Cpk and process capability calculator

What capability indices measure

A specification says where a measurement is allowed to fall. A process has its own natural spread, and for a roughly normal process almost all of its output — 99.73% — lands within three standard deviations either side of the mean. Capability indices put those two widths side by side. If the tolerance band is wider than the ±3σ band, the process can meet the specification; if it is narrower, some output will always fall outside. The indices turn that comparison into a single number that engineers, suppliers and customers can quote.

The formulas

Cp = (USL − LSL) ÷ 6σ Cpu = (USL − x̄) ÷ 3σ Cpl = (x̄ − LSL) ÷ 3σ Cpk = min(Cpu, Cpl)

USL and LSL are the upper and lower specification limits, x̄ is the process mean and σ its standard deviation. Cp is the ratio of the specification width to the process width. Cpu and Cpl each measure the distance from the mean to one limit in units of 3σ, and Cpk is whichever of those is smaller — the side the process is closest to violating.

Take the defaults in the calculator: a mean of 10.2, a standard deviation of 0.5, and a specification of 8.5 to 11.5.

Cp = (11.5 − 8.5) ÷ (6 × 0.5) = 3 ÷ 3 = 1.000 Cpu = (11.5 − 10.2) ÷ (3 × 0.5) = 1.3 ÷ 1.5 = 0.867 Cpl = (10.2 − 8.5) ÷ (3 × 0.5) = 1.7 ÷ 1.5 = 1.133 Cpk = min(0.867, 1.133) = 0.867

The specification is exactly as wide as the process, so Cp is 1.00. But the mean sits 0.2 above the midpoint of 10.0, which pushes the upper tail out of spec: the upper limit is only 2.6σ away, and the area beyond z = 2.6 is about 4,661 ppm. The lower limit is 3.4σ away and contributes about 337 ppm. Roughly 5,000 parts per million — one in two hundred — would be out of specification.

Why Cp ignores centring and Cpk does not

Cp only ever sees the two limits and the standard deviation. Move the mean anywhere you like and Cp does not change, which makes it a measure of potential: what the process could achieve if it were perfectly centred. Cpk brings the mean back in. Because it takes the smaller of the two one-sided indices, any drift away from the middle of the tolerance band lowers it.

Two consequences follow directly from the definitions. Cpk can never exceed Cp, and the two are equal only when the mean sits exactly at the midpoint of the specification. In the example above, Cp is 1.00 and Cpk is 0.87; the gap between them is the cost of being 0.2 off centre. Recentring the process would lift Cpk to 1.00 without reducing variation at all, and that is usually the cheaper fix.

The benchmarks

The thresholds in common use are conventions that grew out of the automotive quality standards of the 1980s, not statistical laws. They are widely shared, though, and a customer who asks for "Cpk 1.33" will mean the same thing whichever industry they are in.

• 1.00 — just capable. The nearer limit is 3σ from the mean. About 0.13% of output falls beyond it, and any drift makes things worse immediately.
• 1.33 — the usual minimum for an established process. The nearer limit is 4σ away, leaving a full standard deviation of margin.
• 1.67 — commonly required for critical, safety-related or expensive characteristics, and for new processes before they have a history.
• 2.00 — the nearer limit is 6σ away: a six sigma process in the short term.

Cpk, z and expected ppm

Multiplying Cpk by three gives the z-score of the nearer specification limit, and the normal tail beyond that z is the expected fraction out of spec on that side. The table gives the one-sided figures; if both limits are equally close, double them.

The steepness of that curve is the point. Going from Cpk 1.00 to 1.33 cuts defects by a factor of forty; going from 1.33 to 1.67 cuts them by a further hundred. Small improvements in centring or spread pay off disproportionately once the process is already near capable.

Cpk versus Ppk

The formulas for Ppk are identical to those for Cpk. What changes is the standard deviation. Cpk is built on the short-term σ — the within-subgroup variation, estimated from the average range or by pooling the subgroup standard deviations. That captures the common-cause noise the process shows from moment to moment, and nothing else. Ppk uses the overall σ, the ordinary standard deviation of every measurement in the study, which also absorbs any shifts, drifts and tool changes between subgroups.

Because the overall σ contains everything the within-subgroup σ contains plus the between-subgroup movement, Ppk is normally the lower of the two. When the process is stable the gap is small. When Cpk is comfortably above 1.33 and Ppk is well below it, the process has the potential to be capable but is not being held still — and the remedy is control, not a tighter machine.

Sigma level and the 1.5σ shift

The short-term sigma level is the z-score at which a normal distribution has the same total tail area as the process has out of spec. For a centred process it is simply 3 × Cpk. The calculator reports this as Z bench and then adds 1.5 to give the sigma level "with shift".

That 1.5 is a convention from the six sigma programme. The observation behind it is that a process mean tends to wander by up to about 1.5σ over the long run, so a process whose limits sit 6σ away in a short study behaves, over months, like one whose nearer limit is 4.5σ away. The famous "3.4 defects per million" of a six sigma process is the one-sided tail at z = 4.5, not at z = 6 — the tail at z = 6 is about 0.001 ppm. Treat the shifted figure as a way of speaking the same language as six sigma reports, not as a prediction; nothing in your data tells the calculator whether your process actually drifts by 1.5σ.

Check the assumptions first

Every number on this page rests on two assumptions: the process is stable, and its output is roughly normal. If the mean is drifting, a single Cpk describes a process that no longer exists. If the distribution is skewed or has heavy tails, the ppm figures — which are normal tail areas — can be wrong by an order of magnitude even when the index itself is computed correctly.

So before quoting a capability index, plot the data. A control chart of the subgroups will show whether the process was in control during the study. A histogram of the individual measurements, with a normal curve drawn over it, will show whether the normal tail is a fair description of what the process actually produces. Ten minutes with those two plots is worth more than three decimal places of Cpk.