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Non-Parametric Statistics: Types, Tests, and Examples

  • Pragya Soni
  • May 12, 2022
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Statistics, an essential element of data management and predictive analysis, is classified into two types, parametric and non-parametric. 


Parametric tests are based on the assumptions related to the population or data sources while, non-parametric test is not into assumptions, it's more factual than the parametric tests. Here is a detailed blog about non-parametric statistics.



What is the Meaning of Non-Parametric Statistics?


Unlike, parametric statistics, non-parametric statistics is a branch of statistics that is not solely based on the parametrized families of assumptions and probability distribution. Non-parametric statistics depend on either being distribution free or having specified distribution, without keeping any parameters into consideration.


Non-parametric statistics are defined by non-parametric tests; these are the experiments that do not require any sample population for assumptions. For this reason, non-parametric tests are also known as distribution free tests as they don’t rely on data related to any particular parametric group of probability distributions.


In other terms, non-parametric statistics is a statistical method where a particular data is not required to fit in a normal distribution. Usually, non-parametric statistics used the ordinal data that doesn’t rely on the numbers, but rather a ranking or order. For consideration, statistical tests, inferences, statistical models, and descriptive statistics.


Non-parametric statistics is thus defined as a statistical method where data doesn’t come from a prescribed model that is determined by a small number of parameters. Unlike normal distribution model, factorial design and regression modeling, non-parametric statistics is a whole different content.


Unlike parametric models, non-parametric is quite easy to use but it doesn’t offer the exact accuracy like the other statistical models. Therefore, non-parametric statistics is generally preferred for the studies where a net change in input has minute or no effect on the output. Like even if the numerical data changes, the results are likely to stay the same.


Also Read | What is Regression Testing?



How does Non-Parametric Statistics Work?


Parametric statistics consists of the parameters like mean, standard deviation, variance, etc. Thus, it uses the observed data to estimate the parameters of the distribution. Data are often assumed to come from a normal distribution with unknown parameters.


While, non-parametric statistics doesn’t assume the fact that the data is taken from a same or normal distribution. In fact, non-parametric statistics assume that the data is estimated under a different measurement. The actual data generating process is quite far from the normally distributed process.


Types of Non-Parametric Statistics


Non-parametric statistics are further classified into two major categories. Here is the brief introduction to both of them:


1. Descriptive Statistics


Descriptive statistics is a type of non-parametric statistics. It represents the entire population or a sample of a population. It breaks down the measure of central tendency and central variability.


2. Statistical Inference


Statistical inference is defined as the process through which inferences about the sample population is made according to the certain statistics calculated from the sample drawn through that population.



Some Examples of Non-Parametric Tests


In the recent research years, non-parametric data has gained appreciation due to their ease of use. Also, non-parametric statistics is applicable to a huge variety of data despite its mean, sample size, or other variation. As non-parametric statistics use fewer assumptions, it has wider scope than parametric statistics.


Here are some common examples of non-parametric statistics:


  1. Consider the case of a financial analyst who wants to estimate the value of risk of an investment. Now, rather than making the assumption that earnings follow a normal distribution, the analyst uses a histogram to estimate the distribution by applying non-parametric statistics.


  1. Consider another case of a researcher who is researching to find out a relation between the sleep cycle and healthy state in human beings. Taking parametric statistics here will make the process quite complicated. 


So, despite using a method that assumes a normal distribution for illness frequency. The researcher will opt to use any non-parametric method like quantile regression analysis.


  1. Similarly, consider the case of another health researcher, who wants to estimate the number of babies born underweight in India, he will also employ the non-parametric measurement for data testing.


  1. A marketer that is interested in knowing the market growth or success of a company, will surely employ a non-statistical approach.


  1. Any researcher that is testing the market to check the consumer preferences for a product will also employ a non-statistical data test. As different parameters in nutritional value of the product like agree, disagree, strongly agree and slightly agree will make the parametric application hard.


  1. Any other science or social science research which include nominal variables such as age, gender, marital data, employment, or educational qualification is also called as non-parametric statistics. It plays an important role when the source data lacks clear numerical interpretation.


Also Read | Applications of Statistical Techniques



What are Non-Parametric Tests?

 Types of Non-Parametric Tests:1. Wilcoxon test 2. Mann-Whitney test 3. Kruskal Wallis test 4. Friedmann test

Types of Non-Parametric Tests

 Here is the list of non-parametric tests that are conducted on the population for the purpose of statistics tests :


  1. Wilcoxon Rank Sum Test


The Wilcoxon test also known as rank sum test or signed rank test. It is a type of non-parametric test that works on two paired groups. The main focus of this test is comparison between two paired groups. The test helps in calculating the difference between each set of pairs and analyses the differences.


