Geology Reference
In-Depth Information
2
, is the average squared difference of the observed values
from their mean and is given by:
The variance,
1
n
2
2
=
zm
(2)
i
n
i
1
The standard deviation,
, is simply the square root of the variance. This
shows the spread of the values from the mean. The coefficient of skewness
showing the symmetry of the distribution is defined as:
1
n
3
zm
i
n
i
1
Coeff. of Skewness =
(3)
3
The coefficient of variation (CV) is a statistics that is often used as an
alternative to skewness to describe the shape of the distribution. It is defined
as the ratio of the standard deviation to the mean:
CV =
m
(4)
Some Remarks
The mean is quite sensitive to erratic extreme values. If the 145 ppm value
in the above data set had been 1450 ppm, the mean would change to 110.6
ppm. The median, however, would be unaffected by this change because it
depends only how many values are above and below it.
The variance, since it involves squared differences, is also quite sensitive
to the extreme values. The variance of the 100 values is 688. Standard
deviation is often used instead of the variance since its units are the same
as the units of the variable.
The coefficient of skewness suffers even more than the mean and variance
from sensitivity to erratic extreme values. A coefficient of variation greater
than one indicates the presence of some erratic high sample values that may
have a significant impact on the final estimates. The CV for the above data
is 0.269, which reflects the fact that the histogram does not have a long tail
of high values.
Bivariate Distribution
In the field of hydrogeology, due to scarcity of data, we always prefer to
work on multivariable information. A typical data set for a two variable
system is given in Table 3.
In the data set in Table 3, values of two variables are available at all the
100 locations. Values of the first variable are written above the circle (denoting
the locations of measurement) and the values of the second variable are
written below the circle in Italics. The basic statistics of the two sets of data
are given in Table 4 for comparison.
