5930 
Mathematics. — “Logarithmic Frequency Distribution”. By Dr. M. J. 
VAN Uven. (Communicated by Prof. J. C. Kaprryn). 
(Communicated in the meeting of September 30, 1916). 
When the frequency-distribution of some measured quantity # does 
not follow the normal law of Gauss: 
52 
de) „—h2E? dé, 
§—w—NX being the deviation soni the arithmetical mean X and 
Wz 2 the probability that this deviation is found between the limits 
&, and &, (the measured quantity between #,— X+§, and a, = 
= X-+ §,), this need not be a reason to drop the law of Gauss, 
as Prof. J. C. Kapreyn has shown.') On the contrary the “skewness” 
of the frequeney-curve may in many cases be explained by merely 
supposing that, instead of 2, another quantity Z—= F(x) connected 
with x is distributed according to this law, so that it is only due 
to the wrong choice of the quantity measured, that the normal 
distribution has not come out. Then it is interesting to deduce the 
normal function Z= He) from the given skew frequency-distribution. 
Let this normal function 7 have the value J/ for its arithmetical 
mean, so that the deviations $= Z— MV are spread round the mean 
value zero according to the normal law, and thus satisfy the equation 
5e hf 22 
Wij= fe PE ae. 
rs Vx 
Zi 
Among the quantities &—= Flr)—M, which apparently are also 
functions of the observed quantity w, there is one, viz. z=/h¢ = 
=h h(x) — Wi = f(z), which answers to the formula 
: Bnn 
We = 7 ff dn. 
nr 
21 
This z has 4=1 for its modulus of precision and consequently 
1 
&, = —— for its (quadratic) mean value. 
V2 
1) J. C. Kapreyn: Skew Sa. Curves in Biology and Statistics; Groningen, 
1903, Noordhoff. 
