Nonparametric tests do not assume a normal distribution and are more robust, require smaller samples, and can be applied with fewer assumptions. They are effective for questions involving frequencies and proportions. Nonparametric tests avoid the assumption of normality and examine data by categorizing or sorting it. A common nonparametric test is the Chi-squared test, which is used to compare observed frequencies or proportions. Another nonparametric test is the Wilcoxon rank sum test, which examines the rank of each value. Nonparametric testing is growing and can be applied in any field where conventional statistics have been used.
A nonparametric test is a type of statistical hypothesis test that does not assume a normal distribution. For this reason, nonparametric tests are sometimes referred to as distribution-free. A nonparametric test is more robust than a standard test, generally requires smaller samples, is less likely to be influenced by external observations, and can be applied with fewer assumptions. On the other hand, nonparametric tests can be less efficient than their standard counterparts, particularly if the population is truly normally distributed. Nonparametric tests are particularly effective for questions involving frequencies and proportions.
Standard hypothesis testing compares a sample from a test population with a sample from a control population to determine whether the test population is statistically comparable to the control population. If the difference between the sample parameter or parameters, usually the mean and/or variance, is large enough, the test sample can be judged to be distinct from the control population. Such a parametric test requires that the parameters come from a normal distribution.
It has been mathematically proven that a sample size of 30 or more will behave approximately like a normal distribution, so this requirement is generally assumed. If the hypothesis is not justified, however, the test results may be invalid. Nonparametric tests avoid this assumption.
Instead, nonparametric hypothesis testing commonly examines data by categorizing or sorting it. If the sample and control populations are the same and if the data was collected correctly, any differences between their categories or rankings are strictly due to chance. If the probability that these differences could have occurred by chance, also called a p-value, is less than a chosen significant probability, usually 5% or 1%, then the tester rejects the hypothesis that the sample and populations of check they are the same and concludes that they are different.
A common nonparametric test is a Chi-squared test, used to compare observed frequencies or proportions. When only one set of frequencies is examined, this is often called a goodness-of-fit test and is used to determine whether the observed frequencies are within the expected range. For example, a goodness-of-fit test might be used to determine whether a roulette table was rigged by comparing the table’s results to the results predicted by probability theory, or to determine whether a headache drug is effective by comparing the proportion of people whose headaches improved with the medicine to the proportion of people whose headaches improved when they took a placebo. If two frequencies are examined, the nonparametric Chi-square test can be used to test for correlation or independence between factors. Political pollsters often look for a correlation between social, economic, or demographic factors and political beliefs—for example, to see if there is a correlation between a person’s education and whether they approve of an elected official’s behavior.
Another nonparametric test is the Wilcoxon rank sum test, which is generally used in the same situations as the standard parametric hypothesis test. Instead of examining the mean of each sample, however, the Wilcoxon test examines the rank of each value if the two samples are ordered from least to greatest. If the two samples are equal, each group should be evenly distributed in the ranking. If a group is clustered at the lower or upper end of the ranking, this indicates that the two groups are different.
For example, suppose someone wants to determine whether animated movies are longer or shorter than non-animated movies. For a standard test, determine the average length of a sample of animated films and a sample of action films and compare the difference with the variance of the samples. For Wilcoxon’s nonparametric test, the film times are ranked from least to greatest, and the ranks of the animated film times are summed.
The person could calculate the probability that the sum of ranks is that size or smaller by determining the number of possible sorts with a given sum of ranks and the total number of possible sorts, a calculation that is simple given enough brute-force computational strength . With two small samples of six films each, there are already 924 possible ranking arrangements, a number that rapidly grows much larger as more films are added. Alternatively, there are published tables that provide probabilities corresponding to certain rank sums for certain sample sizes. These can be found in statistical texts or online.
Nonparametric testing is a growing field. It can be applied in any field where even more conventional statistics have been used. However, applications are particularly common in the social sciences and medicine, particularly when the normal distribution cannot be applied.
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