J. F. Tocher 301 
whether there are more marked deviations with long series generally, or in the case 
of the insane, from this form. In the latter case, such woul(i probably arise if 
the bulk of the insane were characterised by two or more special head shapes ; for 
example, if the insane had special tendencies to macrocephaly and microcephaly. 
As stated in the introductory paragraph, certain individuals have been excluded 
from the general analysis, because they were considered by their medical attendants 
to have characters affected by special causes, not characteristic of insanity in 
general. The differences arising in the frequency distributions, according as these 
cases — throughout this memoir, termed exceptionals — are included or excluded, 
will now be noted. The constants have been calculated for the " entire insane " 
population — 4436 males and 3951 females — and also for the "general insane" 
population, i.e. without the exceptionals — 4381 males and 3925 females. The 
values of the constants are given in the following table (Table I.). They show 
that the distributions are more or less skew, and that when the "entire insane" 
population is considered, the extremes in the range affect the goodness of fit 
considerably — in other words, while the skew or normal curves fitted to the 
distributions fairly describe their nature when considered as a whole, the most 
important contributions to defect of fit arise from the " tails." Even the " general 
insane " population shows for most characters excess frequency at the tails, in- 
dicating the probability that the homogeneity of the series is affected by the 
" undetermined residue " referred to above. 
Considering the " entire insane " population first, we see that (a) the skew- 
ness measured by % is probably significant in all cases since it amounts in each 
case to three, or more, times the probable error, (h) the kurtosis, rj, is in all 
cases significant, amounting in every case to many times the probable error. In 
all the distributions, the positive values of r] indicate leptokurtic curves, their 
magnitudes very largely depending on outlying observations, as will presently be 
seen*. Thus the skevvness and leptokurtosis so affect the distributions that their 
frequencies cannot be represented by normal curves. Considering further the 
" entire insane " population, we see that lies between 0 and 1 and therefore 
Type IV. is the actual form needed, but since Ko is in every case very small it is 
clear-f- that the distributions will, if /3i be very small and differ in excess 
significantly from 3, approach closely to Type IV. with symmetry j. This actually 
occurs for all characters but stature, /3i being less than '03 in all cases except one 
{Hf^) when it is only '1 and /Sa= 3 +17, differing very sensibly from 3. Thus it is 
seen that the " entire insane " population approaches to symmetry^ in distribution 
of head characters, and the deviation from normal frequency, accordingly, is not 
* In 10 out of 24 cases, C. D. Fawcett's series shows leptokurtosis, while Macdonell's English series 
shows 14 cases of leptokurtosis out of 26. Biometrika, Vol. i. p. 442, Vol. iii. p. 228. 
t Pearson : Phil. Trans. A. Vol. 197, pp. 443—459. 
J That is to the form: y = ij.2[l + x-l{2cr'^{m.^-i)\]-i'>'2+^), where /3o- 3 = 6/(2)«^- 3). Professor 
Pearson points out to me that there is an unfortunate interchange of and ?7!3 in the memoir, lines 
2, 3, 4 and 5 from bottom. Biometrika, Vol. iv. p. 174. 
§ Not absolute symmetry, because the values of /Sj, although small, do differ sensibly from zero. 
Biometrika v 39 
