Process capability: stability, distribution and limits

Why it matters

A product's dimension must stay between the acceptance limits. The dimensions the process produces, however, vary with a certain location and spread. Process capability investigates how much of the specification range this spread takes up. A narrow spread can be useful, but if the mean is near a limit, product risk can still rise. For this reason, comparing only the largest value with the smallest, or finding all observed products conforming, does not complete a capability review. The calculation must explain which data represents which process conditions. The result is not a firm promise of conformity for every future product; it is an assessment tied to the stated conditions.

Stability is the question of whether the same process behavior continues over time. Capability, on the other hand, looks at the relationship of this behavior to the product specification. In an unstable series, a single mean and a single standard deviation can be a mixture of different states. Even if the ratio for such a mixture comes out high, it does not explain which state will continue in the future. First the reliability of the measurement and the time order, then the distribution structure, and then the capability calculation are addressed. A calculator producing a number does not change this order. A missing stability proof cannot be completed with more decimal places or with a ratio that looks high.

In this lesson the sample standard deviation s is used for the spread estimate. Dividing the specification range by the six-standard-deviation width gives Cp. This ratio summarizes how much of the specification range the spread takes up when the distribution can be centered within it. The mean's distance from the limit is evaluated separately. Dividing the nearer limit distance by three standard deviations gives Cpk. The two numbers answer different questions: two processes with the same spread can have different nearer-limit distances because of different means. The measurement units cancel within the ratio; the result is unitless, not a percentage. Here s is used to estimate the process standard deviation from independent observations of the same stable process; the calculated numbers are point estimates. In a study with subgroups, the within-group spread estimate and the standard deviation of the whole period are not the same thing. Indices produced with different definitions of spread are not compared without explanation.

This interpretation made with six standard deviations rests on the process distribution being suitably represented. One does not move on to tail probability without explaining the approximately normal distribution assumption. A single-peaked look is not sufficient evidence by itself either; product mix and changes over time must be investigated. No nonconforming-product probability is calculated here. The measurement system is deepened in M24, and the choice of a suitable distribution and estimate in later advanced applications. What is expected of this lesson is, besides using the formula correctly, to notice under which condition the formula is not enough to decide. Not giving approval on insufficient evidence is also a measurable engineering decision.

Formula