Four capability indices, and what each one is and is not measuring.
Narrow enough, and in the right place?
Cp = 2.05 — “world class” — and Cpk = 0.57, with a quarter of the sample already out of spec.
n = 40, mean 10.13888, sigma 0.08134, limits 10.0000 to 11.0000
Cpk = 0.5691
Cp is 2.0490 — the spread fits comfortably. The gap of 1.4799 is entirely the mean being off centre.
Cp
2.0490
potential — ignores the mean
Cpk
0.5691
actual — min(Cpu, Cpl)
Pp
2.0490
overall sigma
Ppk
0.5691
overall, including drift
Cpu / Cpl
3.529 / 0.569
upper and lower separately
k (off-centre)
0.7222
fraction of the half-tolerance
Predicted PPM
43,877.9
assuming normality
Observed PPM
25,000
in this sample
Cp is 2.0490 and Cpk is 0.5691. Cp asks only whether the spread fits inside the tolerance, and it does — comfortably. Cpk asks whether the spread fits where the process is actually sitting, and it does not. The mean is 72.2% of the half-tolerance off centre, which is where the whole gap comes from, and this sample already has 2.50% of its parts outside the limits. A supplier quoting Cp has reported nothing about conformance, and the fix here is centring rather than reducing variation.
A Cpk of 0.57 predicts 43,877.86 defects per million on the nearer side, under normality. The conventional benchmarks: Cpk 1.00 is 1,349.9 PPM, 1.33 is 33.04, 1.67 is 0.27 and 2.00 is 0.001. The observed rate in this sample is 25,000PPM. When the two disagree sharply, the normality assumption is what failed — and it fails in the tails, which is exactly where the prediction lives. A capability index on visibly skewed or bounded data is arithmetic rather than a forecast.
This is 1.71 sigma short-term, or 0.21after the conventional 1.5-sigma shift — 417,865.55 PPM. That convention is where “six sigma means 3.4 defects per million” comes from: a Cpk of 2.00 is six sigma, subtracting 1.5 leaves 4.5, and the normal tail beyond 4.5 sigma is 3.40 per million. The 1.5 is an empirical allowance for long-term drift, proposed rather than derived, and it is the reason a “six sigma” process is quoted at 3.4 PPM rather than the 0.002 a genuine six-sigma normal tail would give.
What this tool shows
Cp says nothing about conformance. The first preset here has a Cp of 2.05 — world class by every published benchmark — a Cpk of 0.57, and a quarter of the sample already outside the limits. Cp measures whether the spread fits the tolerance; only Cpk asks whether it fits where the process is actually sitting.
Cp, Cpk, Pp and Ppk on the same data, with Cpu and Cpl separately
k, the offset from centre as a fraction of the half-tolerance
Predicted defects per million, and the rate observed in the sample
The within-subgroup sigma from the average range, with the d₂ constants
A flag when Cp and Pp disagree, which is drift rather than noise
The sigma level with and without the 1.5-sigma shift convention
All four indices Cp vs Cpk Observed vs predicted Drift flagged
Updated 12 September 2026 · Works in any browser, no installation
Capability indices compare a process’s spread against its tolerance. Cp asks whether the spread would fit; Cpk asks whether it fits where the process is running. The difference between them is entirely the mean being off centre, and quoting the first without the second is the most common way a capability figure misleads.
At a glance
Formula shown
Cp = (USL − LSL)/(6σ), so it ignores the mean entirely. Cpk = min((USL − μ)/(3σ), (μ − LSL)/(3σ)), which does not. Pp and Ppk are the same two expressions with the overall sample standard deviation instead of the within-subgroup one, so Cp describes what a stable process could do and Pp what this one did. Cpk ≤ Cp always, with equality exactly when the process is centred.
Scenario support
Supplier qualification, production part approval, monitoring a machining or filling process, deciding whether a tolerance is achievable, and any manufacturing question phrased as “is this process good enough”.
Educational estimate
Planning support from the values you enter — not professional advice.
Cp ignores the mean, and that is the whole problem
The two indices are quoted interchangeably and one of them cannot tell you whether the process makes conforming parts.
Cp is the tolerance divided by six sigma. The mean does not appear in it. A process can have a Cp of 2.00 and produce nothing but scrap, if it is running far enough off centre.
The first preset is exactly that. Cp = 2.05, which every published benchmark calls world class, and Cpk = 0.57 — with 25,000 parts per million already outside the limits in the sample itself.
