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 = = (e y) 
MPQ 
Where « and y are the deviations from the averages of 
the series [and therefore > (# 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. 
l=? 
The probable error = 0°6745 x (7>~ where r = coefficient 
of correlation and « = the number of items in the 
series. 
*King:; Elements of Statistics, p, 213. 
