Process Capability (Cp, Cpk) Calculator

About the Process Capability (Cp, Cpk) Calculator

Paste measurements — from a spreadsheet, a gauge export or typed by hand, as individual values or in subgroups — add the specification limits, and get a full process capability study: the mean, the within-subgroup and overall standard deviation, Cp, Cpk, Pp, Ppk and, with a target, Cpm, the expected and observed parts per million out of specification, Z.bench and the sigma level, and 95 % confidence intervals of Pp and Ppk so you can see how far the data can be trusted.

The page also draws a histogram with both normal curves and the limits, an I-MR chart for individual values or an X̄–R or X̄–S chart for subgroups, flags every point that breaks a Western Electric or trend rule, and checks normality with the Anderson–Darling test. For attribute data it converts defect counts to DPMO and a sigma level, and back. Everything runs in your browser; a CSV or text file you open is read on your device.

Without a pass, the free preview shows your data as a watermarked histogram and run chart and the checks that apply; the indices, the ppm, the control limits, the working, Copy result and the CSV come with a Pro pass.

Common uses

• A capability study for a new part, a first-article run or a supplier’s submission.
• Checking whether a machine can hold a tolerance before a customer asks.
• Comparing Cpk before and after a process change.
• Spotting drift and special causes on an I-MR or X̄–R chart.
• Converting defect counts into DPMO and a sigma level for a Six Sigma project.
• Quality engineering and statistics coursework.

Formulas

• Cp = (USL − LSL) ÷ 6σ_within; CPU = (USL − x̄) ÷ 3σ_within; CPL = (x̄ − LSL) ÷ 3σ_within; Cpk = min(CPU, CPL)
• Pp, PPU, PPL, Ppk: the same with the overall sample standard deviation s
• Cpm = (USL − LSL) ÷ 6√(s² + (x̄ − T)²), with the target T
• σ within: MR̄ ÷ d₂(2) for individual values (moving ranges of two); R̄ ÷ d₂(n) or s̄ ÷ c₄(n) for subgroups of n
• Expected ppm: 10⁶ × [Φ((LSL − x̄) ÷ σ) + 1 − Φ((USL − x̄) ÷ σ)], once with σ within and once with s
• Z.bench = Φ⁻¹(1 − fraction out of specification); sigma level = overall Z.bench + 1.5
• 95 % intervals: Pp × √(χ²₀.₀₂₅,ν ÷ ν) to Pp × √(χ²₀.₉₇₅,ν ÷ ν) with ν = N − 1; Ppk ± 1.96 √(1/9N + Ppk²/2(N − 1))
• Control limits: X̄ ± A₂R̄ and R̄·D₃ to R̄·D₄; X̄ ± A₃s̄ and s̄·B₃ to s̄·B₄; x̄ ± 3MR̄ ÷ d₂(2) and 0 to D₄(2)·MR̄. d₂ and d₃ are the mean and standard deviation of the range of n normal values (A₂ = 3/(d₂√n), D₃ = 1 − 3d₃/d₂, D₄ = 1 + 3d₃/d₂), and c₄ = √(2/(n − 1)) Γ(n/2)/Γ((n − 1)/2).

Cp, Cpk, Pp and Ppk

Cp and Pp compare the tolerance with the spread of the process; Cpk and Ppk also count how close the mean sits to the nearer limit, so they are never higher than Cp and Pp. The C indices use the short-term within variation — the scatter inside subgroups, or between neighbouring values — and show what the process could do if it stayed where it is. The P indices use the overall standard deviation of all the data, including drift between subgroups, and show what it actually did. A big gap between Cpk and Ppk means the process moves around between samples.

A Cpk of 1.00 puts the nearer limit 3σ from the mean (about 1,350 ppm beyond it), 1.33 puts it 4σ away (about 32 ppm), 1.67 5σ (about 0.3 ppm) and 2.00 6σ.

Stable, normal and enough data

• Stable: capability predicts the future only for a process in statistical control. The page checks the control charts with the Western Electric rules — a point beyond 3σ; 2 of 3 beyond 2σ on one side; 4 of 5 beyond 1σ on one side; 8 in a row on one side — and the two trend rules the NIST handbook lists beside them, 6 in a row rising or falling and 14 in a row alternating up and down, and flags every point that breaks one.
• Normal: the ppm and the indices assume a normal distribution. The Anderson–Darling statistic A*², adjusted for the sample size, above 0.752 is evidence at the 5 % level that the data are not normal.
• Enough data: the NIST handbook asks for at least 25 values before its confidence intervals hold and about 100 for a capability study. The page shows the 95 % interval of Ppk so the uncertainty is visible.

Sigma level and DPMO

By the Six Sigma convention a process’s sigma level is the z of its long-term defect rate plus a 1.5σ shift, so the usual table reads: 1σ = 691,462 DPMO, 2σ = 308,538, 3σ = 66,807, 4σ = 6,210, 4.5σ = 1,350, 5σ = 233, 5.5σ = 32 and 6σ = 3.4. DPMO = defects ÷ (units × opportunities per unit) × 1,000,000. Set the shift to 0 for the plain z.

Zero defects give no finite sigma level, so with no defects counted the page gives a 95 % lower bound instead: the defect rate at which N opportunities would show no defect only 5 % of the time, 1 − 0.05^(1/N) (about 3 ÷ N, the rule of three). 18,000 opportunities without a defect put the sigma level at 5.09 σ or more.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods: process capability indices and their confidence limits, X̄ and R charts, individuals charts, the Western Electric and trend rules and the Anderson–Darling test
• ISO 22514-2: process capability and performance; ISO 7870-2: Shewhart control charts
• D’Agostino and Stephens, Goodness-of-Fit Techniques: the small-sample adjustment of A²

Limitations

• The indices and ppm assume normally distributed, stable data. For clearly non-normal data (flatness, roundness, other one-sided characteristics) use a transformation or a non-normal capability method; this page does not fit other distributions.
• Subgroups must all have the same size (2 to 25). Up to 50,000 values per study.
• The confidence intervals are those of Pp and Ppk from the overall standard deviation; the C indices use a within-subgroup σ whose sampling error is larger than its count of values suggests.
• The Western Electric and trend rules are applied to the individuals or means chart; the range, moving-range and standard-deviation charts are checked against their limits only.

Privacy

On this device: nothing is sent

Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.