Bates Distribution: Definition

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What is the Bates Distribution?

The Bates Distribution (also called the rectangular mean distribution) is the distribution of the mean of n independent standard uniform variates (from 0 to 1).

The distribution was derived by Grace Bates (1955) for testing the null hypothesis that a particular distribution is uniform distribution, with the alternate hypothesis being that it is a truncated exponential distribution.

Shape of the Distribution

bates distribution
PDF for the Bates distribution.

The distribution can resemble a bell curve; although the shape of the PDF is heavily dependent on how many items are in the sample— the shape can be triangular (for n = 2), uniform (for n = 1) or unimodal (for n > 2).

Standardized Bates Distribution

A standardized Bates distribution is commonly found in statistical software packages and has the following characteristics: mean = 0; standard deviation = 1; sample size = 12 (Kotz & Dorp, 2004).
An important historical use of the standardized Bates distribution was that it generated standard normal variables in computing.

Similar Distributions

For n = 1, the Bates is an exact match for a uniform distribution. For n = 2, it matches a triangular distribution.

The Bates distribution is similar to the Irwin–Hall distribution. The difference is that while the Bates is the distribution of the mean, the Irwin-Hall is the distribution of the sum.


Bates,G.E. (1955) “Joint distributions of time intervals for the occurrence of successive accidents in a generalized Polya urn scheme”, Annals of Mathematical Statistics, 26, 705–720.
Kotz, S. & Dorp, J. (2004). Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. World Scientific.
Wolfram. Bates Distribution. Retrieved July 9, 2019 from:

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