The Wilcoxon test is classified as a statistical hypothesis test and is used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean rank is different or not.



  1. Mann- Whitney U Test


The Mann-Whitney U test also known as the Mann-Whitney-Wilcoxon test, Wilcoxon rank sum test and Wilcoxon-Mann-Whitney test. It is a non-parametric test based on null hypothesis. It is equally likely that a randomly selected sample from one sample may have higher value than the other selected sample or maybe less.


Mann-Whitney test is usually used to compare the characteristics between two independent groups when the dependent variable is either ordinal or continuous. But these variables shouldn’t be normally distributed. For a Mann-Whitney test, four requirements are must to meet. The first three are related to study designs and the fourth one reflects the nature of data.



  1. Kruskal Wallis Test


Sometimes referred to as a one way ANOVA on ranks, Kruskal Wallis H test is a nonparametric test that is used to determine the statistical differences between the two or more groups of an independent variable. The word ANOVA is expanded as Analysis of variance.


The test is named after the scientists who discovered it, William Kruskal and W. Allen Wallis. The major purpose of the test is to check if the sample is tested if the sample is taken from the same population or not.



  1. Friedman Test


The Friedman test is similar to the Kruskal Wallis test. It is an alternative to the ANOVA test. The only difference between Friedman test and ANOVA test is that Friedman test works on repeated measures basis. Friedman test is used for creating differences between two groups when the dependent variable is measured in the ordinal.


The Friedman test is further divided into two parts, Friedman 1 test and Friedman 2 test. It was developed by sir Milton Friedman and hence is named after him. The test is even applicable to complete block designs and thus is also known as a special case of Durbin test.



  1. Distribution Free Tests


Distribution free tests are defined as the mathematical procedures. These tests are widely used for testing statistical hypotheses. It makes no assumption about the probability distribution of the variables. An important list of distribution free tests is as follows:


  •  Anderson-Darling test: It is done to check if the sample is drawn from a given distribution or not.


  • Statistical bootstrap methods: It is a basic non-statistical test used to estimate the accuracy and sampling distribution of a statistic.


  • Cochran’s Q: Cochran’s Q is used to check constant treatments in block designs with 0/1 outcomes.


  • Cohen’s kappa: Cohen kappa is used to measure the inter-rater agreement for categorical items.


  • Kaplan-Meier test: Kaplan Meier test helps in estimating the survival function from lifetime data, modeling, and censoring.


  • Two-way analysis Friedman test: Also known as ranking test, it is used to randomize different block designs.


  • Kendall’s tau: The test helps in defining the statistical dependency between two different variables.


  • Kolmogorov-Smirnov test: The test draws the inference if a sample is taken from the same distribution or if two or more samples are taken from the same sample.


  • Kendall’s W: The test is used to measure the inference of an inter-rater agreement.


  • Kuiper’s test: The test is done to determine if the sample drawn from a given distribution is sensitive to cyclic variations or not.


  • Log Rank test: This test compares the survival distribution of two right-skewed and censored samples.


  • McNemar’s test: It tests the contingency in the sample and revert when the row and column marginal frequencies are equal to or not.


  • Median tests: As the name suggests, median tests check if the two samples drawn from the similar population have similar median values or not.


  • Pitman’s permutation test: It is a statistical test that yields the value of p variables. This is done by examining all possible rearrangements of labels.


  • Rank products: Rank products are used to detect expressed genes in replicated microarray experiments.


  • Siegel Tukey tests: This test is used for differences in scale between two groups.


  • Sign test: Sign test is used to test whether matched pair samples are drawn from distributions from equal medians.


  • Spearman’s rank: It is used to measure the statistical dependence between two variables using a monotonic function.


  • Squared ranks test: Squared rank test helps in testing the equality of variances between two or more variables.


  • Wald-Wolfowitz runs a test: This test is done to check if the elements of the sequence are mutually independent or random.


Also Read | Factor Analysis



Advantages and Disadvantages of Non-Parametric Tests


The benefits of non-parametric tests are as follows:


  1. It is easy to understand and apply.


  1. It consists of short calculations.


  1. The assumption of the population is not required.


  1. Non-parametric test is applicable to all data kinds


The limitations of non-parametric tests are:


  1. It is less efficient than parametric tests.


  1. Sometimes the result of non-parametric data is insufficient to provide an accurate answer.



Applications of Non-Parametric Tests


Non-parametric tests are quite helpful, in the cases :


  1. Where parametric tests are not giving sufficient results.


  1. When the testing hypothesis is not based on the sample.


  1. For the quicker analysis of the sample.


  1. When the data is unscaled.


The current scenario of research is based on fluctuating inputs, thus, non-parametric statistics and tests become essential for in-depth research and data analysis.

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