FLOOD08610

• County: Statewide
• Basin: Statewide
• Title: Techniques of Water-Resources Investigations of the US Geological Survey Some Statistical Tools in Hydrology
• Date: 1/1/1969
• Prepared By: USGS
• Floodplain - Doc Type: Educational/Technical/Reference Information

28

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

5000
4000

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MEAN FLOW (G)). IN CUBIC FEET PER SECONO
PER SQUARE MILE

Figure 20.-GlOphical regression in which one variable is used twice.

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100

100

1000
ORAINAGE AREA (A), IN SQUARE MILES

General methods

All linear equations in two variables are of
the form

Y =a+bX,

and thiB general form is the equation of a
straight line on rectangular graph paper. The
linear form on log paper is

log Y=log a+b log X,

which, when expressed in the original variables,
is the power equation

Y=aX",

A straight line on semilog paper has the linear
form

log Y=log a+bX,

which reduces to the exponential equation,
Y=a(lO)oi,

If b=c log k in the above equation, then
Y=ak"",

Occasionally, points plotted on log paper

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define a gentle curve rather than a straight line.
The locus of the points can sometimes be made
linear by adding or subtracting a constant from
one of the variables. The relation would be of
the form

.

log Y=loga+b log (X+c),

or

Y=a(X+c)',

To determine the equation of any linear two-
variable relation, compute the slope of the line,
b, as vertical distance divided by horizontal
distance, These distances are always measured
in arithmetic units even though the plot is on
log paper (the b values are not transformed),
The scale interval should be the same on both
axes or an appropriate arithmetic adjustment
made. The intercept, a, is usually read off the
graph sheet on the ordinate scale at the appro-
priate value of X. For the relation

Y=a+bX,

Y=a when X=O,

and for the relation

log Y=log a+b log X,

Y=awhenX=1.

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