Two Rectangulars Added Distribution

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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.

two rectangulars added
Entry from Index to the Distributions of Mathematical Statistics [1]

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].


[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 Mathematics54(2), 353–366.

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

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