330 The Basics of financial economeTrics
extremely to the left and half of the observations are extremely to the right
of the mean, thus, leveling out, on average.
Of the three measures of central tendency, the mode is the measure with
the greatest loss of information. It simply states which value occurs most
often and reveals no further insight. This is the reason why the mean and
median enjoy greater use in descriptive statistics. While the mean is sensitive
to changes in the data set, the mode is absolutely invariant as long as the
maximum frequency is obtained by the same value. The mode, however, is
of importance, as will be seen, in the context of the shape of the distribu-
tion of data. A positive feature of the mode is that it is applicable to all data
levels.
Variation
Rather than measures of the center or one single location, we now discuss
measures that capture the way the data are spread either in absolute terms
or relative terms to some reference value such as, for example, a measure
of location. Hence, the measures introduced here are measures of varia-
tion. We may be given the average return, for example, of a selection of
stocks during some period. However, the average value alone is incapable
of providing us with information about the variation in returns. Hence,
it is insufficient for a more profound insight into the data. Like almost
everything in real life, the individual returns will most likely deviate from
this reference value, at least to some extent. This is due to the fact that the
driving force behind each individual object will cause it to assume a value
for some respective attribute that is inclined more or less in some direction
away from the standard.
While there are a great number of measures of variation that have been
proposed in the finance literature, we limit our coverage to those that are
more commonly used in financial econometrics—absolute deviation, stan-
dard deviation (variance), and skewness.
Absolute deviation The mean absolute deviation (MAD) is the average devia-
tion of all data from some reference value (which is usually a measure of
the center). The deviation is usually measured from the mean. The MAD
measure takes into consideration every data value.
Variance and Standard deviation The variance is the measure of variation used
most often. It is an extension of the MAD in that it averages not only the
absolute but the squared deviations. The deviations are measured from the
mean. The square has the effect that larger deviations contribute even more
to the measure than smaller deviations as would be the case with the MAD.