Home / Shared Math Details / Shared Estimators and Tests / Box-Cox Transformation

Box-Cox Transformation¶

This page defines the Box-Cox power transformation that Quantum XL uses to bring skewed positive data closer to a normal distribution before capability or control chart math runs on the transformed values.

Notation¶

Term	Description
\(x_i\)	an original (untransformed) data value; must be positive
\(y\)	a transformed value
\(\lambda\)	Box-Cox power parameter, limited to \([-5, 5]\)
\(g\)	geometric mean of the data, \(g = \left(\prod_i x_i\right)^{1/N}\)
\(W_i(\lambda)\)	scaled transform used only to select \(\lambda\)

The transformation¶

\[ y = \begin{cases} x^{\lambda} & \lambda \ne 0 \\[4pt] \ln x & \lambda = 0 \end{cases} \]

Quantum XL uses the simple power form (not the scaled form \((x^{\lambda} - 1)/\lambda\)), so the transformed values are directly comparable to transformed specification limits. The transformation requires every value to be positive, at least 3 non-missing values, and non-zero variation. \(\lambda\) is either supplied by the user or fitted automatically; in both cases it is limited to \([-5, 5]\).

Selecting lambda¶

The automatic \(\lambda\) minimizes the spread of the scaled transform

\[ W_i(\lambda) = \begin{cases} \dfrac{(x_i/g)^{\lambda} - 1}{\lambda} & \lambda \ne 0 \\[8pt] \ln(x_i/g) & \lambda = 0 \end{cases} \]

where \(g\) is the geometric mean of the data (dividing by \(g\) makes spreads comparable across \(\lambda\)). The objective depends on the data structure:

Subgrouped data: minimize the pooled within-subgroup standard deviation of \(W(\lambda)\).
Individual observations: minimize the moving-range sigma of \(W(\lambda)\).

The minimum is found with Brent's bounded minimization after a coarse scan brackets it.

Specification limits and the target¶

Specification limits and the target are transformed with the same \(y = x^{\lambda}\) (or \(\ln x\)) rule so capability is evaluated consistently in the transformed domain. A negative \(\lambda\) reverses order, so the transformed LSL and USL swap roles; Quantum XL performs this swap internally and reports results in the user's original orientation.

Used by¶

Cpk/Histogram Math Details: the Box-Cox transformation option; capability indices, PPM, and the histogram are computed on the transformed data against the transformed limits.
The variable control charts offer the same transformation: Xbar-R and Xbar-S, I-MR, I-MR-R/S and EWMA.

References¶

Box, G. E. P., and Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society: Series B (Methodological), 26(2), 211-252.
Brent, R. P. (1973). Algorithms for Minimization Without Derivatives. Prentice-Hall.
Sakia, R. M. (1992). The Box-Cox transformation technique: A review. Journal of the Royal Statistical Society: Series D (The Statistician), 41(2), 169-178.
Rodriguez, R. N. (1992). Recent developments in process capability analysis. Journal of Quality Technology, 24(4), 176-187.