Karl Pearson 
289 
TABLE II. 
Generalised Tchehycheff Limit applied to the Prohahility that an association of two 
variables lies inside a r/iven contour ^^n" of a normal frequency surface. 
Actual 
Minimum value 
Probability 
of P 
4 
•8647 
•5000 
(A) 
5 
•9179 
•6800 
(Id 
6 
•9502 
•7778 
7 
•9698 
•8600 
8 
•9817 
•9062 
ih) 
9 
•9889 
•9415 
10 
•9933 
•9616 
ih) 
12 
•9975 
•9846 
(/-,) 
(/«) 
14 
•9991 
•9939 
IG 
•9997 
•9976 
(/t) 
18 
•9999 
•9991 
ih) 
20 
•99995 
•99964 
(^) 
Here as in the case of a single variate the generalised Tchehycheff limit is not 
very useful for low values of But if in any particular type of observation we 
consider it desirable to look with suspicion on an observation which has occurred 
and yet the odds against which are greater than 50 to 1, the Tchehycheff limit may 
be of value. As illustration, suppose two variates are correlated with intensity "7, 
what suspicion should we cast on an observation which gave the deviation of one 
variate 3^8 times its standard deviation and of the other 3"2 times ? Here 
2 1 /a^ _ 2rxi/ y-\ 
= imr- - r4 (3-8) (3-2) + (3-2)=} 
= 15 "01, or say 15. 
2' ('7 l"* 
Then P > 1 - 4^ > '9962, 
15' 
or the odds are greater than 250 to 1 against it. Actually the probability of the 
occurrence of anything as unusual as or more unusual than this is "9994, or the 
actual odds 1700 to 1 about. For many pui-poses the odds of 250 to 1 would 
amply suffice to mark suspicion, although of course in the case of normal fre- 
quency it would be as easy or even easier to calculate the real probability as the 
generalised Tchehycheff limit. 
The chief interest of the investigation thus far is to show that iinless we use an 
/, of a high order the Tchehycheff limit is unlikely to be of very much service. We 
can obtain it in the case of material following a normal distribution, but then we 
know the exact result and do not need it ! 
