CHAPTER VIL
AVERAGES.
1. Necessity for quantitative definition of the characters of a frequency-
distribution—2. Measures of position (averages) and of dispersion—s3.
The dimensions of an average the same as those of the variable—4.
Desirable properties for an average to possess—5. The commoner forms
of average—6-13. Thearithmetic mean : its definition, calculation, and
simpler properties—14-18. The median : its definition, calculation, and
simpler properties—19-20. The mode: its definition and relation to
mean and median—21. Summary comparison of the preceding forms
of average—22-26. The geometric mean: its definition, simpler pro-
perties, and the cases in which it is specially applicable—27. The
harmonic mean : its definition and calculation,
1. IN § 2 of the last chapter it was pointed out that a classification
of the observations in any long series is the first step necessary
to make the observations comprehensible, and to render possible
those comparisons with other series which are essential for any
discussion of causation. Very little experience, however, would
show that classification alone is not an adequate method, seeing
that it only enables qualitative or verbal comparisons to be made.
The next step that it is desirable to take is the quantitative
definition of the characters of the frequency-distribution, so that
quantitative comparisons may be made between the corresponding
characters of two or more series. It might seem at first sight
that very difficult cases of comparison could arise in which, for
example, we had to contrast a symmetrical distribution with a “J-
shaped ” distribution. As a matter of practice, however, we seldom
have to deal with such a case; distributions drawn from similar
material are, in general, of similar form. When we have to
compare the frequency-distributions of stature in two races of
man, of the death-rates in English registration districts in two
successive decades, of the numbers of petals in two races of the
same species of Ranunculus, we have only to compare with each
other two distributions of the same or nearly the same type.
9. Confining our attention, then, to this simple case, there are
two fundamental characteristics in which such distributions may
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