Environmental Engineering Reference
In-Depth Information
Taking the expected value of the random variable
S
2
yields
E S
x
This estimate of
σ
X
50
is appropriate if the population
distribution of
x
is normal or approximately normal, and
for
N
≥ 30, the sampling distribution of the median is
very nearly normal.
(
2
, which is related to the population variance,
σ
2
, by the relation
E S
x
(
2
)
= σ
2
(10.81)
x
10.7.5 Coefficient of Variation
Based on this result, the standard deviation of the
population can be estimated by
S
x
, and it can be shown
that the standard deviation of,
S
x
, is given by
The coefficient of variation,
C
v
, is defined as the ratio of
the standard deviation to the mean in the population
from which the data is drawn. Hence,
σ
v
=
σ
x
σ
=
(10.82)
x
C
µ
(10.86)
S
x
2
N
x
The magnitude of
C
v
is commonly used as a measure
of the relative amount of variation in the population.
Many types of water-quality data sampled from differ-
ent locations show increased variance corresponding
increased means, with the coefficient of variation
remaining approximately constant. This is particularly
true for bacteria concentrations. Values of
C
v
are also
an indicator of skewness, particularly when only positive
values of
x
are admissible and
C
v
> 1. Sample estimates
of
C
v
can be derived from sample data using the statistic
CoV, where (Sokal and rohlf, 2012)
Equation (10.82) neglects the skewness of the popu-
lation distribution.
10.7.3 Coefficient of Skewness
The sample skewness,
C
s
, of
N
realizations of the random
variables,
X
i
,
i
∈ [1,
N
] is defined by the equation
N
1
N
∑
C
s
=
⋅
(
X X
−
)
3
(10.83)
i
3
S
(
N
−
1
)(
N
−
2
)
x
i
=
1
1
4
N
S
X
and
S
x
is the sample standard deviation derived using
Equation (10.80). The skewness coefficient estimated
using Equation (10.83) is an approximately unbiased
estimate of the population skewness coefficient,
g
x
, with
the exact bias depending on the underlying distribution
of the random variable (Bobée and robitaille, 1975).
The standard error of
C
s
derived using Equation (10.83),
σ
C
s
, can be estimated by
x
COV =
1
+
(10.87)
and the expected value of CoV is
C
v
. The standard error
of CoV,
σ
CoV
, is given by
C
1 2
2
+
C
2
(10.88)
v
v
σ
COV
=
N
6
N N
(
−
1
)
This estimate of
σ
CoV
is appropriate if the population
distribution is normal or nearly normal and
N
≥ 100.
(10.84)
σ
C
s
=
(
N
+
1
)(
N
+
2
)(
N
+
3
)
EXAMPLE 10.13
10.7.4 Median
The median,
X
50
, is defined as the middle value of a data
set, such that 50% of the data are greater than the
median, and 50% of the data are less than the median.
The median is less sensitive to the presence of outliers
in the data, and in cases where the distribution of the
data is skewed, the median is a better measure of the
the central tendency of the data. The standard error,
σ
X
50
, of the median is given by (McBean and rovers,
1998)
using the sample data given in Example 10.9, calculate
the expected values and the standard errors of the
mean, standard deviation, coefficient of skewness,
median, and coefficient of variation.
Solution
From the given data,
N
= 50 and sample statistics are
calculated as follows:
N
1
1
50
1
∑
(10.85)
C
=
C
i
=
(
233 5
. )
=
4 67
. mg/L
σ
=
σ
X
50
x
N
2
N
i
=
1
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