XIIL.—SIMPLE SAMPLING OF ATTRIBUTES. 259
Mean M = 2-000, standard-deviation c=1296. Actual proportion
of successes 2:00/12 =0-1667, agreeing with the theoretical value
to the fourth place of decimals. Of course such very close
agreement is accidental, and not to be always expected.
(3) (G. U. Yule.) The following may be taken as an illustra-
tion based on a smaller number of observations. Three dice were
thrown 648 times, and the numbers of 5s or 6’s noted at
each throw. p=1/3, ¢=2/3. Theoretical mean 1. Standard-
deviation, 0-816.
Frequency-distribution observed :—
Successes, Frequency.
“ 179
i 298
2 141
o 30
Total 648
M=1'034, 0=0823. Actual proportion of successes 0:345.
For other illustrations, some of which are cited in the questions
at the end of this chapter, the student may be referred to the
list of references on p. 273. The student should notice that in
all the distributions given a range of six times the standard-
deviation includes either all, or the great bulk of, the observations,
as in most frequency-distributions of the same general form. We
shall make use of this rule below, § 13.
8. In deducing the formule (1) and (2) for the standard-
deviations of simple sampling in the cases with which we have
been dealing, only one condition has been explicitly laid down as
necessary, viz. the independence of the several drawings, tossings,
or other events composing the sample. But in point of fact this
is not the only nor the most fundamental condition which has
been explicitly or implicitly assumed, and it is necessary to realise
all the conditions in order to grasp the limitations under which
alone the formule arrived at will hold. Supposing, for example,
that we observe among groups of 1000 persons, at different times
or in different localities, various percentages of individuals
possessing certain characteristics —dark hair, or blindness, or
insanity, and so forth. Under what conditions should we
expect the observed percentages to obey the law of sampling
that we have found, and show a standard-deviation given by
equation (2)?
(a) In the first place we have tacitly assumed throughout the
preceding work that our dice or our coins were the same set or
