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Mean Comparisons
The term mean comparisons refers to the comparison of the average of one or more continuous variables over one or more categorical variables. It is a general term that can refer to a large number of different research questions and study designs. For example, one can compare the mean from one sample of data to a hypothetical population value, compare the means on a single variable from multiple independent groups, or compare the means for a single variable for one sample over multiple measurement occasions. In addition, more complex research designs can employ multiple continuous dependent variables simultaneously, as well as a combination of multiple groups and multiple measurement occasions. Overall, mean comparisons are of central interest in any experimental design and many correlational designs when there are existing categorical variables (e.g., gender).
Two primary questions must be asked in any mean comparison: Are the means statistically different, and how big are the differences? The former question can be answered with a statistical test of the difference in means. The latter is answered with a standardized measure of effect size. Together, these more accurately characterize the nature of mean differences.
Statistical Differences
Testing for statistical differences attempts to answer the question of whether the observed differences, however large or small, are due to some real effect or simply random sampling error. Depending on the nature of the data and the specific research question at hand, different statistical tests must be employed to properly answer the question of whether there are statistical differences in the means.
z Test
The z test is employed when a researcher wants to answer the question, Is the mean of this sample statistically different from the mean of the population? Here, the researcher would have mean and standard deviation information for the population of interest on a particular continuous variable and data from a single sample on the same variable. The observed mean and standard deviation from the sample would then be compared with the population mean and standard deviation. For example, suppose an organization, as part of its annual survey process, had collected job satisfaction information from all its employees. The organization then wishes to conduct a follow-up study relating to job satisfaction with some of its employees and wishes to make sure the sample drawn is representative of the company. Here, the sample mean and standard deviation on the job satisfaction variable would be compared with the mean and standard deviation for the company as a whole and tested with a z test. This test, however, has limited applications in most research settings, for the simple fact that information for the population is rarely available. When population information is unavailable, different statistical tests must be used.
t Test
Unlike the z test, the t test is a widely used and applicable statistical test. In general, the t test is used to compare two groups on a single continuous dependent variable. Generally, this statistical test answers the question, Is the mean of this sample statistically different from the mean of this other sample? By removing the assumption that there is population information available, the t test becomes a much more flexible statistical technique. The t test can be used with experimental studies to compare two experimental conditions, with correlational studies to compare existing dichotomous groups (e.g., gender), and with longitudinal studies to compare the same sample over two measurement occasions. An important limiting factor of the t test is that it can compare only two groups at a time; investigations of mean differences in more complex research designs require a more flexible analytic technique.
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