Statistical tests are used to determine whether a hypothesized relationship between variables has statistical significance. Typically, the test will measure the degree to which the variables either correlate or differ. Parametric tests are those that rely on the central tendencies of the variables and assume a normal distribution. Non-parametric tests do not make assumptions about the population distributions.
The t-test is a parametric test that compares the means of the samples and populations involved. There are several varieties of t-tests. A one-sample t-test compares the mean of a sample with a hypothesized mean. An independent samples t-test looks at whether the means of two different samples have similar values. A paired sample t-test is used when there are two observations to compare for each subject in the sample. The t-test is designed for numeric data that has a normal distribution.
Ordinal data is derived data that describes the relative values of each unit in the sample. For example, ordinal data of the heights of 10 students in a classroom would simply be the numbers 1 through 10, where 1 might represent the shortest student and 10 might represent the tallest student. No students would have the same value unless they had exactly the same height. Measures of central tendency are meaningless with ordinal data.
Inappropriateness of T-test
T-tests are not appropriate to use with ordinal data. Because ordinal data has no central tendency, it also has no normal distribution. The values of ordinal data are evenly distributed, not grouped around a mid-point. Because of this, a t-test of ordinal data would have no statistical meaning.
Other Appropriate Tests
There are three tests of statistical significance that are appropriate to use with ordinal data. Spearman’s rank-order correlation is appropriate to use when there are only two variables involved, and their relationship is monotonic, though not necessarily linear. In monotonic relationships, as the first variable increases, there is no change in the direction of the second variable. The Kruskal-Wallis test is designed for instances where there are more than two samples, and the data is not normally distributed. It is similar to a one-way analysis of variance. The Friedman analysis of variance by ranks can be used when there are three or more observations of a single variable in a single group.