< 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.

- Triangular distribution: n = 2.
- Uniform distribution: n = 1.
- Unimodal (normal) distribution: n > 2.

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.

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.

## 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.

## References

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

[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.