Showing papers by "Maurice G. Kendall published in 1939"

The Distribution of Spearman's Coefficient of Rank Correlation in a Universe in which all Rankings Occur an Equal Number of Times:

Abstract: 3. Certain simple properties of this distribution are obtainable immediately. (a) Any value of S(d2) must be even. For S(d) = 0, being the difference of the sums of the first n natural numbers; hence the number of odd values of d is even, and so is the number of odd values of d2. (b) The possible values of S(d2) range from 0 to 4(n3 -n) and hence there are 1 (n3-n) + I of them. (c) The distribution is symmetrical, about a central value if i,(n3-n) is even, or about two adjacent central values if J(n3 n) is odd. This follows from the fact that to any given value ofp corresponding to a permutation P there will correspond a negative value of p of the same absolute value arising from P inverted. For, if the permutation P is X1, X2, ..., Xn the inverted permutation is Xn) Xn) ...,) X,1. S(d2) calculated from P is then S(Xi-i)2 and that from P inverted is S(X, n + 1 + i)2. The sum of these two is S(X2) + S(i2) 2S(Xi i) + S(X;) + S(n + i)2 -2S{Xi(n + 1i)}.