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

11

. 12
'" .
w
J:1O
u
;:
;: 8

-1
~ 6
'"
~
~ 4
"

20
'"
w
J: . .
~ 10 .
-8 .
;: .
:d 5
'"
~ . .
~ 3
"
2
4 6 0.3 0.5 1 2 3 5
INCHES RUNOFF, IN INCHES

2
o 2

RUNOFF, IN

Figure 11.-Data from U.S. Geological Survey (19491 p.
488) plotted on natural and log scales showing the
achievement of equal variance about the regression 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

.

Yo", Runoff I PreclpitB- XY X' yo
(Y) tion~ (X)
1928._______u___ 125 110
1929.____________ 87 73
1930__.___.m___ 08 ,.
193L______n____ 71 01
1932.____________ 118 158
1933.____________ 144 130
1934____h_______ 160 152
193~L____________ 188 134
1936_____________ 102 "
1937.____________ 01 90
1938.____________ 125 110
1939____00_______ 87 77
1940_____________ " 100
194:L.___________ " "
1942..___________ 7\l 85
1943_____________ 124 115
1944_____________ 52 70
1945.____________ 87 .,
-
%__u_u_________ l.m 1,SOl 192,042 189,291 197,373
Mean_hh_______ 99.4.4 100.06

I Annualrunoff in percent of mean (B~ping River near Nile, Wash.).
2 Annual rainfall in percent of mean (a.t Bumping Lake, Wash_).

.

. Simple linear regression

Computation 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
products, 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.
2,'W---622 0-67-3

L;XY
b-

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

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

L;X'

b

192,042 (1,801) (1,799)
18
189 291- (1,801)'
, 18

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-squares line

(L;X)'
N
N-l

189.291

(1,801)'
18

L;X'
S'-
,-

17
= 534.76.

Variance of X

L;Y'- (L;Y)'
N
~-- N-l

(1,799)'
18

197,373
17
=1,033.71. Variance of Y

"" _N-l [S' b2S,]_17[1 7
o"'-N_2 ,- '-16 ,033.1
- (1.325)'( 534.76)]= 100.8.

S,.,=1O.0.

Standard error of estimate of Y

bS, ( 2) (23.13)
r~S;= 1.3 5 32.15 ~0.95.
Correlation coefficient

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

&. 8;.3:
. L;(x')

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

0.056.

Testing the hypothesis that fJ=O,

tn_2

b-fJ
S,

1.325-0
0.236 5.6.