Process Capability

Process capability compares stable process variation with specification limits.
It is not a control chart.
Use it only after statistical stability is established.^[1]

Prerequisites

Prerequisites: control charts, sample mean, sample standard deviation.

Process Context

Use capability indices when a process is in control and the question is whether output fits customer, engineering, or regulatory requirements.

Definition

For a two-sided specification with lower specification limit $L S L$ , upper specification limit $U S L$ , process mean $μ$ , and process standard deviation $σ$ :

C_{p} = \frac{U S L - L S L}{6 σ}

C_{p k} = min (\frac{U S L - μ}{3 σ}, \frac{μ - L S L}{3 σ}) .

$C_{p}$ measures potential capability if centered. $C_{p k}$ includes off-centering.

Assumptions / Requirements

The process is stable.
Observations are independent enough for the estimate used.
The distribution model is appropriate; the classical formulas assume approximate normality.
Specification limits are real requirements, not control limits.

Notation

Symbol	Meaning
$U S L$	Upper specification limit
$L S L$	Lower specification limit
$T$	Target value
$μ, σ$	Process mean and standard deviation
$\bar{x}, s$	Sample estimates

Procedure

Verify control-chart stability.
Estimate $\bar{x}$ and $s$ from representative data.
Check distribution shape and measurement system adequacy.
Calculate capability.
Improve centering or variation if capability is insufficient.

Worked Example

A shaft has $L S L = 9.90$ , $U S L = 10.10$ , $\bar{x} = 10.03$ , and $s = 0.025$ .

{\hat{C}}_{p} = \frac{10.10 - 9.90}{6 (0.025)} = 1.33

{\hat{C}}_{p k} = min (\frac{10.10 - 10.03}{3 (0.025)}, \frac{10.03 - 9.90}{3 (0.025)}) = 0.93 .

The spread could be adequate if centered, but the process mean is too close to the USL.

Common Mistakes

Calculating capability from unstable data.
Reporting only $C_{p}$ when the process is off-center.
Confusing LSL/USL with LCL/UCL.
Assuming normal capability formulas are valid for strongly nonnormal data.

Connections

Related note	Use
Control Limits and Specification Limits	Required distinction
Control charts	Stability prerequisite
Quadratic loss function	Target-centered quality
Common-Cause and Special-Cause Variation	Stability interpretation

References

NIST/SEMATECH, e-Handbook of Statistical Methods, "What is Process Capability?", https://www.itl.nist.gov/div898/handbook/pmc/section1/pmc16.htm ↩︎