Miscellanea 
425 
§ 3. Professor Edgeworth lias on several occasions advocated the value of Wk generalized 
frequency curve, and endeavoured to illustrate its numerical advantages. On the present 
occasion he has introduced what appears to be an erroneous formula for testing its "goodness 
of fit." In his paper he gives no theoretical discussion of the test. It would be well if Professor 
Edgeworth tried on some future occasion to give a demonstration of its validity. Any such attempt 
would have to account for the departure from Bernoulli's law that accuracy varies as the square 
root of the number of observations. According to Professor Edgeworth's test results obtained 
from 100 observations and from 1,000,000 would be equally accurate provided the relative 
frequencies were the same*! However taking him on his own ground we find his curves give 
a very poor fit compared with Pearson's well-known types, and offer no easy systematical methods 
of treating frequency. He escapes what is the real hard work of curve fitting by adopting the 
higher moments calculated by the biometricians and then appears to say how very much shorter 
his methods are! We are prepared on any occasion to race him in fitting a frequency curve 
ah initio and getting a better fit as a result. 
Added April \2th, 1917. Professor Edgeworth in reply to a letter in which I suggested that 
there might be an error in his numerical values quoted in my Table IV, has replied that owing 
to a clerical error the frequency in the last column is wrong. On using his amended relative 
frequency, that entry becomes 18'35 instead of 28-44. With this alteration becomes 1 -297 
instead of 7-035, so that the corresponding value of P is -733. This of course denotes a good 
fit, but one which is still very inferior to the Pearsonian Type II curve for which P=-999. 
Ill, Relation of the Mode, Median and Mean in Frequency Curves. 
By ARTHUR T. DOODSON, M.Sc. 
§ 1. It is well known that in frequency curves of a moderate degree of asymmetry the distance 
from the mode to the median is approximately two-thirds of the distance from the mode to the 
mean. So far as the author knows there has not been published any investigation of this property 
except in the case of one type of curve where 
y = 2/oa:"e-'. 
Professor Karl Pearson, in his memoir on "Skew Variation in Homogeneous Material" (Phil. 
Trans. 1895), remarks on this property of the curve and he gives an expression for the ratio of the 
two distances in the form 
c = -6691 + -0094/;; (1). 
This was obtained from a few particular cases by the method of least squares. 
§ 2. Let y = yo<f> (x) represent a frequency curve which has a maximum at a; = 0 and which 
is zero at a; = - and at a; = oo- As a rule there is only one maximum in frequency curves of 
this type. On these assumptions Laplace has given a method of approximating to the definite 
integral, the limits of integration lying between x = - and x = a^. Following Laplace, we 
assume 
y = 2/o'^>(-c) = 2/0 (2), 
where ?/o is the maximum ordinate, or the mode. Since x vanishes with t we may assume 
x = B^l + + BJ^ + (3). 
* Cf. on the other hand Pearson's test. Let there be two series of observations with populations 
iVj.iVj and same relative frequency leading to S [e^/u) =X(,~. The two values of would be Xi' = ^^iXo'> 
X2^ = N.,Xo^ so that the probabilities would be proportional to e^1-l\/^i)- and e^^/'^-^^)', comparing with 
standard deviations of form 1/v^-^i a-nd l/^yiVj. 
Biometrika, 28 
