The Use of Probability Paper for the Graphical Analysis of Polymodal Frequency Distributions

Abstract

The mathematical analysis of bimodal distributions is very complex. Karl Pearson (1894) investigated the problem and developed equations for the purpose; but found them unsolvable as the ‘majority [of the relations] lead to exponential equations the solution of which seems more beyond the wit of man than that of a numerical equation even of the ninth order’. He did indeed evolve an equation of this order and used it to analyse a few bimodal distributions, but the arithmetic involved was very laborious. Later he (Pearson, 1914) gives a table for ‘Constants of normal curve from moments of tail about stump ’which, as he describes in the introduction, occasionally permits a rough analysis of a distribution which is known to be bimodal. This method is much more rapid than the solution of the nonic equation, but ‘owing to the paucity of material in tails and corresponding irregularity there will be large probable errors’. Gottschalk (1948) discusses the question and shows that inthe special case where the bimodal distribution is symmetrical comparatively simple solutions can be found.