ON GENERALISED TCHEBYCHEFF THEOREMS IN THE 
MATHEMATICAL THEORY OF STATISTICS. 
By KARL PEARSON, F.R.S. 
(1) Single Variate. 
Let !/ = 4> (x) be any law of frequency and let the limits of the distribution be 
a and b, then if N be the total frequency, 
N = I (f) {a:) dx, 
J a 
and if x be the mean value of the variate, 
Nx = xcj> (x) die. 
J a 
Generally, if /iig be the sth moment-coefficient about the mean, 
iV/Zj =1 (x — xy <f) (x) dx. 
J a 
1 /-i _ " 
Now consider /x.^,^ = y I {x — xf' (j) (x) dx, 
and let e be any value of x — x, then 
Hsle-" = j — — cj) {x) dx. 
Now pick out all the values for which x — x is greater than e, and let us suppose 
b > a; then 
1 
and therefore yu-as/f"* > ivr f 4' (*') 
since {x — ,r)/e is always greater than unity. 
1 r* ... 
But I (j) (x) dx is the chance of an individual occmmng with a deviation 
greater than e from the mean = 1 — P where P is the chance of an individual 
occurring with a deviation less than e. Hence 
Now let e = Xa, where a = V/Xo is the standard deviation of the distribution. 
P>1 
