unless the coefficients of variation be constant for local races, it is impossible th at the coefficients of correlation can be constant for indices. In other words, the hypothesis of the constancy for local races of correlation, and th at of the constancy for local races of variation, stand on exactly the same footing.
The conclusions of this paper although applied to organic correlation are equally valid so far as concerns the use of indices in judging the correlation of either physical or economic phenomena. I t was, indeed, a difficulty arising from my discussion of personal judgments —a spurious correlation between the judgments of different observers —which first drew my attention to the m atter.
Note, January 13, 1897.—The result described by Professor Pearson evidently affects the value of the correlation coefficients determined by me in Crangon and Carcinus ( ‘ Roy. Soc. Proc.,’ vols. 51 and 54), because I have always expressed the size of the organs measured in terms of body length.
In order to show the effect of this, I have lately performed, at Professor Pearson’s suggestion, the following experim ent: It happens th at my measures of Plymouth shrimps are recorded in a book, in the order in which they were measured, and therefore at random as regards carapace length or other characters. I constructed from these records 420 “ spurious” shrimps, in the following way: the total length of the first shrimp in the book was associated with the carapace length of the tenth shrimp and the “ post-spinous len g th ” of the twentieth, and so throughout. Evidently these three measures were associated at random, and we might expect that these spurious shrimps would show no organic correlation; but when the carapace lengths and “ post-spinous lengths ” of these spurious shrimps were divided by the body length, and the correlation between the resulting indices was determined, the value of was found to be 0'38, the value for real shrimps being 0-81, or the correlation due to the use of indices forms 47 per cent, of the observed value. W. E. R. W eldon.
I send this note to serve as a kind of appendix to the memoir of Professor K. Pearson, believing that it may be useful in enabling others to realise the genesis of spurious correlation. It is important though rather difficult to do so, because the results arrived at in the memoir, which are of serious interest to practical statisticians, have at first sight a somewhat paradoxical appearance.
