Process Capability

Overall Process Capability Ppk

Overall process capability based on total variation (population σ, ddof = 0).

Statusvalidated

Version1.0.0

Minitab equivalentStat > Quality Tools > Capability Analysis > Normal (Overall)

Description

Pp and Ppk are the 'long-term' counterparts to Cp and Cpk. They use the population standard deviation (ddof = 0), which captures between-subgroup variation as well — not just within-subgroup variation. For a process without drift, Pp ≈ Cp and Ppk ≈ Cpk; if they diverge significantly, the process is unstable across subgroups. Ppk = min(PPU, PPL) and, like Cpk, penalises off-center processes. Ppk ≥ 1.33 is generally considered capable.

Formulas

P_{p} = \frac{U S L - L S L}{6 σ}

Overall process capability (σ = population standard deviation, ddof = 0)

Ppk

P_{p k} = min (\frac{U S L - x ˉ}{3 σ}, \frac{x ˉ - L S L}{3 σ})

Minimum of upper and lower overall capability indices

PPU

P_{P U} = \frac{U S L - x ˉ}{3 σ}

Upper overall capability index

PPL

P_{P L} = \frac{x ˉ - L S L}{3 σ}

Lower overall capability index

CI(Ppk)

C I (P_{p k}) = P_{p k} \pm z_{1 - α /2} \cdot \frac{1}{9 n} + \frac{P _{p k}^{2}}{2 ( n - 1 )}

Approximate confidence interval for Ppk (Kushler-Hurley normal approximation)

Assumptions

Data is approximately normally distributed
Dataset represents the long-term process (contains between-subgroup variation)
Measurements are independent

Limitations

Pp (two-sided) only defined with both LSL and USL
Divergence between Pp/Ppk and Cp/Cpk is an indicator of instability — do not ignore it
Requires sufficient sample size (n ≥ 100 for a robust long-term statement)

References

Montgomery, Introduction to Statistical Quality Control, 8th Ed. — Chapter 7
AIAG — Statistical Process Control (SPC) Reference Manual, 2nd Edition
Kushler, R. & Hurley, P. (1992), Confidence Bounds for Capability Indices, Journal of Quality Technology, 24(4), 188–195