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

4

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

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Statistical inference

We have 54 years of record on the Rappahan-
nock River of Virginia and might ask two ques-
tions about the mean flow. First, what is the
mean flow for the period of record? This is a
unique value which can easily be computed.
The second question, what is the mean flow of
the stream?, cannot be answered definitely. We
Can only assume that the mean of the 54-year
sample is an estimate of the true (population)
mean. In other words, we infer the population
charact~ristics from those of a sample from that
population.
Statistical inference is based on the theory
of sampling. From a population of known char-
acteristics many samples are drawn (either
actually or conceptually), and the relation of
the sample characteristics to the population
characteristics is defined.
Sampling theory requires use of the concept
of a probability distribution. Assume that the
distribution of some random variable is normal
with mean 1', and standard deviation, q, as
shown in figure 4. (The term "random," as

J.t

Figure 4,-Normal distribution.

used here, means that the probability of drawing
anyone item of the population is the same as
for any other.)
Now suppose we take many samples of size
N from tills distribution, compute the mean of
each of these samples, and compute the mean
and variance of these sample means. The dis-
tribution of the means of samples of size N is
superposed on the original distribution in figure
5. It Can be shown that the distribution of
the means is centered at I' and that the stan-
dard deviation of the distribution of means is
<r/m. Therefore, the mean of the means of

Distribution of means
of size N

Original distribution

J.L

Figure 5,-Oistribution of means of samples from a norma-
distribution.

samples of size N is an unbiased estimate of 1'.
Furthermore, the mean of one sample is an
unbia~ed estimate of 1'. Consequently, we infer
that the sample mean, X, is an estimate of the
population mean. Obviously, if we used other
samples we would obtain d:i1ferent estimates
of the population mean.
From a single sample, we can appraise the
reliability of the estimate, X, of the population
mean. The distribution of means of values
drawn from a normal distribution is normal.
Consequently, two-thirds of the values should
fall witilln one standard deviation (q/~N)
on each side of the mean. However, we do not
know q so we have to substitute S for it (where
S is the standard deviation computed from the
sample). The distribution of X having a
standard deviation of S/.,fli.f is known as the
Student's t distribution, values of which are
tabulated in statistics texts for various sizes
of N.
Suppose now that we have K samples of
size N and have defined K different sampling
distributions of the mean of size N. For each
sampling distribution, we can define a mean
and a range of reliability, and we. are interested
in whether such a range includes the true mean
1'. Considering the range as a random interval,
we may state that the probability (P) that
the random interval includes I' is 1-e, where e
is the level of significance. Mathematically, for
e=0.32,

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P[(X -o-/../N) <I'<(X +q/~N)]=1-e=0.68.

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