036: On a Distribution Yielding the Error Functions of Several Well Known Statistics.

Abstract

1. THEORETICAL DISTRIBUTIONS The idea of an function is usually introduced to students in connection with experimental errors; the normal curve itself is often introduced almost as if it had been obtained experimentally, as an function. The student is not usually told that little or nothing is known about experimental errors, that it is not improbable that every instrument, and every observer, and every possible combination of the two has a different curve, or that the functions of experimental errors are only remotely related to the functions which are in practical use, because these are applied in practice not to single observations but to the means and other statistics derived from a number of observations. Many statistics tend to be normally distributed as the data from which they are calculated are increased indefinitely; and this I suggest is the genuine reason for the importance which is universally attached to the normal curve. On the other hand some of the most important statistics do not tend to a normal distribution, and in many other cases, with small samples, of the size usually available, the distribution is far from normal. In these cases tests of Significance based upon the calculation of a probable error or standard error are inadequate, and may be very misleading. In addition to tests of significance, tests of goodness of fit also require accurate functions; both types of test are constantly required in practical research work; the test of goodness of fit may be regarded as a kind of generalized test of significance, and affords an a posteriori justification of the curves employed in other tests. Historically, three distributions of importance had been evolved by students of the theory of probability before the rise of modern statistics; they are