An Example of Spurious Correlation

Abstract

IF I am not mistaken, the first method of forecasting the summer season proposed by Mr. A. B. MacDowall in NATURE of September 16, 1909 (vol. lxxxi., p. 335), is based upon a spurious correlation. If we take a series of departures from normal of a meteorological element and tabulate the sums of consecutive groups of thirty, there will always be a relationship between these sums, although tiie original departures may be entirely independent, and hence the relationship between the sums cannot be utilised for forecasting an individual term of the original series. That such sums of independent departures are not independent may be seen in the following way. If we denote the original independent departures by d₁, d₂,..., and the sum ot thirty quantities beginning with d_p by s_p, the correlation coefficient between such quantities as s_p and s_p+1, as given by statistical methods, will be the mean value of a long series of products s_p s_p+1 divided by the product of the square roots of the mean values of s²_p and of s²_p+1 Now as d_p, d_q are independent the mean value of the product d_p, d_q will be zero; and it is easily seen that the correlation coefficient in question is the mean value of (d²_p+1 + d²_p+2 + + d²_p+29) divided by the product of the square roots of the mean values of (d²_p+...+a² _p+29) and of (d²_p+1 + + d²_p+30); if we denote the mean value of d²_q by m², this becomes 29m/30m², or 29/30. Thus the thirty-year sums of independent annual departures will tend to vary closely together, and the dots in a diagram like that of p. 335 would tend to lie on a straight line.