• Introduction
• Landmark Sources
• General Overviews
• Critical Evaluations
• Reference Works
• Textbooks
• Encyclopedias and Dictionaries
• Bibliographies
• Statistical Journals
• Online Statistical Resources

# Nonparametric Statistical Analysis in PsychologybyGregory J. Privitera, James J. GillespieLAST MODIFIED: 23 June 2023DOI: 10.1093/obo/9780199828340-0221

## Introduction

While nonparametric testing was first introduced in the early 1700s in a paper that utilized a version of the sign test, most nonparametric tests utilized today were developed later in the twentieth century, primarily since the late 1930s. Nonparametric testing has three unique characteristics that make it advantageous for analysis: (a) it can be used to analyze data that are not scaled, that is, data on a nominal or an ordinal scale of measurement; (b) it generally does not require assumptions about population parameters; and (c) it generally does not require that the distribution in a given population is normal, often referred to as “distribution free” tests. In terms of computation, the analysis of nonparametric tests are achieved without needing the value of a sample mean and sample variance, thereby making it possible for these tests to evaluate “effects” in populations with any type of distribution, that is, these tests can be computed without assumptions related to variability in a population. Two critical concerns highlight the need for the elucidation of nonparametric testing in terms of its role in psychology. First, much of human behavior and performance does not conform to a normal distribution, which drives the need for “distribution free” tests to comprehensively study human behavior. Second, the disclosure and reporting of statistical testing can be problematic. To some extent, a disconnect exists in the peer-review literature between the reporting of parametric tests and the corresponding assumptions for those tests, that is, assumptions often are not mentioned at all to justify the use of parametric testing. It is largely left to the reader to accept that all assumptions were satisfied. Further evidence exists of misreported statistical outcomes in the peer-review psychological literature tending to favor the researchers’ expectations of an outcome. These discrepancies highlight the need for a broader understanding of the role and utility of nonparametric statistical alternatives in null hypothesis significance testing. This article aims to provide resources, both in text and online, for introducing and explaining nonparametric statistics and advanced nonparametric methodologies in psychology.

## Landmark Sources

This section presents landmark studies that introduced the nonparametric techniques most often employed in the field of psychology. In the early 1700s, the sign test, or a version of it, was the first nonparametric test to be introduced, found in the paper Arbuthnott 1710. Wolfowitz 1942 was written by the first researcher to formally coin the term nonparametric as an alternative framework to parametric statistics. Early advances in nonparametric testing were most commonly applied to analyze the frequency of nominal or categorical data, thus the chi-square test was first evaluated in Pearson 1900 to calculate the goodness of fit for frequency distributions. To analyze correlational data, Spearman 1904a and Spearman 1904b introduced rho as a nonparametric alternative to the Pearson correlation coefficient. Among the nonparametric tests most commonly applied in the psychological sciences to analyze ordinal data was the Friedman test, found in Friedman 1937 and Friedman 1939, as a nonparametric alternative to the one-way repeated measures analysis of variance; the Kruskal-Wallis H test in Kruskal and Wallis 1952 as a nonparametric alternative to the one-way between-subjects analysis of variance; the Wilcoxon Signed-Ranks T test in Wilcoxon 1945, as a nonparametric alternative to the paired samples t test; and the Mann-Whitney U test in Mann and Whitney 1947 as a nonparametric alternative to the independent-samples t test. The authors of these works helped pioneer the growth and development of nonparametric testing as an alternative framework to parametric statistics.

• Arbuthnott, J. 1710. An argument for divine providence, taken from the constant regularity observed in the births of both sexes. Philosophical Transactions 27.328: 186–190.

This is the first article known to introduce a nonparametric test, the sign test, to assess differences in births between two groups, males and females.

• Friedman, M. 1937. The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association 32.200: 675–701.

This is the first article to introduce a nonparametric alternative to a one-way repeated measures analysis of variance, the Friedman test.

• Friedman, M. A. 1939. A correction: The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association 34.205: 109.

DOI: 10.2307/2279169

This article adds a correction to a formula published in Friedman 1937 on p. 695, placing the denominator of the fraction under a square sign.

• Kruskal, W. H., and W. A. Wallis. 1952. Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association 47: 583–621.

DOI: 10.2307/2280779

This is the first article to introduce a nonparametric alternative to a one-way between-subjects analysis of variance, the Kruskal-Wallis H test.

• Mann, H. B., and D. R. Whitney. 1947. On a test whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics 18.1: 50–60.

This is the first article to introduce a nonparametric alternative to an independent-samples t test, the Mann-Whitney U test.

• Pearson, K. 1900. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine, Series 5 50.302: 157–175.

This is the first article to investigate a nonparametric test to analyze frequency data for nominal variables, the chi-square test.

• Spearman, C. E. 1904a. The proof and measurement of association between two things. American Journal of Psychology 15.1: 72–101.

DOI: 10.2307/1412159

This is the first article to introduce a nonparametric alternative to a Pearson correlation coefficient, the Spearman rho.

• Spearman, C. E. 1904b. “General intelligence,” objectively determined and measured. American Journal of Psychology 15.2: 201–292.

DOI: 10.2307/1412107

• Wilcoxon, F. 1945. Individual comparisons by ranking methods. Biometrics 1.6: 80–83.

DOI: 10.2307/3001968

This is the first article to introduce a nonparametric alternative to a paired samples t test, the Wilcoxon Signed-Ranks T test.

• Wolfowitz, J. 1942. Additive Partition Functions and a Class of Statistical Hypotheses. Annals of Mathematical Statistics 13.3: 247–279.