XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 295 
If p=q the effect of increasing m is to raise the mean and 
increase the dispersion. If p is not equal to ¢, however, not 
only does an increase in mn raise the mean and increase the 
dispersion, but it also lessens the asymmetry; the greater 
n, for the same value of p and ¢, the less the asymmetry. 
Thus if we compare the first distribution of the above table 
with that given by »=100, we have the following :— 
B.—Terms of the Binomial Series 10,000 (0'9 + 0-1), (Figures given 
to the nearest unit.) 
Number Number Number 
01 Frequency. of Frequency. of Frequency. 
Successes. Successes. Successes. 
y — : 1148 15 193 
3 1304 17 106 
16 1 1319 83 54 
59 1 1199 | 13 2. 
IX E1508 [& £12 988 | 2) I 
339 | 13 743 21 
596 1% 513 24 
’29 £7 : 
The maximum frequencies now occur for 9 and 10 successes, 
and the two “tails” are much more nearly equal. If, on the 
other hand, n is reduced to 2, the distribution is— 
Number of Successes, Frequency. 
8100 
1800 
100 
and the maximum frequency is at one end of the range. What- 
ever the values of p and ¢, if » is only increased sufficiently, the 
distribution may be treated as sensibly symmetrical, the necessary 
condition being (we state this without proof) that p —¢ shall be 
small compared with the standard-deviation npg. It is left 
to the student to calculate as an exercise the theoretical distribu- 
tions corresponding to the experimental results cited in Chapter 
XIII. (Question 1). 
4. The property of the binomial series used in the scheme of 
§ 2 for deducing the series with exponent » from that with 
exponent n-1 leads to two interesting methods—graphical and 
mechanical — for constructing approximate representations of