From Five-Number Summary to Absolute Heterogeneity: Recent Methodological Advances in Meta-Analysis With Continuous Outcomes

Abstract

Meta-analysis with continuous outcomes presents a range of methodological challenges. Among these, two issues have received increasing attention: (i) integrating studies that report only the five-number summary (such as the median, interquartile range, and range) rather than the sample mean and standard deviation (SD), and (ii) accurately quantifying between-study heterogeneity. This review first summarizes recent advances in estimating the sample mean and SD from the five-number summary, covering both normality- and non-normality-based estimation methods. We also review recently developed skewness tests that help determine when normality-based estimators are appropriate and present a practical flow chart for integrating studies with five-number summaries into meta-analysis. Building on this, we discuss methods for quantifying the heterogeneity, focusing on the widely used relative heterogeneity statistic I² and its limitations, particularly its dependence on study sample sizes. We then review the absolute heterogeneity statistic I_A^2, which quantifies population-level variation across studies and is invariant to study sample sizes, thus complementing traditional measures. By synthesizing these methodological developments and providing practical guidelines and tools, this review aims to support more rigorous and transparent meta-analytic practice for continuous outcomes, especially in the presence of nonstandard reporting formats and varying degrees of heterogeneity.

Keywords

absolute heterogeneity / continuous outcomes / five-number summary / mean / meta-analysis / standard deviation

References

[1] M. Egger, S. Ebrahim, and G. D. Smith, “Where Now for Meta-Analysis?,” International Journal of Epidemiology 31, no. 1 (2002): 1–5.

[2] G. Glass, “Primary, Secondary, and Meta-Analysis of Research,” Educational Researcher 5, no. 10 (1976): 3–8.

[3] K. Pearson, “Report on Certain Enteric Fever Inoculation Statistics,” Bmj 5, no. 2 (1904): 1243–1246.

[4] R. Fisher, Statistical Methods for Research Workers (Oliver and Boyd, 1925).

[5] H. Beecher, “The Powerful Placebo,” JAMA 159, no. 17 (1955): 1602–1606.

[6] L. Hedges and I. Olkin, Statistical Methods for Meta-Analysis (Academic Press, 1985).

[7] L. Hedges and I. Olkin, Statistical Methods for Meta-Analysis, 2nd ed. (Academic Press, 2014).

[8] R. DerSimonian and N. Laird, “Meta-Analysis in Clinical Trials,” Controlled Clinical Trials 7, no. 3 (1986): 177–188.

[9] L. Bero and D. Rennie, “The Cochrane Collaboration: Preparing, Maintaining, and Disseminating Systematic Reviews of the Effects of Health Care,” Jama 274, no. 24 (1995): 1935–1938.

[10] S. Hozo, B. Djulbegovic, and I. Hozo, “Estimating the Mean and Variance From the Median, Range, and the Size of a Sample,” BMC Medical Research Methodology 5, no. 1 (2005): 13.

[11] J. Shi, D. Luo, X. Wan, et al., “Detecting the Skewness of Data From the Five-Number Summary and Its Application in Meta-Analysis,” Statistical Methods in Medical Research 32, no. 7 (2023): 1338–1360.

[12] T. Tong, Estimating the Sample Mean and Standard Deviation (SD) From the Five-Number Summary [Internet] (Hong Kong: Hong Kong Baptist University, 2023), https://www.math.hkbu.edu.hk/~tongt/median2mean.html.

[13] J. Higgins and S. Thompson, “Quantifying Heterogeneity in a Meta-Analysis,” Statistics in Medicine 21, no. 11 (2002): 1539–1558.

[14] J. Higgins, S. Thompson, J. Deeks, and D. Altman, “Measuring Inconsistency in Meta-Analyses,” Bmj 327, no. 7414 (2003): 557–560.

[15] M. Borenstein, L. Hedges, J. Higgins, and H. Rothstein, Introduction to Meta-Analysis (John Wiley & Sons, 2009).

[16] M. Borenstein, “Avoiding Common Mistakes in Meta-Analysis: Understanding the Distinct Roles of Q, I-Squared, Tau-Squared, and the Prediction Interval in Reporting Heterogeneity,” Research Synthesis Methods 15, no. 2 (2024): 354–368.

