Semi-Normal Distribution

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According to Haight (1958), the “Semi-Normal distribution” is another name for the Helmert Distribution [1], which is itself another name for the chi-square distribution. The name “semi-normal” was coined by Steffensen [2] in a 1937 article titled “On the semi-normal distribution” which appeared in volume 20 of Skandinavisk Aktuarietidskrift.

Haight’s entry for Semi-normal distribution.

Steffensen’s work found the distribution of the quotient of the semi-normally distributed random variables;  he suggested the use of the distribution of a multiple of a chi random variable (i.e., cχv) where v is “sufficiently large” [3].

Steffenson’s work on the semi-normal distribution was recognized in Journal of the Royal Statistical Society’s Recent Advances in Mathematical Statistics [4].


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

[2] Steffensen, J. F. (1937). On the semi-normal distribution. Skandinavisk Aktuarietidskrift, 20: ½, pp. 60-74.

[3] [2] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[4] Hartley, H. O. (1939). Recent Advances in Mathematical Statistics. Journal of the Royal Statistical Society102(3), 406–444.