MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 38095 
circles, or parabolas of various degrees which give results extremely close to any given 
set of observations. 
For example, taking the range of frequency to be sensibly 7 times the standard 
deviation, we have the following close expression for the error function by harmonic 
analysis 
z 2a 3 
y = Y {899 + “482 cos + 109 cos + “009 cos Ff 
Here y, is the maximum ordinate, x any deviation, and o the standard deviation. 
A couple of wave curves* will thus very frequently give us a close approximation to 
a set of statistical measurements, quite as close as statistical practice shows the error 
curve to be. 
The above expression further allows the normal curve to be constructed by aid of 
scale and compasses—geonietrically, or its ordinates calculated from a table of cosines. 
Another example of the fitting of a point-binomial will be found in Part 2, § 34, 
Pauper Percentages. 
(7.) Consider the point-bimomial e xX ($+ 4)", where e is any constant, and 
suppose a polygon formed by plotting up the terms of the binomial at distance ¢ 
from each other. 
Then, corresponding to x, = rc, we have 



DO. WG Ac os (Ge ae) pany 
Yo = € [p—1 (3)” 
and 
Yror a Yr ee Ae (1 ate 2) ian (a + @r 41) Ure (wy We, wy 4) 
£ (Yrs, + Yn) Xe sate I AGED Se 
if w’, = a, — 4c(n + 2). 
Now (y,+, — y;)/e is the slope of the polygon corresponding to the mean ordinate 
4(Y,2,+ ¥,), or, writingt P® =} x t(n4+ 1)2e, 
slope of polygon 2 x mean abscissa 

mean ordinate 20° 
* Jt is often sufficient to take 
ae 2x 
y= (8 + 7 COS ~~ + 7 Cos ar 
+ The divergence of this value of o? from the ordinary value 4 x 4 x nis to be noted. The two agree 
sensibly if n be great. [Drawing on a large scale, however, the point-binomial ($ + 4)!° and the two 
normal curves with standard deviations of 1°5811 and 1°6533, I find that the latter has a mean percentage 
error of only 1°76 as compared with 5:1 of the former. Thus it would appear that the normal curve 
corresponding to V(n + 1) pq fits the point-binomial closer than one with the standard deviation Vv apq 
usually adopted. | 
22 
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