THEORY OF STATISTICS. 
can be drawn freehand, or by aid of a curve ruler, through the 
tops of the ordinates so determined. The logarithms of # in the 
table on p. 303 are given to facilitate the multiplication. The only 
point in which the student is likely to find any difficulty is 
in the use of the scales: he must be careful to remember 
that the standard-deviation must be expressed in terms of the 
class-interval as a wnat in order to obtain for y, a number of 
observations per interval comparable with the frequencies of his 
table. 
The process may be varied by keeping the normal curve 
drawn to one scale, and redrawing the actual distribution 
80 as to make the area, mean, and standard-deviation the 
same. Thus suppose a diagram of a normal curve was printed 
once for all to a scale, say, of y,=5 inches, o=1 inch, and 
it were required to fit the distribution of stature to it. 
Since the standard-deviation is 2-57 inches of stature, the 
scale of stature is 1 inch =2'57 inch of stature, or 0:389 inches 
=1 inch of stature ; this scale must be drawn on the base of the 
normal-curve diagram, being so placed that the mean falls 
at 67-46. As regards the scale of frequency-per-interval, this 
is given by the fact that the whole area of the polygon showing 
the actual distribution must be equal to the area of the 
normal curve, that is 5 «/2r=1253 square inches. If, therefore, 
the scale required is n= observations per interval to the inch, 
we have, the number of observations being 8585, 
8585 
nx 2:57 RRs 
which gives n= 266-6. 
Though the second method saves curve drawing, the first, 
on the whole, involves the least arithmetic and the simplest 
plotting. 
15. Any plotting of a diagram, or the equivalent arithmetical 
comparison of actual frequencies with those given by the 
fitted normal distribution, affords, of course, in itself, only a 
rough test, of a practical kind, of the normality of the given 
distribution. The question whether all the observed differences 
between actual and calculated frequencies, taken together, 
may have arisen merely as fluctuations of sampling, so that the 
actual distribution may be regarded as strictly normal, neglecting 
such errors, is a question of a kind that cannot be answered in 
an elementary work (cf. ref. 22). At present the student is in 
a position to compare the divergences of actual from calculated 
frequencies with fluctuations of sampling in the case of single 
class-intervals, or single groups of class-intervals only. If the 
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