Miscellanea 
311 
Or, if p be taken = 
12 
Thus 
which is in all practical cases of no importance. 
Hence n,. = hx. u,., or the frequencies are given by the ordinates of a normal curve for which 
the standard deviation is o--+^/r, but if Nv.^hc the 'raw' second moment about the mean we 
have seen above that (t' = v.,- It". Thus (t^- = v.,., very closely. Or, we conclude that : 
If the ordinates of a normal curve he calculated from the raw second moment value of the 
standard deviation, these ordinates will more closely represent the actual frequencies than do the 
ordinates of the true normal curve, which ham to he corrected hy the factor 
to ohtain the actual frequencies. 
If therefore our sole object is to compare observed and calculated frequencies for a definite 
series of groups, there are advantages in using the raw second moment in the equation to the 
curve. Such a curve has been termed by Sheppard a ' spurious curve of frequency.' Generally 
if the raw moments be used to calculate such a sjiurious curve its ordinates will give the 
frequencies, but the constants of such a ciu've are not the constants of the true frequency 
distribution, but functions of h the unit of grouping. 
As we want as a rule both the constants of the true frequency distribution (for comparative 
purposes), and the theoretical frequencies to compare with the observed frequencies, we are in 
some difficulty in the case of curves, which unlike the normal curve, have not their areas tabled. 
We have cither to calculate the spurious curve as well as the true curve, or devise some process 
by which the areas of the true curve may be foiuid from its ordinates for small ranges li. 
We have used for some time past the following formulae for finding areas in terms of 
mid-ordinates in the case of skew curves of frequency : 
(i) Normal Curve: Origin at mode. 
Area of strip on base h 
(ii) Curve of type: 
Origin at mode and p = ay. 
Area of strip on base Ii 
{p must be > 2, if this is to be of value near the terminal of the cur\'e). 
(iii) Curve of type: 
Origin at mode. 
