178 Oh Criteria for the Existence of Differential Deathrates 
We have m = ~ >S\« x , 
where must be given the sign of ^-^ — The advantage of this method is 
that the value of has usually been tabled as a stage to the finding of 
= Six' {Q^). In the present case we deduce: 
Arithmetic Value of Wi 
Loudon and County Boroughs ... ... ... ... ... ... I'OO 
London and Urban Districts ... ... ... ... ... ... 1-94 
London and Rural Districts ... ... ... ... ... ... 1"94 
County Boroughs and Urban Districts ... ... ... ... ... 0-06 
County Boroughs and Rural Districts ... ... ... ... ... 0-49 
Urban Districts and Rural Districts 0-26 
We can now calculate the probability that each pair are random samples of 
the same population by aid of our four tests. We place the pairs of districts in 
order of their improbability as random samples of the same population, taking 
Xo^ as our standard test. 
Probability P of (he Districts being Samples of the same Pofulation. 
Paired Districts compared 
for Cancer Mortality 
Xo^ Test, or 
Goodness of Fit 
Test 
Test from 
Difference of 
Corrected 
Deatlirates 
[M' -M)lffj^^,_j^ 
Test from 
Distribution of 
Squares of 
Deviations 
Test from 
Mean of 
Deviations 
London and Rural 
Districts 
•9625/1030 
rsoai/io^" 
1^9628/10i« 
•3121 
County Boroughs and 
Rural Districts 
■5414/10" 
•7929/1034 
1^9385/10i<" 
•0262 
London and Urban 
Districts 
1-1740/10" 
1-7497/10" 
•2555/1023 
•0262 
County Boroughs and 
Urban Districts 
1^6355/108 
■6606/10" 
•5133/10» 
•4761 
London and County 
Boroughs 
r4996/10' 
1-6586/1012 
•1927/106 
•1587 
Urban Districts and 
Rural Districts 
■4214/10' 
•6327/10" 
•1962/105 
•3974 
The probability* in the first of these tests is deduced from the Goodness of 
Fit Tables ; in the remaining three tests from the Tables of the Probability 
Integral. It will be seen that the first three tests give absolutely the same order 
of significance for the six pairs of differences. The fourth test is irregular and 
* The very high improbabilities given are only rough approximations, sufficient, however, for our 
present purposes. Since we enter the Goodness of Fit Tables with n = ten (i.e. u + \), we must use the 
first value of P in Equation (xxix). Tables for Statisticians, p. xxxi. The integral / may then be 
calculated by the first term of the Schlomilch formula (ihid. p. xxxiii), as this integral wUl only affect 
the fourth figure in the decimals. Table IV (ihid. p. 11) has been used to approximate to the extreme 
tails of the probability integral. Very useful work could be done by extending this Table between 5 
and 50 to the first decimal place in the argument. 
