304 WHEAT PRODUCTION IN NEW ZEALAND 



these deviations; (4) calculating the average of these squares; 

 and (5) extracting the square root of this average. The next 

 step is to find the sum of the products of the deviations from 

 the averages in each series. Finally, the standard deviations 

 are multiplied together and then multiplied by the number of 

 items in the series. This gives the divisor by which to divide 

 the sum of the products of the deviations in order to get the 

 coefficient, which always lies between + 1 and - 1. 



The above explanation may be summarised in the following 

 formula: 



Coefficient of correlation = ^ & 3/) 



npq 



Where x and y are the deviations from the averages of 

 the series [and therefore 2 (x y) is the sum of the products 

 of these deviations], p and q are the standard deviations of 

 the series, and n is the number of items in the series. 



The coefficient of correlation will be positive if the series 

 move together, e.g., with the yield of wheat and the rainfall. 

 It will be negative when the series move in the opposite 

 directions, e.g., supply and price. In the former case the 

 correlation is said to be direct, and in the latter it is inverse. 

 The nearer the coefficient approaches unity, (either positively 

 or negatively), the more evident is it that correlation does 

 exist between the two series. 



But correlation cannot be established by reference to the 

 magnitude of the coefficient alone. "If we find that two 

 variables fluctuate together in two or three different instances, 

 it by no means follows that this is a proof of the existence of 

 correlation any more than would the fact of throwing double 

 sixes with a pair of dice three times in succession prove that 

 there was any connection between the dice. Such coincidences 

 are likely to be entirely due to chance."* Thus if, of 100 

 pairs of deviations, about 50 were concurrent and the remain- 

 der divergent the element of chance would be great. But if 

 a large majority of the deviations were concurrent then it 

 would be reasonable to suppose that the element of chance is 

 small. But it is generally present to some degree, and it 

 therefore involves a probable error. The coefficient of corre- 

 lation must be considered in relation to this error. Mathe- 

 maticians have worked out the law of probable error, and it 

 is necessary here to state only the formula, which is generally 

 used. 



1-7-2 



The probable error = 0-6745 x '^~^~ where r = coefficient 



of correlation and n - the number of items in the 

 series. 



*King : Elements of Statistics, p. 213. 



