Semi-Triangular Distribution

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semi-triangular distribution
Graph of semi-triangular distribution for a = 3 (red) compared the the “full” triangular distribution (extended with dashed red line).

Haight’s 1958 Index to the Distributions of Mathematical Statistics [1] lists the following formula for the semi-triangular distribution (p. 106):

For 0 ≤ xa [2].

With grouping corrections.

The formula originated from an article by Kupperman in an article titled
On Exact Grouping Corrections to Moments and Cumulants 
(pp. 429-434). Kupperman considered the effect of grouping on the mean and variance of a new twist on the rectangular distribution that he dubbed the “semi-triangular distribution”. This new distribution has a frequency curve shaped like the right half of the “regular” triangular distribution’s frequency curve.  

Kupperman gave the following properties for the semi-triangular distribution:

  • Mean =  (1/3) a
  • Variance = (1/18)a2.
  • Second moment about the origin = (1/6) a2.

The Rarity of the Semi-Triangular Distribution

Following Kupperman’s publication in Biometrika, the distribution makes a few sparse entries in the literature afterwards. Entries tend to be of a bibliographical nature rather than a discussion of the formula or properties. For example, Kupperman’s article is mentioned in another publication by the National Bureau of Standards in 1970.

Kupperman noted that grouping the range into a number of equal intervals “overstates the mean and understates the variance”, which may be one reason why the distribution never took off in a practical sense. Another reason may be that the “full” triangular distribution, from which the semi-triangular distribution was developed, has limited practical use—it is usually used when little information is known about outcomes except the most likely one (this creates the peak in the center of the distribution).   


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

[2] Kupperman, M. On Exact Grouping Corrections to Moments and Cumulants. Biometrika 39, (pp. 429-434).

[3] National Bureau of Standards. An Author and Permuted Title Index to Selected Statistical Journals

1970). Special Publication 321. Online: