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

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SOME STATISTICAL TOOLS IN HYDROLOGY

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limits are quite wide for samples of 30 items
or less, unless the correlation coefficient is very
large. For example, a chart shown by Bennett
and Franklin (1954, p. 275) indicates that a
correlation coefficient of +0.8 computed from
a sample of 20 items would have a confidence
belt extending from 0.6 to 0.9 for 95-percent
probability. Because of this uncertainty, com-
parison of two correlation coefficients differing
only by a few hundredths cannot be meaning-
fully interpreted. There also seems to be no
justification for reporting correlation coefficients
to more than two significant figures.
If the data can reasonably be assumed to
be drawn from a normal bivariate distribution,
then both correlation and regression analyses
are appropriate. It is under this assumption
that most of the examples in statistics texts
are analyzed. However, regression is also
appropriate under certain other conditions
when correlation is not. The only assumptions
required for regression are:
1. The deviations of the dependent variable
about the regression line (for any fixed X)
are normally distributed, and the same
varian~e exists throughout the range of
definition.
2. Values of the independent variable are
known without error. The dependent vari-
able is considered as an observation on a
random variable, and the independent
variable as some known constant as-
sociated with this random variable.
3. Observed values of the dependent variable
are uncorrelated random events.
4. Each of the variables is homogeneous;
that is, all individual values of a variable
measure tbe same thing. Data are consid-
ered homogeneous if any subgroup to which
certain of these data may be logically as-
sigued has the same expected mean and
variance as any other subgroup of the
population. Neither variable need have a
probability distribution in regression (but,
of course, Y values corresponding to a
fixed X are assumed to be normally dis-
tributed).
The end products of a regression analysis
are two equations, Y=j(X) and X=j(Y)
(usually only one is computed), because re-

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gression is directional. In contrast, correlation
gives one index of the relation between variables.
The regression equation gives the average
amount of change in the dependent variable
corresponding to a unit change in the independ-
ent variable. Thus it gives more specific in-
formation than correlation. The regression
coefficient can be tested to determine whether
it is significantly different from zero, and this
test is identical to the test of significance of the
correlation coefficient (providing the data are
drawn from a bivariate normal distribution).
The reliability of a regression is measured by
the standard error, which is the standard devia-
tion of the distribution (assumed normal) of
residuals about the regression line (fig. 8 shows
distribution of residuals). By definition, the
standard error is the same throughout the range
of X. This standard error was called the
standard error of estimate by Ezekiel (1950,
p. 131). It is also referred to as the standard
error of regression and as the standard deviation
from regression.

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x

Figure S.-Normal distribution of plotted points about the
regression line.

The standard error of a prediction from
regression is made up of three parts: the error
of the mean, the error of the slope of the line,
and the standard error of estimate. All three
may be expressed in terms of the standard error
of estimate so that the standard error of a
prediction (8.) is

_ II! (X-X)'_.
8.-S,-y + n + L.;(X-X)'