Statistics: An Introduction

# Making Predictions Using Inferential Statistics

Inferential statistics are used to draw conclusions and make predictions based on the descriptions of data. In this section, we explore inferential statistics by using an extended example of experimental studies. Key concepts used in our discussion are probability, populations, and sampling.

### Experiments

A typical experimental study involves collecting data on the behaviors, attitudes, or actions of two or more groups and attempting to answer a research question (often called a hypothesis). Based on the analysis of the data, a researcher might then attempt to develop a causal model that can be populations.

A question that might be addressed through experimental research might be "Does grammar-based writing instruction produce better writers than process-based writing instruction?" Because it would be impossible and impractical to observe, interview, survey, etc. all first-year writing students and instructors in classes using one or the other of these instructional approaches, a researcher would study a sample – or a subset – of a population. Sampling – or the creation of this subset of a population – is used by many researchers who desire to make sense of some phenomenon.

To analyze differences in the ability of student writers who are taught in each type of classroom, the researcher would compare the writing performance of the two groups of students.

#### Dependent Variables

In an experimental study, a variable whose score depends on (or is determined or caused by) another variable is called a dependent variable. For instance, an experiment might explore the extent to which the writing quality of final drafts of student papers is affected by the kind of instruction they received. In this case, the dependent variable would be writing quality of final drafts.

#### Independent Variables

In an experimental study, a variable that determines (or causes) the score of a dependent variable is called an independent variable. For instance, an experiment might explore the extent to which the writing quality of final drafts of student papers is affected by the kind of instruction they received. In this case, the independent variable would be the kind of instruction students received.

### Probability

Beginning researchers most often use the word probability to express a subjective judgment about the likelihood, or degree of certainty, that a particular event will occur. People say such things as: "It will probably rain tomorrow." "It is unlikely that we will win the ball game." It is possible to assign a number to the event being predicted, a number between 0 and 1, which represents degree of confidence that the event will occur. For example, a student might say that the likelihood an instructor will give an exam next week is about 90 percent, or .9. Where 100 percent, or 1.00, represents certainty, .9 would mean the student is almost certain the instructor will give an exam. If the student assigned the number .6, the likelihood of an exam would be just slightly greater than the likelihood of no exam. A rating of 0 would indicate complete certainty that no exam would be given(Shoeninger, 1971).

The probability of a particular outcome or set of outcomes is called a p-value. In our discussion, a p-value will be symbolized by a p followed by parentheses enclosing a symbol of the outcome or set of outcomes. For example, p(X) should be read, "the probability of a given X score" (Shoeninger). Thus p(exam) should be read, "the probability an instructor will give an exam next week."

### Population

A population is a group which is studied. In educational research, the population is usually a group of people. Researchers seldom are able to study every member of a population. Usually, they instead study a representative sample – or subset – of a population. Researchers then generalize their findings about the sample to the population as a whole.

### Sampling

Sampling is performed so that a population under study can be reduced to a manageable size. This can be accomplished via random sampling, discussed below, or via matching.

Random sampling is a procedure used by researchers in which all samples of a particular size have an equal chance to be chosen for an observation, experiment, etc (Runyon and Haber, 1976). There is no predetermination as to which members are chosen for the sample. This type of sampling is done in order to minimize scientific biases and offers the greatest likelihood that a sample will indeed be representative of the larger population. The aim here is to make the sample as representative of the population as possible. Note that the closer a sample distribution approximates the population distribution, the more generalizable the results of the sample study are to the population. Notions of probability apply here. Random sampling provides the greatest probability that the distribution of scores in a sample will closely approximate the distribution of scores in the overall population.

### Matching

Matching is a method used by researchers to gain accurate and precise results of a study so that they may be applicable to a larger population. After a population has been examined and a sample has been chosen, a researcher must then consider variables, or extrinsic factors, that might affect the study. Matching methods apply when researchers are aware of extrinsic variables before conducting a study. Two methods used to match groups are:

#### Precision Matching

In precision matching, there is an experimental group that is matched with a control group. Both groups, in essence, have the same characteristics. Thus, the proposed causal relationship/model being examined allows for the probabilistic assumption that the result is generalizable.

#### Frequency Distribution

Frequency distribution is more manageable and efficient than precision matching. Instead of one-to-one matching that must be administered in precision matching, frequency distribution allows the comparison of an experimental and control group through relevant variables. If three Communications majors and four English majors are chosen for the control group, then an equal proportion of three Communications major and four English majors should be allotted to the experiment group. Of course, beyond their majors, the characteristics of the matched sets of participants may in fact be vastly different.

Although, in theory, matching tends to produce valid conclusions, a rather obvious difficulty arises in finding subjects which are compatible. Researchers may even believe that experimental and control groups are identical when, in fact, a number of variables have been overlooked. For these reasons, researchers tend to reject matching methods in favor of random sampling.