Introduction: Two Types of t Tests

A t test is appropriate when your analysis involves a single predictor variable that is measured on a nominal scale and assumes only one of two values (e.g., sex), and a single criterion variable that is measured on an interval or ratio scale (e.g., GRE scores). This statistical procedure is usually viewed as a test of group differences. For example, when experimental condition is the predictor variable and scores on an attitude scale are the criterion variable, you might want to know whether there is a significant difference between the experimental group and the control group with respect to their mean attitude scores. A t test can determine this.

There are actually two types of t tests that are appropriate for different experimental designs. First, the independent-samples t test is appropriate if the observations obtained under one treatment condition are independent of (i.e., unrelated to) the observations obtained under the other treatment condition. For example, imagine that you draw a random sample of participants and randomly assign each participant to either Condition 1 or Condition 2 in your experiment. After manipulating the independent variable, you determine scores on the attitude scale for participants in both conditions, and use an independent-samples t test to determine whether the mean attitude score is significantly different for the participants in Condition 1 as compared to those in Condition 2. The independent-samples t test is appropriate here because the observations (attitude scores) in Condition 1 are completely unrelated to the observations in Condition 2 (i.e., Condition 1 consists of one group of people, and Condition 2 consists of a different group of people who are not related to, or affected by, the people in Condition 1).

The second type of test is the paired-samples t test. This statistic is appropriate if each observation in Condition 1 is paired in a meaningful way with a corresponding observation in Condition 2. There are several ways that this pairing can be achieved. For example, imagine that you draw a random sample of participants and decide that each participant will provide two attitude scores: one score after being exposed to Condition 1; and a second score after being exposed to Condition 2. In a sense, you still have two samples of observations (the sample from Condition 1 versus that from Condition 2), but the observations from the two samples are now related (i.e., from the same set of participants at two points in time). For example, this means that if a given participant scored relatively high on the attitude scale under Condition 1, it is likely that participant will also score relatively high under Condition 2. When analyzing the data, it therefore makes sense to pair each participant’s score from Condition 1 with his or her score from Condition 2. Because of this pairing, a paired-samples t statistic is calculated differently than an independent-samples t statistic.

This chapter has two major sections. The first deals with the independent-samples t test, and the second deals with the paired-samples t test. These sections provide additional examples of situations in which the two procedures might be appropriate.

Earlier, you read that a t test is appropriate when the analysis involves a nominal-scale predictor variable and an interval/ratio-scale criterion. A number of additional assumptions should also be met for the test to be valid (e.g., responses to the dependent variable are normally distributed). These assumptions are summarized in an appendix at the end of this chapter. When these assumptions are violated, consider using a nonparametric statistic instead. For help, see the SAS/STAT User’s Guide.