Yellow Belt · 15 min
Capable, or just in control?
After this you can
- calculate Cp and Cpk from a set of measurements and state the conditions that make them meaningful.
- explain the difference between Cpk and Ppk and say which one answers a customer's question.
Assumes you have done Control limits are not specification limits.
The problem
A supplier was asked for a Cpk and sent back 1.67. The customer accepted the part. Eleven weeks later the reject rate at goods-in was 2%, against an implied 0.6 parts per million. The Cpk had been computed from thirty consecutive parts made on one afternoon, on one machine, by one operator — a genuinely excellent afternoon. Nobody had asked whether the process was stable, whether the data was normal, or how much of the year that afternoon represented. Every number in the calculation was arithmetically correct.
The idea
Control asks: is the process behaving consistently? Capability asks: is consistent good enough? They are different questions and the second one only makes sense after the first is answered yes.
A capability index compares the voice of the process (its spread) with the voice of the customer (the specification).
The four indices, and why there are four
Cp = (USL − LSL) / 6σ — the potential. It asks whether the process spread would fit inside the tolerance if it were centred. It ignores where the process actually sits.
Cpk = min[(USL − μ), (μ − LSL)] / 3σ — the actual. It measures to the nearer limit, so a perfectly capable but off-centre process scores badly, correctly. Cpk can never exceed Cp, and the gap between them is entirely centring.
Pp and Ppk are the same two formulae with a different σ, and that is the whole distinction:
| Sigma used | What it describes | |
|---|---|---|
| Cp, Cpk | Within-subgroup: mean moving range ÷ d₂, or R̄ ÷ d₂ | Short-term spread — how the process behaves within a moment |
| Pp, Ppk | Overall: the sample standard deviation, dividing by n − 1 | Long-term spread — including every drift between moments |
Cpk answers "what is this process capable of?". Ppk answers "what did the customer receive?" When a customer asks for a capability index, Ppk is nearly always the honest answer, and Cpk is nearly always what gets sent.
Cpk above Ppk means the process drifts between the moments you sampled it. That gap is the number worth talking about.
The three conditions, which are the actual content of this lesson
A capability index is a statement about a stable, roughly normal process with enough data behind it. Every calculator on the internet will return a number without checking any of that.
Stable. On an unstable process there is no single σ to estimate — the spread is a different quantity at different times — so Cpk does not describe anything that will still be true tomorrow. Check with a control chart, before computing anything.
Roughly normal. The expected parts-per-million figures come straight from the normal distribution. On skewed data — and times, flatness, concentricity and roundness are all routinely skewed — they can be wrong by an order of magnitude, always in the optimistic direction. Check with Anderson–Darling or a probability plot.
Enough data. Thirty measurements is a floor, not a recommendation; AIAG's initial studies use 25 subgroups of 5. Below that the confidence interval on Cpk is wide enough to span the decision you are trying to make.
Fail any of the three and the index is still computable. It is simply not a statement about your process.
What the numbers mean
Cpk of 1.33 is the common minimum in manufacturing, and 1.67 for safety-critical characteristics. Both are conventions, not derivations — they exist because somebody had to pick a number, and they are worth knowing precisely so you can say that out loud when somebody quotes one as though it were physics.
Worked example
Thirty bottles, fill volume in ml, specification 9.7 to 10.3.
Step 1 — check stability, before anything else. An individuals-and-moving-range chart on the same thirty values: no point outside the limits, no rule violations. Stable.
Step 2 — check normality. Anderson–Darling gives p = 0.43, comfortably consistent with a normal distribution. The expected-ppm figures will mean something.
Step 3 — n = 30, which meets the floor. All three conditions pass, so the indices are worth computing. Note the order: the conditions came first, and if any had failed the honest answer would have been "not yet" rather than a number.
Step 4 — the two sigmas.
- Within (MR̄ / d₂): 0.0773 ml
- Overall (sample SD, n − 1): 0.0803 ml
Step 5 — the indices.
| Value | Reading | |
|---|---|---|
| Cp | 1.293 | The spread would fit the tolerance with room, if centred |
| Cpk | 1.227 | It is slightly off centre — the mean is 9.985 against a midpoint of 10.0 |
| Pp | 1.245 | |
| Ppk | 1.181 | What the customer actually experienced |
Cp = (10.3 − 9.7) / (6 × 0.0773) = 0.6 / 0.4640 = 1.293. Cpk uses the nearer limit: the mean 9.985 is closer to the lower limit, so (9.985 − 9.7) / (3 × 0.0773) = 0.285 / 0.232 = 1.227.
Reading the gap between Cp and Cpk. 1.293 against 1.227 is small — the process is nearly centred. Centring it perfectly would buy 0.066 of Cpk and no more, so the remaining work is reducing spread rather than shifting the mean. That decision falls straight out of the two numbers and cannot be made from either alone.
Reading the gap between Cpk and Ppk. 1.227 against 1.181. Close, because within and overall sigma are close, which says the process is not drifting much between the moments sampled. On a process that does drift, this gap opens up — and Ppk is the number the customer's rejects will agree with.
