FLOOD01900

• County: Statewide
• Community: Nationwide
• Title: Techniques of Weather-Resources Investigations of the USGS Book 4, Chapter A1
• Date: 1/1/1968
• Floodplain - Doc Type: Educational/Technical/Reference Information

SOME STATISTICAL TOOLS IN HYDROLOGY

Referring again to figure 2, probability is related
to the continuous distribution in the following
way: The area under the curve represents the
sum of all probabilities and therefore must
equal one. Because every item was used in
defining the distribution for which the total
area is one, then the probability that any item
will fall in the distribution is one and the
probability that an item will fall in any segment
of the distribution is the ratio of the area of
that segment to the total area.
The distributions just described, both
discrete and continuous, are called relative-
frequency distributions, probability distribu-
tions, or just distributions. However the
,
probability interpretation is valid only if the
data used are random. For example, the daily
mean flow of a stream is closely related to the
flows of previous days, so the distribution of
daily means is not one to which the proba-
bility inte,pretation strictly applies. It is
possible also to approximate a distribution
which merely describes the sample. For instance,
the distribu lion of grain sizes of a sample of
a streambed is measured to characterize the
material; there is no interest in the probability
of obtaining a grain in a particular size range
by additional sampling. Here the sample is
not the individual grain but an aggregate of
grains of various sizes.
Only a few standard theoretical distributions
are widely used. Samplinl! theory and inference
are based largely on the normal distribution
with which the reader is assumed to be familiar.
Other theoretical distributions will be intro-
duced in this and succeeding chapters as
appropriate.

.

.

Cumulative distributions

'.

Suppose we know the probability density
curve (probability distribution) for a variable
and are interested in the probability of a
random event being greater than some par-
ticular value E. This probability can be ob-
tained by measuring or computing the propor-
tion of the total area above the base value.
For instance, the left curve of figure 3 shows an
area under the curve to the right of E of 0.1,
that is, P=O.1. Thus the probability of a
random event exceeding E is 0.1.
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E

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E
MAGNITUDE

0.1
PROBABILITY OF
EXCEEDANCE

Figure 3.~robobility density curve (left) and its cumulative
form (right).

Another form of the probability curve can
be prepared by cumulating the probabilities
from one end of the curve and plotting each
of these cumulated probablities against the
magnitude of its appropriate event. The
cumulation is usually done mathematically.
The result is the right-hand curve of figure 3.
Cumulative distributions are commonly plotted
to a probability scale such that the theoretical
curve is a straight line. Such a scale can be
devised for any two-parameter distribution.
Normal probability plotting paper is widely
known and used. Gumbel plotting paper is
used in many hydrologic frequency analyses.
(Although the Gumbel extreme-value distribu-
tion is a three-parameter distribution, one
parameter, the skew, is constant for the form
used and permits the construction of a scale
which gives a straight-line plot). Both the
normal and Gumbel probability plotting papers
are available with either arithmetic or log-
arithmic ordinate scales. Thus plotting papers
for four distributions-normal, log-normal,
Gumbel, and log-Gumbel-are available.
When the probability density curve is cumu-
lated from the right end, the probabilities of
exceeding the various magnitudes are obtained.
If cumulated from the left, probabilities of not
exceeding those magnitudes are obtained. The
appropriate cumulative curve, commonly called
a frequency curve in hydrology, depends on
the desired use.
The various theoretical cumulative distri-
butions used in hydrology and methods of
estimating their parameters from a sample of
data are discussed in another chapter of this

series.