Process Capability: Cp and Cpk

Process capability is simply a measure of whether a process's normal scatter fits comfortably inside the spec limits. The two indices are short ratios. $C_p = \frac{\text{USL} - \text{LSL}}{6\sigma}$ is the room you're given (the spec width) over the room you need (about $6\sigma$, the spread of nearly all parts). $C_{pk}$ is the same idea measured from wherever the mean actually sits to the nearer limit, over $3\sigma$. Bigger is better, and 1.33 is a common minimum. Two ratios, and a z table to turn them into scrap rates.

In control isn't the same as good. A control chart says a process is stable: the same distribution day after day. It doesn't say whether that distribution fits inside the spec. A drill that's perfectly stable but scatters holes over ±0.08 mm will never hold a ±0.05 mm tolerance, however long you chart it. Capability compares the voice of the customer (the spec width) with the voice of the process (its natural spread).

Cp: does the spread fit? A normal process puts 99.73% of its output within $\mu \pm 3\sigma$, a band $6\sigma$ wide. $C_p$ is the spec width divided by that band. At $C_p = 1$ the band exactly fills the spec, and even a centered process makes 0.27% scrap (2,700 ppm). $C_p = 1.33$ leaves a $1\sigma$ cushion on each side, so each limit is $4\sigma$ away (63 ppm when centered). Many customers require at least that, and 1.67 for safety-critical features.

Cpk: does it fit where it's running? $C_p$ never looks at the mean, so a process sitting right on the upper limit can boast a fine $C_p$ while scrapping half its output. $C_{pk}$ measures the distance from the mean to the nearer limit in units of $3\sigma$, so $C_{pk} \le C_p$, with equality only when the process is centered. To see how centering enters, let $m$ be the middle of the spec and $k = |\mu - m|\big/\tfrac12(\text{USL} - \text{LSL})$ the offset as a fraction of the half-width. The nearer limit is $\tfrac12(\text{USL} - \text{LSL})(1 - k)$ from the mean; divide by $3\sigma$ and you get $C_{pk} = C_p(1 - k)$.

From an index to ppm. $C_{pk}$ is a scaled distance, so it converts straight into a tail area. The nearer limit is $3C_{pk}$ standard deviations from the mean, so the fraction beyond it is $\Phi(-3C_{pk})$. The two distances add up to the whole spec width, $6\sigma C_p$, so the far limit is $3(2C_p - C_{pk})$ standard deviations away, which gives the last form of the third formula. It's the normal-table work of Using the Normal Distribution, aimed at scrap.

What "six sigma" means. A six sigma process has its spec limits $6\sigma$ from the target: $C_p = 2$. Centered, that's $2\Phi(-6) \approx 0.002$ ppm. Real means wander, though, so the famous figure budgets a long-term drift of $1.5\sigma$. The nearer limit is then $4.5\sigma$ away, $C_{pk} = 1.5$, and $\Phi(-4.5) = 3.4 \times 10^{-6}$: the "3.4 defects per million" quoted everywhere.

Where σ comes from. Use the control chart's $\hat\sigma = \bar R/d_2$ from X̄ and R Control Charts: it measures the short-term, within-subgroup variation the process is capable of. The overall standard deviation of all the data, which includes drift between subgroups, gives the long-term indices $P_p$ and $P_{pk}$. If the process isn't in control, neither number predicts anything, because there's no single distribution to predict from.

Cautions.

• Tail areas beyond about $4\sigma$ extrapolate the normal model. Check the shape with a probability plot before quoting 3 ppm.
• An index from 30 parts is an estimate. A χ² interval for $\sigma$, as in Intervals for a Mean and for Variability, turns into an interval for $C_p$.
• Never use the $\bar x$ chart's spread: single parts vary $\sqrt n$ times more than subgroup means.

Fixing a poor index. Low $C_{pk}$ with a good $C_p$: re-center, often with a single machine offset. Low $C_p$: the spread itself has to shrink, which takes engineering (a designed experiment finds the factors that drive it), or the tolerance has to widen, justified by a stack-up analysis.

Customers write capability into contracts: automotive and aerospace suppliers routinely have to show $C_{pk} \ge 1.33$ (often 1.67 for safety-critical features) before production is approved, and keep showing it. Engineers use capability to choose between machines, to decide whether a tolerance can be tightened and to set inspection: a feature with $C_{pk} = 2$ may need only audit checks, while one with $C_{pk} = 0.8$ needs 100% inspection until the process is fixed.