Environmental Engineering Reference
In-Depth Information
calculated statistic would be exceeded given that the
(null) hypothesis that the sample distribution was drawn
from the proposed population distribution is true. In
hypothesis testing, the significance level, α , generally rep-
resents the risk of falsely rejecting the null hypothesis.
m a
+ −
P X x
(
>
)
=
,
m
=
1
,
,
N
(10.52)
X
m
N
1 2
a
where a is a parameter that depends on the population
distribution. For the normal/log-normal and gamma dis-
tributions, a = 0.375, and a = 0.40 is a good compromise
for the usual situation in which the exact distribution is
unknown (Bedient and Huber, 2002). Visual compari-
son of the sample probability distribution with various
theoretical population distributions gives a preliminary
idea of which theoretical distributions might provide
the best fit to the observed data.
10.5.1 Sample Probability Distribution
To assist in identifying a theoretical population proba-
bility distribution that adequately describes observed
data, it is generally useful to graphically compare the
probability distribution of observed data with various
theoretical probability distributions. The first step in
plotting the sample probability distribution is to rank
the data, such that for N observations, a rank of 1 is
assigned to the observation with the largest magnitude,
and a rank of N is assigned to the observation with the
lowest magnitude. The exceedance probability of the
m -ranked observation, x m , denoted by P X ( X > x m ), is
commonly estimated by the relation
EXAMPLE 10.7
Monthly grab samples at a monitoring station along a
river over a 30-month period have yielded the results
shown in Table 10.1. It is proposed that the samples are
drawn from a log-normal distribution with a natural-log
mean of 1.20 and a standard deviation of 0.80. Compare
the observed and theoretical cumulative distributions
and make a visual assessment of the
m
N
(10.50)
P X x
(
>
)
=
1 ,
m
=
1
,
,
N
X
m
+
level of
agreement.
or as a CDF
Solution
m
N
1 (10.51)
P X x
(
<
)
= −
1
1 ,
m
=
,
,
N
X
m
From the given data: N = 30, μ y = 1.20, and σ y = 0.80,
where y = ln x and x represents the measured data.
Taking a = 0.40, Equation (10.52) gives the sample CDF,
F ( c ), of the concentration, c , as
+
Equation (10.50) is called the Weibull formula
(Weibull, 1939), and is widely used in practice. The main
drawback of the Weibull formula for estimating the
cumulative probability distribution from measured data
is that it is asymptotically exact (as the number of obser-
vations approaches infinity) only for a population with
an underlying uniform distribution, which is relatively
rare in nature. To address this shortcoming, Gringorten
(1963) proposed that the exceedance probability of
observed data be estimated using the relation
m a
+ −
0 40
30 1 2 0 40
+ −
m
.
( .
F c
( )
= −
1
= −
1
N
1 2
a
)
(10.53)
m
0.40
30.2
= −
1
=
1 013 0.033 m
.
where m is the rank of the data. ranking the measured
data and applying Equation (10.53) yields the results
shown in Table 10.2.
TABLE 10.1. Concentrations in Monthly Grab Samples at a Monitoring Station
Sample
Concentration (mg/l)
Sample
Concentration (mg/l)
Sample
Concentration (mg/l)
1
1.02
11
2.72
21
0.78
2
1.63
12
14.30
22
1.10
3
14.30
13
4.71
23
4.28
4
1.69
14
1.93
24
4.00
5
2.20
15
2.67
25
1.34
6
2.89
16
4.91
26
2.48
7
2.52
17
1.17
27
3.29
8
1.83
18
2.52
28
0.39
9
15.49
19
4.88
29
3.26
10
11.57
20
3.14
30
6.41
 
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