Cp says the tolerance could fit. Cpk says whether it does.

The two questions

Every capability conversation is two questions asked at once, and most arguments about capability happen because somebody answered one of them and thought they had answered both. Could this process fit inside the tolerance at all? That is about spread, and it does not care where the process sits. Does it fit, where it is actually running? That is about spread and position — because a process can be narrow enough and still be making scrap, by being narrow in the wrong place.

Cp answers the first. Cpk answers the second. Reporting Cp on its own is the commonest way a capability figure flatters a process, and the honest version costs nothing, because both numbers come from exactly the same data.

Cp: tolerance width against process width

Cp is the specification width divided by six standard deviations of the process: (USL − LSL) ÷ 6σ. Six sigma, because that is very nearly the whole of a normal distribution — so Cp asks how many process widths would fit inside the tolerance.

A Cp of 1.00 means the process is exactly as wide as the tolerance, with nothing spare either way. Below 1.00 it cannot hold the tolerance however well it is centred, and adjusting the mean will not fix it — that is a variation problem, belonging to the tool, the material or the machine rather than to the setter.

What Cp never tells you is whether the process is near the middle: the mean does not appear in the formula. A process running hard against the upper limit and one sitting on nominal produce identical Cp values, which is why Cp alone is not a capability claim.

Cpk: the same process, with the centring counted

Cpk takes the distance from the process mean to the nearer specification limit and divides it by three sigma — the smaller of (USL − μ) ÷ 3σ and (μ − LSL) ÷ 3σ. The word doing the work is smaller: Cpk is deliberately pessimistic, because the limit you are closest to is the one you breach first. Two consequences fall out of that. Cpk can never exceed Cp; and when the process is perfectly centred the two are equal. The gap between them is not noise. It is a measurement of how far off centre you run.

A worked example on one dimension

Take a moulded feature specified at 10.00 mm ± 0.10 mm, so the tolerance runs 9.90 to 10.10 and the width is 0.20 mm. The process standard deviation, estimated from the control chart, is 0.025 mm, so six sigma is 0.15 mm.

The process did not get wider. Nothing changed about the tool, the material or the machine. The mean moved 0.04 mm and capability fell by 40%. The relationship is exact: 0.04 mm out of a 0.10 mm half-tolerance is 0.4 off centre, and 1.33 × 0.6 = 0.80.

That is the whole argument for reporting both. Cp 1.33 with Cpk 0.80 tells a setter to move the mean, which is usually a morning’s work. Cp 0.95 with Cpk 0.94 tells an engineer the tolerance cannot be held as things stand, which is a much more expensive conversation. One number cannot tell those two cases apart, and quoting only the flattering one is how a capability report stops being evidence.

The sigma estimator, and why it belongs on the screen

Here is the part most explanations leave out, and it is why two systems handed the same measurements can return different capability figures without either being wrong. σ is not measured. It is estimated, and there is more than one legitimate way to estimate it.

• Within-subgroup. Derived from the variation inside each subgroup — the average range or average subgroup standard deviation, scaled by the standard constants your quality engineer already has in a folder. It captures short-term, common-cause variation: what the machine does over five consecutive cycles, not what it does across a fortnight. This is the estimator conventionally behind Cp and Cpk.
• Overall. The ordinary standard deviation of every measurement in the study, ignoring subgroup structure. It absorbs everything — short-term variation plus drift, shift-to-shift differences, tool changes, a new material lot. This is the estimator behind Pp and Ppk.

On a genuinely stable process the two converge. On a process that drifts, the within-subgroup estimate is smaller, so Cpk comes out higher than Ppk. The gap between Cpk and Ppk measures everything happening between your subgroups, and it is often the most useful number on the report. A Cpk of 1.5 beside a Ppk of 0.9 is not a contradiction: it is a process that is capable minute to minute and wandering between samples, telling you to go and look at changeovers, warm-up, or the point in the week a tool comes off preheat.

So a capability figure quoted without its estimator is not a figure. Given one number labelled “Cpk” and no statement of how sigma was arrived at, you cannot tell whether you are looking at short-term capability, long-term performance, or a spreadsheet that ran STDEV down the column and chose the friendlier label. SmartFactory reports Cp and Cpk alongside Pp and Ppk with the estimator named on screen for each — two honest numbers rather than one convenient one.

