J. F. Tocher 
143 
(5) Methods Employed to Determine Significant Differences. 
In making a survey of the measurable physical characters of a population one 
has not only to ascertain the type and variability of each character but also to 
consider the relationship of each local group to the general population*. Thus, in 
the recent investigation on the inmates of asylums it is shown that several 
physical types exist among the Scottish insane, and that, whether they differ or 
not from the sane population, local asylum groups generally do not resemble the 
general insane population. But non-measurable characters can scarcely yet be 
dealt with in the same way. It has not been found possible up to the present 
time, for instance, to determine the value of the character, hair colour, just 
because no quantitative scale based on experience has yet been devised on which 
to plot the observations in an orderly way indicating increase or decrease of 
intensity of colour. It is not clear whether such a scale is possible. Experimental 
work has just been undertaken by the writer which may throw some light on this 
point. But while hair colour cannot yet be represented on a scale of intensity of 
colour such as stature or head length, it can be quite properly dealt with under 
well defined classes or categories. As already explained, the limits of these classes 
have been defined in the analytical table given in each schedule. What statisti- 
cians have here to consider therefore are the frequencies of the various classes 
individually and collectively without reference as to whether the classes can be 
arranged on a scale showing grades of intensity of colour. This has been done on 
a moderate scale for adults f, and it may be well to restate here the methods 
employed before proceeding to state the results of the analysis. 
A population of iV individuals is to be considered, each of which possesses the 
character X. The character X is not measurable but can be divided into m 
classes. Let s, .So.-.Sjn, be the classes and let the class frequencies for the whole 
population be respectively yg,, t/s^-'-ys^- The population is divided into groups 
of magnitude n, and each group is observed and classed with respect to the 
character X. In making the observations, the probability that any person 
observed (if the operation is a random one) belongs to class s is y^jN = p, and the 
probability of the person not belonging to that class, but to one of the others is 
(1 —p) = q. If the groups are samples drawn from the general population purely 
at random, the frequency for the class s for each of the groups is therefore equal 
to nyajN = np = yg, which is thus for the class s the most probable number likely 
to be drawn in this way ; or is, shortly, the theoretical class frequency. It is 
necessary to consider what would happen if the whole population was observed in 
unselected groups at random for the following reason. If the observed class 
frequencies in the various geographical areas actually differed insignificantly from 
the theoretical class frequencies then it would be clear that the population was 
evenly distributed with respect to the character. Thus, so far as this character is 
* Tocher: Biometrika, Vol. v. Part iii. pp. 315 et seq. 
t Tocher : Biometrika, Vol. v. Part iii. pp. 335 et seq. 
