Process Capability: Cp, Cpk and What Six Sigma Actually Claims

Cp: the specification against the spread

Cp is a ratio of two widths. The numerator is the width of the specification, USL − LSL, the room the drawing gives you. The denominator is 6σ, the width the process occupies — six standard deviations, because ±3σ covers 99.73% of a normal distribution and 6σ is the conventional shorthand for "the whole of the process". So Cp = (USL − LSL) ÷ 6σ, and it answers exactly one question: could this process fit inside this tolerance if it were perfectly aimed? A Cp of 1.00 means the process exactly fills the tolerance with nothing to spare. A Cp of 1.33 means the tolerance is a third wider than the process needs.

Work one. A shaft is specified between 9.80 and 10.20 inches — a spec width of 0.40 — and the process has a standard deviation of 0.05 inches. Cp = 0.40 ÷ (6 × 0.05) = 0.40 ÷ 0.30 = 1.333. Note carefully what did not enter that calculation: the mean. Cp is blind to where the process is aimed, and every consequence in this article follows from that blindness.

Cpk: the penalty for being off-centre

Cpk repairs that blindness by measuring each side separately and keeping the worse one. Cpk = min[(USL − μ) ÷ 3σ, (μ − LSL) ÷ 3σ]. The 3σ in each denominator is half of the 6σ in Cp, because each term now measures one side of the distribution rather than the whole of it. When the process sits exactly at the midpoint of the tolerance, the two terms are equal and Cpk equals Cp. Any drift away from the midpoint makes one term smaller, so Cpk is always less than or equal to Cp, and the difference between them is a pure measure of miscentring.

The pair is worth keeping together because they answer different management questions. Cp asks whether the machine and the method are capable at all — whether the variation is small enough for the job. Cpk asks whether the process is currently set up correctly. A low Cp is an engineering problem: you need a better machine, a better fixture or a tighter method. A high Cp with a low Cpk is a setting problem: the equipment can do it, somebody just needs to move the aim, and that is usually cheap and fast.

Same spread, 359 times the scrap

Take the same specification and the same σ = 0.05, so Cp is 1.333 in both cases, and run two means through it. Centred at 10.00, each specification limit sits (0.20 ÷ 0.05) = 4 standard deviations away, so Cpk = 4 ÷ 3 = 1.333, equal to Cp. The fraction outside is Φ(−4) on each side, which is 31.671 per million per tail and 63.34 per million in total — one part in 15,787.

Now move the mean to 10.10, two standard deviations high, without touching the spread. Cp is still 1.333 — the process is exactly as tight as it was. But the upper limit is now only (10.20 − 10.10) ÷ 0.05 = 2 standard deviations away while the lower limit is 6 away, so Cpk = min(2 ÷ 3, 6 ÷ 3) = 0.667. The scrap rate is Φ(−2) above the upper limit, 22,750.13 per million, plus a completely negligible Φ(−6) = 0.000987 per million below the lower one. Total: 22,750 defects per million, 2.275%, one part in 44.

That is the whole reason both indices exist. Two processes with an identical Cp of 1.333 — identical machines, identical variation, identical capability in the engineering sense — produce 63 defects per million and 22,750 defects per million, a factor of 359. A capability report that quotes Cp alone has told you nothing about scrap. If you are only going to look at one number, look at Cpk; if you are going to act on it, look at both, because they point at different fixes.

3.4 per million, and the 1.5σ shift that produces it

"Six sigma" as a quality level means the nearest specification limit sits six standard deviations from the mean, which for a centred process is a Cp and a Cpk of 2.00. Compute the defect rate from the normal distribution directly and you get 2 × Φ(−6) = 0.00197 defects per million. That is roughly two parts per billion. It is not 3.4, and it is not close to 3.4 — the two figures differ by a factor of 1,722.

The 3.4 comes from an added assumption. The convention is that a process mean does not stay put: over hours, shifts, tool changes and batches of raw material it drifts, and the accepted allowance for that long-term drift is 1.5σ. Assume the mean sits 1.5σ off target and the near limit is only 6 − 1.5 = 4.5 standard deviations away. Compute that tail: Φ(−4.5) = 3.398 per million. The far limit is now 7.5σ away and contributes 0.0000000319 per million, which rounds to nothing. So the number is 3.398, quoted as 3.4 defects per million opportunities. It is arithmetically exact — given the shift.

The same convention rewrites the whole table. At a three-sigma level, the centred normal gives 2,699.80 defects per million; with the shift it gives 66,810.6, twenty-five times more. At four sigma, 63.34 becomes 6,209.68. At five sigma, 0.573 becomes 232.6. The gap grows with the sigma level because you are removing a fixed 1.5σ from an increasingly steep tail. That is why the shift matters most exactly where the marketing claims are loudest.

Where the shift came from, and how contested it is

The 1.5σ allowance did not come out of a measurement of any particular process. It descends from statistical tolerancing work in the 1970s — Bender's rule of thumb for stacking tolerances allowed a factor for the fact that means wander — and it was adopted at Motorola in the 1980s as a standard correction between short-term capability, measured on a run of consecutive parts, and long-term performance across weeks of production. The intent was reasonable: a capability study run in one shift with one operator and one bar of material genuinely does understate the variation a customer will see.

What is contested is the universality of the number. There is no reason a well-controlled continuous process should drift by exactly 1.5σ, and no reason a poorly controlled batch process should drift by only that much. The statistical arguments for 1.5 as a general figure — including the best-known one, which derives it from the probability that a control chart on small subgroups fails to detect a shift of that size — are constructions after the fact, and they have been argued over in the quality literature for decades. Treat 1.5σ as what it is: a documented convention that makes published sigma levels comparable, not an empirical property of your process.

