In statistics, a power transform is a family of functions applied to create a monotonic transformation of data using power functions. It is a data transformation technique used to stabilize variance, make the data more normal distribution-like, improve the validity of measures of association (such as the Pearson correlation between variables), and for other data stabilization procedures.

Power transforms are used in multiple fields, including multi-resolution and wavelet analysis, statistical data analysis, medical research, modeling of physical processes, geochemical data analysis, epidemiology and many other clinical, environmental and social research areas.

The power transformation is defined as a continuous function of power parameter λ, typically given in piece-wise form that makes it continuous at the point of singularity (λ = 0). For data vectors (y₁,..., y_n) in which each y_i > 0, the power transform is

where

is the geometric mean of the observations y₁, ..., y_n. The case for ${\displaystyle \lambda =0}$ is the limit as ${\displaystyle \lambda }$ approaches 0. To see this, note that ${\displaystyle y_{i}^{\lambda }=\exp({\lambda \ln(y_{i})})=1+\lambda \ln(y_{i})+O((\lambda \ln(y_{i}))^{2})}$ - using Taylor series. Then ${\displaystyle {\dfrac {y_{i}^{\lambda }-1}{\lambda }}=\ln(y_{i})+O(\lambda )}$, and everything but ${\displaystyle \ln(y_{i})}$ becomes negligible for ${\displaystyle \lambda }$ sufficiently small.

The inclusion of the (λ − 1)th power of the geometric mean in the denominator simplifies the scientific interpretation of any equation involving ${\displaystyle y_{i}^{(\lambda )}}$, because the units of measurement do not change as λ changes.

Box and Cox (1964) introduced the geometric mean into this transformation by first including the Jacobian of rescaled power transformation

with the likelihood. This Jacobian is as follows: