Statistics Review - Mathematics and Statistics for the Environment - page 109

Our inferences may be incorrect if we are given inaccurate probabilities. Obviously, reliable
computation of these probabilities requires knowledge of how the variable we are dealing with
is distributed (that is, what the probability is of the chance occurrence of different values of the
variable). Accordingly, if we know that the number of beetles caught in light traps follows what is
called the Poisson distribution we can compute the probability of catching X or more beetles. But,
if we assume that this variable follows the Poisson distribution when it actually follows the negative
binomial distribution, then our computed probabilities may be in error.
Even with reliable probabilities, statistical tests can lead to the wrong conclusions. We will some-
times reject a hypothesis that is true. If we always test at the 0.05 level, we will make this mistake
on the average of 1 time in 20. We accept this degree of risk when we select the 0.05 level of testing.
If we are willing to take a bigger risk, we can test at the 0.10 or the 0.25 level. If we are not willing
to take this much risk, we can test at the 0.01 or 0.001 level.
Researchers can make more than one kind of error. In addition to rejecting a hypothesis that is
true (a Type I error), one can make the mistake of not rejecting a hypothesis that is false (a Type II
error). In crapshooting, it is a mistake to accuse an honest shooter of cheating (Type I error—reject-
ing a true hypothesis), but it is also a mistake to trust a dishonest shooter (Type II error—failure to
reject a false hypothesis).
The difficulty is that for a given set of data, reducing the risk of one kind of error increases the
risk of the other kind. If we set 15 straight passes as the critical limit for a crapshooter, then we
greatly reduce the risk of making a false accusation (probability about 0.00025). But in doing so we
have dangerously increased the probability of making a Type II error—failure to detect a phony. A
critical step in designing experiments is the attainment of an acceptable level of probability of each
type of error. This is usually accomplished by specifying the level of testing (i.e., probability of an
error of the first kind) and then making the experiment large enough to attain an acceptable level of
probability for errors of the second kind.
It is beyond the scope of this topic to go into basic probability computations, distribution theory,
or the calculation of Type II errors, but anyone who uses statistical methods should be fully aware
that he or she is dealing primarily with probabilities (not necessarily lies or damnable lies) and not
with immutable absolutes. Remember, 1-in-20 chances do actually occur—about one time out of
t went y.
7.2 MEASURE OF CENTRAL TENDENCY
When we talk about statistics, it is usually because we are estimating something with incomplete
knowledge. Maybe we can only afford to test 1% of the items we are interested in and we want to
say something about the properties of the entire lot, or perhaps we must destroy the sample to test
it. In that case, 100% sampling is not feasible if someone is supposed to get the items back after we
are done with them. The questions we are usually trying to answer are “What is the central tendency
of the item of interest?” and “How much dispersion about this central tendency can we expect?”
Simply, the average or averages that can be compared are measures of central tendency or central
location of the data.
7.3 SYMBOLS, SUBSCRIPTS, BASIC STATISTICAL TERMS, AND CALCULATIONS
In statistics, symbols such as X , Y , and Z are used to represent different sets of data. Hence, if we
have data for five companies, we might let
X = company income
Y = company materials expenditures
Z = company savings