398 
Francis Galton's 
These will give the corrective terms in the brackets close enough, even if n be as small as 10. 
The terms outside the brackets will need determining by (xxvii) instead of (xxvi) if n be less 
than 30, say. We see that (xxx) gives us the average difference between the mediocre individuals 
and (xxxi) the difference between two individuals at the quartile. Roughly the differences in the 
two cases are as 5 to 6. But if we compare the extreme individuals' difference for n = 100, we 
have 
Xi = -3611xs, X2="2020xs, X25= '0315 x s, X50="0251xs. 
Thus the interval between extreme individuals is more than ten times the interval between 
mediocre individuals. 
Now, of course, the normal distribution in a general sort of way indicates that the differences 
between modal, or what the biologists term ' normal,' individuals are very small. But Mr Galton's 
difference problem enables us for the first time to quantitatively appreciate how much wider the 
differences are between the extreme (or biologists' 'abnormal' individuals) and modal (or normal) 
individuals. Now the range of a distribution being somewhat about Qs, we see that extreme 
individuals may be sej^arated by as much as of the range, while modal individuals have only a 
difference of of the range, and even individuals at the quartile only a difference of T^^^tYi of 
the range. 
It is not possible to pass over the general bearing of such results on human relations. If we 
define ' individuality ' as difference in character between a man and his immediate compeers, we 
see how immensely individuality is emphasised as we pass from the average or modal individuals 
to the exceptional man. Differences in ability, in power to create, to discover, to rule men, do 
not go by uniform stages. We know this by experience, but we see it here as a direct conse- 
quence of statistical theory, flowing from a characteristic and familiar chance distribution. We 
ought not to be surprised, as we frequently are, at the results of competitive examination, where 
the difference in marks between the first men is so much greater than occurs between men 
towards the middle of the list. In the same way the individuality of imbeciles and criminals at 
the other end of the intellectual and moral scales receives its due statistical appreciation. 
We stand in a better position to judge the pathological from the merely exceptional, mere 
isolation no longer leads us to doubt the position of au extreme outlying error, observation or 
individual*. 
In short Galton's difference problem leads us to look upon samples of populations and even 
on populations themselves, no longer as arrays of continuously varying individuals, but as 
systems of discrete units. We see discontinuity in every sample and in every population. We 
obtain a new and most valuable conception of a normal or standard population. It is one in 
which each individual is separated from his immediate neighbovu's, when the whole is arranged 
according to any character, by definite calculable intervals. These intervals are, of course, the 
average intervals which would be found by taking the mean of many such samples or populations, 
but they are none the less of extreme suggestiveness. Just as the continuous representation by a 
frequency curve is only an ideal representation of the observed facts, so we now reach an ideal 
representation of the actual discontimdty in the given population. As in the case of many 
physical investigations, so we find in statistical theory both continuous and discontinuous repre- 
sentations of the phenomena equally important and equally valid within the legitimate limits of 
interpretation. 
(8) As a last illustration I propose to investigate the value of x when n = % and p = \. We 
easily find : 
^ ^ 1 
m = 0, y,„=— =, 
V27r 
* I propose on another occasion to consider the applicatioa of Galton's problem to a new theory for 
the rejection of outlying individuals. 