[17] J. P. T. Higgins and J. A. Lopez-Lopez, “Reflections on the I-Squared Index for Measuring Inconsistency in Meta-Analysis,” Research Synthesis Methods 17, no. 3 (2026): 389–402.

[18] G. Rücker, G. Schwarzer, J. Carpenter, and M. Schumacher, “Undue Reliance on I² in Assessing Heterogeneity May Mislead,” BMC Medical Research Methodology 8, no. 1 (2008): 79.

[19] K. Yang, E. Lin, W. Xu, L. Zhu, and T. Tong, “An Alternative Measure for Quantifying the Heterogeneity in Meta-Analysis,” Statistics in Medicine 44, no. 10–12 (2025): e70089.

[20] C. Hong, A. Wong, and C. Hong, “PAUL® Glaucoma Implant-Efficacy and Safety: A Systematic Review and Meta-Analysis,” preprint, medRxiv 11. 25339321, November 04, 2025.

[21] X. Wan, W. Wang, J. Liu, and T. Tong, “Estimating the Sample Mean and Standard Deviation From the Sample Size, Median, Range, and/or Interquartile Range,” BMC Medical Research Methodology 14, no. 1 (2014): 135.

[22] D. Luo, X. Wan, J. Liu, and T. Tong, “Optimally Estimating the Sample Mean From the Sample Size, Median, Mid-Range, and/or Mid-Quartile Range,” Statistical Methods in Medical Research 27, no. 6 (2018): 1785–1805.

[23] J. Shi, D. Luo, H. Weng, et al., “Optimally Estimating the Sample Standard Deviation From the Five-Number Summary,” Research Synthesis Methods 11, no. 5 (2020): 641–654.

[24] A. De Livera, L. Prendergast, and U. Kumaranathunga, “A Novel Density-Based Approach for Estimating Unknown Means, Distribution Visualisations and Meta-Analyses of Quantiles,” preprint, arXiv:2411.10971, November 17, 2024.

[25] N. Balakrishnan, J. Rychtář, and D. Taylor, “Estimating Sample Skewness From Sample Data Summaries and Associated Evaluation of Normality,” Mathematical Methods of Statistics 32, no. 4 (2023): 260–273.

[26] N. Balakrishnan, J. Rychtář, D. Taylor, and S. Walter, “Unified Approach to Optimal Estimation of Mean and Standard Deviation From Sample Summaries,” Statistical Methods in Medical Research 31, no. 11 (2022): 2087–2103.

[27] M. Bland, “Estimating Mean and Standard Deviation From the Sample Size, Three Quartiles, Minimum, and Maximum,” International Journal of Statistics in Medical Research 4, no. 1 (2015): 57–64.

[28] S. Cai, J. Zhou, and J. Pan, “Estimating the Sample Mean and Standard Deviation From Order Statistics and Sample Size in Meta-Analysis,” Statistical Methods in Medical Research 30, no. 12 (2021): 2701–2719.

[29] D. Kwon and I. Reis, “Simulation-Based Estimation of Mean and Standard Deviation for Meta-Analysis via Approximate Bayesian Computation (ABC),” BMC Medical Research Methodology 15, no. 1 (2015): 61.

[30] S. McGrath, X. Zhao, R. Steele, B. D. Thombs, A. Benedetti, and Collaboration DESD, “Estimating the Sample Mean and Standard Deviation From Commonly Reported Quantiles in Meta-Analysis,” Statistical Methods in Medical Research 29, no. 9 (2020): 2520–2537.

[31] J. Shi, T. Tong, Y. Wang, and M. Genton, “Estimating the Mean and Variance From the Five-Number Summary of a Log-Normal Distribution,” Statistics and Its Interface 13 (2020): 519–531.

[32] X. Tang, T. Tong, X. Zhang, and H. Chu, “Minimum Distance Estimation of Mean and Standard Deviation From Reported Quantiles,” Research Synthesis Methods (2026), https://doi.org/10.1017/rsm.2026.10090.

[33] S. Walter and X. Yao, “Effect Sizes Can Be Calculated for Studies Reporting Ranges for Outcome Variables in Systematic Reviews,” Journal of Clinical Epidemiology 60, no. 8 (2007): 849–852.

