Descriptive Statistics 331

This is of particular interest if deviations from the mean are more harmful
the larger they are. In the conext of the variance, one often speaks of the
averaged squared deviations as a risk measure.
The sample variance is defined by

=−∑

s

n

xx

1

()i

i

n

22

1

(A.4)

using the sample mean. If, in equation (A.4) we use the divisor n – 1 rather
than just n, we obtain the corrected sample variance.
Related to the variance is the even more commonly stated measure of
variation, the standard deviation. The reason is that the units of the stan-
dard deviation correspond to the original units of the data whereas the units
are squared in the case of the variance. The standard deviation is defined to
be the positive square root of the variance. Formally, the sample standard
deviation is

= ∑
−

−

=

s

n

xx

1

1

()i

i

n

2

1

(A.5)

Skewness The last measure of variation we describe is skewness. There
exist several definitions for this measure. The Pearson skewness is defined as
three times the difference between the median and the mean divided by the
standard deviation.^4 Formally, the Pearson skewness for a sample is

=

−

s

mx

s

3( )

P

d

where m denotes the median.
As can be easily seen, for symmetrically distributed data, skewness is
zero. For data with the mean being different from the median and, hence,
located in either the left or the right half of the data, the data are skewed.
If the mean is in the left half, the data are skewed to the left (or left skewe)
since there are more extreme values on the left side compared to the right
side. The opposite (i.e., skewed to the right, or right skewed), is true for data
whose mean is further to the right than the median. In contrast to the MAD
and variance, the skewness can obtain positive as well as negative values.

(^4) To be more precise, this is only one of Pearson’s skewness coefficients. Another one
not presented here employs the mode instead of the mean.