Geoscience Reference
In-Depth Information
It is most important that the strata are defined without taking any notice of
the values of observations because otherwise bias will be introduced into the
variance calculation. For example, if the strata are chosen to ensure that large
observations only occur in some of the strata, then this will tend to lead to
the within-strata variances being too small.
If the number of sample points or the area is not the same within each
of the strata, then the estimated mean from the stratified sampling equa-
tions will differ from the simple mean of all of the observations. This is to
be avoided because it will be an effect that is introduced by a more or less
arbitrary system of stratification. The estimated variance of the mean from
the stratified sampling equations will inevitably depend on the stratification
used, and under some circumstances, it might be necessary to show that all
reasonable stratifications give about the same result.
Another alternative to treating a systematic sample as a simple random sam-
ple involves joining the sample points with a serpentine line that joins neigh-
boring points and passes only once through each point. This method was
described by Manly (2009, Section 2.9) and usually seems to give results that
are rather similar to the stratified sampling approach. Another possibility for
the analysis of a systematic sample involves estimating the mean over an area
using a geostatistical method, as reviewed by Manly (2009, Chapter 9). This
will require some specialized computer software to perform the calculations.
EXAMPLE 2.7 Concentrations of Trichloroethylene in Groundwater
Kitanidis (1997, p. 15) gives the values of trichloroethylene (TCE) in
groundwater samples in a fine-sand surficial aquifer. Here, 40 of these
observations in a semisystematic rectangular grid are considered, as
shown in Figure 2.4.
Treating the data as coming from a random sample of size 40, the sample
mean is
y
= 5609.1 ppb, with a sample standard devia
ti
on of
s
= 12,628.9
and an estimated standard error for the mean of
SE() 40 1996.8
.
An approximate 95% conide
nc
e interval f
o
r the true mean concentration
of TCE over the area is then
y
± 1.96.SE(
y
), which gives the range from
1695.3 to 9522.8. This range is wide because of the large amount of varia-
tion in the TCE values.
ys
=
=
-40
-50
-60
-70
-80
-90
10
10
6
542
346
191
850
30
12
23
187
164
701
2130
1860
622
40
42
655
21600
67700
38900
584
2540
660
663
3750
8760
14000
16100
12800
763
11
1160
4270
9870
7320
4030
273
190
0
50
100
X (Feet)
150
200
FIGURE 2.4
Concentrations of TCE (ppb, parts per billion) measured at 40 locations in a fine-sand surficial
aquifer.
