Environmental Engineering Reference
In-Depth Information
EXAMPLE 10.10
N
1
ˆ µ x
=
x
(10.56)
i
N
Monthly grab samples from a contaminated lake are
collected over a 30-month period, and the measured
concentrations in mg/l of a substance of interest are as
follows:
i
=
1
N
1
ˆ
(
ˆ
)
2
σ
2
=
x
µ
(10.57)
x
i
x
N
1
i
=
1
N
(
)
ˆ
3
x
µ
i
x
14.44
2.56
3.44
2.90
1.80
(10.58)
N
ˆ
i
=
1
g
=
1.44
7.25
7.34
3.43
2.16
x
ˆ
(
N
1
)(
N
2
)
σ
3
x
1.75
1.74
3.91
1.33
36.56
1.04
3.54
11.25
3.33
2.32
where ˆ µ x , ˆ σ 2 , and g x are unbiased estimates of the mean
( μ x ), variance ( σ 2 ), and skewness ( g x ) of the population
distribution from which the N samples of x , denoted by
x i , are drawn. unbiased estimates of the variance, ˆ σ 2 ,
and skewness, g x , are generally dependent on the under-
lying population distribution. The accuracy of the esti-
mated skewness, g x , is usually of most concern, since it
involves the summation of the cubes of deviations from
the mean and is therefore subject to larger errors caused
by inaccurate outliers.
Estimates of the mean, variance, and skewness
given by Equations (10.56-10.58) are based on samples
of a random variable and are therefore random
variables themselves. The standard deviation of an
estimated parameter is commonly called the standard
error , and the standard errors of the estimated mean,
variance, and skewness are given by the following
relations
7.91
6.28
4.72
2.75
1.31
2.70
0.38
1.46
6.83
1.20
Estimate the mean, standard deviation, and skewness
of the population from which these samples were
drawn and assess the uncertainty of the parameter
estimates.
Solution
The measured concentrations are represented by c i ,
i = 1, N , where N = 30. using Equations (10.56-10.58)
yields the following parameter estimates:
N
1
1
30
ˆ
µ c
=
c
=
(
149 1
. )
=
4 97
.
mg/L
i
N
i
=
1
N
1
1
30 1
ˆ
(
ˆ
) =
2
σ
=
c
µ
(
1330 3
. )
=
6 77
.
mg/L
ˆ
σ
c
i
c
N
1
x
S
=
(10.59)
ˆ
i
=
1
µ
x
N
N
ˆ
(
ˆ
)
3
1 0 75
2
.
g
c
µ
x
(10.60)
ˆ
i
c
S
=
σ
N
ˆ
σ
x
x
N
ˆ
i
=
1
g
=
c
ˆ
(
N
1
)(
N
2
)
σ
3
c
= [
]
0 5
.
A B
log
(
N
/
10
)
(10.61)
S g
10
10
30
30 1 30 2
32087 1
6 7
.
ˆ
x
=
= .
3 82
3
(
)(
)
.
7
where A and B are given by
The uncertainties in the parameter estimates can be
measured by their standard errors as given by Equa-
tions (10.59-10.61), where
ˆ
ˆ
0.33 0.08
+
g
if
g
0.90
x
x
A
=
(10.62)
ˆ
ˆ
0.52 0.30
+
g
if
g
>
0.90
x
x
ˆ
and
σ
6 77
30
.
c
S
=
=
=
1 24 mg/L
.
µ
ˆ
c
N
0.94 0.26
g
ˆ
if
g
g
ˆ
ˆ
>
1.50
x
x
B
=
(10.63)
ˆ
1 0 75
2
+
.
g
1 0 75 3 82
2 30
+
.
( .
)
0.55 if
1.50
ˆ
c
S
=
σ
=
6 77
.
=
1 72 mg/L
.
x
ˆ
σ
c
c
N
(
)
By combining Equations (10.56-10.63), it can be
shown that for a given sample size, the relative accuracy
of the skewness is much less than the relative accuracies
of both the mean and standard deviation, especially for
small sample sizes.
A = −
0 52 0 30 3 82
.
+
.
.
=
0 62
.
B = 0.55
= [
]
= [
]
0 5
.
0 5
.
S g
10
A B
log
(
N
/
10
)
10
0 62 0 55
.
.
log
(
30 10
/
)
=
1
.
52
10
10
ˆ
c
 
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