Microsoft Office Tutorials and References
In Depth Information
6.1 Introduction
In Chap. 3, we introduced several statistical techniques for the analysis of data,
most of which were descriptive or exploratory. But, we also got our ﬁrst glimpse
of another form of statistical analysis known as Inferential Statistics . Inferential
statistics is how statisticians use inductive reasoning to move from the speciﬁc, the
data contained in a sample, to the general, inferring characteristics of the population
from which the sample was taken.
Many problems require an understanding of population characteristics; yet, it can
be difﬁcult to determine these characteristics because populations can be very large
and difﬁcult to access. So rather than throw our hands into the air and proclaim
that this is an impossible task, we resort to a sample : a small slice or view of a
population. It is not a perfect solution, but we live in an imperfect world and we
must make the best of it. Mathematician and popular writer John Allen Paulos sums
it up quite nicely—“Uncertainty is the only certainty there is, and knowing how to
live with insecurity is the only security.”
So what sort of imperfection do we face? Sample data can result in measurements
that are not representative of the population from which they are taken, so there is
always uncertainty as to how well the sample represents the population. We refer
to these circumstances as sampling error : the difference between the measurement
results of a sample and the true measurement values of a population. Fortunately,
through carefully designed sampling methods and the subsequent application of
statistical techniques, statisticians are able to infer population characteristics from
results found in a sample. If performed correctly, the sampling design will provide
a measure of reliability about the population inference we will make.
Let us carefully consider why we rely on inferential statistics:
1. The size of a population often makes it impossible to measure characteristics
for every member of the population—often there are just too many members of
populations. Inferential statistics provides an alternative solution to this problem.
2. Even if it is possible to measure characteristics for the population, the cost can be
prohibitive. Accessing measures for every member of a population can be costly.
3. Statisticians have developed techniques that can quantify the uncertainty associ-
ated with sample data. Thus, although we know that samples are not perfect,
inferential statistics provides a reliability evaluation of how well a sample
measure represents a population measure.
This was precisely what we were attempting to do in the survey data on the
four webpage designs in Chap. 5; that is, to make population inferences from the
webpage preferences found in the sample. In the descriptive analysis we presented
a numerical result. With inferential statistics we will make a statistical statement
about our conﬁdence that the sample data is representative of the population. For the
numerical outcome, we hoped that the sample did in fact represent the population,
but it was mere hope. With inferential statistics, we will develop techniques that
allow us to quantify a sample’s ability to reﬂect a population’s characteristics, and
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