XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 313 
Example iv.—The diagram of fig. 49 shows that the number of 
statures recorded in the group “62 in. and less than 63” is 
markedly less than the theoretical value. Could such a difference 
occur owing to fluctuations of simple sampling; and if so, how 
often might it happen ? 
The actual frequency recorded is 169. To obtain the theoreti- 
cal frequency we may either take it as given roughly by the 
ordinate in the centre of the interval, or, better, use the integral 
table. Remembering that statures were only recorded to the 
nearest % in., the true limits of the interval are 6115-6212 or 
61:94-62'94, mid-value 62:44. This is a deviation from the 
mean (67°46) of 5°02. Calculating the ordinate of the normal 
curve directly we find the frequency 197-8. This is certainly, as 
is evident from the form of the curve, a little too small. The 
interval actually lies between deviations of 4:52 in. and 552 
in., that is, 17590 and 2:1480. The corresponding fractions of 
area are 0'96071 and 0-98418, difference, or fraction of area 
between the two ordinates, 0:02347. Multiplying this by the 
whole number of observations (8585) we have the theoretical 
frequency 201-5. 
The difference of theoretical and observed frequencies is therefore 
32:5. But the proportion of observations which should fall into 
the given class is 0023, the proportion falling into other classes 
0-977, and the standard error of the class frequency is accordingly 
0-023 x 0977 x 8585 =14'0. As the actual deviation is only 
2:32 times this, it could certainly have occurred as a fluctuation of 
sampling. 
The question how often it might have occurred can only be 
answered if we assume the distribution of fluctuations of sampling 
to be approximately normal. It is true that 2 and gq are very 
unequal, but then =z is very large (8585)—so large that the 
difference of the chances is fairly small compared with npg 
(about one-fifteenth). Hence we may take the distribution of 
errors as roughly normal to a first approximation, though a 
first approximation only. The tables give 0-990 of the area 
below a deviation of 232s, so we would expect an equal or 
greater deficiency to occur about 10 times in 1000 trials, or once 
in a hundred. 
REFERENCES. 
The Binomial Machine. 
(1) GavroN, FraNcis, Natural Inheritance ; Macmillan & Co. London, 1889, 
(Mechanical method of forming a binomial or normal distribution, 
SREP ya p. 63; for Pearson’s generalised machine, see below, 
rel. .