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. Definition The power transformation is defined as a continuously varying function, with respect to the power parameter λ, in a piece-wise function form that makes it continuous at the point of singularity (λ = 0). For data vectors (y1,..., yn) in which each yi > 0, the power transform is where is the geometric mean of the observations y1, ..., yn. The case for is the limit as approaches 0. To see this, note that . Then , and everything but becomes negligible for 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 , 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: This allows the normal log likelihood at its maximum to be written as follows: From here, absorbing into the expression for produces an expression that establishes that minimizing the sum of squares of residuals from is equivalent to maximizing the sum of the normal log likelihood of deviations from and the log of the Jacobian of the transformation. The value at Y = 1 for any λ is 0, and the derivative with respect to Y there is 1 for any λ. Sometimes Y is a version of some other variable scaled to give Y = 1 at some sort of average value. The transformation is a power transformation, but done in such a way as to make it continuous with the parameter λ at λ = 0. It has proved popular in regression analysis, including econometrics. Box and Cox also proposed a more general form of the transformation that incorporates a shift parameter. which holds if yi + α