K. Pearson 
176 
to show that in the neighbourhood of the origin yjy^ is always greater than yjy^ 
for the same value of x. In other words, the first curve is flatter topped than the 
second, and both lie on different sides of the corresponding Gaussian curve. The 
first curve type is platykurtic and the second leptokurtic. 
Now there is nothing to prevent us fitting curves of the above types to any 
series of frequency observations. Supposing those observations are truly normal, 
then m, or will be so large that 13^= 'i within the error of random sampling. 
Now the probable error of /S^ for a Gaussian distribution of total frequency N* : 
= -07449 y^^, 
and if /S^ differs from 3 by several times this probable error, it is absolutely 
impossible to treat the system as mesokurtic. In any such case one or other of 
the above curves most give a truer representation than the Gaussian curve. It is 
easy to show that for leptokurtic distributions the maximum frequency is greater 
than that given by the normal curve and for platykurtic distributions it is less. 
The Gaussian curve compels us to assert that the product of the maximum 
frequency into the standard deviation is a constant (i.e. y^a — NI\/2it). This 
condition of mesokurtosis is unfulfilled — within the limits of random sampling — 
for a great variety of frequency distributions. 
Further it is absolutely certain that divergencies from the Gaussian or normal 
curve are not exclusively in the direction of either platykurtic or leptokurtic 
distributions. Thus the symmetrical binomial is essentially leptokurtic, i.e. yS^ < 3, 
and therefore cannot be used for a great variety of distributions. In general all 
skew binomials with p>'2118 and < '7887 are leptokurtic; outside these limits 
they are platykurtic. 
The test whether a curve satisfies the mesokurtic condition has nothing to do 
with my particular views on frequency, it is merely deduced from the general 
principles of probability and is a test of normal distribution. Of course there are 
• many other conditions to be satisfied, e.g. ft^n should equal (2n— l)/i^,j_2. But as 
I have shown elsewhere the probable errors of the high moments increase so 
rapidly, that it becomes easier and easier to satisfy such conditions within the 
errors of random sampling, and without very large numbers they are of little 
practical value. 
The following are significantly platykurtic distributions : 
The Maximum Breadth in English skulls, 
The Nasal Breadth in English j/" skulls. 
The Gephalic Index of Altbaierisch skulls, 
The Auricular Height in % Naqada skulls. 
* Pearson : Phil. Trans. Vol. 198 A, p. 278. 
