On finite sequences of real numbers

Abstract The material of the science of statistics consists in the main of finite sets of numbers, which, since they are recorded in order, are sequences. They may be regarded as functions of the rank of their elements, and theorems concerning real functions hold good for them. In particular they are capable of partition, orthogonal or other, of development analogous to expansion in series and of transformation; sets of such sequences may likewise be subjected to substitution. These considerations bring certain parts of the theory of statistics into line with general mathematical theory and suggest interesting possibilities. The so-called theory of “ factors,” put forward by Spearman in connection with his researches into intelligence, has become very important in recent years. It is an outgrowth of the theory of partial correlation. Mathematically speaking the “factors” of a sequence are components resulting from orthogonal partition, orthogonal sequences being those which have zero correlation. In considering how far the “factors” or other components of a sequence of measurements represent physical realities, the statistician has to make a careful study of questions of distribution and sampling, but from what has been said it will be seen that the general theory of the partition and development of sequences, which is the subject of this paper, does not involve these questions. A brief discussion of the application of this method was given in a letter in ‘Nature’ in which the following results were indicated.