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MR. R. A. FISHER OX THE MATHEMATICAL 
4. Criteria op Estimation. 
The common-sense criterion employed in problems of estimation may be stated thus :— 
That when applied to the whole population the derived statistic should be equal .to the 
parameter. This may be called the Criterion of Consistency. It is often the only test 
applied : thus, in estimating the standard deviation of a normally distributed population, 
from an ungrouped sample, either of the two statistics— 
and 
(Mean error) 
(Mean square error) 
will lead to the correct value, a, when calculated from the whole population. They both 
thus satisfy the criterion of consistency, and this has led many computers to use the 
first formula, although the result of the second has 14 per cent, greater weight (7), and 
the labour of increasing the number of observations by 14 per cent, can seldom be less 
than that of applying the more accurate formula. 
Consideration of the above example will suggest a second criterion, namely :—That in 
large samples, when the distributions of the statistics tend to normality, that statistic 
is to be chosen which has the least probable error. 
This may be called the Criterion of Efficiency. It is evident that if for large samples 
one statistic has a probable error double that of a second, while both are proportional 
to n~*, then the first method applied to a sample of 4 n values will be no more accurate 
than the second applied to a sample of any n values. If the second method makes use 
of the whole of the information available, the first makes use of only one-quarter of it, 
and its efficiency may therefore be said to lie 25 per cent. To calculate the efficiency of 
any given method, we must therefore know the probable error of the statistic calculated 
by that method, and that of the most efficient statistic which could be used. The 
square of the ratio of these two quantities then measures the efficiency. 
The criterion of efficiency is still to some extent incomplete, for different 
methods of calculation may tend to agreement for large samples, and yet differ for 
all finite samples. The complete criterion suggested by our work on the mean 
square error (7) is 
That the statistic chosen should summarise the whole of the relevant information 
supplied by the sample. 
This may be called the Criterion of Sufficiency. 
In mathematical language we may interpret this statement by saying that if 6 be 
the parameter to be estimated, a statistic which contains the whole of the information 
as to the value of 0, which the sample supplies, and b, any other statistic, then the 
