On an Extension of the Concept of Moment with Applications to Measures of Variability, General Similarity, and Overlapping
where Wj denotes the weight corresponding to the particular origin Xj, and the sunmnation is over a field 4. In particular, if 4 is the set of all values assumed in D by the variate Xi, and if Wj = F1, we shall call the quantity the rth complete total moment of D. If, on the contrary, Wj is the frequency F' of the value X} in a second frequency distribution D': [X, F1 and 4' is the set of all values assumed by the variate X} in D', PMr will be called the rth aggregate moment of D and D'. A modification of this procedure leads to what we shall call the moment of transvariation of D and D'. The consideration of complete moments draws attention to certain previously known measures of variability which are independent of the origin selected, and also provides simple methods of computation which are useful for data given in the form of a frequency distribution. The investigation of aggregate moments and moments of transvariation gives rise to certain measures of general similarity betAveen two distriblutions, as well as measures of the amount of overlapping.