294 THEORY OF STATISTICS, 
p and ¢ may be interchanged without altering the value of 
any “term, and consequently terms equidistant from either 
end of the series are equal. If » and ¢ are unequal, on the 
other hand, the distribution is asymmetrical, and the more 
asymmetrical, for the same value of 7, the greater the inequality 
of the chances. The following table shows the calculated 
distributions for m=20 and values of p, proceeding by 0.1, 
from 0.1 to 0.5. When p=0.1, cases of two successes are the 
A. — Terms of the Binomial Series 10,000 (q+ p)? for Values of p 
from 0-1 to 0°5. (Ligures given to the nearest unit.) 
Number of p=0:1 p=0:2 »=0'3 p=0408 p=05 
Successes. q=0°9 g=0-8 g=07 g=0'0lg=05 
0 1216 115 8 — = 
2702 576 68 5 — 
92852 1369 278 s1 | 2 
Bi 1901 2054 716 ol 11 
§98 2182 1304 850 46 
319 1746 1789 746 148 
89 1091 1916 | 1244 370 
20 545 1643 1659 739 
4 222 1144 1797 1201 
1 74 654 1597 1602 
Y 0 308 1171 1762 
120 710 1602 
’ 355 1201 
146 739 
19 370 
Py he Nl 148 
a 3 46 
17 - — 11 
13 - — 2 
19 = = 
20 = 
most frequent, but cases of one success almost equally frequent : 
even nine successes may, however, occur about once in 10,000 
trials. As p is increased, the position of the maximum 
frequency gradually advances, and the two tails of the distribution 
become more nearly equal, until p=0.5, when the distribution 
is symmetrical. Of eourse, if the table were continued, the 
distribution for p=0.6 would be similar to that for ¢=0.6 
but reversed end for end, and so on. Since the standard- 
deviation is (npg)! and the maximum value of pg is given by 
p=g¢q, the symmetrical distribution has the greatest dispersion. 
a.