Lognormal Distribution

Many quality characteristics that engineers assume are normal are actually lognormal. Surface roughness (Ra), particle contamination counts, assembly clearances, and chemical purity measurements often exhibit the characteristic right skew of a lognormal distribution. The telltale sign: values cannot go below zero, the distribution tails off gradually to the right, and the data looks normal on a log scale.

The practical problem is that computing Cpk assuming normality on lognormal data produces misleading results. For right-skewed data, the normal model underestimates the upper tail probability (risk of exceeding the upper spec) and overestimates the lower tail probability (risk of going below the lower spec). A process that appears capable (Cpk = 1.4 under normality) might actually produce significant upper-spec-limit exceedances.

The traditional workaround is to log-transform the data and compute Cpk on the transformed scale. But this changes the interpretation — Cpk on log-transformed data does not directly correspond to the fraction nonconforming on the original measurement scale. Engineers must back-transform to get meaningful defect rate estimates, and this step is often skipped or done incorrectly.

EntropyStat handles lognormal data without transformation. The EGDF fits the distribution in the original measurement space, capturing the right skew directly. Capability indices are computed from the actual distribution shape, so the Cpk value reflects the true fraction nonconforming without requiring log transformation or back-transformation.

This eliminates a common source of confusion and error. When engineers log-transform data for capability analysis, specification limits must also be transformed — and asymmetric specifications on the original scale become asymmetric in a different way on the log scale. The EGDF works directly on the raw measurements with the original specification limits, producing results that are immediately interpretable.

The EGDF is also more robust than lognormal fitting for data that is "approximately lognormal." Real manufacturing data rarely follows any theoretical distribution exactly. A process might be mostly lognormal but with a few measurements suggesting a second mode (perhaps from a different material lot). The EGDF captures this deviation naturally, while forcing a lognormal fit would smooth over it — potentially hiding actionable process information.