Karl Pearson 
285 
Thus the chance of a deviation being of less magnitude than \a is 
73 > 1 _ 2 . (i). 
If we put s = 1, the chance P of a deviation less than \a is limited by 
1 
P > 1 
.(ii). 
This special case is Tchebycheff's Theorem *. 
Inequality (i) gives our first generalisation for a single variate of Tchebycheff's 
Theorem in (ii)i". We can now compare the accuracy of (i) and (ii) by supposing 
them applied to a normal distribution of frequency for tlie cases of deviations 
1, 2, 3 and 4 times the standard deviation. In this case 
TABLE I. 
Values Of Lower Limit jor P given by \ — -~ — . 
s 
X=:l-5 
X = 2 
X = 3 
X = 4 
1 •, 
•5556 
•7500 
•8889 
•9375 
2 
•4074 
•8125 
•9630 
•9883 
3 
- -3169 
•765G 
•9794 
•9963 
4 
•9840 
.•9984 
5 
•9840 
•9991 
6 
•9804 
•9994 
7 
■99950 
8 
•99953 
9 
•99950 
10 
•99940 
Actual value 
of P 
•86G4 
•9545 
•9970 
•99994 
Clearly the maximum for any A. will be found by making (2.s- 1)/X.^ equal to 
unity, or if X,^ = an odd number, .s = ^ (X- + 1) and h (\- + 1) — 1 will give equal limits. 
If X" be an even number then .s = l A.- will give the highest limit. 
* It was first proved in the Eeciieil dcs sciences matheniatiques, T. ii, according to Liouville, but 1 
cannot trace this reference at all. It was translated from Bussian into French in Liouville's Journal dc 
mathematiques, Vol. xii, pp. 177 — 184, Paris, 1867. The proof there given is somewhat lengthy and at 
first sight the result might appear more general than (ii) ; but this is not so. Assume x = u + v + w+ ... 
and suppose u, v, lu uncorrelated, so that o-j.''' = <r„''' + cr„- + a-,,,- + . . . then we have with minor differences 
of notation and terminology (especially the use of the words "mathematical expectation" for our 
moments) TchebycheiFs own phrasing of his theorem. The remark of Dr Anderson (Biometriha, 
Vol. X, p. 269) with regard to the neglect of the theory of "mathematical expectation" by the 
English statistical school seems based on a misunderstanding of the moment method. 
t This generalised form of Tchebycheff's Theorem was given by me in a paper for the Honours 
degree of the University of London in Statistics, October, 1915. 
