Geoscience Reference
In-Depth Information
7.11
SAMPLING MEASUREMENT VARIABLES
7.11.1 s imple r andom s ampling
Most environmental practitioners are familiar with simple random sampling. As in any sampling
system, the aim is to estimate some characteristic of a population without measuring all of the
population units. In a simple random sample of size n , the units are selected so that every possible
combination of n units has an equal chance of being selected. If sampling is without replacement,
then at any stage of the sampling each unused unit should have an equal chance of being selected.
7.11.1.1 Sample Estimates of the Population Mean and Total
From a population of N = 100 units, n = 20 units were selected at random and measured. Sampling
was without replacement—once a unit had been included in the sample it could not be selected
again. The unit values were
10 9 10 9 11 16 11 7 12 12 11 3 5 11 14 8 13 12 20 10
Sum of all 20 random units = 214
From this sample we estimate the population mean as
X
214
20
X
=
=
=
10 .
n
A population of N = 100 units having a mean of 10.7 would then have an estimated total of
ˆ
TN == =
100 10 7
(.)
1070
7.11.1.2 Standard Errors
The first step in calculating a standard error is to obtain an estimate of the population variance (σ 2 )
or standard deviation (σ). As noted in a previous section, the standard deviation for a simple random
sample (like our example here) is estimated by
(
)
2
X
n
2
214
20
2
2
2
2
X
10
+++−
16
10
s
=
=
=
13 4
. 842
= .
3 672
n
1
19
For sampling without replacement, the standard error of the mean is
2
s
n
n
N
13 4842
20
.
20
100
=
=
s
x =
1
1
05
. 39368
= .
0 734
From the formula for the variance of a linear function we can find the variance of the estimated total:
2
2
2
s
=
Ns
x
T
The standard error of the estimated total is the square root of this, or
s
==
s
100 0 734
(.
)
=
73 4
.
ˆ
x
T
 
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