[34] X. Yang, A. Hutson, and D. Wang, “A Generalized Blue Approach for Combining Location and Scale Information in a Meta-Analysis,” Journal of Applied Statistics 49, no. 15 (2022): 3846–3867.

[35] Q. Zhang and Q. Li, “Neural Network-Based Estimation of Sample Mean and Standard Deviation From Some Quartiles,” Journal of Systems Science and Complexity (2026), https://doi.org/10.1007/s11424-026-5481-4.

[36] D. Kwon, R. Reddy, and I. Reis, “ABCMETAapp: R Shiny Application for Simulation-Based Estimation of Mean and Standard Deviation for Meta-Analysis via Approximate Bayesian Computation,” Research Synthesis Methods 12, no. 6 (2021): 842–848.

[37] W. Viechtbauer, “Conducting Meta-Analyses in R With the Metafor Package,” Journal of Statistical Software 36 (2010): 1–48.

[38] S. McGrath, X. Zhao, O. Ozturk, S. Katzenschlager, R. Steele, and A. Benedetti, “Metamedian: An R Package for Meta-Analyzing Studies Reporting Medians,” Research Synthesis Methods 15, no. 2 (2024): 332–346.

[39] S. McGrath, S. Katzenschlager, A. J. Zimmer, A. Seitel, R. Steele, and A. Benedetti, “Standard Error Estimation in Meta-Analysis of Studies Reporting Medians,” Statistical Methods in Medical Research 32, no. 2 (2023): 373–388.

[40] A. Whitehead, Meta-Analysis of Controlled Clinical Trials (John Wiley & Sons, 2002).

[41] J. Higgins, S. Thompson, and D. Spiegelhalter, “A Re-Evaluation of Random-Effects Meta-Analysis,” Journal of the Royal Statistical Society Series A: Statistics in Society 172, no. 1 (2009): 137–159.

[42] K. Rice, J. Higgins, and T. Lumley, “A Re-Evaluation of Fixed Effect(s) Meta-Analysis,” Journal of the Royal Statistical Society Series A: Statistics in Society 181, no. 1 (2018): 205–227.

[43] A. Veroniki and J. McKenzie, “A Brief Note on the Common (Fixed)-Effect Meta-Analysis Model,” Journal of Clinical Epidemiology 169 (2024): 111281.

[44] M. Borenstein, L. Hedges, J. Higgins, and H. Rothstein, “A Basic Introduction to Fixed-Effect and Random-Effects Models for Meta-Analysis,” Research Synthesis Methods 1, no. 2 (2010): 97–111.

[45] N. M. Laird and F. Mosteller, “Some Statistical Methods for Combining Experimental Results,” International Journal of Technology Assessment in Health Care 6 (1990): 5–30.

[46] L. Hedges and J. Vevea, “Fixed-and Random-Effects Models in Meta-Analysis,” Psychological Methods 3, no. 4 (1998): 486–504.

[47] R. Turner, J. Davey, M. Clarke, S. Thompson, and J. Higgins, “Predicting the Extent of Heterogeneity in Meta-Analysis, Using Empirical Data From the Cochrane Database of Systematic Reviews,” International Journal of Epidemiology 41, no. 3 (2012): 818–827.

[48] R. Bender, T. Friede, A. Koch, et al., “Methods for Evidence Synthesis in the Case of Very Few Studies,” Research Synthesis Methods 9, no. 3 (2018): 382–392.

[49] K. Yang, H. Kwan, Z. Yu, and T. Tong, “Model Selection Between the Fixed-Effects Model and the Random-Effects Model in Meta-Analysis,” Statistics and Its Interface 13, no. 4 (2020): 501–510.

[50] G. Gariépy, S. Danna, L. Hawke, J. Henderson, and S. Iyer, “The Mental Health of Young People Who Are Not in Education, Employment, or Training: A Systematic Review and Meta-Analysis,” Social Psychiatry and Psychiatric Epidemiology 57, no. 6 (2022): 1107–1121.

[51] J. Higgins and S. Green, Cochrane Handbook for Systematic Reviews of Interventions (John Wiley and Sons, 2008).

