IDEAS home Printed from https://ideas.repec.org/a/spr/stmapp/v25y2016i4d10.1007_s10260-016-0353-z.html
   My bibliography  Save this article

Methods to test for equality of two normal distributions

Author

Listed:
  • Julian Frank

    (Karlsruhe Institute of Technology (KIT))

  • Bernhard Klar

    (Karlsruhe Institute of Technology (KIT))

Abstract

Statistical tests for two independent samples under the assumption of normality are applied routinely by most practitioners of statistics. Likewise, presumably each introductory course in statistics treats some statistical procedures for two independent normal samples. Often, the classical two-sample model with equal variances is introduced, emphasizing that a test for equality of the expected values is a test for equality of both distributions as well, which is the actual goal. In a second step, usually the assumption of equal variances is discarded. The two-sample t test with Welch correction and the F test for equality of variances are introduced. The first test is solely treated as a test for the equality of central location, as well as the second as a test for the equality of scatter. Typically, there is no discussion if and to which extent testing for equality of the underlying normal distributions is possible, which is quite unsatisfactorily regarding the motivation and treatment of the situation with equal variances. It is the aim of this article to investigate the problem of testing for equality of two normal distributions, and to do so using knowledge and methods adequate to statistical practitioners as well as to students in an introductory statistics course. The power of the different tests discussed in the article is examined empirically. Finally, we apply the tests to several real data sets to illustrate their performance. In particular, we consider several data sets arising from intelligence tests since there is a large body of research supporting the existence of sex differences in mean scores or in variability in specific cognitive abilities.

Suggested Citation

  • Julian Frank & Bernhard Klar, 2016. "Methods to test for equality of two normal distributions," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 25(4), pages 581-599, November.
  • Handle: RePEc:spr:stmapp:v:25:y:2016:i:4:d:10.1007_s10260-016-0353-z
    DOI: 10.1007/s10260-016-0353-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10260-016-0353-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10260-016-0353-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Loughin, Thomas M., 2004. "A systematic comparison of methods for combining p-values from independent tests," Computational Statistics & Data Analysis, Elsevier, vol. 47(3), pages 467-485, October.
    2. E. George & G. Mudholkar, 1983. "On the convolution of logistic random variables," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 30(1), pages 1-13, December.
    3. Lingyun Zhang & Xinzhong Xu & Gemai Chen, 2012. "The Exact Likelihood Ratio Test for Equality of Two Normal Populations," The American Statistician, Taylor & Francis Journals, vol. 66(3), pages 180-184, August.
    4. Marco Marozzi, 2012. "A combined test for differences in scale based on the interquantile range," Statistical Papers, Springer, vol. 53(1), pages 61-72, February.
    5. Murdoch, Duncan J. & Tsai, Yu-Ling & Adcock, James, 2008. "P-Values are Random Variables," The American Statistician, American Statistical Association, vol. 62, pages 242-245, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Marco Marozzi, 2012. "A combined test for differences in scale based on the interquantile range," Statistical Papers, Springer, vol. 53(1), pages 61-72, February.
    2. Matvei Khoroshkin & Andrey Buyan & Martin Dodel & Albertas Navickas & Johnny Yu & Fathima Trejo & Anthony Doty & Rithvik Baratam & Shaopu Zhou & Sean B. Lee & Tanvi Joshi & Kristle Garcia & Benedict C, 2024. "Systematic identification of post-transcriptional regulatory modules," Nature Communications, Nature, vol. 15(1), pages 1-21, December.
    3. Feld, Jan & Zölitz, Ulf, 2022. "The effect of higher-achieving peers on major choices and labor market outcomes," Journal of Economic Behavior & Organization, Elsevier, vol. 196(C), pages 200-219.
    4. Alexander Kaever & Manuel Landesfeind & Kirstin Feussner & Burkhard Morgenstern & Ivo Feussner & Peter Meinicke, 2014. "Meta-Analysis of Pathway Enrichment: Combining Independent and Dependent Omics Data Sets," PLOS ONE, Public Library of Science, vol. 9(2), pages 1-12, February.
    5. Day, Brett & Bateman, Ian & Binner, Amy & Ferrini, Silvia & Fezzi, Carlo, 2019. "Structurally-consistent estimation of use and nonuse values for landscape-wide environmental change," Journal of Environmental Economics and Management, Elsevier, vol. 98(C).
    6. Jan Feld & Ulf Zölitz, 2017. "Understanding Peer Effects: On the Nature, Estimation, and Channels of Peer Effects," Journal of Labor Economics, University of Chicago Press, vol. 35(2), pages 387-428.
    7. Lan Cheng & Xuguang Simon Sheng, 2017. "Combination of “combinations of p values”," Empirical Economics, Springer, vol. 53(1), pages 329-350, August.
    8. Xuan Ye & Heng Li, 2023. "A Non-parametric Test Based on Local Pairwise Comparisons of Patients for Single and Composite Endpoints," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 15(2), pages 419-429, July.
    9. Rukhin, Andrew L., 2016. "Confidence regions for comparison of two normal samples," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 273-280.
    10. Doyle, John R. & Chen, Catherine H., 2013. "Patterns in stock market movements tested as random number generators," European Journal of Operational Research, Elsevier, vol. 227(1), pages 122-132.
    11. Henrik Hansen & John Rand, 2006. "On the Causal Links Between FDI and Growth in Developing Countries," The World Economy, Wiley Blackwell, vol. 29(1), pages 21-41, January.
    12. Xuguang Sheng & Jingyun Yang, 2013. "Truncated Product Methods for Panel Unit Root Tests," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 75(4), pages 624-636, August.
    13. Judith H. Parkinson-Schwarz & Arne C. Bathke, 2022. "Testing for equality of distributions using the concept of (niche) overlap," Statistical Papers, Springer, vol. 63(1), pages 225-242, February.
    14. Husam Awni Bayoud, 2016. "Testing the similarity of two normal populations with application to the bioequivalence problem," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(7), pages 1322-1334, July.
    15. Yoav Benjamini & Ruth Heller, 2008. "Screening for Partial Conjunction Hypotheses," Biometrics, The International Biometric Society, vol. 64(4), pages 1215-1222, December.
    16. Dominika Polko-Zając, 2019. "On Permutation Location–Scale Tests," Statistics in Transition New Series, Polish Statistical Association, vol. 20(4), pages 153-166, December.
    17. Polko-Zając Dominika, 2019. "On Permutation Location–Scale Tests," Statistics in Transition New Series, Polish Statistical Association, vol. 20(4), pages 153-166, December.
    18. Moraes, Fernando & Cavalcante-Filho, Elias & De-Losso, Rodrigo, 2021. "Unskilled fund managers: Replicating active fund performance with few ETFs," International Review of Financial Analysis, Elsevier, vol. 78(C).
    19. Kechris Katerina J & Biehs Brian & Kornberg Thomas B, 2010. "Generalizing Moving Averages for Tiling Arrays Using Combined P-Value Statistics," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 9(1), pages 1-31, August.
    20. Chen, Zhongxue & Nadarajah, Saralees, 2014. "On the optimally weighted z-test for combining probabilities from independent studies," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 387-394.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:stmapp:v:25:y:2016:i:4:d:10.1007_s10260-016-0353-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.