of-freedom non-central χ² on the pooled sigma. The scatter figure shows the parts inside the zone circle, the mean position and the 3-sigma (overall) ellipse.

Assembly yield and ranking

The probability that a part meets all k characteristics is the product of the k yields when the characteristics vary independently, Y_asm = Π Y_i; the tool prints it with both sigmas and, for the wide format, the lot's observed first-pass yield as well. Correlated characteristics (two dimensions cut in one set-up from one datum) make the true yield higher than the product, so the product is the conservative figure. The ranking orders the characteristics from the lowest Cpk upward and the bar chart draws them against the target: the first rows are the ones to work on, and for each the gap between C_p and C_pk says how much would be recovered by recentring alone against how much needs the spread reduced.

Normality

Every index and every ppm figure rests on the normal model. The Shapiro–Wilk test (Royston's AS R94 algorithm) is run on each characteristic and a p-value below 0.05 is flagged. Geometric characteristics bounded at zero (flatness, runout, roughness) are seldom normal: their distribution is folded, so the Cpk of such a characteristic is indicative only and the observed ppm is the better guide. Skewed data need a transformation or a fitted distribution (the Distribution Fitting tool) before a ppm figure means anything.

Worked examples on the page

The default case is an illustrative machined part with six characteristics: four bilateral lengths (one of them, the bore, running off-centre toward its upper limit with Cpk about 0.57), a flatness with a maximum only, and the position of a hole pattern with twelve x y pairs in a ⌀0.25 zone. The second example in the select is Montgomery's hard-bake flow-width data (7th ed., Table 6.1, 25 subgroups of five), for which the book prints σ̂ = R̄/d₂ = 0.1398, C_p = 1.192 for the specification 1.50 ± 0.50 and about 350 ppm out; the page reproduces them when the subgroup size is 5 and the range estimator is chosen. The test script checks the library against these, against NIST/SEMATECH 6.1.6 (Cp 1.00, Cpk 0.667, Cpm 0.707), Montgomery's Table 8.2 ppm values, Example 8.3 (Cpm 0.63) and Example 8.5 (Bissell interval 0.88 to 1.78), the Shapiro–Wilk 1965 example, the Rayleigh and non-central χ² closed forms and a numpy Monte Carlo of the position yields.

Limits of the tool

• Stability is assumed, not tested: run the SPC and Capability tool's control charts on a characteristic whose within and overall sigma disagree.
• Independent normal x and y deviations for position; a correlated or skewed pattern (a rotated plate) needs the bivariate covariance, which is not modelled.
• The assembly yield assumes independent characteristics.
• Measurement error is not separated out; a gauge R&R (SPC and Capability tool) should precede the study.
• Bissell's bound is an approximation; for n below about 30 it is somewhat optimistic.
• Composite position tolerances, profile of a surface and datum shift are not handled; enter the resulting deviations as max-only characteristics.