Process Capability Calculator

What the calculation does

Indices. With a mean μ, a standard deviation σ and the limits, Cp = (USL − LSL) ÷ 6σ, Cpu = (USL − μ) ÷ 3σ and Cpl = (μ − LSL) ÷ 3σ, and Cpk is the smaller of Cpu and Cpl. These are the definitions in section 6.1.6 of the e-Handbook. With one limit only, Cp cannot be formed and Cpk is the one-sided index, a wider use of the name that the handbook's glossary notes. The same formulas with the overall standard deviation give Pp, Ppu, Ppl and Ppk.

Within-subgroup σ. Single readings: the average of the moving ranges between consecutive values, divided by 1.128. Subgroups: the average range divided by d2, or the average subgroup standard deviation divided by c4, both from section 6.3.2.1; or the pooled standard deviation, the square root of the subgroup variances averaged with weights n − 1, without a bias correction. Subgroups are formed from consecutive values in the order pasted, so the order matters.

Overall σ. The sample standard deviation of all values, with n − 1 in the denominator.

Fraction outside. The normal probability below (LSL − μ) ÷ σ plus the probability above (USL − μ) ÷ σ, for each σ. The glossary of the handbook states that a fraction out of specification can be estimated this way when the process is stable and normal. Percentages are shown to two decimals; below 0.1% the result is given in parts per million to three significant figures.

Interval for Ppk. Ppk ± 1.96 × √(1 ÷ 9n + Ppk² ÷ 2(n − 1)), the approximation the handbook describes as common in practice for an index computed with the sample standard deviation of n values. It is shown from 25 values, the minimum the handbook gives for it, and not for Cpk, whose within-subgroup σ behaves differently.

Stability screen. Individuals: limits at the mean ± 3 × average moving range ÷ 1.128, as in section 6.3.2.2. That section gives no limits for the moving ranges themselves; each is a range of two values, so we compare it with 3.267 times their average, the D4 factor that section 6.3.2.1 prints for subgroups of two. Subgroups with the range method: X̄ ± A2 × average range, ranges between D3 and D4 times the average range. Otherwise: X̄ ± 3 × average s ÷ (c4 √n), and the subgroup standard deviations against s̄ ± 3 × (s̄ ÷ c4) × √(1 − c4²). The handbook writes that lower limit without a floor; where it comes out negative we show zero, which changes no flag.

Rounding. Indices are shown to two decimals and measurement-scale figures to five significant digits; the calculation keeps full precision.

A2, D3 and D4 are as printed in section 6.3.2.1. The handbook pages we read print d2 only for n = 2, as 1.128; the other d2 values are the expected range of n normal values, which we computed by numerical integration and checked against the handbook's A2 = 3 ÷ (d2 √n). The c4 values come from the formula in section 6.3.2; the handbook gives 0.9727 for n = 10.

Worked examples from the handbook

• Capability index example (section 6.1.6). With limits of 8 and 20, a mean of 16 and a standard deviation of 2, the handbook finds Cp = 1.0 and, because the mean is 2 above the midpoint of 14, k = 0.3333 and Cpk = 0.6667, the same as Cpu; Cpl is 1.3333. The calculator gives 1.00, 0.67, 0.67 and 1.33. Under a normal model it also estimates 2.28% above the upper limit and 31.7 ppm below the lower one, a figure the handbook does not give. The handbook's s is the sample standard deviation of the values, which this page would call the overall figure; we load it as the within-subgroup one to keep the handbook's names, and typed as the overall figure it gives the same numbers as Pp, Ppk, Ppu and Ppl.
• Rejects for a centred process (section 6.1.6). The handbook tabulates a tolerance of 6, 8, 10 and 12 standard deviations, Cp 1.00, 1.33, 1.66 and 2.00, with 0.27%, 64 ppm, 0.6 ppm and 2 ppb outside, and 100, 75, 60 and 50% of the specification used. The normal formula here gives 0.27%, 63.3 ppm, 0.573 ppm and 0.00197 ppm (about 2 ppb); the share used is 100 ÷ Cp. The table prints 1.66 for 10 ÷ 6, which is 1.667.
• Flow-rate individuals chart (section 6.3.2.2). Ten batches average 50.81 with an average moving range of 1.8778, which puts the control limits at 45.8159 and 55.8041 and no point outside them. The calculator opens with these values and shows the same limits to five significant figures, 45.816 and 55.804; the within-subgroup σ is 1.8778 ÷ 1.128 = 1.6647 and the overall standard deviation is 2.0339. The specification limits of 45 and 57 that turn this into a capability result are our illustration.
• The c4 factor (section 6.3.2). The handbook works the fractional factorial 3.5! = 11.632 and states c4 = 0.9727 for n = 10. The calculator's c4 function returns the same value.

The tests for this page reproduce each figure above with the calculator's own functions.

