The statisticians George Box and David Cox developed a procedure to identify an appropriate exponent (Lambda = l) to use to transform data into a “normal shape.” The Lambda value indicates the power to which all data should be raised. In order to do this, the Box-Cox power transformation searches from Lambda = -5 to Lamba = +5 until the best value is found. Table 1 shows some common Box-Cox transformations, where Y’ is the transformation of the original data Y. Note that for Lambda = 0, the transformation is NOT Y (because this would be 1 for every value) but instead the logarithm of Y.
Does Box-Cox Always Work?
The Box-Cox power transformation is not a guarantee for normality. This is because it actually does not really check for normality; the method checks for the smallest standard deviation. The assumption is that among all transformations with Lambda values between -5 and +5, transformed data has the highest likelihood – but not a guarantee – to be normally distributed when standard deviation is the smallest. Therefore, it is absolutely necessary to always check the transformed data for normality using a probability plot.
Additionally, the Box-Cox Power transformation only works if all the data is positive and greater than 0. This, however, can usually be achieved easily by adding a constant (c) to all data such that it all becomes positive before it is transformed. The transformation equation is then:
Y’ = (Y+C)l