Statistical tests are used to test the fit between a hypothesis and the data.[1][2] Choosing the right statistical test is not a trivial task.[1] 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.[3][4][5]

## 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.[3] Nominal scale is also known as categorical.[6] Interval scale is also known as numerical.[6] When categorical data has only two possibilities, it is called binary or dichotomous.[1]
• 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.[7] Non-parametric tests have the advantage of being more resistant to misbehaviour of the data, such as outliers.[7] They also have the disadvantage of being less certain in the statistical estimate.[7]
• Type of data: Statistical tests use different types of data.[1] 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.[1]
• 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

This section needs expansion. You can help by adding to it. (February 2024)
Test name Scaling Assumptions Data Samples Exact Special case of Application conditions
One sample t-test interval normal univariate 1 No[8] Location test
Unpaired t-test interval normal unpaired 2 No[8] Location test Homoscedasticity[9]
Welch's t-test interval normal unpaired 2 No[8] 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[10]
Mann–Whitney ${\displaystyle U}$ test ordinal non-parametric unpaired 2 Kruskal-Wallis test[11]
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
${\displaystyle \chi ^{2))$ test nominal[1] non-parametric[12] No Contingency table,
sample size > ca. 60,[1]
any cell content ≥ 5,[13]
marginal totals fixed[13]
Pearson's ${\displaystyle \chi ^{2))$ test nominal/ordinal non-parametric No ${\displaystyle \chi ^{2))$ test
Median test ordinal non-parametric No Pearson's ${\displaystyle \chi ^{2))$ test
Multinomial test nominal non-parametric univariate 1 Yes Location test
McNemar's test binary non-parametric[14] paired 2 Yes Cochran's ${\displaystyle Q}$ test[15]
Cochran's ${\displaystyle Q}$ 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[13] Yes Contingency table,
marginal totals fixed[13]
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[16]
Kolmogorov–Smirnov test interval 1 Normality test distribution parameters known[16]
Shapiro-Francia test interval univariate 1 Normality test Simpliplification of Shapiro–Wilk test
Lilliefors test interval 1 Normality test

## References

1. Parab, Shraddha; Bhalerao, Supriya (2010). "Choosing statistical test". International Journal of Ayurveda Research. 1 (3): 187–191. doi:10.4103/0974-7788.72494. ISSN 0974-7788. PMC 2996580. PMID 21170214.
2. ^ "Entscheidbaum" (in German). Retrieved 8 February 2024.
3. ^ a b Nayak, Barun K; Hazra, Avijit (2011). "How to choose the right statistical test?". Indian Journal of Ophthalmology. 59 (2): 85–86. doi:10.4103/0301-4738.77005. ISSN 0301-4738. PMC 3116565. PMID 21350275.
4. ^ Lewis, Nancy D.; Lewis, Nigel Da Costa; Lewis, N. D. (2013). 100 Statistical Tests in R: What to Choose, how to Easily Calculate, with Over 300 Illustrations and Examples. Heather Hills Press. ISBN 978-1-4840-5299-0.
5. ^ Kanji, Gopal K. (18 July 2006). 100 Statistical Tests. SAGE. ISBN 978-1-4462-2250-8.
6. ^ a b "What is the difference between categorical, ordinal and interval variables?". stats.oarc.ucla.edu. Retrieved 10 February 2024.
7. ^ a b c Huth, R.; Pokorná, L. (1 March 2004). "Parametric versus non-parametric estimates of climatic trends". Theoretical and Applied Climatology. 77 (1): 107–112. Bibcode:2004ThApC..77..107H. doi:10.1007/s00704-003-0026-3. ISSN 1434-4483. S2CID 121539673.
8. ^ a b c de Winter, J.C.F. (2019). "Using the Student's t-test with extremely small sample sizes". Practical Assessment, Research, and Evaluation. 18. doi:10.7275/e4r6-dj05.
9. ^ "t-Test für unabhängige Stichproben". Hochschule Luzern (in German). Retrieved 10 February 2024.
10. ^ Choi, Won; Lee, Jae Won; Huh, Myung-Hoe; Kang, Seung-Ho (11 January 2003). "An Algorithm for Computing the Exact Distribution of the Kruskal–Wallis Test". Communications in Statistics - Simulation and Computation. 32 (4): 1029–1040. doi:10.1081/SAC-120023876. ISSN 0361-0918. S2CID 123037097.
11. ^ McKight, Patrick E.; Najab, Julius (30 January 2010). "Kruskal-Wallis Test". The Corsini Encyclopedia of Psychology. Wiley. p. 1. doi:10.1002/9780470479216.corpsy0491. ISBN 978-0-470-17024-3.
12. ^ McHugh, Mary L. (15 June 2013). "The Chi-square test of independence". Biochemia Medica. 23 (2): 143–149. doi:10.11613/BM.2013.018. PMC 3900058. PMID 23894860.
13. ^ a b c d Warner, Pamela (1 October 2013). "Testing association with Fisher's Exact test". Journal of Family Planning and Reproductive Health Care. 39 (4): 281–284. doi:10.1136/jfprhc-2013-100747. ISSN 1471-1893. PMID 24062499.
14. ^ Károly, Héberger; Róbert, Rajkó (1999). Pair-Correlation Method with parametric and non-parametric test-statistics for variable selection. Description of computer program and application for environmental data case studies. szef. pp. 82–91.
15. ^ Carpi, Angelo; Rossi, Giuseppe; Coscio, Giancarlo Di; Iervasi, Giorgio; Nicolini, Andrea; Carpi, Federico; Mechanick, Jeffrey I.; Bartolazzi, Armando (2010). "Galectin-3 detection on large-needle aspiration biopsy improves preoperative selection of thyroid nodules: a prospective cohort study". Annals of Medicine. 42 (1): 70–78. doi:10.3109/07853890903439778. ISSN 1365-2060.
16. ^ a b Ahmad, Fiaz; Khan, Rehan Ahmad (8 September 2015). "A power comparison of various normality tests". Pakistan Journal of Statistics and Operation Research. 11 (3): 331–345. doi:10.18187/pjsor.v11i3.845. ISSN 1816-2711.