And this is a better limit in such cases. _Taking
28 as the limit of error on 100 instances_ and proportionally increasing
the others so that _the mean error becomes 7.8 and the probable error
5.6_, we may now calculate the answer without gross mistake.
The probable variation on the 1000 deaths by accident will be 18, the
mean variation will be 24.6, and the limits of variation 88.5. One such
table showing in five years a mean number of deaths of about 1120 per
annum gives an annual deviation of about 50 up or down of this. It will
be seen at once that an improvement of 30 or 40 in any one year would be
without meaning, but that an improvement of from 100 to 200 would
indicate some change for the better in the circumstances of the
industry. Before applying these principles to the elucidation of some of
the problems of sampling it will be well to give Poisson's formula (in a
modified form) and to illustrate its working.
Let _x_ equal the number of cases of one sort, _y_ the cases of the
other sort, and _z_ the total. In the example, _z_ will be the 700,000
engaged in the industry; _x_ will be the 1000 killed by accidents, and
_y_ will be the 699,000 who did not so die. The limit of deviation or
error calculated by Poisson's formula will be the square root of
8_xy_/_z_. Replacing _x_, _y_ and _z_ by the figures of the example we
get the square root of (8?1000?699000)/700,000, which works out to the
square root of 7988.
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