Process Capability

Process Capability Cp

Measures process spread relative to the tolerance width, ignoring centering.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Quality Tools > Capability Analysis > Normal

Description

Cp compares the tolerance width (USL − LSL) to process spread (6s) and answers: does the natural process variation even fit within the tolerance? Cp uses the sample standard deviation (ddof = 1) and is interpreted as 'short-term capability'. Cp ≥ 1.33 is considered capable, Cp ≥ 1.67 highly capable. Cp says nothing about process location — that is what Cpk is for. Cp is only defined for two-sided specifications.

Formulas

C_{p} = \frac{U S L - L S L}{6 s}

Process capability ignoring centering (s = sample standard deviation, ddof = 1)

CPU

C_{P U} = \frac{U S L - x ˉ}{3 s}

Upper capability index

CPL

C_{P L} = \frac{x ˉ - L S L}{3 s}

Lower capability index

CI(Cp)

C I (C_{p}) = C_{p} \frac{χ _{α /2, n - 1}^{2}}{n - 1}, C_{p} \frac{χ _{1 - α /2, n - 1}^{2}}{n - 1}

Exact confidence interval for Cp based on the chi-squared distribution (α = 1 − confidence level)

Assumptions

Data is approximately normally distributed
Process is in statistical control
Two-sided specification (both LSL and USL required)

Limitations

Insensitive to location shifts — a strongly off-center process can be incapable despite a high Cp (use Cpk)
Requires a two-sided specification
Sample size should be ≥ 30 for the confidence intervals to be meaningful

References

Montgomery, Introduction to Statistical Quality Control, 8th Ed. — Chapter 7
Kotz, S. & Johnson, N. L. (1993), Process Capability Indices, Chapman & Hall