Geoscience Reference
In-Depth Information
2.3 Estimation of Mean Values
Assume that a simple random sample of size n is taken from a population
of N units, and that the variable of interest Y has values y 1 , y 2 , . . . , y n for the
sampled units. Then, sample statistics that are commonly computed are the
sample mean
n
yyy
=+
(
+
y
)/
n
=
y
/
n
,
(2.1)
1
2
n
i
i
=
1
the sample variance
n
2
2
s
=
(
y
y
) /(
n
1)
,
(2.2)
i
i
=
1
where s is the sample standard deviation (the square root of the variance),
and
() · is the estimated coefficient of variation. Note the use of a
caret ^ to indicate a sample estimate. This is a common convention in statis-
tics that is used frequently in this chapter.
The coefficient of variation is often just refe r red to as the CV. Also, it is
often expressed as a percentage because
CV ysy
=
100 /
sy
is the standard deviation as
a percentage of the mean.
The sample mean is an estimator of the population mean μ, where this is
the mean of Y for all N units in the population. The difference − y is the
sampling error; this will vary from sample to sample if the sampling process
is repeated. It can be shown theoretically that if the random sampling pro-
cess is repeated many times, then the sampling error will average out to zero.
Therefore, the sample mean is an unbiased estimator of the population mean.
It can also be shown theoretically that the distribution of y that is obtained
by repeating the process of simple random sampling has variance
2
Var( ){/}{1 /}
y
n nN
,
(2.3)
where σ 2 is the variance of the Y values for the N units in the population.
In this equation, the factor {1 n / N } is called the finite population correc-
tion. The square root of
Var( ) is commonly called the standard error of the
sample mean, which is denoted by
y
SE() .
Equation (2.3) might not be familiar to those who have taken a standard
introductory course in statistics because it is usual in such courses to assume
that the population size N is infinite, leading to n / N = 0 and
y
2
Var(
y
) /
n
.
 
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