# Bates Distribution

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 .

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  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

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 :

• 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.
• 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 .

## References

Image: By Shiyu Ji – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=59354219

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

 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.

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

 Wolfram Research (2010), BatesDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BatesDistribution.html (updated 2016).

Scroll to Top