# Nonparametric Statistics: Overview, Types, and Examples

## What Are Nonparametric Statistics?

Nonparametric statistics, often known as distribution-free statistics, is a field of statistics that deals with data that does not fit into any of the standard distributions. Nonparametric statistics, in contrast to parametric statistics, which assumes that the data follows a normal or other specified distribution, merely implies that the data has certain features, such as being continuous, discrete, or ordinal.

When the assumptions of parametric statistics are not satisfied, such as when the sample size is small or the data is not regularly distributed, nonparametric statistics are utilised. We will present an overview of nonparametric statistics, explore the many types of nonparametric tests, and provide examples of its applications in this article.

## Overview of Nonparametric Statistics

Nonparametric statistics is a statistical analysis approach that makes no assumptions about the underlying distribution of the data. It is instead based on the ranking or ordering of the data. Nonparametric techniques are particularly useful when the data does not conform to the normal distribution assumptions or when the sample size is restricted.

Nonparametric techniques can be used to compare more than two groups, test the difference between two groups, test the correlation between two variables, and evaluate the independence of two variables. When the sample size is high and the normal distribution assumptions are fulfilled, nonparametric techniques are less powerful than parametric methods.However, they are typically more robust and can provide correct results even in the presence of outliers or non-normal data.

## Types of Nonparametric Tests

There are various types of nonparametric tests that are frequently used in statistical analysis. These tests may be classified into two categories: location tests and independence tests. Location tests are used to compare the location, or central tendency, of two or more groups. To examine the relationship between two variables, tests of independence are utilised. Tests of independence are used to test the association between two variables.

## Tests of Location

1. Mann-Whitney U test: The Mann-Whitney U test is used to compare the medians of two independent samples. The test involves ranking the data and calculating the U statistic, which is used to determine whether the two samples come from the same distribution.
2. Wilcoxon rank-sum test: The Wilcoxon rank-sum test is similar to the Mann-Whitney U test, but it is used to compare the medians of two independent samples when the sample sizes are small. The test involves ranking the data and calculating the sum of the ranks for each sample.
3. Wilcoxon signed-rank test: The Wilcoxon signed-rank test is used to compare the medians of two related samples. The test involves ranking the differences between the two samples and calculating the signed-rank statistic.
4. Kruskal-Wallis test: The Kruskal-Wallis test is used to compare the medians of three or more independent samples. The test involves ranking the data and calculating the sum of the ranks for each sample. The test is a nonparametric equivalent of the one-way ANOVA.

## Tests of Independence

1. Chi-square test: The chi-square test is used to test the independence of two categorical variables. The test involves comparing the observed frequencies to the expected frequencies under the assumption of independence.
2. Fisher’s exact test: Fisher’s exact test is used to test the independence of two categorical variables when the sample size is small. The test involves calculating the exact probability of obtaining the observed data under the assumption of independence.
3. Kendall’s tau: Kendall’s tau is a nonparametric measure of correlation that is used to test the association between two ordinal variables.
4. Spearman’s rank correlation coefficient: Spearman’s rank correlation coefficient is a nonparametric measure of correlation that is used to test the association between two interval or ordinal variables. The test involves ranking the data and calculating the correlation coefficient between the ranks.
1. Kruskal-Wallis H test: The Kruskal-Wallis H test is a nonparametric equivalent of the one-way ANOVA, used to test for differences in medians among three or more independent samples.

## Examples of Nonparametric Tests

1. A scientist wants to evaluate the efficacy of two different therapies for a certain medical problem. The researcher assigns participants to one of two therapy groups at random and examines the end variable, which is symptom reduction. The data is out of the ordinary, and the sample sizes are modest. The researcher chooses to compare the medians of the two treatment groups using the Wilcoxon rank-sum test.
2. A corporation want to compare the levels of customer satisfaction among three distinct items. The firm picks a sample of consumers who have tried each product at random and asks them to score their satisfaction on a scale of 1 to 10 on a scale of 1 to 10. The data is out of the ordinary, and the sample sizes are modest. The corporation chooses to compare the medians of the three goods using the Kruskal-Wallis test.
3. A researcher wants to explore the relationship between age and income in a sample of 100 people. The researcher examines each participant’s age and income and discovers that the data is not regularly distributed. To assess the relationship between the two variables, the researcher decides to employ Spearman’s rank correlation coefficient.
4. An organization wants to see if the proportion of customers who favour a certain brand of product is consistent across the country. The corporation chooses a random sample of clients from each location and asks them to identify their prefered brand. The information is classified, and the sample size is modest. Fisher’s exact test is used by the firm to assess the independence of brand choice and geography.

Nonparametric statistics is an effective technique for analysing data that does not fit into a predefined distribution. Nonparametric approaches have a wide range of applications, including comparing more than two groups, evaluating the difference between two groups, testing the correlation between two variables, and assessing the independence of two variables. Researchers and analysts can generate reliable statistical judgements using nonparametric tests even when the conditions of normality and homogeneity are not satisfied.

What exactly is nonparametric statistics?
Nonparametric statistics is a type of statistical procedure that does not make assumptions about the data’s underlying distribution. When the data does not fit a given distribution or the sample size is too small to allow for normalcy assumptions, nonparametric approaches are applied.

When should I use nonparametric statistics?
When the data being analysed does not match the assumptions of normality or homogeneity of variance, or when the data is ordinal or categorical, nonparametric statistics should be employed. When the sample size is limited, nonparametric approaches are also beneficial.

What are some examples of common nonparametric tests?
The Wilcoxon rank-sum test, the Mann-Whitney U test, the Kruskal-Wallis test, the Friedman test, the Spearman rank correlation coefficient, and Fisher’s exact test are all examples of common nonparametric tests.

How do I interpret nonparametric test results?
Nonparametric test results are interpreted similarly to parametric test findings. If the p-value is less than the significance level, the null hypothesis is rejected, and the alternative hypothesis is accepted. Nonparametric tests, on the other hand, often look for differences in medians rather than means.

Is it possible to apply nonparametric statistics with huge sample sizes?
Nonparametric approaches can be applied to any sample size, although they may be less powerful than parametric methods when applied to large samples.

Can nonparametric approaches be used with continuous data?
Nonparametric approaches can be employed with continuous data, but only when the data is non-normal or the sample size is limited.

What is the difference between parametric and nonparametric methods?
The decision between parametric and nonparametric approaches is determined by the type of data being analyzed, the data assumptions that may be made, and the research question being answered. With high sample sizes, parametric methods are often more powerful than nonparametric approaches, but nonparametric methods are more resilient when the assumptions of normality or homogeneity of variance are not satisfied.