THE CORRELATIONS OP AREAS OF MATURED CROP AND THE RAINFALL. 357 



places themselves in the case of the rainfall of April to September, and is quite 

 comparable with it for the rainfall of October to March. That the ratio of the root- 

 mean-square of the differences to the standard deviation is much larger for the 

 summer than for the winter rains appears to indicate that a smaller error will be 

 made in taking the rainfall measured at one point as a measure of the true rainfall at 

 another neighbouring point in the latter case than in the former. 



The four places above have been entered in the table in the order of their near- 

 ness to Sialkot, the direct distances from Sialkot being approximately, Daska 16 

 miles, Pasrur 18 miles, Zafarwal 28 miles, and Raya 36 miles. 



It does not appear that within the limits of these distances that the root-mean- 

 square of the differences varies much with the distance. No doubt it would diminish 

 if the rainfall were measured at a near point to Sialkot, but its variation within 

 the limits of 16 to 36 miles appears not to be sensible, at least so far as 

 the above results go. The matter is interesting, but cannot be pursued further 

 here. 



So far, then, it might seem quite beside the mark to attempt to correlate 

 the growth of crops in one neighbourhood with the rainfall at some point whose 

 distance from that neighbourhood is of the order of the above distances. But the 

 high correlations existing between the rainfall at the places named shows that the 

 expectation of some such correlation is not an idle one. 



The correlations between the rainfalls at the above places can be very simply de- 

 duced from the data above. 



For let x l} x. 2 be the rainfalls at two points, and put &=x x -x 2 . 



Then a 2 = Xi—2x 1 x 2 -\-x 2 % . 



Whence by summation for each of the years in which the rainfall is measured, 



2(a 2 ) = 2(% 2 )-22(a; 1 % 2 ) + 2(a; 2 2 ) 

 and dividing by the number of years 



ff ' 2 A = °x? ~ 2° x x ° Xl fxw+Vxi+ix, - X 2 f 



where " denotes the standard deviation, x the mean, and r the correlation between 

 x 1} x 2 . 



Vx\X-2 ~ 



= \ °**i +° i x i - ct2 a + A* I where a = x, - x 2 . 



In the present instances we know <r Xl) <r Xl) <r A) £ and the correlations follow 

 at once. The correlations are with the Sialkot rainfall. 



