Bates Distribution

< List of Probability Distributions

The Bates Distribution (or rectangular mean distribution) resembles a normal distribution although it can also resemble several other distributions depending on how many items are in the sample.

More formally, it is the probability distribution of the mean of n independent standard uniform variates.

The distribution was is named after American mathematician Grace Bates [1] who tested the null hypothesis that a particular distribution is a uniform distribution with the alternate hypothesis that it is a truncated exponential distribution.


PDF of the Bates distribution with a = 0, b = 1.

The following formula shows the probability density function (PDF) for a Bates random variable X on the interval (0, 1), where sgn denotes the sign function.

PDF for a Bates random variable X on the interval (0, 1).

Standardized Bates Distribution

The standardized Bates distribution is found in many statistical software packages. It is characterized by [2]:

  • Mean = 0; 
  • Standard deviation = 1.
  • Sample size = 12.

An important historical use of the standardized Bates distribution was that it generated standard normal variables in computing.

Distributions similar to the Bates distribution

As noted above, when the sample size is 1, the Bates distribution is equal to the uniform distribution and for a sample size of 2 it is equal to a triangular distribution.

The Bates distribution is also similar to the Irwin-Hall distribution but while the Irwin-Hall is the distribution of the sum, the Bates is the distribution of the mean.


Image: By Shiyu Ji – Own work, CC BY-SA 4.0,

[1] 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.
[2] Kotz, S. & Dorp, J. (2004). Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. World Scientific.