Biology Reference
In-Depth Information
a symmetric histogram as in Figure 6-4(A). Note that the mean and the
median coincide. Now suppose that we take a data point that is to
the right of the center and move it farther to the right [Figure 6-4(B)]. The
mean moves to the right, but the median says the same. The data in
Figure 6-4(B) are skewed to the right.
This example illustrates the following ideas:
1. The mean is sensitive to extreme values, but the median is less so.
2. If a body of data has a long tail to the right but not to the left, the
data is skewed to the right.
3. If the data is skewed to the right, then the mean will lie to the right
of the median.
One way to quantify the balance between the left and right tails of a
histogram is to use its skewness. The skewness
g
of a distribution is
defined as the quotient of the third moment
m
3
about the mean E(X)
and the third power of the SD
s
and is given by the expression
3
g ¼
m
3
s
E
fð
X
E
ð
X
ÞÞ
g
3
¼
2
:
The skewness for symmetric distributions
2
3
=
½
E
fð
X
E
ð
X
ÞÞ
g
is zero; it is positive if the distribution develops a longer tail to the right;
and it is negative if the distribution develops a longer tail to the left.
Accordingly, distributions skewed to the right are called positively skewed
and distributions skewed to the left are called negatively skewed.
In Figure 6-5(A), we have a symmetric distribution. The degree of
variability is quantified by the SD - reduced variability corresponds to
Median
Mean
A
Median
Mean
B
FIGURE 6-4.
A schematic representation of skewness as illustrated by the relative position of the mean versus the
median. In panel A, the mean and median coincide, whereas in panel B the mean is to the right of the
median.
