Process Capability for Non-Normal Data

Standard Cp and Cpk assume normality. When a process clearly does not follow a normal distribution, you need an alternative approach before reporting capability to a customer.

Three approaches

Method	When to use	How it works
Box-Cox transformation	Data is skewed but can be transformed	Find λ to transform y → y^λ; compute Cpk on transformed data
Percentile method	Any distribution	Use 0.135th and 99.865th percentiles in place of μ ± 3σ
Johnson transformation	Any distribution, more flexible	Fits one of four Johnson families; transforms to near-normal

Percentile method formula

Ppk equivalent (non-normal) = min[ (USL − P₅₀) / (P₉₉.₈₆₅ − P₅₀), (P₅₀ − LSL) / (P₅₀ − P₀.₁₃₅) ] Where P₅₀ = median, P₀.₁₃₅ and P₉₉.₈₆₅ are the 0.135th and 99.865th sample percentiles.

Common non-normal distributions in manufacturing

Weibull: tool wear, fatigue failure times
Lognormal: particle size, concentrations, cycle times
Exponential: time between events (if constant failure rate)

CQE exam tips

Before using non-normal methods, verify the data is truly non-normal (Anderson-Darling or Shapiro-Wilk test).
Box-Cox λ = 0 is a log transform; λ = 0.5 is a square root transform.
The percentile method makes no distribution assumption and is the safest for unknown distributions.

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