K. Pp: ARSON 
fundamental deviations of any distribution from the Gaussian curve and deter- 
mining whether these deviations are significant or not. Looked at solely from this 
standpoint — which I am very far from accepting — my curves provide an empirical 
series which accurately measures the deviations from the Gaussian law and enables 
the enquirer to determine how far that law is applicable. Each one of them passes 
into the Gaussian curve if that curve is the better fit to the observations. This is 
not true of many of the other remedies which have been proposed to svipplement 
what I venture to call the universally recognised inadequacy of the Gaussian law. 
They cannot as we shall see in the sequel effectively describe the chief deviation^ 
from the Gaussian distribution. 
The chief physical differences between actual frequency distributions and the 
Gaussian theoretical distribution are : 
(i) The significant separation between the mode or position of maximum 
frequency and the average or mean character. 
(ii) The ratio of this separation between mean and mode to the variability 
of the character — a quantity I have termed the sketvness. 
(iii) A degree of flat-toppedness which is greater or less than that of the 
normal curve. Given two frequency distributions which have the same variability 
as measured by the standard deviation, they may be relatively more or less 
flat-topped than the normal curve. If more flat-topped I term them platykiirtic, 
if less flat-topped leptokurtic, and if equally flat-topped mesokurtic. A frequency 
distribution may be symmetrical, satisfying both the first two conditions for 
normality, but it may fail to be mesokurtic, and thus the Gaussian curve cannot 
describe it. : . ..3 ■ - :■' 
The Gaussian curve is usually fitted from the mean square deviation, but it 
may also be fitted from the piol)able error, or the mean error, or again from the 
mean fourth power of the deviations — fi^ in my notation. Wiiichever method is 
adopted we ought to get the same result within the errors of random sampling. 
When I first began to describe frequency data by the normal curve, I was startled 
to find the very large number of cases in which these different processes led to 
Gaussian curves, differing widely from one another, i.e. beyond all the limits of 
probable error. I was soon led to see that in actual statistics two distributions 
might have equal total frequency, be sensibly symmetrical, and have the same 
standard deviation and yet differ largely in their flat-toppedness. The mesokurtosis 
of the Gaussian curve is not a universal characteristic of frequency distributions. 
When we test a theoretical distribution of frequency against observation, we 
may find an excellent fit for the total distribution and yet the distinction between 
mode and mean, the skewness, and the deviation from mesokurtosis may be most 
significant. The reason for this is that the test for goodness leaves a margin of 
variation which may be due to random sampling, or to the non-normal character 
of an important constant of the distribution. For example, 10 coins are tossed 
a hundred times, and the proportion of cases with five and more heads is somewhat 