Back to the supplier in the Hook. Their 1.67 was probably a real Cpk from a real afternoon. What it was not is a statement about the year, because nobody checked stability across shifts, and a Ppk over a longer window would have shown the drift the goods-in inspector eventually found.
Dataset: ds-fill-volume-capability — the same data loads in the tool below, so you can reproduce every figure here yourself.
Your turn
The tool opens with the fill-volume data and puts the three conditions above the indices.
- Confirm Cpk 1.227 and Ppk 1.181, and that the gate reports the conditions met.
- Change the last value to 10.9 — a single fat-fingered entry. Two checks flip, not one: stability AND normality, because one point that far out is both a special cause and a tail the normal model does not have. The gate withdraws the warrant while the indices remain on screen underneath. That is the design: hiding them would send you to a calculator that does not check.
- Delete everything after the twentieth value. The sample-size condition fails at n = 20 and the gate says so. Note that the indices move as well — Cpk 1.227 to 1.112, about 9% — which is the point rather than a complication: the formula is unchanged, but ten fewer observations is a wider interval around the same process, not a different process.
- Set the LSL to 9.9 and leave the USL. Cpk collapses, because it measures to the nearer limit — and the process mean has not moved at all. This is the number that catches an off-centre process, and Cp will not.
- Remove the USL entirely. Cp and Pp both become undefined — there is no tolerance width to divide — while Cpk and Ppk both survive as one-sided indices. A tool that returned a Cp here would be inventing a specification limit the customer never gave it.
Process Capability· fill volume
practiceThe conditions for a capability index are met.
- Stable
- yes
- Normal
- yes
- n ≥ 30
- yes(n = 30)
Cp · within
1.293
Cpk · within
1.227
Pp · overall
1.245
Ppk · overall
1.181
- n
- 30
- Mean
- 9.9847
- σ within
- 0.0773
- σ overall
- 0.0803
Cp/Cpk use within-subgroup sigma (within:mrbar-over-d2); Pp/Ppk use overall sigma (overall:sample-sd-n-minus-1).
The two sigma estimates are close, so short-term and long-term variation are much the same here. The process is behaving consistently between the moments you sampled it.
Anderson–Darling A² = 0.367, p = 0.432. Consistent with a normal distribution, which is what the expected-PPM figures assume.
| Expected ppm out of spec | Below LSL | Above USL | Total |
|---|---|---|---|
| Within (short-term)the sigma behind Cp and Cpk | 116.3 | 22.8 | 139.1 |
| Overall (long-term)the sigma behind Pp and Ppk | 197.4 | 43.3 | 240.7 |
These are MODEL outputs for fill volume, not counts of what happened: they are the normal distribution’s tails beyond each limit, so they are only as good as the normality check above. The within row is what the process could deliver if it held still; the overall row is what the customer received. Which tail is larger says which way the mean has to move, and the total says nothing about that.
One per line. Order matters — stability is checked from it.
How this is calculated
- Within-subgroup sigma = mean moving range ÷ d₂ (d₂ = 1.128 for n = 2), the short-term variation. Feeds Cp and Cpk.
- Overall sigma = the sample standard deviation, dividing by n − 1. Feeds Pp and Ppk. Dividing by n understates the spread and inflates every index — it is the most common defect in hand-built capability spreadsheets.
- Cp = (USL − LSL) / 6σ; Cpk = min[(USL − μ), (μ − LSL)] / 3σ.
- Stability is measured from an individuals-and-moving-range chart on the same data, not asserted. Normality by Anderson–Darling at α = 0.05.
Source: Montgomery, D.C., Introduction to Statistical Quality Control 7e, Ch. 8 (process capability) and §6.4 (individuals charts); AIAG, Statistical Process Control 2nd ed., for the within/overall distinction.
Check yourself
No hints. Wrong answers are explained, not softened.
A process has Cp = 2.0 and Cpk = 0.8. What is the situation, and what should you do first?
A customer asks for your capability. Your Cpk is 1.52 and your Ppk is 0.94. What do you send, and what does the gap mean?
Cpk comes out at 1.41 on data whose control chart shows two points beyond the limits. What is the correct report?
Worth remembering
What is the difference between Cp and Cpk?
Cp is potential — would the spread fit the tolerance if centred. Cpk measures to the nearer limit, so it also captures centring. Cpk can never exceed Cp, and the gap between them is entirely centring.
Cpk or Ppk for a customer?
Ppk. Cpk uses within-subgroup sigma and describes a moment; Ppk uses overall sigma and describes what the customer received. A large gap between them means the process drifts between the moments you sampled.
What three conditions must hold before a capability index means anything?
The process is stable, the data is roughly normal, and there is enough of it (30 is a floor). Fail any one and the index is still computable — it is simply not a statement about your process.
Can you do this now?
Rate yourself honestly. We compare your rating with how you actually answered — the gap is more useful than either number alone.
I can calculate Cp and Cpk from a set of measurements and state the conditions that make them meaningful.
I can explain the difference between Cpk and Ppk and say which one answers a customer's question.
Your rating is recorded alongside your drill results. Neither alone marks the competency as met.