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46 HANDBOOK OF PORTFOLIO MATHEMATICS
worse than tenth place were recorded as a tenth-place finish. Thus, we had
an open distribution.
The third measure of central tendency is themode—the most frequent
occurrence. The mode is the peak of the distribution curve. In some distri-
butions there is no mode and sometimes there is more than one mode. Like
the median, the mode can often be regarded as a superior measure of cen-
tral tendency. The mode is completely independent of extreme outlier val-
ues, and it is more readily obtained than the arithmetic mean or the median.
We have seen how the median divides the distribution into two equal
areas. In the same way a distribution can be divided by threequartiles(to
give four areas of equal size or probability), or ninedeciles(to give 10 areas
of equal size or probability) or 99percentiles(to give 100 areas of equal
size or probability). The 50th percentile is the median, and along with the
25th and 75th percentiles give us the quartiles. Finally, another term you
should become familiar with is that of aquantile.A quantile is any of the
N−1 variate-values that divide the total frequency into N equal parts.
We now return to the mean. We have discussed the arithmetic mean
as a measure of central tendency of a distribution. You should be aware
that there are other types of means as well. These other means are less
common, but they do have significance in certain applications.
First is thegeometric mean, which we saw how to calculate in the first
chapter. The geometric mean is simply the Nth root of all the data points
multiplied together.G=
(N
∏
i= 1Xi) 1 /N
(2.02)
where: G=The geometric mean.
Xi=The ith data point.
N=The total number of data points in the distribution.The geometric mean cannot be used if any of the variate-values is zero or
negative.
Another type of mean is theharmonic mean.This is the reciprocal of
the mean of the reciprocals of the data points.1 /H= 1 /N
∑N
i= 11 /Xi (2.03)where: H=The harmonic mean.
Xi=The ith data point.
N=The total number of data points in the distribution.