Page 197 - Statistics for Environmental Engineers

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23

Sizing the Experiment

KEY WORDS arcsin, binomial, bioassay, census, composite sample, confidence limit, equivalence of
means, interaction, power, proportions, random sampling, range, replication, sample size, standard devi-
ation, standard error, stratified sampling, t-test, t distribution, transformation, type I error, type II error,
uniform distribution, variance.

Perhaps the most frequently asked question in planning experiments is: “How large a sample do I need?”
When asked the purpose of the project, the question becomes more specific:
What size sample is needed to estimate the average within X units of the true value?
What size sample is needed to detect a change of X units in the level?
What size sample is needed to estimate the standard deviation within 20% of the true value?
How do I arrange the sampling when the contaminate is spotty, or different in two areas?
How do I size the experiment when the results will be proportions of percentages?

There is no single or simple answer. It depends on the experimental design, how many effects or
parameters you want to estimate, how large the effects are expected to be, and the standard error of the
effects. The value of the standard error depends on the intrinsic variability of the experiment, the precision
of the measurements, and the sample size.
In most situations where statistical design is useful, only limited improvement is possible by mod-
ifying the experimental material or increasing the precision of the measuring devices. For example, if
we change the experimental material from sewage to a synthetic mixture, we remove a good deal of
intrinsic variability. This is the “lab-bench” effect. We are able to predict better, but what we can predict
is not real.

Replication and Experimental Design

Statistical experimental design, as discussed in the previous chapter, relies on blocking and randomization
to balance variability and make it possible to estimate its magnitude. After refining the experimental
equipment and technique to minimize variance from nuisance factors, we are left with replication to
improve the informative power of the experiment.
The standard error is the measure of the magnitude of the experimental error of an estimated statistic
(mean, effect, etc.). For the sample mean, the standard error is σ / n, compared with the standard
deviation σ. The standard deviation (or variance) refers to the intrinsic variation of observations within
individual experimental units, whereas the standard error refers to the random variation of an estimate
from the whole experiment.
Replication will not reduce the standard deviation but it will reduce the standard error. The standard
error can be made arbitrarily small by increased replication. All things being equal, the standard error
is halved by a fourfold increase in the number of experimental runs; a 100-fold increase is needed to
divide the standard error by 10. This means that our goal is a standard error small enough to make

© 2002 By CRC Press LLC