---
title: "Box–Cox distribution"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Box–Cox_distribution"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:21:41.144483+00:00"
instance: "kb-cron"
---

In statistics, the Box–Cox distribution (also known as the power-normal distribution) is the distribution of a random variable X for which the Box–Cox transformation on X follows a truncated normal distribution. It is a continuous probability distribution having probability density function (pdf) given by

  
    
      
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    {\displaystyle f(y)={\frac {1}{\left[1-I(f<0)-\operatorname {sgn}(f)\Phi (0,m,{\sqrt {s}})\right]{\sqrt {2\pi s^{2}}}}}\exp \left\{-{\frac {1}{2s^{2}}}\left({\frac {y^{f}}{f}}-m\right)^{2}\right\}}
  

for y > 0, where m is the location parameter of the distribution, s is the dispersion, ƒ is the family parameter, I is the indicator function, Φ is the cumulative distribution function of the standard normal distribution, and sgn is the sign function.


== Special cases ==
ƒ = 1 gives a truncated normal distribution.


== References ==
Freeman, Jade; Reza Modarres. "Properties of the Power-Normal Distribution" (PDF). U.S. Environmental Protection Agency.