FOUNDATIONS OF THEORETICAL STATISTICS. 
317 
surface of distribution of pairs of values of 0 1 and 0 2 , for a given value of 0, is such that 
for a given value of 0 1} the distribution of 0 2 does not involve 6. In other words, when 
Oi is known, knowledge of the value of 0 2 throws no further light upon the value of 6. 
It may be shown that a statistic which fulfils the criterion of sufficiency will also 
fulfil the criterion of efficiency, when the latter is applicable. For, if this be so, the 
distribution of the statistics will in large samples be normal, the standard deviations 
being proportional to n~'K Let this distribution be 
l _i_ i 6>, — e 2 _ -ji-Bt-e ft,-» 9.,- i 
df — ~ -77=7 e *' ^ J dOide 3 , 
'z,Trcr l a 2 V I —T 
then the distribution of 0 X is 
1 _ 
df = — — 7 = e ao-i* r j0 u 
°r \ r 
so that for a given value of Q x the distribution of Q., is 
d f = 1 ' 7— 
r 2 V 2-JT v 1 — V“ 
e 21 -r 
/ 
-0 0,-0 1 
<*! 
*2 
0| 2 
dO 2 
and if this does not involve 0, we must have 
V'O‘2 — CTi 
showing that cri is necessarily less than o- 2 , and that the efficiency of 0-. is measured by 
r 2 , when r is its correlation in large samples with 0 j. 
Besides this case we shall see that the criterion of sufficiency is also applicable to finite 
samples, and to those cases when the weight of a statistic is not proportional to the 
number of the sample from which it is calculated. 
5. Examples of the Use of the Criterion of Consistency. 
In certain cases the criterion of consistency is sufficient for the solution of problems 
of estimation. An example of this occurs when a fourfold table is interpreted as repre¬ 
senting the double dichotomy of a normal surface. In this case the dichotomic ratios 
of the two variates, together with the correlation, completely specify the four fractions 
into which the population is divided. If these are equated to the four fractions into 
which the sample is divided, the correlation is determined uniquely. 
In other cases where a small correction has to be made, the amount of the correction 
is not of sufficient importance to justify any great refinement in estimation, and it is 
sufficient to calculate the discrepancy which appears when the uncorrected method is 
applied to the whole population. Of this nature is Sheppard’s correction for grouping, 
2 Y 
VOL. CCXXII.—A. 
