XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 299 
as before, and the four streams that result will bear the propor- 
tions ¢%: 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 arith- 
metical 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