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

32

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

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sample may be resolved into component varia-
tions due to independent factors. It is closely
related to correlation but is applicable to
problems where some of the factors can be
described only by classes, not as numerical
variates.
The analysis depends on the additive
characteristic .of variances. Its purpose is to
test whether several means are alike or not.
The basic features of the process are (1) the
measurement of variance among experimen tal
data by the sum of the squared deviations of
the observations from their mean, (2) the par-
titioning of the total sum of squared deviations
into independent parts, each part associated
with some physical feature of the experiment,
(3) the estimation of parameters in the distribu-,
tions postulated to underlie the data, and (4)
tests of significance regarding these parameters.
Results of the test give the probability of there
being a significant difference between the effects
of a factor or factors at different levels.
A very simple example of an analysis of
variance concerns whether the mean runoffs
for two periods of record at a gaging station
are estimates of the same population mean. The
annual runoffs are given below:

Period 1 Period J
17.3__________________ 6.4
21.9__________________ 15.2
13.6__________________ 9.7
10.8__________________ 4.4
19.7__________________ 9.9
20.7___ _ __ _ _ __ __ __ __ __ 11. 9
16.3__________________ 11.9
16.2__________________ 15.4
12.5__________________ 9.4
11.3__________________ 7.0
14.0__________________ 16.0
16.5__________________ 17.0
15.3__________________ 11. 2
19.2__________________ 13.2
13.000________________ 11. 5

238.3

170.1

Computations are as follows:

Grand total=T=238.3+170.1=408.4.
Total number of items=N=30.

Number of items in each period=n= 15.

T'/N= (408.4)'/30=5,559.7.

Sum of square of all individuals
=L;Y,'.=6,067.3.
(Sum of squares of sums) /n= L;T;/n
= [(238.3)'+ (170.1)'1/15=5,714.7.

Between-periods sum of squares= L; T;/n
- T'/N=5,714.7-5,559.7=155.0.

Within-periods sum of squares= L;Y.'
- L;T;/n=6,067.3-5,714.7=352.6.

Total sum of squares=L;Yf,-T'/N=6,067.3
-5,559.7=507.6.

The analysis of variance table is

Source

Sum of Degrees of l\-1ean Average
squares freedom square mean
square

Between
periods _ _ _ 155.0 1 **155 cr2+n a~
Within
periods _ _ _ 352.6 28 12.6 ..
Tot.aLu 507. 6 29

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"Statistical significance above the 0.01 level.

The degrees of freedom, D.F., are one less
than the number of periods, p, for the between-
periods sum of squares and N -1 for the total.
Thus the degrees of freedom associated with the
within-periods source is N-p. Mean square is
obtained by dividing the sum of squares by
D.F.
The last column in the analysis of variance
table shows expected values of the mean
squares. If the means for the periods are alike,
the term nul would be zero. Estimates of the
ratio (u'+null/u' may be greater than one,
because of chance or because there is a real
difference. This ratio has the F distribution and
can be tested statistically. The ratio in the
above table is 155/12.6=12.3. The value of F
for 1 and 28 degrees of freedom and a proba-
bility of 0.01 is F1,2B.O'Ol =7.6 from a table of
F distribution. Because the sample ratio
exceeds the tabular ratio, we conclude that
there is a real difference between periods; that
is, the probability is less than 0.01 that such
a difference in means would have occurred by

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