Irwin–Hall Distribution.

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The Irwin-Hall distribution, also known as the Uniform Sum Distribution, is a powerful mathematical tool with many practical applications. Named after proofs provided by Irwin and Hall in 1927 [1,2], it helps to determine sums of random variables within problems such as statistics or probability distributions. As one of its most useful features, this type of distribution offers simplicity for mathematicians tackling various issues across disciplines like data science and finance.

A related distribution is the Bates distribution — which is the distribution of mean values of n — instead of the sum. The two rectangulars added distribution is connected to the Irwin-Hall distribution, but its exact properties remain a mystery.

Irwin-Hall distribution properties

The Irwin-Hall distribution is the distribution of the sum of values taken from the uniform distribution on the interval (0, 1).

Probability density function (PDF):

irwin-hall distribution PDF

Mean: n/2

Variance: n/12

Kurtosis: ≈ 3 [3]

Or, more precisely: 3 – (6/5n)

Irwin-Hall distribution usage

Many problems in applied mathematics require the calculation of the distribution of the sum of independent random uniform variables, including aggregating scaled values, change point analysis and working with data drawn from measurements with different precision levels.

The Irwin-Hall distribution is also used to approximate a normal distribution and demonstrate the central limit theorem: as n increases, the distribution will rapidly start to look like a normal distribution.

As n increases, the distribution starts to look like the normal distribution. Credit: Thomasda: Wikimedia Commons

The generation of pseudo-random numbers  having an approximately normal distribution is sometimes accomplished by computing the sum of a number of pseudo-random numbers having a uniform distribution; usually for the sake of simplicity of programming. Rescaling the Irwin–Hall distribution provides the exact distribution of the random variates being generated.


Image: Thomasda | Wikimedia Commons CC BY-SA 3.0 File:Irwin-hall-pdf.svg

[1] Hall, P. (1927). The distribution of means for samples of size n drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika, Vol. 19, No. 3/4. pp. 240–245.

[2] Irwin J.(1927). 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.

[3] Marengo, J et al. (2017). A Geometric Derivation of the Irwin-Hall Distribution. International Journal of Mathematics and Mathematical Sciences.

Two rectangulars added distribution

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 Irwin-Hall distribution, which is continuous.

Connection to the Irwin-Hall distribution

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 Irwin-Hall distribution. 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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