Before you read an index

Stability first. Process capability compares an in-control process with its specification, so the control chart comes before the index; see statistical process control. Control limits describe what the process does and specification limits what the product needs, and the handbook keeps the two apart. An index from a process that shifted part-way through describes a mixture and predicts nothing. The handbook quotes Shewhart's rule of thumb that a state of control should not be claimed before some 25 subgroups of four have been in control.

Normality. Cp, Cpk and the fraction outside assume a normal distribution. For data that are not normal the handbook suggests a transformation such as Box-Cox or percentile-based indices; this calculator does neither. A fraction in parts per million is a statement about the far tails of the model, which a sample of a few hundred values cannot confirm.

Sample size. An index from 30 values is itself uncertain. The handbook's glossary puts the 95% interval for Cp at roughly plus or minus 25% when at least 33 repetitions are used; the interval shown for Ppk makes the same point for your data.

Measurement. The spread you enter includes the error of the gauge, so an imprecise measurement system lowers every index. Check the measurement system before the process; the handbook's glossary calls this a metrology or gauge capability study, which estimates repeatability and reproducibility.

Why the page has no pass mark

The statistics do not set a minimum. The handbook's glossary calls a process capable when Cpk exceeds 1.0, and notes that other criteria are sometimes promoted, among them 1.33, 1.5 and 2.0, which it says are usually kept for calling a process manufacturable. Its worked example also asks for a Cpk of at least 1.0, while the 2026 review cited below says a Cpk above 1.33 is generally considered acceptable. These figures are conventions, not requirements. A value you must meet comes from a customer, a contract or a sector manual, and applies to named characteristics with an agreed method and amount of data. Record it with the characteristic, as below, and compare the result with it yourself.

Where the formulas and constants come from

• NIST/SEMATECH e-Handbook of Statistical Methods, section 6.1.6, What is Process Capability?: the definitions of Cp, Cpk, Cpu and Cpl, the k factor, the normality assumption, sample sizes, the confidence-interval approximations, the rejects table, the worked example and the remedies for non-normal data.
• Section 6.3.2, What are Variables Control Charts?: c4 and its formula, control limits against specification limits, and Shewhart's rule of thumb on the number of samples.
• Section 6.3.2.1, Shewhart X-bar and R and S Control Charts: σ from the average range and d2 or the average standard deviation and c4, the X̄, R and s chart limits, and the A2, D3 and D4 table.
• Section 6.3.2.2, Individuals Control Charts: moving ranges, d2 = 1.128 and the flow-rate example.
• The handbook's glossary: the entries on the capability index Cpk, on its precision with 33 repetitions, on a capable process (the 1.0 criterion and the other values sometimes promoted) and on a metrology study.
• Fei Jiang and Lei Yang, Practical Process Capability Indices Workflows, arXiv:2604.15544, April 2026: the within and overall standard deviations behind Cp and Cpk against Pp and Ppk, the pooled standard deviation, and a table of d2 and c4 for n = 2 to 10, read in its HTML version.

All of these were opened on 22 October 2026. They describe the methods; they do not validate this calculator or any result.

What we could not verify (22 October 2026)

• The e-Handbook sections listed above, and its glossary, do not define Pp and Ppk or the pooled standard deviation, and a web search of the NIST site did not turn up a section that does. We took both from the arXiv review, a preprint; the journal version its record links to was not opened.
• Two sold documents cover capability and performance studies: the AIAG & VDA SPC Manual (first edition, announced on 1 July 2026) and ISO 22514-2:2026, Process capability and performance of time-dependent process models (edition 3, February 2026). We opened only their catalogue pages and AIAG's press release, so this page does not claim to follow either, and their definitions, estimators or required values may differ from those here. ISO's summary says the standard covers processes that do not remain in statistical control, which this calculator does not handle.
• Statistical software differs in details such as bias corrections to the pooled or overall standard deviation and the handling of uneven subgroups. We did not compare any package, so its results can differ slightly from these.
• The handbook pages we read print d2 only for subgroups of two; an older version of its X-bar and R page (section 6.3.1.1) refers readers to textbooks for the rest. Our values for 3 to 10 reproduce its A2 table to three decimals and agree with the review's table, but we found no handbook page that lists them.
• The handbook's rejects table gives 64 ppm for a tolerance of eight standard deviations, where the normal formula gives 63.3 ppm. We did not find the reason for the difference.
• The handbook's chi-square limits for Cp and the exact Cpk variance of Zhang, Stenback and Wardrop are not computed here.

Recording the requirement and the result

A capability requirement belongs with the characteristic it applies to, the measurement method and the sampling plan. The free quality plan template has rows for objectives with acceptance criteria and for inspection and test points, where the required index, the data behind the result and the date can be kept together. The glossary pages on process capability and statistical process control define the terms used here.