Inferential Statistics

Two basic types of reasoning underlie all of science—deductive reasoning and inductive or inferential reasoning. Deduction involves reasoning from a set of starting assumptions or general principles to a specific future observation. Induction requires reasoning from a limited set of observations to draw conclusions about things that have not been observed or to derive general principles. That is, induction (inference) is the process by which researchers attempt to learn what the general principles or scientific laws are.

Deduction will fail if one or more of the general assumptions is false. If one has the following starting assumptions:

• All cows give milk.
• Ferdinand is a cow.

The deduction that “Ferdinand gives milk” follows logically. If one were to observe Ferdinand and find out that he does not give milk, one of the two starting assumptions must be false. In this case, the most likely reason for the failure of the prediction is that Ferdinand is not a cow but a bull (but note that the first assumption might also not be true).

Science uses deduction to make predictions about future observations based on past observations. Suppose a researcher has a theory (general assumption) that verbal praise will increase the reading level of second graders. The researcher uses deduction to make a prediction about what will happen if verbal praise is used as part of the program of reading instruction for a sample of second graders. The researcher predicts that their reading scores will rise more than the scores of a control group that does not receive verbal praise. That is, the researcher uses [Page 525]deduction to reason from a theory or set of beliefs to a hypothesis or prediction about future observations.

Now the researcher wants to test the hypothesis that students who receive verbal praise will show more growth in reading scores than students who do not receive praise. Because the researcher cannot administer the treatment to all second graders, he or she must reason from a limited set of observations to a general conclusion about the effectiveness of praise on learning to read. Here, the researcher must use inferential or inductive reasoning to go from a specific set of observations (researcher's data) to derive or support a general principle. Unfortunately, for reasons discussed shortly, if the researcher observes higher reading scores in a group of students who have received verbal praise, he or she cannot proceed directly to the conclusion that praise is effective.

Inferential statistics, as a branch of applied mathematics, provides a way to reason inferentially with quantitative information such as reading test scores. But before discussing inferential statistics in detail, it is necessary to distinguish between two contexts in which one might want to use quantitative data. The basic issue is whether one has data on all of the entities with which he or she is concerned. If, for example, the Board of Education of the state of Texas wants to know the level of mathematics achievement of eighth-grade students in the state in the current year, it would be theoretically possible to test the mathematics achievement of every eighth-grade student and compute the mean, standard deviation, and other statistics. These statistics describe the group that was tested and, disregarding the possibility of measurement or computational errors, the results accurately reflect the mathematics achievement of Texas eighth-grade students. The mean and other statistics computed on the group that has been tested are called descriptive statistics. If the Board has no interest in going beyond this specific group, then no inference is involved. However, it may not be practical to test every student for financial or other reasons. In this case, the entire group about which the Board wants to make a statement is called a population and the Board could select a subgroup from this population, called a sample, for testing and compute the mean, standard deviation, and so on of the sample. The results computed on the sample would be descriptive statistics for the sample, but the Board's objective is to reason or generalize from the sample to the population. Therefore, because they want to use the results from the sample to draw conclusions about the population, the Board must use inferential statistics. Whenever one uses the results from a sample to draw conclusions about a larger population, he or she is engaged in statistical inference. Notice that in statistical inference, descriptive statistics from the sample are used to reach conclusions about the population.

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