What capability indices compare
A capability index is a ratio of two widths. The numerator is the tolerance the customer allows; the denominator is the spread the process actually produces, taken as six standard deviations because that interval covers 99.73% of a normal distribution. Cp = 1.00 means the process spread exactly fills the tolerance with nothing to spare. Cp = 2.00 means the tolerance is twice as wide as the process needs.
Cp alone is not enough, because it says nothing about where the process sits. A process with half the tolerance width can still make scrap if it is centred near a limit. Cpk fixes this by measuring the distance from the mean to each limit separately, in units of three standard deviations, and reporting the smaller of the two. The nearest limit is the one that generates defects, so it is the one that defines capability.
The relationship between them is exact: Cp − Cpk = |x̄ − m| / 3σ, where m is the midpoint of the tolerance. Cpk can therefore never exceed Cp, and the two are equal only when the mean sits exactly at the midpoint. That single identity tells you how to read any capability report: the gap between the two numbers is centring, and whatever Cp itself falls short of is variation.
The distinction drives what you do next. If Cp is high but Cpk is low, adjust the process setting — usually a fast, cheap change. If Cp itself is low, no amount of centring will help; you have to reduce variation, which means tooling, method, material or measurement work.
Cp/Cpk against Pp/Ppk: which sigma goes in the denominator
The four indices use identical arithmetic and differ only in which standard deviation they divide by, and that difference carries the whole meaning.
Cp and Cpk use within-subgroup sigma, the short-term variation estimated from the control chart as R̄/d₂ or s̄/c₄. Subgroups are deliberately formed so that only common-cause noise appears inside them, so this figure describes what the process is capable of when nothing shifts — its best behaviour.
Pp and Ppk use overall sigma, the ordinary sample standard deviation of every individual reading in the study. It contains the within-subgroup noise and any drift, tool wear, batch changes or shift-to-shift differences that occurred over the period. It describes what the customer actually received.
For a process in statistical control the two sigmas are close, and the four indices nearly coincide. A large gap between Cpk and Ppk is a signal in itself: it says the process moved during the study, and the capability numbers are being propped up by a subgroup structure that hides the movement. The AIAG SPC reference manual is explicit that capability indices are meaningful only for a process in statistical control — which is why a capability study always follows a control chart rather than replacing one.
The X-bar and R chart limits calculator produces the within-subgroup sigma this page expects, and the standard deviation calculator produces the overall one.
Worked example: a shaft diameter running 4 units high
A shaft diameter is specified as 100 ± 10, so LSL = 90 and USL = 110. A capability study of 25 subgroups of 5 gives a grand mean of 104 and a within-subgroup standard deviation of 2.0.
- Cp. (110 − 90) / (6 × 2.0) = 20 / 12 = 1.667. The tolerance is 1.667 times wider than the process spread — comfortable.
- Cpu. (110 − 104) / (3 × 2.0) = 6 / 6 = 1.000.
- Cpl. (104 − 90) / (3 × 2.0) = 14 / 6 = 2.333.
- Cpk. min(1.000, 2.333) = 1.000.
- Check the identity. The midpoint is 100, so |104 − 100| / (3 × 2.0) = 4/6 = 0.667, and 1.667 − 0.667 = 1.000. The entire shortfall is centring.
- Expected defect rate. Φ(−3 × 1.000) = Φ(−3) = 0.00134990 above the upper limit, and Φ(−3 × 2.333) = Φ(−7) ≈ 1.3 × 10⁻¹² below the lower — negligible. Total 1,349.9 PPM, or 0.135% of production.
- Z bench. The single tail area matching that total is 0.00134990, whose normal quantile is 3.000 sigma.
Now suppose you re-centre the process at 100 without changing the variation at all. Cpu and Cpl both become 1.667, Cpk rises to 1.667, and the expected defect rate falls from 1,349.9 PPM to 2 × Φ(−5) × 10⁶ = 0.573 PPM. (A ±10 tolerance with σ = 2 gives Cpk = 10/6 = 1.6667 exactly, a shade above the 1.67 benchmark in the table below, which is why this figure is 0.573 rather than 0.54.) A machine-setting adjustment that costs an hour has cut expected scrap by a factor of 2,355, which is why capability reports separate Cp from Cpk in the first place.
What the numbers mean in practice
Cpk = 1.00 puts the nearest specification limit exactly three standard deviations from the mean. For a centred, normal, in-control process that is 2,700 PPM defective — roughly one part in 370. Almost no supply agreement accepts it, because any drift immediately produces scrap.
