2 
XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 307 1 
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more general still, without introducing the concepti mot ina olek 
binomial at all, by founding the curve on more or less complex 
cases of the theory of sampling for variables instead of ¥ aftri- £5 
butes. If a variable is the sum (or, within limits, some slightly(i~ > 
more complicated function) of a large number of other varia 
then the distribution of the compound or resultant variable is 
normal, provided that the elementary variables are independent, 
or nearly so (¢f. ref. 6). The forms of the frequency-distribu- 
tions of the elementary variables affect the final distribution less 
and less as their number is increased: only if their number is 
moderate, and the distributions all exhibit a comparatively high 
degree of asymmetry of uniform sign, will the same sign of 
asymmetry be sensibly evident in the distribution of the compound 
variable. On this sort of hypothesis, the expectation of normality 
in the case of stature may be based on the fact that it is a highly 
compound character--depending on the sizes of the bones of the 
head, the vertebral column, and the legs, the thickness of the 
intervening cartilage, and the curvature of the spine—the elements 
of which it is composed being at least to some extent independent, 
v.e. by no means perfectly correlated with each other, and their 
frequency-distributions exhibiting no very high degree of asym- 
metry of one and the same sign. The comparative rarity of 
normal distributions in economic statistics is probably due in part 
to the fact that in most cases, while the entire causation is 
certainly complex, relatively few causes have a largely predominant 
influence (hence also the frequent occurrence of irregular 
distributions in this field of work), and in part also to a high 
degree of asymmetry in the distributions of the elements on which 
the compound variable depends. Errors of observation may in 
general be regarded as compounded of a number of elements, due 
to various causes, and it was in this connection that the normal 
curve was first deduced, and received its name of the curve of 
errors, or law of error. 
14. If it be desired to compare some actual distribution 
with the normal distribution, the two distributions should be 
superposed on one diagram, as in fig. 49, though, of course, on 
a much larger scale. When the mean and standard-deviation 
of the actual distribution have been determined, Y, 1s given by 
equation (5); the fit will probably be slightly closer if the 
standard-deviation is adjusted by Sheppard’s correction (Chap. 
XI. § 4). The normal curve is then most readily drawn by plot- 
ting a scale showing fifths of the standard-deviation along the 
base line of the frequency diagram, taking ‘the mean as origin, 
and marking over these points the ordinates given by the figures 
of the table on p. 303, multiplied in each case by ,- The curve