Cpk

Going deeper

What the index measures

Take the distance from the process mean to the nearer of the two specification limits, and divide it by three standard deviations of the process. That is Cpk. A value of 1.00 means exactly three standard deviations fit on the tight side, which for a centred, normally distributed process leaves roughly 2700 parts per million outside the limits – not a comfortable place to be.

The companion index Cp uses the full width of the specification against six standard deviations and ignores where the mean sits. The pair is informative together: Cp near Cpk means the process is centred, and Cp much larger than Cpk means a process that is tight enough but aimed wrongly – which is usually a much easier problem to fix. Most industries take 1.33 as a working minimum, at which about 30 parts per million fall outside on the tight side, and demand 1.67 or more for anything critical.

What it assumes

A capability index describes a distribution, and that presupposes there is one. If the process is not in statistical control – if it drifts, jumps between states, or responds to something nobody has identified – then the sample of parts measured is not drawn from a stable distribution, and the number computed from it describes nothing and predicts nothing. Control comes first, capability second, always in that order.

The second assumption is normality, and it matters far more than it looks, because the index is used to make statements about tails. A distribution with slightly heavier tails than normal has the same Cpk and many times the defect rate. The third is often forgotten entirely: the measurement system has its own variability, and if the gauge is not substantially better than the process, part of the measured spread is the instrument. Capability studies that skip that check overstate the problem or hide it.

Why the bar is so high for electronics

The numbers above are for consumer manufacturing, where a few tens of parts per million is acceptable because the parts are counted in millions. A processor contains billions of transistors, and every one of them has to work. At that scale the useful statistic is not the index but the extreme tail: a defect rate of one part per million per transistor would leave no working chips at all, which is why semiconductor process control is expressed in defects per billion and why redundancy is designed in wherever it can be.

This is the gap between the current state of 2D electronics and a technology. The best variability studies on monolayer transistors cover hundreds of devices, which is enough to establish a distribution’s centre and rough width and nowhere near enough to say anything about a tail at one in a billion. Getting from one to the other is not a matter of better champion devices; it is the entire discipline that Cpk, control charts and designed experiments belong to.