K. Pearson 
181 
in the least approximately normal. Accordingly if we wish to get a good 
interpolation curve to determine the distance between the mode and the mean, 
we may assume 
I dij —X / ■ w ■ 
- / = :7^^- \ , 
II ax cr^j- ( 1 + «j X + a-,x-) 
In this case we discover with the previous notation, that 
is the distance between mean and mode, and that v = i'^,^^-^^--}, is the ratio 
^ 5,»o — bpi — 9 
of this distance to the variability, or what I term the skewness, or the asymmetry 
relative to the variability. 
If we leave out o., we Kud the skewness given by % = ^^^A and the distance 
between mean and mode d = \\l^^o-. In practice these give fairly closely the 
same values as the fuller expressions above, and the fuller expressions are not 
numerically much modified if we include 03. Shortly we have got a very fair 
mathematical process of determining the position of the mode and the degree of 
asymmetry. 
Now the constants of such a curve as (vii) his are absolutely determined by a 
knowledge of a, /S^, and or looked at inversely they suffice to hx 0-, yS^, ami /S^. 
In other words the degree of kurtosis (/3^ — 3), the skewness and the distance 
between mean and mode — all most detinite physical constants — are at once fixed 
by a knowledge of the constants of the curve, or on the other hand, being known, 
they fix those constants. It is of course allowable to replace any one of the three 
by the variability of the system. The actual position of the mode and the total 
magnitude of frequency suffice to fix the position and size of the curve. I have 
already called ^., — 3 = 7) the degree of kurtosis; I call d the modal divergence. 
Then unless 
t], the degree of kurtosis be zero, subject to probable error, ■67449 V24/i\^, 
X, the skewness be zero, subject to probable error, "67449 Vl -.5 /iV, 
d, the modal divergence be zero, subject to probable error, ■67449o- Vl-.D/iV, 
no distribution can be legitimately described as normal or Gaussian. 
It would be of interest to know how far Ranke and Greiner have applied such 
tests to any series containing a large number N of individuals. I think if they 
had done so, they must have come to the same conclusion as the majority of 
statisticians that the normal curve has only a limited range of aj^plication. 
Of course if N be small, as in most craniological series, we find our probable 
errors so large, that it is not possible to say more than that for short series the 
Gaussian curve may roughly describe the result. But for long series in economics, 
sociology, zoology, botany and anthropometry the Gaussian curve over and over 
again fails. If in all these cases Ranke and Greiner assert that the material is 