[52] A. Whitehead and J. Whitehead, “A General Parametric Approach to the Meta-Analysis of Randomized Clinical Trials,” Statistics in Medicine 10, no. 11 (1991): 1665–1677.

[53] W. Cochran, “The Combination of Estimates From Different Experiments,” Biometrics 10, no. 1 (1954): 101–129.

[54] R. Paule and J. Mandel, “Consensus Values and Weighting Factors,” Journal of Research of the National Bureau of Standards 87, no. 5 (1982): 377–385.

[55] R. DerSimonian and R. Kacker, “Random-Effects Model for Meta-Analysis of Clinical Trials: An Update,” Contemporary Clinical Trials 28, no. 2 (2007): 105–114.

[56] R. Hardy and S. Thompson, “A Likelihood Approach to Meta-Analysis With Random Effects,” Statistics in Medicine 15, no. 6 (1996): 619–629.

[57] W. Viechtbauer, “Bias and Efficiency of Meta-Analytic Variance Estimators in the Random-Effects Model,” Journal of Educational and Behavioral Statistics 30, no. 3 (2005): 261–293.

[58] J. Bowden, J. Tierney, A. Copas, and S. Burdett, “Quantifying, Displaying and Accounting for Heterogeneity in the Meta-Analysis of RCTs Using Standard and Generalised Q Statistics,” BMC Medical Research Methodology 11, no. 1 (2011): 41.

[59] D. Jackson, R. Turner, K. Rhodes, and W. Viechtbauer, “Methods for Calculating Confidence and Credible Intervals for the Residual Between-Study Variance in Random Effects Meta-Regression Models,” BMC Medical Research Methodology 14, no. 1 (2014): 103.

[60] E. Kontopantelis and D. Reeves, “Performance of Statistical Methods for Meta-Analysis When True Study Effects Are Non-Normally Distributed: A Simulation Study,” Statistical Methods in Medical Research 21, no. 4 (2012): 409–426.

[61] D. Langan, J. Higgins, and M. Simmonds, “Comparative Performance of Heterogeneity Variance Estimators in Meta-Analysis: A Review of Simulation Studies,” Research Synthesis Methods 8, no. 2 (2017): 181–198.

[62] D. Langan, J. Higgins, D. Jackson, et al., “A Comparison of Heterogeneity Variance Estimators in Simulated Random-Effects Meta-Analyses,” Research Synthesis Methods 10, no. 1 (2019): 83–98.

[63] H. Michael, S. Thornton, M. Xie, and L. Tian, “Exact Inference on the Random-Effects Model for Meta-Analyses With Few Studies,” Biometrics 75, no. 2 (2019): 485–493.

[64] K. Hanada and T. Sugimoto, “Inference Using an Exact Distribution of Test Statistic for Random-Effects Meta-Analysis,” Annals of the Institute of Statistical Mathematics 75 (2023): 281–302.

[65] T. Huedo-Medina, J. Sánchez-Meca, F. Marin-Martinez, and J. Botella, “Assessing Heterogeneity in Meta-Analysis: Q Statistic or I² Index?,” Psychological Methods 11, no. 2 (2006): 193–210.

[66] E. Kulinskaya, M. Dollinger, and K. Bjørkestøl, “Testing for Homogeneity in Meta-Analysis I. The One-Parameter Case: Standardized Mean Difference,” Biometrics 67, no. 1 (2011): 203–212.

[67] J. IntHout, J. P. Ioannidis, M. M. Rovers, and J. J. Goeman, “Plea for Routinely Presenting Prediction Intervals in Meta-Analysis,” BMJ Open 6, no. 7 (2016): e010247.

[68] C. Partlett and R. D. Riley, “Random Effects Meta-Analysis: Coverage Performance of 95% Confidence and Prediction Intervals Following REML Estimation,” Statistics in Medicine 36, no. 2 (2017): 301–317.

[69] B. Takkouche, C. Cadarso-Suarez, and D. Spiegelman, “Evaluation of Old and New Tests of Heterogeneity in Epidemiologic Meta-Analysis,” American Journal of Epidemiology 150, no. 2 (1999): 206–215.

