As shown by Hill (1) there is a simple method of determining Kendall’s Rank correlation coefficient, z, by drawing lines between corresponding points in the two rankings. z is then given by:

where c = number of crossings

The drawings aid the understanding of the correlation coefficient. When there are no crossings, all lines being parallel (Fig. la) c = 0 and t = 1, i.e. there is complete positive correlation. When the number of crossings is half of the maximum possible (Fig. 1b) t = 0, i.e. no correlation. When the number of crossings is equal to the maximum possible, there is complete reversal of the order (Fig. 1c) and t = — 1, i.e. there is a complete negative correlation, a result which is as significant as complete positive correlation. Fig. 1d has been drawn to facilitate the counting of the number of crossings implied in Fig. 1c. (Note that for half of all values of n it is not possible to obtain t = 0 exactly because (n/2)(n — 1) is odd and 2c is even.)

Windscale, Cumbria
 
 

Reference

1. Hill, I. D. (1974). Association Football and Statistical Inference, Applied Statistics 23, No. 2, p. 203.

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