Control

Control Chart (Box-Cox transformed)

I-MR with Box-Cox transform for non-normal data

Box-Cox Transformed I-MR

Classic Shewhart charts assume approximate normality. For right-skewed data (lifetime, cost, defect rate) the limits become biased: too many false alarms above the mean, too few below. This module brings the data onto a normal scale via Box-Cox, computes I-MR there, and back-transforms the limits to the original scale for display.

Box-Cox transform: y = (x^λ − 1) / λ for λ ≠ 0, y = ln(x) for λ = 0. λ is found via Anderson-Darling optimisation (grid [-3, 3], step 0.05) or set manually.

Chart and limits: The chart plots on the transformed scale — there UCL/LCL and zones are statistically meaningful. A table shows limits on both scales so operators can read action limits in the original units.

Prerequisite: all values must be strictly positive. For zero or negative values either shift (x + |min| + 1) or use a different transform family (Yeo-Johnson, Log+1).

Pitfalls

Auto-λ without sanity check: The Anderson-Darling search always finds a minimum — even when the data cannot be normalised by Box-Cox at all. Inspect the distribution first (histogram, P-P plot).

Back-transformed limits are asymmetric: Re-projected limits are NOT symmetric about the centre line — that is correct: right-skewed originals leave more room above than below. Anyone expecting symmetry has misread the distribution.

λ drifts with new data: In live mode λ adapts to new data. For Phase-II monitoring freeze λ (manual mode with the Phase-I λ).

Examples

This module ships with the following example datasets — load any of them in the app with a single click.

Box-Cox I-MR chart: lognormal lifetimes — Project template

50 lognormally distributed lifetimes. A plain I-MR chart raises false alarms because of the right skew. After Box-Cox transformation (λ ≈ 0) the residuals become approximately normal.

Available in the following cycles

DMAIC: Control
DMADV: Verify
8D: D6 — Implementation