Cpk takes the worse of the two sides. min(Cpu, Cpl), so it measures the distance to whichever limit the process is closer to. Here Cpu is 3.53 and Cpl is 0.57, and only the second matters.
Cpk ≤ Cp always, with equality only when centred. The gap between them is a pure measure of how far off centre the process is, which is what k reports directly.
And the two suggest different fixes. A low Cp means the process is too variable and needs engineering. A high Cp with a low Cpk means it needs adjusting, which is usually much cheaper — so the pair is more actionable than either number alone.
Cp against Pp is a stability check
The second pair of indices is usually presented as a naming convention. It is a diagnostic.
Cp uses the within-subgroup sigma; Pp uses the overall one. The within-subgroup estimate comes from the average range inside small consecutive samples, so it captures only the variation present moment to moment.
If the process is stable those are the same number. A process in statistical control has no between-subgroup variation beyond what is within, so Cp and Pp agree.
When they disagree, the process drifted. The drift preset has a within-subgroup sigma of 0.033 and an overall sigma of 0.078 — Cp 5.08 against Pp 2.13. Tight moment to moment, wandering over a shift.
And capability indices are not meant for an unstable process. Every index estimates a long-run defect rate from a short-run spread, which assumes the spread is the whole story. Under drift it is not, and the honest order of work is to achieve control first and compute capability second.
The tool flags the gap rather than leaving it to be spotted. With no subgrouping the two are identical by construction — which is itself worth knowing, because software that prints all four on ungrouped data is printing two numbers twice.
The defect rate assumes normality, in the tails
Every capability table converts an index into parts per million. That conversion is the least reliable part of the whole exercise.
The conversion is a normal tail probability. Cpk 1.00 gives 1,349.9 PPM on the nearer side, 1.33 gives 33.04, 1.67 gives 0.27 and 2.00 gives 0.001.
And it lives entirely in the tails. At Cpk 1.33 the prediction is about what happens four standard deviations out, which is precisely where a normal assumption is least testable and least likely to hold.
A bounded or skewed process breaks it badly. Flatness, runout, concentricity and anything else that cannot go below zero is skewed by construction, and a normal-based PPM for it can be wrong by orders of magnitude in either direction.
So the tool prints the observed rate beside the predicted one. A large disagreement is the normality assumption failing, and it is visible immediately rather than after someone questions the figure.
A normality check is not optional here. The index itself is descriptive and survives non-normality; the PPM does not. Where the data is clearly not normal, a transformation or a distribution-specific method is the alternative, and reporting the index without the PPM is the honest minimum.
Where 1.33 and 1.67 come from
The benchmarks are quoted as requirements. They are conventions, and knowing their origin tells you when to depart from them.
Cpk ≥ 1.33 is the common minimum for an existing process. It corresponds to four sigma to the nearer limit, and about 33 defects per million on that side.
Cpk ≥ 1.67 is typical for a new process or a critical characteristic. Five sigma, 0.27 PPM. The extra margin is an allowance for the process degrading after qualification.
Cpk ≥ 2.00 is the six-sigma target. And “six sigma means 3.4 defects per million” comes from subtracting a conventional 1.5-sigma long-term drift: six minus 1.5 is 4.5, and the normal tail beyond 4.5 sigma is 3.40 per million. A genuine six-sigma normal tail would be 0.002.
The 1.5 is an empirical allowance, not a derivation. It was proposed as a typical long-term shift observed in practice, and it is applied by convention rather than measured for the process in front of you — which is exactly what Cp against Pp measures directly.
None of these thresholds knows what your part does. A 1.33 on a decorative dimension and a 1.33 on a safety-critical one carry different consequences, and the benchmark is silent on both.
What the sample has to look like
A capability index is an estimate from a sample, and three properties of that sample matter more than the arithmetic.
The measurements must be in production order. Subgrouping only means anything if consecutive parts really were consecutive; shuffling the data destroys the within-subgroup estimate and silently turns Cp into Pp.
The sample must cover the sources of variation you care about. Thirty parts from one hour of one shift on one machine estimates that hour’s capability, not the process’s. Long-run capability needs data spanning shifts, operators, batches and tool changes.
Estimates from small samples are very uncertain. A Cpk computed from 30 parts has a wide confidence interval — easily ±0.3 — and comparing a 1.30 against a 1.35 from samples that size is comparing noise.
Measurement error is included in the sigma. If the gauge contributes meaningfully to the spread, the capability index is measuring the gauge as much as the process. A measurement systems analysis comes before a capability study for that reason.
