XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 299
as before, and the four streams that result will bear the proportions
 ¢%: 39% : 3gp?: p®. The final set, at the heads of the
vertical strips, will give the streams proportions ¢*: 4¢3%p : 69%? :
49p®: p*, and these streams will accumulate between the strips
and give a r~presentation of the binomial by a kind of histogram,
as shown. Of course as many rows of wedges may be provided
as may be desired.
This kind of apparatus was originally devised by Sir Francis
Galton (ref. 1) in a form that gives roughly the symmetrical
binomial, a stream of shot being allowed to fall through rows of
nails, and the resultant streams being collected in partitioned
spaces. The apparatus was generalised by Professor Pearson,
who used rows of wedges. fixed to movable slides, so that they
could be adjusted to give any ratio of g:p. (Ref. 13.)
6. The values of the mean and standard-deviation of a binomial
distribution may be found from the terms of the series directly,
as well as by the method of Chap. XIII. (the calculation was
in fact given as an exercise in Question 8, Chap. VII., and
Question 6, Chap. VIIL). Arrange the terms under each other
as in col. 1 below, and treat the problem as if it were an arithmetical
 example, taking the arbitrary origin at 0 successes: as
XV is a factor all through, it may be omitted for convenience.
(1) oh (4)
Frequency f. Dev. &. JE
qn Te es —y
nglp nq" 1p nq" 1p
-1
i 3 — n(n —-1)g—2p? 2n(n —- 1)g"—2p?
mn-1)n-2) | n(n—1)(n-2) 3n(n—-1)(n-2
23 7 Ml fy peg BEE D
The sum of col. 1 is of course unity, i.e. we are treating IV as
unity, and the mean is therefore given by the sum of the terms
in col. (3). But this sum is
n—-1)(n-2
np | "+ (n- Nye Tai) 1 Js 4 Sot ;
=np(q +p)" =np.
That is, the mean J/ is mp, as by the method of Chap. XIII