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3.2 Empirical Distributions 31

Measures of Central Tendency

Parameters of central tendency or location represent the most important
measures for characterizing an empirical distribution (Fig. 3.2). These val-
ues help to locate the data on a linear scale. They represent a typical or best
value that describes the data. The most popular indicator of central tendency
is the arithmetic mean, which is the sum of all data points divided by the
number of observations:

The arithmetic mean can also be called the mean or the average of an uni-
variate data set. The sample mean is often used as an estimate of the popula-
tion mean µ for the underlying theoretical distribution. The arithmetic mean
is sensitive to outliers, i.e., extreme values that may be very different from
the majority of the data. Therefore, the median as often used as an alterna-
tive measure of central tendency. The median is the x-value which is in the
middle of the data, i.e., 50% of the observations are larger than the median
and 50% are smaller. The median of a data set sorted in ascending order is
defi ned as

Symmetric Distribution Skew Distribution
15
Median 50 Median
Mean Mode
Mean Mode
40
10
f(x) f(x) 30

20
5
10 Outlier
0 0
8 10 12 14 16 0 2 4 6 8
x x
a b
Fig. 3.2 Measures of central tendency. a In an unimodal symmetric distribution, the mean,
median and mode are identical. b In a skew distribution, the median is between the mean and
mode. The mean is highly sensitive to outliers, whereas the median and mode are not much

influenced by extremely high and low values.