And the index is not a prediction about individual parts. It describes the process that produced the sample, under an assumption of continued stability — which the Cp-against-Pp comparison is the only part of the output that actually tests.
Reading a capability report without being misled
Capability figures arrive in supplier reports and qualification packs. Five questions separate a real one from a flattering one.
Is Cpk quoted, or only Cp? Cp alone is the single most common way to make a non-conforming process look excellent, and the preset here shows the size of the gap available.
Do Cp and Pp agree? If only Cp and Cpk are given, the stability question has been skipped. A capability index for a process that is not in control is not interpretable.
How many parts, and over what period? Thirty consecutive parts from one setup is a different claim from three hundred across a month, and both get reported as “Cpk = 1.45”.
Was normality checked? The PPM figure depends on it entirely, and for a bounded characteristic it is usually wrong. The observed defect count in the same sample is the sanity check.
Is the specification the real one? Widening a tolerance raises every index without changing anything physical, and a Cpk improvement that coincided with an engineering change notice is worth reading twice.
And is one number standing in for a distribution? A histogram of the measurements against the limits says more than any index, which is why plotting it is the first thing to ask for.
Sources and methodology
References for the indices and the conventions around them.
Method. The within-subgroup sigma is estimated from the average subgroup range using the standard d₂ constants for sizes 2 to 10, which is what Cp and Cpk are defined against; with no subgrouping it falls back to the sample standard deviation, and Cp then equals Pp by construction rather than by coincidence. Predicted defect rates come from the normal tail at the observed mean and overall sigma, and the rate actually observed in the sample is reported beside them so a normality failure is visible. The suite asserts that Cpk never exceeds Cp across 1,500 random processes and that they are equal only when centred, that shifting the mean by three sigma leaves Cp unchanged while collapsing Cpk from 2 to 1, that every index is invariant to the unit of measurement, and it derives the Six Sigma figure rather than quoting it: a Cpk of 2.00 minus the conventional 1.5-sigma shift leaves 4.5 sigma, whose tail is 3.3977 per million. That engine is verified on every change against 39 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Standard DeviationSample and population standard deviation, plus variance, mean, median, quartiles, z-scores, outliers, and confidence intervals.
Normal DistributionProbabilities under a normal curve in all four directions with the region shaded — and the empirical rule given exactly, because two standard deviations is 95.45% and the 95% everyone quotes sits at 1.96σ.
Frequency DistributionA frequency table with relative and cumulative columns, and all six standard bin-count rules computed at once — they disagree by a factor of 6.67 at n = 10,000, where Sturges asks for 15 bins and the square-root rule asks for 100.
Z-ScoreA z-score from your data or from a known mean and SD — with the normal-table percentile checked against the share of your data that actually falls below it, and a warning when they disagree.
Empirical RuleGives the exact 68.27/95.45/99.73 at any k, plus the multiplier for a round 95% — and prints Chebyshev's distribution-free floor beside each figure so the cost of assuming normality is a number.
Moving AverageSimple, exponential, weighted and centred moving averages with the lag each one carries — exactly (n−1)/2 periods for a simple average, which is also the reason α = 2/(n+1) is in every EMA formula.
An educational tool. Capability indices assume a stable, normally distributed process; the predicted parts-per-million figures depend entirely on normality in the tails and can be wrong by orders of magnitude for bounded or skewed characteristics.
Published a capability calculator whose first preset is the argument: Cp = 2.05, which every published benchmark calls world class, with Cpk = 0.57 and a quarter of the sample already outside the limits. Cp measures whether the spread fits the tolerance and says nothing about where the process is sitting, so a supplier quoting Cp has reported nothing about conformance.
Separates Cp from Pp as a stability check rather than a naming convention. Cp uses the within-subgroup sigma from the average range and Pp the overall one, so a gap between them is drift rather than noise — 5.08 against 2.13 on the drift preset — and capability indices are not interpretable for a process that is not in control.
Derives the Six Sigma figure rather than quoting it: a Cpk of 2.00 is six sigma short-term, the conventional 1.5-sigma shift leaves 4.5, and the normal tail beyond 4.5 sigma is 3.3977 per million. It also pins a Cpk of 1.33 at 33.04 PPM one-sided, not the 31.7 that circulates.
Prints the observed defect rate beside the predicted one, because the PPM conversion is a normal tail probability and lives four standard deviations out — exactly where the normality assumption is least testable and least likely to hold, and where a bounded or skewed characteristic breaks it by orders of magnitude.
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