# Box Cox Transformation

Transformations > Box Cox Transformation

## What is a Box Cox Transformation?

A Box Cox transformation is a way to transform non-normal dependent variables into a normal shape. Normality is an important assumption for many statistical techniques; if your data isn’t normal, applying a Box-Cox means that you are able to run a broader number of tests.

The Box Cox transformation is named after statisticians George Box and Sir David Roxbee Cox who collaborated on a 1964 paper and developed the technique.

## Running the Test

At the core of the Box Cox transformation is an exponent, lambda (λ), which varies from -5 to 5. All values of λ are considered and the optimal value for your data is selected; The “optimal value” is the one which results in the best approximation of a normal distribution curve. The transformation of Y has the form:

This test only works for positive data. However, Box and Cox did propose a second formula that can be used for negative y-values:

The formulae are deceptively simple. Testing all possible values by hand is unnecessarily labor intensive; most software packages will include an option for a Box Cox transformation, including:

• R: use the command boxcox(object, …).
• Minitab: click the Options box (for example, while fitting a regression model) and then click Box-Cox Transformations/Optimal λ.
 Common Box-Cox Transformations Lambda value (λ) Transformed data (Y’) -3 Y-3 = 1/Y3 -2 Y-2 = 1/Y2 -1 Y-1 = 1/Y1 -0.5 Y-0.5 = 1/(√(Y)) 0 log(Y)** 0.5 Y0.5 = √(Y) 1 Y1 = Y 2 Y2 3 Y3

**Note: the transformation for zero is log(0), otherwise all data would transform to Y0 = 1.
The transformation doesn’t always work well, so make sure you check your data after the transformation with a normal probability plot.

Reference:
Box, G. E. P. and Cox, D. R. (1964). An analysis of transformations, Journal of the Royal Statistical Society, Series B, 26, 211-252. Available online here.

------------------------------------------------------------------------------

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. If you rather get 1:1 study help, Chegg Tutors offers 30 minutes of free tutoring to new users, so you can try them out before committing to a subscription.

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.