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 HYDROWGY

33

chance if there were no real diflerence between
periods. The double asterisk on the mean
square for between periods (in the analysis-of-
variance table) denotes statistical significance
above the 0.01 level.
Now consider a similar problem, to deter-
mine whether mean annual precipitation at
three stations is diflerent. The data are given
in the following table.

Precipitation, in inches, at-

Year

Site 1

Site 2

Site 3

.
,
1

1945_____________ 40. 6 48. 2 47. 5
1946_ _ ----------- 36.1 40. 2 34- 8
1947_____________ 37. 5 37. 8 42. 2
1948_____________ 52. 3 58.2 59.9
1949_____________ 42. 2 43.3 51. 7
1950_____________ 40. 6 41. 4 42.5
1951_____________ 38.3 42.3 40.5
1952_____________ 45.8 48.2 47. 8
Sums______~ 333.4 359.6 366.9
ynn __ __ _ ____ ___ 41. 7 45.0 45. 9

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From the data in the table we can make the
following calculations:

T=1,059.9
T'/N=46,807.8
~Y:,=47,784.0
~T,'fn=46,885.4

The analysis-of-variance table is

Source

Sum of Degrees
squares of
freedom

Mean
square

Among sites_ _ _ _ _ __
Within sites_____n
TotaL_____

77.6
898.6
976. 2

2 38. 8
21 42.9
23 n___n__

I.

F'.'1=38.8/42.9<1; therefore there is no dif-
ference statistically among the three means.
A perusal of hydrologic literature will turn
up very few applications of analysis of variance.
Most analyses of variance are based on data
from a designed experiment, and it is tms ap-
lication for which the best results are obtained.
Hydrologic data are usually parts of a time
series which may not be stationary. Thus the
individual values may not be entirely inde-
pendent as required for a valid analysis of

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variance. In the example comparing mean run-
offs for two periods of record, it was concluded
that a real diflerence existed between periods.
But there is no physical reason to expect a
change in this basin. The earlier period was
one of high precipitation; the later period in-
cluded the drought of the thirties. It is also
possible that some of the annual runoffs were
serially correlated. Thus the characteristics
of the data tend to discredit the results of this
particular application of the analysis of
variance.
In the last example, the precipitation at site
2 is greater than that at site 1 for every year
shown, yet the analysis of variance shows no
diflerence in means. (An analysis of variance
between site 1 and site 2, only, shows a dif-
ference at a probability level of about 0.25.)
The annual precipitations at a site may be
independent, but the precipitations at the sev-
eral sites for the same year are not. Therefore
the requirements of the method are not met.
and the results must be accepted with
reservation.
The two examples given utilize a very simple
statistical model. For more complicated prob-
lems, several models may be considered. Selec-
tion of the appropriate one is difficult for the
Ifpart-time" statistician. Many statistics texts
treat analysis of variance in detail. See Bennett
and Franklin (1954), Brownlee (1960), and
Dixon and Massey (1957). In general, an
analysis of variance made by someone not
thoroughly familiar with the process should be
reviewed by a statistician for suitability of the
model and correctness of the interpretation.

Analysis of covariance

The analysis of variance of the runoff data
for two periods (see p. 32) indicated that the
population means were probably diflerent, yet
other information, particularly precipitation
records, leads to the opposite conclusion. The
precipitation data can be incorporated in the
analysis by using an analysis of covariance.
This method includes concepts from analysis of
variance and regression and is applicable where
a variable represents a measurement for each
individual as opposed to a variable which can
only be separated into a few categories.