Process Capability

Process Capability Cp

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

Statusvalidated

Version1.1.0

Minitab equivalentStat > Quality Tools > Capability Analysis > Normal

Description

Cp compares the tolerance width (USL − LSL) to process spread (6σ within) and answers: does the natural process variation even fit within the tolerance? σ within is estimated from the average moving range for individuals (MR̄/d2) and from the pooled standard deviation for subgroups (s_p/c4). It holds only short-term variation — that is why Cp is read as 'short-term capability' and exceeds Pp for a drifting process. 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 σ ^ _{w}}

Process capability ignoring centering (σ̂w = σ within)

CPU

C_{P U} = \frac{U S L - x ˉ}{3 σ ^ _{w}}

Upper capability index

CPL

C_{P L} = \frac{x ˉ - L S L}{3 σ ^ _{w}}

Lower capability index

σ̂ innerhalb / within (n = 1)

\overset{σ}{^}_{w} = \frac{M R}{d _{2}}, d_{2} = 1.128

Individuals: average moving range of consecutive values divided by d2 (Minitab default)

σ̂ innerhalb / within (n ≥ 2)

\overset{σ}{^}_{w} = \frac{s _{p}}{c _{4} ( d + 1 )}, s_{p} = \frac{\sum _{i} \sum _{j} ( x _{ij} - x ˉ _{i} ) ^{2}}{d}, d = i \sum (n_{i} - 1)

Subgroups: pooled standard deviation of consecutive subgroups with bias correction c4 (Minitab default)

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)
Values are in production order (for MR̄ and subgroup formation)

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