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

2

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

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frequencies indicates that the binomial distri-
bution is applicable to this problem.

Table

1.-Results of dice-tossing
[After Kendall (19.52)]

experiment

~o. of
sixes

Fre-
quency

Relative
frequency

Theoretical
relative
frequency

0____________ 447 0.109 0.112
L___________ 1, 145 .280 .269
2____________ 1, 181 .2S8 .296
3____________ 796 .194 .197
4____________ 380 .093 .089
5____________ 115 .028 .029
6____________ 24 .006 .007
7 and over____ 8 .002 .001
TotaL-___ __ 4,096 1.00 1. 00

The binomial distribution is a discrete dis-
tribution, that is, it can take values only at
specific points along a scale. In the dice-tossing
experiment, it is possible to obtain an integer
number of sixes only; there is no such thing as
5.5 or 3.2 sixes.
More commonly, a variable may take any
value along a scale. Such a variable and its
distribution are known as continuous. A
variable may be classified as continuous :if it
can take any value along a scale even though
the limitations of measurement restrict the
observations to discrete values. This condition
exists with most natural phenomena.
To aid in understanding a distribution, con-
sider 1,000 tree-ring indices ranging in size from
2 to 240. If these are grouped by six-unit in-
orements of size, a histogram, or frequency
distribution, is obtained (fig. 1). The irregu-
larity of the profile of this distribution is due
to the small (in a statistical sense) number of
indices used in its preparation. The greater the
number used, the smoother would be the profile
of the frequency distribution. If the number of
observations approaches infinity and the size
increment approaches zero, the enveloping line
of the frequency distribution will approach a
smooth curve. Then if the ordinate values are
divided by a number such that the area under
the curve becomes one, the resulting curve is a
probability density curve, or probability dis-
tribution, such as figure 2. The process just
described requires the additional assumption
that the variable can take any value within the

range, that the variable is continuous, not
discrete.
80

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040
0:
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z

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o 30 60 90 120 150 180 210 240
WIDTH INDEX

Figure 1.-Histogram, or frequency distribution, of 1,000
tree-ring indices.

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240

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30 60 90 120 150
WIDTH INDEX

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Figure 2.-Probobility density curve of 1,000 tree-ring
indices.

A theoretical probability distribution de-
scribes the relation between size (or some other
other characteristic) and probability. For this
relation to be valid, the individuals must oceur
randomly or be drawn randomly. The size of
any individual drawn should not depend on
the size of anyone previously drawn. Probabil-
ity, in the concept of frequency distributions,
is defined as relative frequeney. The distribu-
tion of the number of sixes obtained from re-
peated tosses of 12 dice may be illustrated by
plotting the theoretical relative frequencies of
table 1. The relati ve frequencies of each of the
6-unit increments in figure 1 could likewise be
computed. In the first of these examples, a
probability is associated with each possible
outcome. In the second, a probability i~ assoei-
ated with eaeh increment of size; here the
probability is of obtaining not a specific indi-
vidual but any individual within the increment
of size. This interpretation is required for con-
tinuous distributions because there is an infinite
number of possible valnes and, thus, no proba-
bility of occurrence of a particular individual.

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