Geoscience Reference
In-Depth Information
Equation (2.3) is therefore more general than the equation that is often quoted
for the variance of a sample mean.
The variance of the sample mean can be estimated by
·
2
Var( ){/}{1 /}
ys nnN
=
−
.
(2.4)
The square root of this quantity is the estimated standard error of the mean
·
2
SE() {/}{1/}
y
=
s nnN
−
,
(2.5)
··
.
The terms
standard error of the mean
and
standard
deviation are often con-
fused when encountered for the first time. What must be remembered is that
the standard error of the mean is the standard deviation of the mean rather
than the standard deviation of individual observations. More generally, the
term
standard error
is used to describe the standard deviation of any sample
statistic that is used to estimate a population parameter.
The CV of the mean is an index that reflects the precision of estimation rel-
ative to the magnitude of the mean. This can be used to compare the results
of several studies to see which have relatively better precision than others.
Often, this is used to define the required precision of a study. For example, in
estimating the size of the shellfish of a certain species on a beach, it might be
required that the population size be estimated with a CV of less than 20%. In
practice, this would
me
a
n
that after the sampling has taken place, it should
be found that 100
and the estimated CV of the mean is
CV() SE()/
y
=
y
y
SE()/
·
.
As an example of the calculation of the statistics that have just been defined,
suppose that a random sample of size
n
= 5 is taken from a population of size
N
= 100 and the sample values are found to be 1, 4, 3, 5, and 8. Then,
=
yy
<
y
4.20
and
·
Var(
·
= {6.70/5}
s
2
= 6.70, so that
=
s
6.70
=
2.59
,
CV()
y
= 2.59/4
.2
0 = 0.62,
y
CV(
·
= 1.13/4.20 = 0.27. These
calculations have been carried through to two decimal places. As a general
rule, it is reasonable for statistics to be quoted with at least one more decimal
place than the original data, which suggests that results should be given to
at least one decimal place in this example.
The accuracy of a sample mean for estimating the population mean is often
represented by a 100(1 − α)% confidence interval of the form
{1 − 5/100} = 1.27,
SE() 1.27
y
=
=
1.13
, and
y
/2
·
yz y
±
SE()
,
(2.6)
α
where
z
α/2
refers to the value that is exceeded with probability α/2 for the
standard normal distribution. Common values used for the confidence
level with the corresponding values of
z
are
z
0.25
= 0.68 for 50% confidence,
z
0.16
= 1.00 for 68% confidence,
z
0.05
= 1.64 for 90% confidence,
z
0.025
= 1.96 for
