Geoscience Reference
In-Depth Information
Table 13.1Values of the normal distribution function, expressed as [
F
(
x
)
−
0.5] according to Equation (13.36)
y
=
(
x
−
μ
/σ
)
0
0.5
1.0
1.5
2.0
2.5
3.0
5 erf(
y
/
√
2)
.
0
0
0.1915
0.3413
0.4332
0.4772
0.4938
0.4987
mode and mean values coincide, and its skew coefficient equals zero. The distribution
function is the integral of Equation (13.34) or, according to (13.5),
exp
2
dy
y
x
1
σ
√
2
1
2
−
μ
σ
F
(
x
)
=
−
(13.35)
π
−∞
This can also be written more concisely as
√
2
1
2
+
1
2
erf[(
x
F
(
x
)
=
−
μ
)
/
σ
]
(13.36)
in which the error function (cf. Equation (9.57)) is defined as
y
2
√
π
z
2
)
dz
erf(
y
)
=
exp(
−
(13.37)
0
Note that erf(
erf(
y
). The error function cannot be expressed in closed form;
however, it has been tabulated and close approximations are available for computation
(Abramowitz and Stegun, 1964, p. 299); it is also available in most computational soft-
ware. For convenient reference a few values are presented in Table 13.1. As will be seen
below, the values listed in Table 13.2 for zero skew (i.e. for
C
s
=
−
y
)
=−
0) are the quantiles of
(
x
of the normal distribution for the indicated probabilities.
The use of this distribution can be justified by means of the Central Limit Theorem;
this states that, if a random variable is the sum of
n
random, not necessarily independent,
variables, each with its own, not necessarily normal, density function with a finite mean
and variance, then the density of this random variable tends to the normal function
(13.34) as
n
increases. In hydrology the normal distribution is commonly assumed to be
applicable in the description of various types of central tendency observations, such as
mean annual temperatures, annual river discharge rates, among others.
−
μ
)
/σ
Example 13.5. Probability distribution of annual mean streamflows
In Figure 13.8 the annual mean values are plotted on normal probability paper for the
Susquehanna River, as measured near Waverly, NY, for 57 years over the period 1938-
1994, and for the Chemung River, measured at Chemung, NY, for 90 years during 1907-
2000. Both stations are operated by the US Geological Survey. The Susquehanna River
station is located in Bradford County, Pennsylvania, at approximately 41
◦
59
05
N,
76
◦
30
05
W, with the gage at 227 m above sea level, but the drainage area of
12 362 km
2
is in New York State; after correction (Korzoun
et al
., 1977) the long term
