Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an ordinal scale, one of four levels of measurement described by S. S. Stevens in 1946. The ordinal scale is distinguished from the nominal scale by having a ranking. It also differs from the interval scale and ratio scale by not having category widths that represent equal increments of the underlying attribute.
Examples of ordinal data A well-known example of ordinal data is the Likert scale. An example of a Likert scale is:
Examples of ordinal data are often found in questionnaires: for example, the survey question "Is your general health poor, reasonable, good, or excellent?" may have those answers coded respectively as 1, 2, 3, and 4. Sometimes data on an interval scale or ratio scale are grouped onto an ordinal scale: for example, individuals whose income is known might be grouped into the income categories $0–$19,999, $20,000–$39,999, $40,000–$59,999, ..., which then might be coded as 1, 2, 3, 4, .... Other examples of ordinal data include socioeconomic status, military ranks, and letter grades for coursework.
Ways to analyse ordinal data Ordinal data analysis requires a different set of analyses than other qualitative variables. These methods incorporate the natural ordering of the variables in order to avoid loss of power. Computing the mean of a sample of ordinal data is discouraged; other measures of central tendency, including the median or mode, are generally more appropriate.
General Stevens (1946) argued that, because the assumption of equal distance between categories does not hold for ordinal data, the use of means and standard deviations for description of ordinal distributions and of inferential statistics based on means and standard deviations was not appropriate. Instead, positional measures like the median and percentiles, in addition to descriptive statistics appropriate for nominal data (number of cases, mode, contingency correlation), should be used. Nonparametric methods have been proposed as the most appropriate procedures for inferential statistics involving ordinal data (e.g, Kendall's W, Spearman's rank correlation coefficient, etc.), especially those developed for the analysis of ranked measurements. However, the use of parametric statistics for ordinal data may be permissible with certain caveats to take advantage of the greater range of available statistical procedures.
Univariate statistics In place of means and standard deviations, univariate statistics appropriate for ordinal data include the median, other percentiles (such as quartiles and deciles), and the quartile deviation. One-sample tests for ordinal data include the Kolmogorov-Smirnov one-sample test, the one-sample runs test, and the change-point test.
Bivariate statistics In lieu of testing differences in means with t-tests, differences in distributions of ordinal data from two independent samples can be tested with Mann-Whitney, runs, Smirnov, and signed-ranks tests. Test for two related or matched samples include the sign test and the Wilcoxon signed ranks test. Analysis of variance with ranks and the Jonckheere test for ordered alternatives can be conducted with ordinal data in place of independent samples ANOVA. Tests for more than two related samples includes the Friedman two-way analysis of variance by ranks and the Page test for ordered alternatives. Correlation measures appropriate for two ordinal-scaled variables include Kendall's tau, gamma, rs, and dyx/dxy.
Regression applications Ordinal data can be considered as a quantitative variable. In logistic regression, the equation
logit [ P ( Y = 1 ) ] = α + β 1 c + β 2 x {\displaystyle \operatorname {logit} [P(Y=1)]=\alpha +\beta _{1}c+\beta _{2}x}
is the model and c takes on the assigned levels of the categorical scale. In regression analysis, outcomes (dependent variables) that are ordinal variables can be predicted using a variant of ordinal regression, such as ordered logit or ordered probit. In multiple regression/correlation analysis, ordinal data can be accommodated using power polynomials and through normalization of scores and ranks.
Linear trends Linear trends are also used to find associations between ordinal data and other categorical variables, normally in a contingency tables. A correlation r is found between the variables where r lies between -1 and 1. To test the trend, a test statistic:
M 2 = ( n − 1 ) r 2 {\displaystyle M^{2}=(n-1)r^{2}}
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