372 MR. S. M. JACOB ON 



(/?) Rabi harvest. 



For the regression of crop on rain 



C = 2600*9 R-f 28203 



For the regression of rain on crop 



C= 49967 R+3215. 



E c = +2630. 



I will content myself by giving the diagram showing the distribution of 

 rain and crop for the Rabi harvest in Tahsil Sialkot only. This is shown below in 

 Figure 16. 



The diminution in the angle between the regression lines due to the high corre- 

 lation is seen by a comparison with the previously-given diagrams for the Rabi 

 harvest, although their point of intersection is not quite the same. 



§ 6. The prediction of isolated values of cropped areas. 



It has already been pointed out that in predicting the amount of crop from a 

 knowledge of the rainfall, what is actually predicted is the amount of crop which is, 

 if the regression be linear, the mean of the 'array' of values corresponding to the 

 given value of the rainfall. Now the assumption has been made that the regression 

 dealt with is to a first approximation both 'linear' and 'normal,' so that knowing 

 the standard deviation of the whole series of observations and the correlation, namely, 

 °c and r , the standard deviation of the ' array' corresponding to a particular value of the 

 rainfall is v c x / 1 _* t In other words, if we put * = "67449 ' then corresponding to a 

 given rainfall the amount of crop will oscillate about the true mean, and have a 

 probable excess or defect of A ^c x / 1 _ r 9 '. But we do not even know the true mean 

 exactly. The mean found from any number t n > of observations has itself a probable 



A °Ca/ * 

 deviation from the true mean of ~ . 



Where ' n ' is at all large the error produced by an incorrect estimate 

 of the mean is quite negligible as compared with the previous error, but theo- 

 retically, so long as ' n' is finite the latter error does not vanish, and in the 

 present case it produces a certain practical error which has to be taken into 

 account. It is desirable, therefore, to show how these probable errors are to be com- 

 bined, so that the whole probable error in predicting a single isolated value of the crop 

 (which is the practical desideratum) can be obtained. 



Let c denote the true and c' the actually determined regression point. Then let 

 the distribution of the c's about c be such that 



the frequency of c's contained in the interval < *- x . P 



x to x-\-*x is f(x) $x 3 where x is the distance ' • ', ' 8 l 

 from c. , ' 



*-** y/r 



/A — -* fc 