The practical consequence is a labelling rule. Any sigma level or DPMO figure should carry the phrase "with the 1.5σ shift" or "without", because the two differ by three orders of magnitude at the six-sigma end. In the indices themselves, this is the same distinction as Cp and Cpk against Pp and Ppk: the first pair conventionally uses a within-subgroup estimate of σ and describes short-term capability, the second uses the total standard deviation of all the data and describes the performance actually delivered.

Thirty parts is not a capability study

Every figure above assumed σ was known. It is not; it is estimated from a sample, and the estimate is far noisier than people expect. Because (n − 1)s² ÷ σ² follows a chi-square distribution, a 95% confidence interval for Cp runs from Ĉp × √(χ²₀.₀₂₅,ₖ ÷ k) to Ĉp × √(χ²₀.₉₇₅,ₖ ÷ k) with k = n − 1 degrees of freedom. At n = 30 those multipliers are 0.7439 and 1.2556, so a measured Cp of 1.333 gives an interval of 0.992 to 1.674.

Translate the two endpoints into scrap and the problem becomes visible. A centred process at Cp = 0.992 makes 2,920.35 defects per million; at Cp = 1.674 it makes 0.511. That is the same measurement being consistent with defect rates 5,711 times apart. Increasing the sample helps but slowly: at n = 50 the interval is 1.070 to 1.596, at n = 100 it is 1.148 to 1.519 — still a 110-fold span in defect rate — at n = 200 it is 1.202 to 1.464 and at n = 500 it is 1.251 to 1.416.

So a capability index computed on thirty parts should be reported with its interval or not reported at all, and the customary practice of accepting a process on a single Ĉpk of 1.33 from a thirty-piece run is weaker evidence than it looks. It is also worth remembering that a short run underestimates variation for a second, separate reason: thirty consecutive parts share one operator, one tool, one setup and one batch of material, so they measure within-subgroup variation and say nothing about the between-subgroup variation that a customer will meet.

All of it assumes normality — so test it

Converting a capability index into a defect rate uses the normal distribution, and every ppm figure in this article — including the famous 3.4 — is only as true as that assumption. The trouble is that capability indices live in the extreme tails, which is exactly where distributions differ most and where you have the least data to check them.

Quantify a mild departure. Take a Student t distribution rescaled to have exactly the same standard deviation, so the Cp is identical, and compute the tail beyond six standard deviations. With 30 degrees of freedom — a barely detectable excess kurtosis of 0.2 — the tail is 0.3875 per million against the normal's 0.000987, 393 times heavier. With 10 degrees of freedom it is 26.55 per million, 26,900 times heavier. With 5 degrees of freedom it is 286.6 per million, roughly 290,000 times heavier. A distribution you would struggle to reject with a hundred data points destroys the estimate you built on it.

So do three things before you publish a capability number. Confirm the process is stable, with a control chart, because a capability index computed on an out-of-control process describes nothing. Test the distribution rather than assuming it, and if it is genuinely skewed — as flatness, roundness, concentricity and any other characteristic bounded at zero usually are — use the percentile-based indices defined for non-normal data in the ISO 22514 series rather than forcing a normal fit. And report Cpk with its confidence interval and its sample size attached, so the next reader knows how much of it is measurement and how much is noise.

Frequently asked questions

Can Cpk be higher than Cp?

Never. Cpk takes the smaller of the two distances to a limit and divides by 3σ, while Cp divides half the total spec width by 3σ. When the process is exactly centred the two distances are equal and Cpk equals Cp; any drift makes one of them smaller. So Cpk ≤ Cp always, and if a report shows Cpk above Cp, there is an arithmetic error or the two were computed from different data. The gap between them is a clean measure of miscentring on its own.

What is the difference between Cp/Cpk and Pp/Ppk?

The formulas are identical; the estimate of σ is not. Cp and Cpk conventionally use a within-subgroup estimate — from the average range or average standard deviation of small rational subgroups — which captures only the variation present over a few consecutive parts. Pp and Ppk use the overall standard deviation of all the data, which also contains drift between subgroups, tool changes and material lots. Cp and Cpk therefore describe short-term capability and Pp and Ppk describe delivered performance, which is precisely the gap the 1.5σ shift convention was invented to bridge.

Is 3.4 defects per million actually achievable?

It is a defined target, not a physical constant, and whether it is achievable depends entirely on the characteristic. What the number cannot do is be verified by measurement in most factories: demonstrating a rate of 3.4 per million with any confidence requires hundreds of thousands of parts, so in practice it is always inferred from a capability index rather than observed. Given the confidence intervals on Cpk at realistic sample sizes, that inference carries far more uncertainty than the two-decimal figure suggests.

How many parts do I need for a capability study?

More than the thirty that convention suggests, and the right way to choose is to decide how wide an interval you can live with. At n = 30 a measured Cp of 1.333 carries a 95% interval of 0.992 to 1.674; at n = 100 it narrows to 1.148 to 1.519; at n = 200 to 1.202 to 1.464. Precision improves roughly with the square root of the sample, so halving the interval costs four times the parts. Just as important is spreading the sample across shifts, operators, tool changes and material lots, because a large sample taken in one run measures the wrong variation with great precision.

What if my data is not normal?

Then do not convert the index into a defect rate using the normal distribution, because the error lands entirely in the tail you care about. Characteristics bounded at zero — flatness, roundness, runout, surface roughness, contamination counts — are usually skewed by construction and should never be forced into a normal fit. The ISO 22514 series defines percentile-based capability indices for exactly this case, using the 0.135% and 99.865% quantiles of the actual fitted distribution in place of μ ± 3σ. The alternatives are transforming the data or fitting a distribution that suits it; what you must not do is report a Cpk that silently assumed normality.