Bates Distribution

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The Bates Distribution (or rectangular mean distribution) a probability distribution of the mean of a number of independent uniformly distributed random variables on the unit interval [1].

The Bates 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 [2] who tested the null hypothesis that a particular distribution is a uniform distribution [0, 1] with the alternate hypothesis that it is a truncated exponential distribution on [0, 1].

Properties of the Bates distribution

pdf of the bates 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):

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

where sgn denotes the sign function

Other properties include:

  • Mean = ½(a + b)
  • Variance 1/12n (b – a)2
  • Skewness = 0
  • Kurtosis = – 6/5n

Standardized Bates Distribution

The standardized Bates distribution is a single argument form [n] and equals the Bates distribution[n,{0,1}]. Found in many statistical software packages, it is characterized by [3]:

  • 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 [4].

Distributions similar to the Bates distribution

  • As noted above, when the sample size is 1, the Bates distribution is equal to the uniform distribution.
  • For a sample size (n) of 2 it is equal to a triangular distribution.
  • The PDF of a Bates distribution appears visually similar to the PDF of a normal distribution for larger values of n.
  • The Bates distribution is sometimes confused with the Irwin-Hall distribution, but while the Irwin-Hall is the distribution of the sum, the Bates is the distribution of the mean.
  • It is also closely related to the Uniform Sum Distribution, which represents the sum of statistically independent, uniformly distributed random variables — instead of their mean [5].


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

[1] Jonhson, N. L.; Kotz, S.; Balakrishnan (1995) Continuous Univariate Distributions, Volume 2, 2nd Edition, Wiley ISBN 0-471-58494-0(Section 26.9)

[2] 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.

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

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

[5] Wolfram Research (2010), BatesDistribution, Wolfram Language function, (updated 2016).

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