[70] D. Böhning, R. Lerdsuwansri, and H. Holling, “Some General Points on the I² Measure of Heterogeneity in Meta-Analysis,” Metrika 80, no. 6 (2017): 685–695.

[71] Y. Wang, N. DelRocco, and L. Lin, “Comparisons of Various Estimates of the I² Statistic for Quantifying Between-Study Heterogeneity in Meta-Analysis,” Statistical Methods in Medical Research 33, no. 5 (2024): 745–764.

[72] P. von Hippel, “The Heterogeneity Statistic I² Can Be Biased in Small Meta-Analyses,” BMC Medical Research Methodology 15, no. 1 (2015): 35.

[73] L. Lin, H. Chu, and J. Hodges, “Alternative Measures of Between-Study Heterogeneity in Meta-Analysis: Reducing the Impact of Outlying Studies,” Biometrics 73, no. 1 (2017): 156–166.

[74] M. Borenstein, J. Higgins, L. Hedges, and H. Rothstein, “Basics of Meta-Analysis: I² Is Not an Absolute Measure of Heterogeneity,” Research Synthesis Methods 8, no. 1 (2017): 5–18.

[75] P. Sangnawakij, D. Böhning, S. A. Niwitpong, S. Adams, M. Stanton, and H. Holling, “Meta-Analysis Without Study-Specific Variance Information: Heterogeneity Case,” Statistical Methods in Medical Research 28, no. 1 (2019): 196–210.

[76] C. Smith, “On the Estimation of Intraclass Correlation,” Annals of Human Genetics 21 (1957): 363–373.

[77] A. Donner, “The Use of Correlation and Regression in the Analysis of Family Resemblance,” American Journal of Epidemiology 110, no. 3 (1979): 335–352.

[78] K. O. McGraw and S. P. Wong, “Forming Inferences About Some Intraclass Correlation Coefficients,” Psychological Methods 1, no. 1 (1996): 30–46.

[79] D. Jackson, “The Power of the Standard Test for the Presence of Heterogeneity in Meta-Analysis,” Statistics in Medicine 25, no. 15 (2006): 2688–2699.

[80] J. Cohen, Statistical Power Analysis for the Behavioral Sciences (Academic Press, 1969).

[81] J. Goulet-Pelletier and D. Cousineau, “A Review of Effect Sizes and Their Confidence Intervals, Part I: The Cohen's D family,” Quantitative Methods for Psychology 14, no. 4 (2018): 242–265.

[82] H. Cooper, L. Hedges, and J. Valentine, The Handbook of Research Synthesis and Meta-Analysis (Russell Sage Foundation, 2019).

[83] L. Hedges, “Distribution Theory for Glass's Estimator of Effect Size and Related Estimators,” Journal of Educational Statistics 6, no. 2 (1981): 107–128.

[84] T. Tong, Computing the Absolute Measure of Heterogeneity in Meta-Analysis [Internet] (Hong Kong Baptist University, 2025), https://www.math.hkbu.edu.hk/~tongt/heterogeneity.html.

[85] J. R. Landis and G. G. Koch, “The Measurement of Observer Agreement for Categorical Data,” Biometrics 30 (1977): 159–174.

[86] H. Lin, R. Zhang, and T. Tong, “When Tukey Meets Chauvenet: A New Boxplot Criterion for Outlier Detection,” Journal of Computational and Graphical Statistics 35, no. 1 (2026): 198–211.

[87] T. Tong, H. Lin, B. Gang, and R. Zhang, “ChauBoxplot and AdaptiveBoxplot: Two R Packages for Boxplot-Based Outlier detection,” Statistical Theory and Related Fields (2026), https://doi.org/10.1080/24754269.2026.2642439.

[88] I. Elmakaty, J. Abdulmajeed, T. Chivese, M. N. Khan, J. C. Stone, and S. A. R. Doi, “Six Entrenched Misconceptions About Meta-Analysis Models,” Journal of Evidence-Based Medicine 19, no. 1 (2026): e70120.

Rights & permissions

2026 The Author(s). Journal of Evidence-Based Medicine published by Chinese Cochrane Center, West China Hospital of Sichuan University and John Wiley & Sons Australia, Ltd.