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

20

SOME STATISTICAL TOOLS IN HYDROLOGY

11

.

1~
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z
~ 8

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~ 4
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0:

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~ ~
o 2 4 6 0,3 0.5 1 ~ 3 5

RUNOFF, IN INCHES RUNOFF, IN INCHES

Figure 11.-Data from U.S. Geological Survey (1949, p.
488) plaNed on natural and 109 scales showing the
achievement of equal variance about the regretsion line
by use of the log transformation.

Only the log transformation has been used
in the above examples because it is by far
the most common and useful. Other trans-
formations such as the square root may be
appropriate for certain data.

Table 2.-Data and computations for example of two-
variable regression

Year Runoff 1 Precipita- XY X' yo
(Y) tlonl(X)
1928____,,____00_ 12.\ 110
. 192IL..__________ 67 73
1930.____________ .. 74
193L____________ 71 91
19W<L____________ 118 108
19311-____________ 1.. 130
1934______h_____ 100 162
1935__.._________ 138 1M
1931L____.____u_ 102 118
1937___..________ 91 90
1938____..___.__+ 125 11.
1939_____________ 87 77
1940_________0___ 54 100
1941.____________ OS 54
1942..___________ 7<l ..
1943_____________ 124 m
llK4. ________un 62 70
19405..___________ 87 91
- - -
z________________ 1.799 1,801 192,.0i2 189,291 197.373
MeBIl...h_______ ..... 100.00

I Annualrunoff In peroont of mean (Bumping River nearNlle. Wash.),
, Annual rainfall In percent of mean (at Bumping Lake, Wash.).

.

Simple linear regression

Comp11tation of a regression equation using
the model Y =a+bX is demonstrated using the
data given in table 2. That table also shows
computations of means, cross products, and
squares. The individual cross products and
squares need not be recorded; the sum of cross
prod11cts, or squares, can be cumulated on a
desk calculator. Such calculations are ordinarily
checked by repeating the operation. The coeffi-
cients a and b in the regression equation, and
the standard error of estimate are computed as
shown below.

b

L;XY
L;X'

L;xL;Y
N
(L;X)'
N

L;XY-NXY
,-,
L;X'- NX'

(1,80l) (1,799)
18
189,291 (1,80l)'
18

192,042
b-

1.325.

Regression coefficient

a= Y -bX=99.94- (1.325) (100.06) = -32.6.

Intercept

Then

Y =a+bX= -32.6+ 1.32X,

or

Y = Y -b(X -X) =99.94- (1.325) (X -100.06),
Y = -32.6+ 1.32X.
Equation of least-equares line

S~

L;X' (L;X)'
N
N-I

189.291- (1,~~I)'

17

=534.76. Variance of X

~

L;Y'- (L;Y)'
N
N-I

197 373 (1,799)'
, 18
17

=1,033.71. Variance of Y

S' _N-l [S' b'S'] 17[ 0 7
"%-N-2 .- % =161,33. 1

- (1.325)'(534.76) J= 100.8.

S..%=1O.0.

Standard error of estimate of Y

bS% ( 32 ) (23.13)
r= S. ~ 1. 5 32.15 ~0.95.
Correlation coefficient

The regression coefficient can be tested for
significance as follows (Bennett and Franklin,
1954, p. 228):

8f. ~,%
, L;(x')

100.8
189,291-(1,801)'/18 0.056.

Testing the hypothesis that /l= 0,

b-/l 1.325-0
1..,- S, - 0.236 -5.6.