{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T13:41:43Z","timestamp":1776346903412,"version":"3.51.2"},"reference-count":38,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,20]],"date-time":"2019-12-20T00:00:00Z","timestamp":1576800000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Ministry of Economy, Industry and Competitiveness","award":["PSI2016-78737-P (AEI\/FEDER, UE)"],"award-info":[{"award-number":["PSI2016-78737-P (AEI\/FEDER, UE)"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Several measures of skewness and kurtosis were proposed by Hogg (1974) in order to reduce the bias of conventional estimators when the distribution is non-normal. Here we conducted a Monte Carlo simulation study to compare the performance of conventional and Hogg\u2019s estimators, considering the most frequent continuous distributions used in health, education, and social sciences (gamma, lognormal and exponential distributions). In order to determine the bias, precision and accuracy of the skewness and kurtosis estimators for each distribution we calculated the relative bias, the coefficient of variation, and the scaled root mean square error. The effect of sample size on the estimators is also analyzed. In addition, a SAS program for calculating both conventional and Hogg\u2019s estimators is presented. The results indicated that for the non-normal distributions investigated, the estimators of skewness and kurtosis which best reflect the shape of the distribution are Hogg\u2019s estimators. It should also be noted that Hogg\u2019s estimators are not as affected by sample size as are conventional estimators.<\/jats:p>","DOI":"10.3390\/sym12010019","type":"journal-article","created":{"date-parts":[[2019,12,23]],"date-time":"2019-12-23T03:23:12Z","timestamp":1577071392000},"page":"19","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":56,"title":["Bias, Precision, and Accuracy of Skewness and Kurtosis Estimators for Frequently Used Continuous Distributions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7991-6668","authenticated-orcid":false,"given":"Roser","family":"Bono","sequence":"first","affiliation":[{"name":"Department of Social Psychology and Quantitative Psychology, Faculty of Psychology, University of Barcelona, Passeig de la Vall d\u2019Hebron, 171, 08035 Barcelona, Spain"},{"name":"Institute of Neurosciences, University of Barcelona, Passeig de la Vall d\u2019Hebron, 171, 08035 Barcelona, Spain"}]},{"given":"Jaume","family":"Arnau","sequence":"additional","affiliation":[{"name":"Department of Social Psychology and Quantitative Psychology, Faculty of Psychology, University of Barcelona, Passeig de la Vall d\u2019Hebron, 171, 08035 Barcelona, Spain"}]},{"given":"Rafael","family":"Alarc\u00f3n","sequence":"additional","affiliation":[{"name":"Department of Psychobiology and Methodology of Behavioral Science, Faculty of Psychology, University of Malaga, Campus Universitario de Teatinos, s\/n, 29071 Malaga, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4046-9308","authenticated-orcid":false,"given":"Maria J.","family":"Blanca","sequence":"additional","affiliation":[{"name":"Department of Psychobiology and Methodology of Behavioral Science, Faculty of Psychology, University of Malaga, Campus Universitario de Teatinos, s\/n, 29071 Malaga, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1027\/1614-2241\/a000057","article-title":"Skewness and kurtosis in real data samples","volume":"9","author":"Blanca","year":"2013","journal-title":"Methodology"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1037\/0033-2909.105.1.156","article-title":"The unicorn, the normal curve, and other improbable creatures","volume":"105","author":"Micceri","year":"1989","journal-title":"Psychol. Bull."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"648","DOI":"10.5209\/rev_SJOP.2011.v14.n2.13","article-title":"A confirmatory factor analysis of the \u201cAutoconcepto Forma 5\u201d questionnaire in young adults from Spain and Chile","volume":"14","author":"Musitu","year":"2011","journal-title":"Span. J. Psychol."},{"key":"ref_4","first-page":"551","article-title":"Autoconcepto en adultos de Espa\u00f1a y Portugal","volume":"18","author":"Musitu","year":"2006","journal-title":"Psicothema"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1111\/bmsp.12026","article-title":"Should we rely on the Kenward\u2013Roger approximation when using linear mixed models if the groups have different distributions?","volume":"67","author":"Arnau","year":"2014","journal-title":"Br. J. Math. Stat. Psychol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"926","DOI":"10.1016\/j.ssci.2010.03.016","article-title":"Estimation of air traffic longitudinal conflict probability based on the reaction time of controllers","volume":"48","year":"2010","journal-title":"Saf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1176\/appi.ajp.161.7.1159","article-title":"Methods for analysis of skewed data distributions in psychiatric clinical studies: Working with many zero values","volume":"161","author":"Deluchi","year":"2004","journal-title":"Am. J. Psychiat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1016\/S0306-4530(00)00022-6","article-title":"Biosocial aspects of domestic violence","volume":"25","author":"Soler","year":"2000","journal-title":"Psychoneuroendocrinology"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1016\/j.jue.2005.03.003","article-title":"Labor income uncertainty, skewness and homeownership: A panel data study for Germany and Spain","volume":"58","year":"2005","journal-title":"J. Urban Econ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.1111\/j.1467-9868.2008.00669.x","article-title":"Non-parametric heteroscedastic transformation regression models for skewed data with an application to health care costs","volume":"70","author":"Zhou","year":"2009","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1602","DOI":"10.3389\/fpsyg.2017.01602","article-title":"Non-normal distributions commonly used in health, education, and social sciences: A systematic review","volume":"8","author":"Bono","year":"2017","journal-title":"Front. Psychol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1080\/01621459.1974.10480225","article-title":"Adaptive robust procedures: A partial review and some suggestions for future applications and theory","volume":"69","author":"Hogg","year":"1974","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1348\/000711005X63755","article-title":"Adaptive robust estimation and testing","volume":"60","author":"Keselman","year":"2007","journal-title":"Br. J. Math. Stat. Psychol."