Modified Normal Distribution

< Probability Distributions List

The term modified normal distribution (or equinormal distribution) may refer to any number of distributions that are similar in appearance to a normal distribution (in other words, it’s simply a normal distribution modified in some way).

For example, the t distribution is one example of a modified normal distribution [1].

modified normal distribution
The t-distribution is an example of a modified normal distribution.

The term can also refer to normal variance mixture distributions described by Romanowski [2] that modify the normal distribution to fit the variation seen in real life data sets. Romanowski called these modulated normal distributions and — as they are particular instances of normal variance mixtures — can better be described as mixture distributions.

The actual term “modified normal” is usually used in a loose sense to describe the non-normal behavior of a particular distribution, rather than to describe a specific distribution. That’s because most “non normal” distributions have their own names. For example, positively skewed, unimodal distributions might better fit a gamma distribution or lognormal distribution.

Other types of modified normal distribution

A zero-modified normal distribution is a normal distribution modified to put extra probability mass at 0. In other words, it’s a kind of mixture distribution where part of the population comes from a normal distribution but the rest of the population is all zeros [3].

An exponentially modified normal distribution is a three-parameter distribution that is a generalization of the normal distribution for skewed cases.


[1] Stats Means Business: Statistics and Business Analytics for Business, Hospitality and Tourism

[2] Romanowski, M., Green, E. Tabulation of the modified normal distribution functions. Bull. Geodesique 78, 369–377 (1965).

[3[ Environmental Statistics with S-PLUS p.192