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32 3 Univariate Statistics

if N is odd and

if N is even. While the existence of outliers have an affect on the median, their

absolute values do not influence it. The quantiles provide a more general way
of dividing the data sample into groups containing equal numbers of observa-
tions. For example, quartiles divide the data into four groups, quintiles divide

the observations in five groups and percentiles define one hundred groups.

The third important measure for central tendency is the mode. The mode
is the most frequent x value or – in case of data grouped in classes – the
center of the class with the largest number of observations. The data have no
mode if there aren·t any values that appear more frequently than any of the
other values. Frequency distributions with one mode are called unimodal,
but there may also be two modes ( bimodal), three modes ( trimodal) or four
or more modes ( multimodal).
The measures mean, median and mode are used when several quantities
add together to produce a total, whereas the geometric mean is often used
if these quantities are multiplied. Let us assume that the population of an
organism increases by 10% in the first year, 25% in the second year, then

60% in the last year. The average increase rate is not the arithmetic mean,
since the number of individuals is multiplied (not added to) by 1.10 in the

first year, by 1.375 in the second year and 2.20 in the last year. The average
growth of the population is calculated by the geometric mean:

{

The average growth of these values is 1.4929 suggesting a ~49% growth
of the population. The arithmetic mean would result in an erroneous value
of 1.5583 or ~56% growth. The geometric mean is also an useful measure
of central tendency for skewed or log-normally distributed data. In other
words, the logarithms of the observations follow a gaussian distribution.
The geometric mean, however, is not calculated for data sets containing
negative values. Finally, the harmonic mean

{