Process Capability (Cpk)⚠ unverified

Understanding the Parameters

• Process mean $\mu$ and the spec limits $LSL$, $USL$ — the spec limits define the acceptable window; the mean says where the process actually sits within it. $C_{pk}$ measures the distance from the mean to the nearer spec limit. If the process drifts off-centre toward one limit, that side's margin shrinks and $C_{pk}$ falls — even if the spread is unchanged.

• Standard deviation $\sigma$ — the process spread (Normal Distribution PDF). It divides both terms, so a tighter process (smaller $\sigma$) scores higher. The $3\sigma$ in the denominator is why capability is measured in "sigmas": $C_{pk} = 1$ means the nearer spec limit is exactly $3\sigma$ from the mean.

• The minimum operator — the defining feature. $C_{pk} = \min(C_{pu}, C_{pl})$ takes the worse of the upper and lower capabilities, so it cannot be fooled by a comfortable margin on one side. This distinguishes $C_{pk}$ from $C_p = (USL-LSL)/6\sigma$, which measures only the spread relative to the spec width and ignores centering. When the process is perfectly centred, $C_{pk} = C_p$; off-centre, $C_{pk} < C_p$.

• The output $C_{pk}$ — interpreted against industry benchmarks: $C_{pk} \ge 1.33$ is generally "capable" ($\sim 4\sigma$, $\sim 63$ ppm defective if centred); $C_{pk} \ge 1.5$ is a common automotive requirement; $C_{pk} = 2.0$ is the "six sigma" target ($\sim 3.4$ ppm allowing for a $1.5\sigma$ long-term drift). $C_{pk} < 1$ means the process spills outside its limits and produces significant scrap.

Derivation (Approaching a Proof)

Assume the process output is normally distributed (Normal Distribution PDF) with mean $\mu$ and standard deviation $\sigma$. "Capability" asks how many standard deviations of room lie between the process centre and each spec limit — because for a normal distribution the fraction outside a limit is set entirely by its distance in $\sigma$'s.

Define the one-sided capabilities as the distance to each limit measured in units of $3\sigma$ (the $3\sigma$ convention ties $C_{pk} = 1$ to the limit sitting at the $\pm3\sigma$ point, where only $\sim0.13\%$ falls outside):

$$C_{pu} = \frac{USL - \mu}{3\sigma}, \qquad C_{pl} = \frac{\mu - LSL}{3\sigma}.$$

A part is defective if it exceeds either limit, so the process is governed by whichever side is worse — the smaller capability. Hence

$$C_{pk} = \min(C_{pu}, C_{pl}). \qquad\blacksquare$$

The link to defect rate: if $C_{pk} = k$, the nearer spec limit is $3k$ standard deviations from the mean, and the fraction beyond it is $1 - \Phi(3k)$ (the normal tail). So $C_{pk} = 1 \Rightarrow 3\sigma \Rightarrow 1350$ ppm on that side; $C_{pk} = 1.33 \Rightarrow 4\sigma \Rightarrow 32$ ppm; $C_{pk} = 2 \Rightarrow 6\sigma \Rightarrow$ 2 ppb — the quantitative meaning of "six sigma."

Dimensional check. $USL$, $\mu$, $\sigma$ share the units of the measured characteristic, so each ratio $(USL-\mu)/(3\sigma)$ is dimensionless, and $C_{pk}$ is a pure number. $\checkmark$

History and Development

• Statistical process control. Capability indices grew out of Walter Shewhart's 1920s statistical process control at Bell Labs and were codified in Japanese and then American quality practice. $C_p$ (spread only) came first; $C_{pk}$ added the centering correction that made the index honest.

• The six sigma movement. Motorola (1980s) popularised "six sigma" — targeting processes so tight that the spec limits are $6\sigma$ from the mean ($C_p = 2$). The famous "$3.4$ defects per million" figure builds in a $1.5\sigma$ allowance for long-term process drift, distinguishing short-term capability ($C_{pk}$) from long-term performance ($P_{pk}$).

• Ubiquity in manufacturing. $C_{pk}$ is now a standard deliverable in automotive (PPAP), aerospace, and electronics supply chains — a single number certifying that a process reliably makes parts within tolerance, underpinning the tolerance and reliability analyses elsewhere in this category.

Related Concepts: Normal Distribution PDF, Expanded Uncertainty, Coverage Factor, Relative Uncertainty, Gage Rr Percent, Weibull Distribution PDF

Notes: Registry calculator six-sigma-capability (unverified). $C_{pk} = \min(C_{pu}, C_{pl})$ — correct as shipped; accounts for centering (unlike $C_p$). Assumes a normal process. Benchmarks: $\ge1.33$ capable, $2.0$ = six sigma. $C_{pk}=1 \Rightarrow$ nearer limit at $3\sigma$. Returns $+\infty$ if $\sigma=0$.