144 Pigmentation Survey of School Children in Scotland 
concerned, it would be a homogeneous population. Heterogeneity must be sought 
for in other characters. If all the physical characters showed homogeneity then it 
would be clear that one had a common race to deal with. But if, with respect to 
the character X, the observed and theoretical class frequencies appeared to differ 
significantly, then the population would not be evenly distributed with respect to 
X. Instead, there would be excess frequencies in some classes and frequencies 
falling quite short of theory {i.e. the proportional even distribution) in others in 
various localities or groups. One would then have to ascertain whether the 
significant differences were racial or due to other influences. The question now 
is : How can one determine whether any difference between observation and 
theory is significant or not ? In other words, if ?//' = observed class frequency, 
how can one measure the significance of 3//' — 2// ? Pearson* has pointed out 
that the distribution of such differences as ys' — ys, if occurring at random, takes 
the foi'm of the hypergeometrical series 
pN{pN-l)...{pN-n + l) ( qN 
N {N-l)...{N-n + \) \ pN-n+\ 
w(n-l) qNigN-l) 
1 . 2 {pN-n+\){pN-n + 2) ' 
and he has shown that the standard deviation of the distribution is given by 
The areas on either side of the ordinate which divides the distribution at the 
abscissal value (y/' — ys')l'^npq {N — n)/(N — 1), are proportional to the probabilities 
of greater or lesser values than the particular value found occurring in future 
samples. The areas can be determined when the form of curve is known. In the 
great majority of cases in this survey, the values of n although fairly large are but 
small fractions of iV, and p is not very small. In such cases the hypergeometrical 
distribution closely approximates the normal curve, the constants ^1 and /Sj 
being respectively 0 and 3 within the limits of their probable errors. The modal 
(qN + 1) (n + 1) 
value of the distribution is the nearest whole integer to ~ — ^ , which 
differs insignificantly from the mean, nq. Thus the asymmetry and leptokurtosis 
are insignificant and therefore the probability of greater or lesser values than that 
found occurring in future samples can be determined from the tables of the 
71/ 
probability integral. In certain cases the fraction is an appreciable one, and in 
these the asymmetry and leptokurtosis are both significant. 
In certain other cases p is rather small. In these cases the interpretation of 
the value of the standard deviation given, which in itself is correct, requires 
considerable modification because the hypergeometrical series can be no longer 
* Pearson : Biometrika, Vol. v. pp. 173 — 175, 
