An Asymmetric Probability Function

In engineering problems that require statistical analysis, such as studies of rainfall and run-off, the data are frequently too meager to allow the use of the elaborate methods developed by Karl Pearson, or the Danish statisticians; and yet, the asymmetries in the frequency distributions associated with these problems are usually sufficiently marked to place the investigation in a field definitely outside that of the simple Gaussian theory. Unfortunately, where the basic probability function is other than the “normal curve”, the determination of the constants involved, as well as the graduation of the data, generally becomes a difficult task. Furthermore, unless the data are very complete, the final results will seldom have meaning because of the magnitude of the probable errors of the parameters involved.

The purpose of this paper is to introduce a function that differs as little as it seems possible to let it differ from the “normal” in its general characteristics, while allowing it an unlimited degree of “skewness”. At the same time, it is an easy curve to apply, as such things go, and one for the application of which existing tables may be used. The paper is subdivided into the follow in g parts:

Section I contains a critical discussion of the various methods in use at present in the analysis of frequency distributions. The purpose of this section is, first of all, to contrast and to compare the function introduced with those already in use so that both its merits and defects will be exposed at the beginning. It will thus be seen to contain a justification for the introduction of a new function in a field that literally teems with such studies. It is hoped that this paper will also help the engineer to a better understanding of a subject the approach to which is barricaded by so many mathematical hurdles.

The requirements of a function that is to be one degree more general than the Gaussian, are discussed in Section II. This function is introduced in Section II and is subjected to detailed mathematical analysis, and the constants of the curve are expressed in terms of the moments of the distribution which it is to fit. Finally, it is proved that the curve is a true generalization of the Gaussian, becoming identical with it, in fact, when the skewness parameter vanishes.

In Section III the formulas derived in Section II are collected and a procedure for their use is outlined. A number of examples are given which serve both to illustrate the manipulation of the new curve and to compare the results obtained by its use with those obtained by the use of other functions.

In Section IV, finally, the most general homograde function is discussed. Its parameters are expressed in terms of the bounds and standard deviation of the statistics, and a procedure for its application is outlined.