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. Whatever
 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 distributions
 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