Geoscience Reference
In-Depth Information
suitable small value. This then represents an absolute level of precision.
Alternatively, the accuracy might be specified in terms of the CV of the mean,
=
CV() 100 SE()/
CV()
should be no more than
20%. This then represents the precision relative to the population mean. The
CV approach is particularly useful when a number of different quantities
are being estimated and the true population means may v
ar
y considerably.
To obtain a 100(1 − α)% confidence interval for the mean of
yd
, it is required
from Equation (2.6) that
y
y
µ
, so that, for example,
y
/2
·
z
SE()
y
=
d
, so that
zsnnNd
{/ }{1/}
−
=
. Solving
α
α
/2
this equation for
n
yields
n
=
z
α/2
2
s
2
/(
d
2
+
z
α/2
2
s
2
/
N
).
(2.12)
Because the standard deviation
s
is not known in advance, it is necessary to
guess what this might be, for example, using the value from a previous sample.
If the population size
N
is large, then Equation (2.12) simplifies to
n
≈ (
z
α/2
s
/
d
)
2
.
(2.13)
This is a conservative equation in the sense that, for all population sizes
N
,
Equation (2.13) gives a larger value of
n
than Equation (2.12).
If a specific CV is desired rather than a specific confidence interval for the
mean, then this requires that, for the sample obtained,
·
SE()/
yyr
=
, where
r
2
2
2
is the required CV. Then, squaring both sides gives
{/}{1/}/
sn nNyr
−
=
,
2
2
2
so that
nsyr sy N
=
(/)/{
+
( /)/}
, or
··
.
n
=
CV()/{
y
2
r
2
+
CV()/}
y
2
N
(2.14)
This equation can be applied with an estimated or guessed value for the CV
of the population in place of the unknown
·
CV()
y
. For a large population size
N
,
it reduces to
·
{CV( )/ }
2
n
=
y
r
,
(2.15)
which always gives a larger value than Equation (2.14).
Equations (2.12) to (2.15) should be regarded as giving no more than a
rough indication of adequate sample sizes when they are used with guessed
values for standard deviations or CVs. Nevertheless, as a general principle it
is true that any effort spent in determining appropriate sample sizes is better
than no effort at all.
All of the discussion so far about sample sizes has been in terms of just
one variable of interest in a study, but in most studies, several different vari-
ables have to be considered at the same time. If all variables require sample
sizes of about the same magnitude, then the size used can be the maximum
