Scenario
You calculated the Cpk for surface roughness as 1.45, which looks good. But you performed a normality test, and the data is severely right-skewed, with P < 0.005. You know the traditional Cpk formula assumes a normal distribution—does that 1.45 still count?
No, it doesn't; you must recalculate.
Which Process Data Are Inherently Non-Normal
Right-skewed:
- Surface roughness (cannot be negative, but can be very large)
- Burr height
- Leak rate
- Wear amount
- Cycle time (has a lower bound but no upper bound)
Left-skewed:
- Purity percentage (upper limit 100%)
- Yield (upper limit 100%)
Bimodal Distribution:
- Data from two mixed production lines
Three Methods for Non-Normal Cpk
Method One: Transformation Method (Box-Cox / Johnson)
Transform non-normal data into an approximately normal distribution, then apply the standard Cpk formula.
Pros: Results are still presented as Cpk, easy to communicate
Cons: Specification limits also need to be transformed, making communication complex; not all data can be successfully transformed
Applicable: When a suitable transformation function can be found for the data
Method Two: Percentile Method
Calculate directly using the actual percentiles of the data, without assuming any distribution.
Pp (non-normal) = (USL - LSL) / (P₉₉.₈₆₅ - P₀.₁₃₅)
Uses the 0.135 and 99.865 percentiles, corresponding to ±3σ of a normal distribution.
Pros: Does not require assuming a distribution type, intuitive
Cons: Requires sufficient data (at least 100 points) to be reliable
Applicable: When sample size is large and distribution type is uncertain
Method Three: Fitting Known Distributions
Identify which distribution the data fits (Weibull, lognormal, Gamma, etc.), directly calculate the defect rate using the CDF of that distribution, then back-calculate the equivalent Cpk.
Pros: Most mathematically rigorous
Cons: Requires verification of good fit to be reliable
Applicable: When there's a theoretical basis for the data conforming to a known distribution
Selection Logic
Data is not normal → Sample size ≥ 100?
- Yes → Try the Percentile Method first
- No → Try the Transformation Method first → Normal after transformation?
- No → Fit an appropriate distribution
Practical Recommendations
- First, perform a normality test (Anderson-Darling) to confirm if non-normal methods are needed
- Sample size < 100: Insufficient data, any method will be unreliable; collect more data first
- Communicate the method with the client: Different methods yield different numbers; confirm with the client beforehand which method is acceptable
Golden Quote
"Forcing a normal distribution formula to calculate Cpk for non-normal data is like using a weighing scale to measure temperature—you get a number, but it's not what you think it is at all."