Transformations to Normality Using Fractional Powers of the Variable

IN MANY forms of analysis of nonnormal variables we seek some transformation of the variate involved in order to make it approximately a normal variable. This enables us to utilize the normal distribution, which has been exhaustively investigated and tabulated, to obtain percentage points, ordinates, and so on for our original variable. If this is not done we may find it necessary to perform much laborious work in the tabulation of the variate itself, and functions of it, perhaps with only limited uses in view for the completed work. A second reason for a transformation is that sometimes the tables of the exact distribution are not available and it behooves us to find a quick and simple test. As an example of the latter consider the standard deviation, s = 2 (x-) 2/n, in a sample of n drawn from a normal population with standard deviation o,. We know that ns2/o-2 is exactly distributed as a x2 distribution but as an approximation we can assume s to be normally distributed with mean a. [1(1/4n) ] and variance c2/2n thus enabling us, at any rate for n larger than about 15, to utilize the normal distribution for a test of significance. Of course, in many cases, where observations fall into a number of groups, it is more important to equalize the variance of these groups than it is to bring the variate into normality. Thus in the analysis of variance, although both items are desirable, it is of more importance, except possibly for a one-way classification, to stabilize the variance otherwise it would never be known if one effect was really significant or not. This present paper examines the properties of a certain class of transformations under the assumption that they are designed to transform the variate, X, into some form of normal distribution. The class of transformat-ions considered is that of