# List of statistical tests

Statistical tests are used to test the fit between a hypothesis and the data. Choosing the right statistical test is not a trivial task. The choice of the test depends on many properties of the research question. The vast majority of studies can be addressed by 30 of the 100 or so statistical tests in use.

## Explanation of properties

* Scaling of data: One of the properties of the tests is the scale of the data, which can be interval-based, ordinal or nominal. Nominal scale is also known as categorical. Interval scale is also known as numerical. When categorical data has only two possibilities, it is called binary or dichotomous.
* Assumptions, parametric and non-parametric: There are two groups of statistical tests, parametric and non-parametric. The choice between these two groups needs to be justified. Parametric tests assume that the data follow a particular distribution, typically a normal distribution, while non-parametric tests make no assumptions about the distribution. Non-parametric tests have the advantage of being more resistant to misbehaviour of the data, such as outliers. They also have the disadvantage of being less certain in the statistical estimate.
* Type of data: Statistical tests use different types of data. Some tests perform univariate analysis on a single sample with a single variable. Others compare two or more paired or unpaired samples. Unpaired samples are also called independent samples. Paired samples are also called dependent. Finally, there are some statistical tests that perform analysis of relationship between multiple variables like regression.
* Number of samples: The number of samples of data.
* Exactness: A test can be exact or be asymptotic delivering approximate results.

## List of statistical tests

| Test name | Scaling | Assumptions | Data | Samples | Exact | Special case of | Application conditions |
| One sample t-test | interval | normal | univariate | 1 | No | Location test | |
| Unpaired t-test | interval | normal | unpaired | 2 | No | Location test | Homoscedasticity |
| Welch's t-test | interval | normal | unpaired | 2 | No | Location test | |
| Paired t-test | interval | normal | paired | 2 | No | Location test | |
| F-test | interval | normal | | 2 | | | |
| Z-test | interval | normal | | 2 | No | | variance is known |
| Permutation test | interval | non-parametric | unpaired | ≥2 | Yes | | |
| Kruskal-Wallis test | ordinal | non-parametric | unpaired | ≥2 | Yes | | small sample size |
| Mann–Whitney $U$ test | ordinal | non-parametric | unpaired | 2 | | Kruskal-Wallis test | |
| Wilcoxon signed-rank test | interval | non-parametric | paired | ≥1 | | Location test | |
| Sign test | ordinal | non-parametric | paired | 2 | | | |
| Friedman test | ordinal | non-parametric | paired | >2 | | Location test | |
| $\chi ^{2}$ test | nominal | non-parametric | | | No | | Contingency table, sample size > ca. 60, any cell content ≥ 5, marginal totals fixed |
| Pearson's $\chi ^{2}$ test | nominal/ordinal | non-parametric | | | No | $\chi ^{2}$ test | |
| Median test | ordinal | non-parametric | | | No | Pearson's $\chi ^{2}$ test | |
| Multinomial test | nominal | non-parametric | univariate | 1 | Yes | Location test | |
| McNemar's test | binary | non-parametric | paired | 2 | Yes | Cochran's $Q$ test | |
| Cochran's $Q$ test | binary | non-parametric | paired | ≥2 | | | |
| Binomial test | binary | non-parametric | univariate | 1 | Yes | Multinomial test | |
| Siegel–Tukey test | ordinal | non-parametric | unpaired | 2 | | | |
| Chow test | interval | parametric | linear regression | 2 | No | | Time series |
| Fisher's exact test | nominal | non-parametric | unpaired | ≥2 | Yes | | Contingency table, marginal totals fixed |
| Barnard's exact test | nominal | non-parametric | unpaired | 2 | Yes | | Contingency table |
| Boschloo's test | nominal | non-parametric | unpaired | 2 | Yes | | Contingency table |
| Shapiro–Wilk test | interval | | univariate | 1 | | Normality test | sample size between 3 and 5000 |
| Kolmogorov–Smirnov test | interval | | | 1 | | Normality test | distribution parameters known |
| Shapiro-Francia test | interval | | univariate | 1 | | Normality test | Simpliplification of Shapiro–Wilk test |
| Lilliefors test | interval | | | 1 | | Normality test | |