Statistical Analysis for Education and Psychology Researchers

You should by now be convinced of the central role that statistical inference has in
quantitative research. Statistical inference is, however, not an end in itself but is simply
an aid to decision making in uncertain circumstances. Information in the form of data is
collected, sample statistics are then calculated and used in a formal way in what is called
estimation and hypothesis testing. The confidence we have in a sample estimate, or the
strength of evidence we have against a hypothesis is evaluated by this formal process.
The researcher can then draw conclusions based on the outcome(s) of statistical test(s)
and the strength of evidence for or against a hypothesis.

Probability: Its Role in Research

The idea of probability is central to the process of statistical inference. Whenever a
sample is selected an element of uncertainty is introduced. This uncertainty is a
consequence of not collecting information from the whole population, but relying instead
on information contained in a sample. This degree of uncertainty is numerically
expressed as a probability or the likelihood of occurrence of an event.
Probability can generally be thought of as the study of patterns of chance events and is
based on the idea that certain phenomenon are random. Statistics are calculated from
sample data and may be used not only to summarize data but also to assess the strength of
evidence provided by sample data in favour of an assertion or statement about
circumstances in the population, the hypothesis. Provided data is produced by a random
process (step 2), then the sample statistics themselves, such as an average or a proportion,
can be thought of as random variables which obey the laws of probability. We can,
therefore, use the language of probability to make statements about the likelihood of
outcomes such as the difference between averages or the strength of a relationship
between two variables. We can in effect say how likely we would be to observe a given
outcome simply by chance.
We actually observe these outcomes in our sample only. However, because sample
statistics are estimators of corresponding population parameters, and also because they
behave in a probabilistic way, we can use statistical inference to draw conclusions about
circumstances in our population of interest.
A sample which has been produced by a random process is called a probability
sample and subjects or sample values will be independently drawn from the population.
This means the chance that one subject has been sampled is not dependent on the chance
that other members of the population have been or would be sampled. A major problem
with many quantitative research designs is that non-probability samples are used.
A non-probability sample can arise when:

• you do not have a sample frame, essentially a list of every subject or value that is a
  member of the target population;


• not every member of the population has an independent chance of being selected;


• there is no underlying probability model specified, that is we cannot work out in
  advance of sampling the chance of any member of the population being selected. It is
  not necessary that all members of the population have an equal chance of being
  selected.

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