Cpk = 1.33 puts the nearest limit 3.99 standard deviations away, at 66 PPM when centred. This is the figure most customer quality agreements name as the minimum for an ongoing process, and the reason is margin: at 1.33 the mean can drift by very nearly a full standard deviation before capability falls to 1.00, since one sigma of drift costs exactly 1/3 of an index point.
Cpk = 1.67 puts the nearest limit 5.01 standard deviations away, at 0.54 PPM. It is commonly required for new processes at launch and for safety-related or regulated characteristics, on the reasoning that a new process will lose some capability once it meets real production conditions.
Cpk = 2.00 is the “six sigma” target, with the nearest limit six standard deviations from the mean. The famous 3.4 defects per million figure attached to Six Sigma is not this number — it comes from assuming the mean drifts by 1.5 standard deviations over the long run, leaving 4.5 sigma of margin, and Φ(−4.5) × 10⁶ = 3.4 PPM. This calculator reports Z bench with no shift applied, so a centred Cpk of 2.00 shows as about 5.89 sigma and 0.002 PPM — the Z bench is a little under 6 because it folds both tails into one.
Read a negative Cpk as an alarm rather than a number. It means the mean itself lies outside a limit, so over half the output is nonconforming, and no statistical refinement matters until the process is brought back inside.
Capability index, defect rate and sigma level
| Cp = Cpk | Distance to nearest limit | PPM defective | Yield | Typical use |
|---|---|---|---|---|
| 0.67 | 2.0σ | 44,431 | 95.557% | Not capable |
| 1.00 | 3.0σ | 2,700 | 99.730% | Marginal, rarely accepted |
| 1.33 | 3.99σ | 66.0 | 99.99340% | Common minimum for ongoing production |
| 1.67 | 5.01σ | 0.54 | 99.999946% | New processes and safety characteristics |
| 2.00 | 6.0σ | 0.002 | 99.9999998% | Six sigma target |
PPM = 2 × Φ(−3Cp) × 10⁶, evaluated at the index shown — so the 1.33 row is 3 × 1.33 = 3.99σ, not a round 4σ. The 3.4 PPM figure quoted in Six Sigma literature applies to a 2.00 index after allowing a 1.5σ long-term mean shift, not to the centred case shown here.
What can invalidate a capability study
- The process is not in statistical control. An index computed on an unstable process describes a period, not a capability, and will not repeat. Chart first; the individuals and moving range chart covers low-volume cases.
- The data are not normal. The PPM conversion is a normal-tail calculation. For skewed or bounded characteristics such as flatness, concentricity or contamination counts, use a transformation or a non-normal capability method — the index itself will otherwise overstate or understate the tails badly.
- Confusing specification limits with control limits. Specifications come from the customer; control limits come from the process. Putting control limits into these fields makes the answer meaningless.
- Using the wrong sigma. The overall standard deviation in Cp or Cpk understates capability; the within-subgroup sigma in Pp or Ppk overstates performance. Keep them straight, and report all four.
- Ignoring measurement variation. The observed spread contains gauge error. If your gage R&R consumes 30% of the tolerance, part of the sigma you are dividing by belongs to the measurement system rather than the process.
- Too little data. Standard practice is at least 25 subgroups, and typically 100 or more individual readings. Cpk from 20 parts carries a very wide confidence interval and should not be quoted to three decimals.
Where capability analysis fits
A capability study is the last step of a sequence, not the first. Establish a measurement system you trust with a gage R&R study, bring the process into statistical control with an X-bar and R chart, and only then compute capability. Running the steps out of order produces indices that change every time you recompute them.
For defect-based rather than measurement-based processes, the parallel metrics are DPMO and the process sigma level, which count nonconformities against opportunities instead of comparing distributions to limits. They are the right choice for transactional and service processes where there is no continuous measurement to take.
The definitions on this page follow the AIAG Statistical Process Control reference manual, which is the standard used across the automotive supply chain and the source of the Cp/Cpk versus Pp/Ppk distinction. ISO 22514 covers the same ground internationally, including methods for non-normal distributions that the basic formulas here do not address.
Key terms
- Within-subgroup sigma
- Short-term variation estimated from a control chart as R̄/d₂ or s̄/c₄. Excludes drift between subgroups.
- Overall sigma
- The ordinary standard deviation of every individual reading, including any drift over the study period.
- Z bench
- The number of standard deviations from the mean to a single tail carrying the same total defect probability as both real tails combined.
- PPM defective
- Expected nonconforming parts per million, computed from the normal tail areas beyond each specification limit.