CHAPTER XV.
THE BINOMIAL DISTRIBUTION AND THE
NORMAL CURVE.
1-2. Determination of the frequency-distribution for the number of successes
in n events: the binomial distribution—3. Dependence of the form
of the distribution on p, ¢ and n—4-5. Graphical and mechanical
methods of forming representations of the binomial distribution—
6. Direct calculation of the mean and the standard-deviation from
the distribution—7-8. Necessity of deducing, for use in many
practical cases, a continuous curve giving approximately, for large
values of n, the terms of the binomial series—9. Deduction of the
normal curve as a limit to the symmetrical binomial—10-11. The
value of the central ordinate—12. Comparison with a binomial dis-
tribution for a moderate value of n—13. Outline of the more general
conditions from which the curve can be deduced by advanced methods—
14. Fitting the curve to an actual series of observations—15, Difficulty
ofa complete test of fit by elementary methods— 16. The table of areas
of the normal curve and its use—17. The quartile deviation and the
‘“ probable error ”—18. Illustrations of the application of the normal
curve and of the table of areas.
1. In Chapters XIII. and XIV. the standard-deviation of the
number of successes in n events was determined for the several
more important cases, and the applications of the results indicated.
For the simpler cases of artificial chance it is possible, however, to
go much further, and determine not merely the standard-deviation
but the entire frequency-distribution of the number of successes.”
This we propose to do for the case of “simple sampling,” in which
all the events are completely independent, and the chances » and
q the same for each event and constant throughout the trials.
The case corresponds to the tossing of ideally perfect coins (homo-
geneous circular discs), or the throwing of ideally perfect dice
(homogeneous cubes).
2. If we deal with one event only, we expect in IV trials, Ng
failures and Np successes. Suppose we how combine with the
results of this first event the results of a second. The two events
are quite independent, and therefore, according to the rule of
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