On Bravais' Formula in the case of Skew Correlation. 477 



" On the Significance of Bravais' Formulae for Regression, &c., 

 in the case of Skew Correlation." By G. UDXY YULE. 

 Communicated by Professor KARL PEARSON, F.R.S. 

 Received December 14, 1896, Read February 18, 1897. 



The only theory of correlation at present available for practical 

 use is based on the normal law of frequency, but, unfortunately, this 

 law is not valid in a great many cases which are both common and 

 important. It does not hold good, to take examples from biology, 

 for statistics of fertility in man, for measurements on flowers, or for 

 weight measurements even on adults. In economic statistics, on the 

 other hand, normal distributions appear to be highly exceptional : 

 variation of wages, prices, valuations, pauperism, and so forth, are 

 always skew. In cases like these we have at present no means of 

 measuring the correlation by one or more " correlation coefficients " 

 such as are afforded by the normal theory. 



It seems worth while noting, under these circumstances, that in 

 ordinary practice statisticians never concern themselves with the 

 form of the correlation, normal or otherwise, but yet obtain results of 

 interest though always lacking in numerical exactness and fre- 

 quently in certainty. Suppose the case to be one in which two 

 variables are varying together in time, curves are drawn exhibiting 

 the history of the two. If these two curves appear, generally 

 speaking, to rise and fall together, the variables are held to be corre- 

 lated. If on the other hand it is not a case of variation with time, 

 the associated pairs may be tabulated in order according to the 

 magnitude of one variable, and then it may be seen whether the 

 entries of the other variable also occur in order. Both methods are 

 of course very rough, and will only indicate very close correlation, 

 but they contain, it seems to me, the point of prime importance at 

 all events with regard to economic statistics. In all the classical 

 examples of statistical correlation (e.g., marriage-rate and imports, 

 corn prices and vagrancy, out-relief and wages) we are only 

 primarily concerned with the question is a large x usually associated 

 with a large y (or small y) ; the further question as to the form of 

 this association and the relative frequency of different pairs of the 

 variables is, at any rate on a first investigation, of comparatively 

 secondary importance. 



Let Ox, Oy be the axes of a three dimensional frequency-surface 

 drawn through the mean of the surface parallel to the axes of 

 measurement, and let the points marked (x) be the means of succes- 

 sive ^-arrays, lying on some curve that may be called the curve of 

 regression of x on y. Now let a line, RR, be fitted to this curve, 



