XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 309 
expected theoretical frequency in a certain interval is f, the 
standard error of sampling is /A(N —f)/N ; and if the divergence 
of the observed from the theoretical frequency exceed some 
three times this standard error, the divergence is unlikely to 
have occurred as a mere fluctuation of sampling. 
It should be noted, however, that the ordinate of the normal 
curve at the middle of an interval does not give accurately the 
area of that interval, or the number of observations within it: it 
would only do so if the curve were sensibly straight. To deal 
strictly with problems as to fluctuations of sampling in the 
frequencies of single intervals or groups of intervals, we require, 
accordingly, some convenient means of obtaining the number of 
observations, in a given normal distribution, lying between any 
two values of the variable. 
16. If an ordinate be erected at a distance z/o from the mean, 
in a normal curve, it divides the whole area into two parts, the 
ratio of which is evidently, from the mode of construction of the 
curve, independent of the values of y, and of o. The calculation 
of these fractions of area for given values of z/s, though a long 
and tedious matter, can thus be done once for all, and a table 
giving the results is useful for the purpose suggested in § 15 and 
in many other ways. References to complete tables are cited at 
the end of this work (list of tables, pp. 357-8), the short table below 
being given only for illustrative purposes. The table shows the 
greater fraction of the area lying on one side of any given ordinate ; 
e.g. 0'53983 of the whole area lies on one side of an ordinate at 
0-1c from the mean, and 046017 on the other side. It will be 
seen that an ordinate drawn at a distance from the mean equal to 
the standard-deviation cuts off some 16 per cent. of the whole 
area on one side ; some 68 per cent. of the area will therefore be 
contained between ordinates at +o. An ordinate at twice the 
standard-deviation cuts off only 2:3 per cent., and therefore some 
954 per cent. of the whole area lies within a range of +20. At 
three times the standard-deviation the fraction of area cut off is 
reduced to 135 parts in 100,000, leaving 997 per cent. within a 
range of +30. This is the basis of our rough rule that a range 
of 6 times the standard-deviation will in general include the 
great bulk of the observations: the rule is founded on, and is only 
strictly true for, the normal distribution. For other forms of 
distribution it need not hold good, though experience suggests 
that it more often holds than not. The binomial distribution, 
especially if p and ¢ be unequal, only becomes approximately normal 
when 7 is large, and this limitation must be remembered in applying 
the table given, or similar more complete tables, to cases in which 
the distribution is strictly binomial.