Geoscience Reference
In-Depth Information
h e most signii cant change observed is in the mean (16.4490), which is
substantially lower due to the presence of the outlier. h is example clearly
demonstrates the sensitivity of the mean to outliers. In contrast, the median
of 16.9722 is relatively unaf ected.
3.4 Theoretical Distributions
We have now described the empirical frequency distribution of our sample.
A histogram is a convenient way to depict the frequency distribution of the
variable
x
. If we sample the variable sui ciently ot en and the output ranges
are narrow, we obtain a very smooth version of the histogram. An ini nite
number of measurements
N
→∞ and an ini nitely small class width produce
the random variable's
probability density function
(PDF). h e probability
distribution density
f
(
x
) dei nes the probability that the variable has a value
equal to
x
. h e integral of
f
(
x
) is normalized to unity, i.e., the total number of
observations is one. h e
cumulative distribution function
(CDF) is the sum
of the frequencies of a discrete PDF or the integral of a continuous PDF. h e
cumulative distribution function
F
(
x
) is the probability that the variable will
have a value less than or equal to
x
.
As a next step, we need to i nd appropriate theoretical distributions that
i t the empirical distributions described in the previous section. h is section
therefore introduces the most important theoretical distributions and
describes their application.
Uniform Distribution
A
uniform
or
rectangular distribution
is a distribution that has a constant
probability (Fig. 3.4). h e corresponding probability density function is
where the random variable
x
has any of
N
possible values. h e cumulative
distribution function is
h e probability density function is normalized to unity
