The error varies as the square root of
the number. Now
21/3.16 = 6.6 = limit of error for 10 in 20.
5.6/3.16 = 1.8 = mean error " " "
4/3.16 = 1.2 = probable error " " "
It will be seen that these calculated results agree fairly well with
those actually obtained. The rule by which these calculations are made
is important and will bear further illustration. To calculate the
number of heads in 3200 throws, we have to find the limit of error on a
true average of 1600 in 3200. This being 16 times the average of 100 in
200, the corresponding errors must be multiplied by 4. This gives
21?4 = 84 = limit of error.
5.6?4 = 22.4 = mean error.
4?4 = 16 = probable error.
The results I have actually obtained with these large numbers are hardly
enough to base much on, but have a value by way of confirmation.
Expecting 1600 heads, the actual numbers were 1560, 1596, 1643, 1557,
1591, 1605, 1615, 1545.
It will be seen that exactly half are within the probable error; but
this, considering the small number of results, must be more or less of
an accident; it is more to the point they are all well within the limits
of error.
I have a large number of other results which with a single exception are
all in accord with those given; and this exception only just overstepped
the limits.
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