},{"key":"ref_14","first-page":"656","article-title":"A two-sample adaptive distribution free test","volume":"70","author":"Hogg","year":"1975","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/0169-2607(95)01708-9","article-title":"Hinge estimators of location: Robust to asymmetry","volume":"49","author":"Reed","year":"1996","journal-title":"Comput. Meth. Programs Biomed."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.frl.2010.12.001","article-title":"Robust estimation of skewness and kurtosis in distributions with infinite higher moments","volume":"8","author":"Bonato","year":"2011","journal-title":"Financ. Res. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/S1544-6123(03)00003-5","article-title":"On more robust estimation of skewness and kurtosis","volume":"1","author":"Kim","year":"2004","journal-title":"Financ. Res. Lett."},{"key":"ref_18","unstructured":"Bowley, A.L. (1920). Elements of Statistics, Scribner\u2019s Sons."},{"key":"ref_19","first-page":"391","article-title":"Measuring skewness and kurtosis","volume":"33","author":"Groeneveld","year":"1984","journal-title":"J. R. Stat. Soc. Ser. D"},{"key":"ref_20","first-page":"25","article-title":"A quantile alternative for kurtosis","volume":"37","author":"Moors","year":"1988","journal-title":"J. R. Stat. Soc. Ser. D"},{"key":"ref_21","first-page":"1","article-title":"A comparative study on the estimators of skewness and kurtosis","volume":"41","author":"Sazak","year":"2017","journal-title":"Ege Univ. J. Fac. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2531","DOI":"10.1080\/03610918208828404","article-title":"On adaptive statistical inferences","volume":"11","author":"Hogg","year":"1982","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"310","DOI":"10.22237\/jmasm\/1036109880","article-title":"A test of symmetry","volume":"1","author":"Othman","year":"2002","journal-title":"JMASM"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1080\/01621459.1973.10481367","article-title":"A class of simple linear estimators of the standard deviation of the normal distribution","volume":"68","author":"Cureton","year":"1973","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1080\/03610917908812125","article-title":"Distribution of asymmetric trimmed means","volume":"8","author":"Hertsgaard","year":"1979","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_26","unstructured":"Wicklin, R. (2013). Simulating Data with SAS, SAS Institute Inc."},{"key":"ref_27","unstructured":"SAS Institute Inc. (2016). SAS\/STAT\u00ae 14.2 User\u2019s Guide, SAS Institute Inc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1111\/j.2005.0906-7590.04112.x","article-title":"The concepts of bias, precision and accuracy, and their use in testing the performance of species richness estimators, with a literature review of estimator performance","volume":"28","author":"Walther","year":"2005","journal-title":"Ecography"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"4279","DOI":"10.1002\/sim.2673","article-title":"The design of simulation studies in medical statistics","volume":"25","author":"Burton","year":"2006","journal-title":"Stat. Med."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2461","DOI":"10.3389\/fpsyg.2018.02461","article-title":"A comparative study on the performance of GSCA and CSA in parameter recovery for structural equation models with ordinal observed variables","volume":"9","author":"Jung","year":"2018","journal-title":"Front. Psychol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1080\/00273171.2014.963194","article-title":"Simulating univariate and multivariate nonnormal distributions through the method percentiles","volume":"50","author":"Koran","year":"2015","journal-title":"Multivar. Behav. Res."},{"key":"ref_32","unstructured":"Sokal, R.R., and Rohlf, F.J. (2012). Biometry: The Principles and Practice of Statistics in Biological Research, Freeman and Company. [4th ed.]."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1037\/1082-989X.6.4.330","article-title":"A comparison of inclusive and restrictive strategies in modern missing data procedures","volume":"6","author":"Collins","year":"2001","journal-title":"Psychol. Methods"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1111\/j.1745-3984.2012.00168.x","article-title":"Standard error of linear observed-score equating for the NEAT design with nonnormally distributed data","volume":"49","author":"Zu","year":"2012","journal-title":"J. Educ. Meas."},{"key":"ref_35","first-page":"183","article-title":"Comparing measures of sample skewness and kurtosis","volume":"47","author":"Joanes","year":"1998","journal-title":"J. R. Stat. Soc. Ser. D"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Islam, T.U. (2019). Ranking of normality tests: An appraisal through skewed alternative space. Symmetry, 11.","DOI":"10.20944\/preprints201905.0190.v1"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"288","DOI":"10.22237\/jmasm\/1036109820","article-title":"Trimming, transforming statistics, and bootstrapping: Circumventing the biasing effects of heteroscedasticity and nonnormality","volume":"38","author":"Keselman","year":"2002","journal-title":"JMASM"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1002\/(SICI)1521-4036(199906)41:3<321::AID-BIMJ321>3.0.CO;2-5","article-title":"Robust one-way ANOVA under possibly non-regular conditions","volume":"41","author":"Babu","year":"1999","journal-title":"Biom. J."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:44:02Z","timestamp":1760190242000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,20]]},"references-count":38,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["sym12010019"],"URL":"https:\/\/doi.org\/10.3390\/sym12010019","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,20]]}}}