Environmental Engineering Reference
In-Depth Information
The results using stratified random sampling can be expressed as x s or 68:80:99 mg/kg,
which compares favorably with 68:712:54 mg/kg if the same raw data were obtained using
simple random sampling. Although the means are very close in this example, the standard
deviation is much smaller using stratified random sampling.
3.4 ESTIMATING SAMPLE NUMBERS: HOW MANY
SAMPLES ARE REQUIRED
The best sample number is the largest sample number possible. Unfortunately, it is
very unlikely to take too many samples due to the limited time and budget resources
for sample collection and analysis. Oftentimes, investigators should avoid taking too
few samples that could make data scientifically unreliable, or even worse, lead to
error conclusions.
In principle, the sample number (n) is a function of the project goal, type of
sampling approaches, environmental variability (s), cost (C), tolerable error,
and other factors. For example, a judgmental sampling intended to determine
the presence or absence of a contaminant requires only a few samples. On the
contrary, a grid sampling requires a much greater sample number to delineate
the extent of contamination. Each of the above probability sampling methods
discussed in the previous section has its own ways of calculating sample numbers.
There is no universal formula to calculate the adequate sample size. Given below
is an example of sample number (n) calculation for simple random sampling. This
could serve as a conservative estimate since simple random sampling usually
results in the maximal number of samples required. Readers are referred to the
references listed at the end of this chapter for further details on other specific
equations and assumptions.
Manly (2001) gave the following equation in relating sample size (n) to the
variability (s) and acceptable error (d) in simple random sampling from a normally
distributed population:
4s 2
d 2
n ¼
ð3
:
where s is the population standar d deviation and d is half of the width of a 95%
confidence interval on the mean (xd). To use this equation, an estimate or best
guess of s must be known. Sources of a preliminary estimate of population variance
(s 2 ) include: a pilot study of the same population, another study conducted with a
similar population, or an estimate based on a variance model combined with separate
estimate for the individual variance components. In the absence of prior information,
the U.S. EPA guideline recommended the following equation to obtain the crude
approximation of standard deviation (
^
s) by dividing the expected range of the
population by 6, that is
Maximum ð expected Þ Minimum ð expected Þ
6
^
ð3
:
10Þ
s ¼
 
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