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

SOME STATISTICAL TOOLS IN HYDROLOGY

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The interval in brackets is called a confidence
interval and the extremes are called confidence
limits. Note that the above relation holds only
if we use cr instead of 8. If we use 8, then l-e
is a function of the S8ID.ple size, and the
appropriate probability statement is

P[(X -t8/{N)<I'< (X + t8/.JN)J=I-e=O,68,

where Student's t is 1.09 for 10 degrees of
freedom, for example. The width of the con-
fidence interval increases as the level of signifi-
cance decreases, For eX81llple, the 95-percent
confidence limits (e=O,05) are

(X -2,Z38/{N)<I'<(X +2.238NN),

.

where 2,23 is from the t table for 10 degrees of
freedom,
The confidence interval described in the
probability statement is a random interval,
not a specific one. The probability statement
(for e=0.05) means that 95 percent of a large
number of intervals similarly obtained wonld
include the true mean, This probability statement
cannot be extended to one specific interval
because a specific interval either contains the
true mean or it does not and the probability
is either one or zero, The true mean is not a
variable, it is unique.
But we are interested in making a probability
statement about one specific interval, We may
say that the probability of our obtaining a
ran dom interval which includes the true mean
is 0,95, or that we have 95-percent confidence
that the interval obtained includes the true
mean. Ordinarily in hydrologic reports it is
only necessary to state the computed confi-
dence interval and its level, not to interpret
the meaning. See Mood (1950, p. 221-222) for
a precise statement of the interpretation of a
confidence interval.
Using the above theory, from a random
sample, we can compute an estimate of the
population mean and a measure of its relia-
bility. This is an example of statistical inference,
Returning to the sampling theory, consider
the distribution of variances of samples of size
1\ from a normal distribution. This distribu-
tion is not centered around cr' but is to the left
of it, as shown by the upper graph of figure 6,
Therefore 8' is known as a biased estimator of

"

.

5

.r, It can be made unbiased by multiplying it
by N/(N-l) as shown in the lower sketch of
figure 6. The standard deviation of the sampling
distribution of 8'(^/N-l) can also be
computed,

S2 u2

S2(N/N-l )
u2

Figure 6.-Distribution of variances of samples.

A further use of inference is in testing hypoth-
eses. One example will be given. Suppose we set
up the null hypothesis, H" that the mean of a
population is zero; that is, we hypothesize that
there is no difference statistically between the
mean and zero. This null hypothesis is written

H,:I'=O.

We draw a sample from this population and
compute the statistics of the sampling distribu-
tion of the mean. We need some estimate of the
hypothetical sampling distribution of means,
and so we define it os normal with mean zero
and standard deviation equal to 81 v'N os com-
puted from the sample (fig. 7).
Now if X (as computed from the sample) lies
within one standard deviation of zero, we would
conclude that there is no basis for doubting
the hypothesis. If, on the other hand, X were
two or three standard deviations away from
zero, we would conclude that it is unlikely that
the mean of the population is zero. For this