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The “**Two rectangulars added distribution**” seems to be lost to history, although it is likely connected to the Irwin-Hall distribution.

## History of Two Rectangulars Added Distribution

The entry for “two rectangulars Added” in Haight’s 1958 *Index to the Distributions of Mathematical Statistics* [1] refers to **two rectangular distributions added together.**

The notation refers to an article published in the journal *Metron *in 1930 by British statistician Joseph Oscar Irwin titled “*On the Frequency distributions of means, etc.” * [2] in which **Irwin gave a distribution of arithmetic means of samples of size n from a rectangular universe** [3].

Three years earlier, Irwin had published “*On the Frequency Distribution of the Means of Samples from a Population Having any Law of Frequency with Finite Moments, with Special Reference to Pearson’s Type II,” *which led to the development of the Irwin-Hall distribution (IHD), which is the sum of *n* independent random variables uniformly distributed from 0 to 1. According to Craig [3], Irwin extended his method of integral equations to samples from Pearson Type I and VII curves. As integrals are mentioned, **it’s possible “two rectangulars added” may be related to the IHD**, which is continuous.

## The Issue with Metron, Volume 8

However, as Volume 8 of *Metron *isn’t anywhere to be found (except perhaps, in an uncatalogued basement in Rome), **the original formula isn’t available**. Therefore, it’s impossible to say for sure that the “two rectangulars added” is another name for the IHD. The fact that [1] also contains a separate entry for the IHD suggests that they are different distributions.

If anyone has access to a copy of Volume 8 of Metron, please let me know.

Irwin is well-known for **other contributions to mathematics**. For example, he independently developed an exact probability test for 2×2 contingency tables which we now call Fisher’s exact probability test [5].

## References

[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[2] Irwin, J.O. (1930). On the Frequency distributions of means, etc. Metron Vol 8, issue 3, pp 58-105.

[3] Craig, A. T. (1932). On the Distributions of Certain Statistics. *American Journal of Mathematics*, *54*(2), 353–366. https://doi.org/10.2307/2371000

[4] Berry, K. et al. (2014). A Chronicle of Permutation Statistical Methods. Springer.

## 2 responses to “Two Rectangulars Added Distribution”

You say “seems” twice

Not any more! Thanks.