How much data before the number means anything

Cpk is an estimate from a sample, and like any estimate it carries an interval around it that nobody prints. That interval is wider than people expect: on a study of around thirty measurements, a calculated Cpk of 1.33 sits in a range that comfortably includes values near 1.0 and values near 1.7. The point estimate is 1.33; what the data supports is “somewhere in that region, probably”.

Two rules keep a study honest, and both are about refusing to produce a number rather than producing one. Below the accepted subgroup count, say so — a study drawn from a handful of subgroups is a preview, and the professional answer is “provisional, twelve of twenty-five subgroups collected”, not a confident-looking 1.41 printed in the same font as a real one. And a short study is a plan, not a result: say what has been collected, what the provisional figure is, and when the study completes. That conversation goes fine. What does not go fine is a number that later moves and nobody can say why.

This is where software either helps or actively harms. A system that will draw a control chart from six points and print a capability index under it is manufacturing confidence out of nothing. SmartFactory applies minimum-data guards throughout and marks a study provisional below the accepted subgroup count.

What an auditor actually asks for

Worth realising before an audit or a customer visit: the index itself is rarely what gets examined. The number takes four seconds to read. The rest of the hour is everything standing behind it. Expect to be asked:

• Which characteristics this covers, and who decided that. Capability is proven on the characteristics your customer designates on the drawing and in the control plan, not the ones convenient to measure. Show the link from the designated characteristic to the study.
• What the measurement system was doing. A capability study inherits the error of the gauge that fed it, so gauge capability is established before, not after. Without an MSA in hand, a flattering Cpk may be describing the gauge as much as the process — which is why an auditor asks for the measurement study first and the index second.
• The raw data and the subgroup structure, not the index alone: sample size, subgroup size, sampling frequency, and the window covered.
• How sigma was estimated — and whether the figure on the certificate is Cpk or Ppk. Getting that wrong in front of a customer is avoidable and memorable.
• Evidence the process was in control across the window: the chart, its limits, and an account of any out-of-control points and what was done about them.
• Which revision of the specification was in force when the measurements were taken. This is the one that catches sites out two years later. SmartFactory stamps every measurement with the revision current at the time, and specification approval runs through a workflow in which the approver cannot be the author — the segregation an IATF 16949 audit expects to see enforced, not promised.
• What happened when a figure fell short. A reaction plan that was followed, containment that was applied, and a record of it. An auditor is far more interested in a low index with a documented response than a high one with nothing behind it.

On the figure itself: the threshold you have to meet is your customer’s to set, and it reaches you through the part approval submission they require and their customer-specific requirements. Different customers expect different things, and they can differ characteristic by characteristic on the same part. Read it off their documents rather than off anyone’s website, including ours.

Habits that survive scrutiny

• Publish the pairs. Cp with Cpk, and Cpk with Ppk. The first gap is centring, usually the cheapest thing on the list to fix; the second is drift, which is a changeover and scheduling problem more often than a machine one.
• Fix the estimator, then leave it alone. A definition that moves between quarters is worse than one you disagree with — the same discipline that decides which side of the line rework falls on in an OEE calculation.
• Rank worst-first. A capability report sorted alphabetically is a filing exercise. SmartFactory keeps a Cpk watchlist ordered worst-first, so the weekly review starts where the risk is rather than where the alphabet does.

Where this sits in a production system

Capability is not a standalone exercise. The measurements come from inspections somebody performed, against a specification revision that was current, on parts belonging to a works order that ran on a particular machine on a particular shift. Held in separate spreadsheets, those links make reconstructing a study two years later a week of work and a leap of faith.

Which is the argument for doing capability inside the same system that records production. The quality and SPC module covers the mechanics: scan-driven inspections, X̄-R and I-MR charts on the standard constants, frozen baseline limits, Nelson rules marked where they occurred, and capability reported with its estimator named. If you are a UK moulder and the practical question is how that lands on a floor of presses and tools, the moulders’ page works the same argument through tools, cycles and changeovers rather than formulae.

And the honest limit, stated plainly: this software will not make a wide process narrow. What it will do is stop you finding out about the width three months after the parts shipped, and give you a capability figure that still means the same thing the second time somebody asks where it came from.