Cpk / Ppk capability

Method

• σ overall is the sample standard deviation (n − 1), without an unbiasing constant.
• σ within, one column: average moving range of length 2 divided by d2(2) = 1.128. An empty cell breaks the moving range.
• σ within, several columns: pooled standard deviation divided by c4(d + 1), d = Σ(nᵢ − 1). Rows may differ in size.
• These match the defaults of Minitab's Normal Capability Analysis.
• 95% CI for Cpk and Ppk: Bissell's approximation, index ± 1.96 · √(1 / (9N) + index² / (2ν)); ν = N − 1 for Ppk, the within degrees of freedom for Cpk.
• Expected PPM assumes a normal distribution; observed PPM counts values strictly outside the limits.
• Anderson-Darling uses the sample mean and σ overall; the p-value comes from the adjusted A² (D'Agostino & Stephens).
• Required Cpk verdict uses the lower 95% bound of Cpk: Capable when the bound reaches the requirement, Not capable when Cpk itself falls short, Inconclusive when only Cpk reaches it.
• Histogram: Freedman-Diaconis bin count, limited to 5–25. Bin widths are whole steps of the measurement resolution and edges lie half a step between possible values, so no value sits on an edge.
• Results use the decimal sign of the pasted data. The results CSV follows it too: decimal comma gives “;” between columns.
• Files: CSV, TSV and TXT are read as if pasted (UTF-8, UTF-16, otherwise Windows-1257). XLSX: choose the sheet and the columns or a cell range; formula cells use the value Excel last saved, dates are kept as text. Files up to 20 MB are read in this browser only and never uploaded.
• XLSX files are read with SheetJS Community Edition 0.20.3, Apache License 2.0 (license text at the end of this file).
• Indices assume a stable process and normal data. Check both before relying on the numbers.

Non-normal data (Distribution)

• Normal (default) is the analysis above, unchanged. The other choices add a Distribution fits table (Anderson-Darling A² and p-value of every candidate) and a probability plot: each ordered value at its median rank (i − 0.3) / (n + 0.4) on a normal probability scale, with the fitted CDF as the line.
• Box-Cox (values and limits > 0): W = Y^λ, ln Y for λ = 0. The optimal λ is the one in −5…5 with the smallest σ within of the standardised variable (Y^λ − 1) / (λ G^λ−1), G the geometric mean (G ln Y for λ = 0); σ within is MR̄ / d2 for one column and the pooled SD / c4 for subgroups, as in Minitab. Grid of 0.01, then a golden-section search. 95% interval for λ: σ(λ) ≤ σ(λ̂) · exp(χ²_{1; 0.95} / 2N). Rounded λ is the nearest of the integers and ±0.5, used only when it lies in that interval (Minitab's rule), otherwise the optimal λ. The data and the limits are transformed and the normal analysis (within and overall, Bissell intervals, PPM) runs on W. For λ < 0, Y^λ reverses the order: LSL* is then above USL*, and each index and PPM is reported for its original side.
• Johnson (Chou, Polansky & Mason 1998): for z = 0.25, 0.26, …, 1.25 the data percentiles at the normal probabilities of −3z, −z, z, 3z (position nP + ½ in the ordered data, interpolated, clamped to the first and last value) give SB (mn/p² < 1), SU (mn/p² > 1) and SL (m/p > 1) fits by Slifker & Shapiro's equations; a fit counts when every value lies inside its support. The fit whose transformed data has the largest normal AD p-value is chosen and used only when that p-value is above the cut-off (0.10, Minitab's default); otherwise no capability is computed. As in Minitab, Johnson gives overall capability only: the transformation is fitted to all data at once, so no subgroup structure is left for σ within. Ppk has a Bissell interval and the Required Cpk verdict reads Ppk.
• Lognormal and Weibull (2-parameter, values > 0) are fitted by maximum likelihood: lognormal μ = mean(ln x), σ = √(Σ(ln x − μ)² / n); Weibull shape β from Σx^β ln x / Σx^β − 1/β = mean(ln x), scale (Σx^β / n)^1/β. 3-parameter fits are not offered. Overall capability only, as Minitab's Nonnormal Capability Analysis. Z-score method (Minitab's default): Z.LSL = Φ⁻¹(P(X < LSL)), Z.USL = Φ⁻¹(P(X ≤ USL)), PPL = −Z.LSL / 3, PPU = Z.USL / 3, Pp = (Z.USL − Z.LSL) / 6. ISO 22514-2 percentile method: PPU = (USL − X_0.5) / (X_0.99865 − X_0.5), PPL = (X_0.5 − LSL) / (X_0.5 − X_0.00135), Pp = (USL − LSL) / (X_0.99865 − X_0.00135). Ppk is the smaller one-sided index; both methods are shown in the fit table. Expected PPM comes from the fitted distribution, observed PPM counts values as before. No confidence interval and no verdict are given for these fits: Bissell's formula assumes normal data, and no verified interval method for them is implemented.
• Goodness of fit: normal, Box-Cox and Johnson use the normal AD of the (transformed) data, lognormal the normal AD of ln x, all with the D'Agostino & Stephens p-value. These p-values do not allow for λ or the Johnson parameters having been estimated from the same data, so they flatter the transformations. Weibull: A² from the fitted CDF, p-value from A* = A² (1 + 0.2/√n) by the curve 1 / (1 + e^{−0.10 + 1.24 ln A* + 4.48 A*}), which passes through Stephens' (1977) table (0.474, 0.637, 0.757, 0.877, 1.038 at 25, 10, 5, 2.5, 1%); outside 1–25% it is shown as “< 0.010” or “> 0.250”.
• Auto: Normal when its AD p ≥ 0.05; otherwise the better of lognormal and Weibull (largest p-value, at least 0.05); otherwise the better of Box-Cox and Johnson (largest p-value, at least 0.05); otherwise the largest p-value of all, with a warning. Distributions come before transformations because the transformations are fitted more freely and their p-values are optimistic. The reason is printed above the chart.
• The histogram shows the fitted density on the original scale (Box-Cox: σ within and σ overall). Not compared with Minitab output: the Box-Cox interval for λ, the Johnson percentile interpolation and tie handling, the Weibull p-value curve between table values, and the Auto rule (Minitab's Individual Distribution Identification leaves the choice to the user).