THEORY OF STATISTICS. 
serve to illustrate the point. During the twenty years 1887-1906 
there were 2107 deaths from explosions of firedamp or coal-dust 
in the coal-mines of the United Kingdom, or an average of 105 
deaths per annum. From § 12 of Chap. XIII. it follows that this 
should be the square of the standard-deviation of simple sampling, 
or the standard-deviation itself approximately 10-3. But the 
square of the actual standard-deviation is 7178, or its value 84-7, 
the numbers of deaths ranging between 14 (in 1903) and 317 
(in 1894). This large standard-deviation, to judge from the 
figures, is partly, though not wholly, due to a general tendency to 
decrease in the numbers of deaths from explosions in spite of a 
large increase in the number of persons employed ; but even if we 
ignore this, the magnitude of the standard-deviation can be 
accounted for by a very small value of the correlation r, expressive 
of the fact that if an explosion is sufficiently serious to be fatal to 
one individual, it will probably be fatal to others also. For if a, 
denote the standard-deviation of simple sampling, ¢ the standard- 
deviation of sampling given by equation (5), we have 
Siero 
"= Dot 
Whence, from the above data, taking the numbers of persons 
employed underground at a rough average of 560,000, 
7073 
= S00 = 105 ~ 0.00012, 
15. Summarising the preceding paragraphs, §§ 9-14, we see 
that if the chances p and ¢ differ for the various universes, 
districts, years, materials, or whatever they may be from which 
the samples are drawn, the standard-deviation observed will be 
greater than the standard-deviation of simple sampling, as 
calculated from the average values of the chances : if the average 
chances are the same for each universe from which a sample is 
drawn, but vary from individual to individual or from one sub- 
class to another within the universe, the standard-deviation 
observed will be less than the standard-deviation of simple 
sampling as calculated from the mean values of the chances: 
finally, if p and ¢ are constant, but the events are no longer 
independent, the observed standard-deviation will be greater or 
less than the simplest theoretical value according as the corre- 
lation between the results of the single events is positive or 
negative. These conclusions further emphasise the need for 
caution in the use of standard errors. If we find that the 
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