108 On, the Probable Error of the Correlation Coefficient 
And hence from (x) going now to third approximations, 
1 - l/\ 
r = r 
3/\ 
1 - 
i-p» (i-^)(i-V) 
1-p 2 _ (l-p 2 ) 9p ; 
3(1~P 2 ) ( 1-p 2 ) 27^ -* 
»' 2n' 2 J 
P 
l , 3(l-p 2 ) , (41 + 23p 2 )(l-p 2 )) 
The excess of f over true p is zero if p = 0 and if p = 1, but if m is small and p 
fairly large the excess may be such as to make the modal value unity or greater 
than unity. If for instauce n is so small as 4, n being thus 3, the above 
approximate equation gives 
r = p + hp (i - p 2 ) + (Hp + Hp 1 ) ( i - p 2 ) 
or r = 2-069p - -750p 3 - 319p s 
= "93 when p = "5 
= 1-05 „ p = "6 
= 1-14 „ p = -7. 
The frequency distribution in the last two cases is of the J type, there being 
no mode within the range. The greatest frequency is at the extremity of the range, 
or at value unity. The interpretation of this result is clearly that such small 
samples as 3, 4 or 5, as might be expected, fail altogether to give by the product 
moment formula an approximation to the correlation coefficient. Under some 
circumstances the points which graphically represent the observed measures are 
more likely to be in a line than to have a configuration represented by any 
specified fractional correlation coefficient. This will happen if the correlation in 
the material has a larger coefficient than "6 (approx.) when samples of four are 
drawn : or a larger coefficient than "3 (approx.) when samples of three are drawn. If 
samples of two are drawn the coefficient of correlation is necessarily unity in the 
sample whatever it may be in the material*. All the distribution is concentrated 
at value unity and r should in this case be infinite for all values of p. Our 
approximation, neglecting terms in 1/n' 3 etc., cannot of course show this if n' = l. 
It gives r greater than unity and so a J type, but fails to show the complete 
concentration at unity. 
It appears from (x) that r will be infinite when X = 3 and r any value other 
than zero, whilst r will be zero if f is zero and X other than 3. If r, and therefore 
* Supposing the material ungrouped. If it is grouped some values will be indeterminate in small 
samples, viz. when all observations fall into the same group. 
