Once a process is statistically stable, its natural variation can be compared with engineering specification limits.

Let upper and lower specification limits be $USL$ and $LSL$. If the process standard deviation is $\sigma$, the potential capability index is

$$C_p=\frac{USL-LSL}{6\sigma}.$$

It compares tolerance width with a six-standard-deviation process spread but ignores where the process mean lies.

The centering-aware capability index is

$$C_{pk}=\min\left(\frac{USL-\mu}{3\sigma},\frac{\mu-LSL}{3\sigma}\right).$$

Suppose a shaft specification is $20.00\pm0.10$ mm, so $LSL=19.90$ and $USL=20.10$ mm. A stable process has $\mu=20.04$ mm and $\sigma=0.02$ mm.

Then

$$C_p=\frac{0.20}{6(0.02)}=1.67,$$

but

$$C_{pk}=\min\left(\frac{20.10-20.04}{0.06},\frac{20.04-19.90}{0.06}\right)=\min(1.0,2.33)=1.0.$$

The process has relatively small spread but is shifted toward the upper limit.

Capability indices are meaningful only when the data represent a sufficiently stable process and the assumed summary of variation is appropriate. They do not replace control charts, measurement-system analysis or direct engineering understanding of the process.