MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION, 353 
As typical samples of mean percentage errors considered by various statisticians to 
give good results, I may note the following, the frequency being about 1,000 or 
upwards :—A1ry, 9 ; MEeRRIMAN, 13°5 ; Gatton (Anthropometric), 7 to 15 ; WELDON 
(Crabs), 6°7, (Shrimps), 8°8; StrepA (Skulls), 7°6; Porrer (School Girls), 7°7; PERozzo 
(Recruits), 6°8 ; BRADLEY’s observations, 5°85 ; PEARSON (Lottery), 6°7, (Tossing), 6°6. 
It is therefore clear that our point-binomials and generalized curve may be con- 
sidered to give good results.* It will be noticed, however, that a little difference in 
the method of calculating the point-binomials leads, without much alteration of the 
percentage error, to a considerable change in their centroid-positions and the magnitude 
of their constants.t Generally speaking we may conclude that in round numbers the 
barometric frequency corresponds to the binomial (‘9 + *1)”, or to the distribution of 
zeros when 20 ten-sided teetotums, marked 0, 1...9, are spun together. There is 
an apparent upper limit to the height of the barometer, and its deviation below the 
mean can be much greater than its deviation above. At the same time within the 
narrower range round the mean, the frequency of a high barometer is greater than 
the frequency of a low barometer; the odds against a “ contributory cause” tending 
to a low barometer being about 9 to 1. I propose to investigate a wider series of 
barometric observations, in order to test how far the conclusions which may be drawn 
from Dr. VENN’s statistics are general.{ 
A rather interesting point may be considered at this stage. Is it always possible 
to fit a point-binomial to a series of observations with a chance frequency ? Can we 
better the normal curve by a point-binomial ? The answer is Yes, if the fundamental 
cubic in y (second footnote, p. 351), has a real positive root. Now for the normal curve 
2 (8p? — by) Ha + 3pyy°, or 6 + 38, — 26, is zero. For the loaded ordinates ¢ will 
only be real if this expression be positive. It may, however, take small negative 
values for the trapezia, in which case y itself will be small and only within narrow 
limits give suitable values for n. 
Hence, for real values of n, p and q, it is impossible to fit a point-binomial to a 
series of observations for which 6 4- 38, — 28, has a large negative value. The normal 
curve, for which p, = 3p”, is nearer to any such observations than a point-binomial. 
For example, by aid of the modified expressions given in this paper, p. 350, we have 
* As another manner of testing, compare the ten-points of the point-binomial for lines with obser- 
vations :— 
WMineomy Eo eG GRO BRS) pale). IS) TL), GP eee 8 0S) 
Observation . .. 57 158 221 188 12 58 23 11 -2 -00 
+ A curve drawn through the 30 points of the three point-binomials would be very close to the obser- 
vations. As a matter of fact, the skew probability curve passes very near to all 30 points. 
+ [Miss A. Lee has since calculated the constants of three years of Hastbourne barometric observations 
for me. While x and c differ widely from the Cambridge values, she finds p = ‘89375, q = (10625, a 
striking and suggestive agreement. | 
MDCCCXCY.—A. 22 
