THEORY OF STATISTICS. 
The frequency-distribution of the number of deaths per army 
corps per annum was 
Deaths, Frequency. 
0 109 
1 65 
2 22 
2 3 
‘ 1 
whence 
o2=0'6079 
o=078 
—an almost exact agreement with the standard-deviation of simple 
sampling. 
13. We may now turn from these verifications of the theoretical 
results for various special cases, to the use of the formule for 
checking and controlling the interpretation of statistical results. 
If we observe, in a statistical sample, a certain proportion of 
objects or individuals possessing some given character—say A’'s— 
this proportion differing more or less from the proportion which 
for some reason we expected, the question always arises whether 
the difference may be due to the fluctuations of simple sampling 
only, or may be indicative of definite differences between the 
conditions in the universe from which the sample has been drawn 
and the assumed conditions on which we based our expectation. 
Similarly, if we observe a different proportion in one sample from 
that which we have observed in another, the question again arises 
whether this difference may be due to fluctuations of simple 
sampling alone, or whether it indicates a difference between the 
conditions subsisting in the universes from which the two samples 
were drawn : in the latter case the difference is often said to be 
significant. These questions can be answered, though only more 
or less roughly at present, by comparing the observed difference 
with the standard-deviation of simple sampling. We know 
roughly that the great bulk at least of the fluctuations of samp- 
ling lie within a range of + three times the standard-deviation ; 
and if an observed difference from a theoretical result greatly 
exceeds these limits it cannot be ascribed to a fluctuation of 
“simple sampling ” as defined in § 8: it may therefore be signifi- 
cant. The “standard-deviation of simple sampling” being the 
basis of all such work, it is convenient to refer to it by a shorter 
name. The observed proportions of A’s in given samples being 
regarded as differing by larger or smaller errors from the true 
proportion in a very large sample from the same material, the 
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