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Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range
Open accessAbstract
The era of big data is coming, and evidence-based medicine is attracting increasing attention to improve decision making in medical practice via integrating evidence from well designed and conducted clinical research. Meta-analysis is a statistical technique widely used in evidence-based medicine for analytically combining the findings from independent clinical trials to provide an overall estimation of a treatment effectiveness. The sample mean and standard deviation are two commonly used statistics in meta-analysis but some trials use the median, the minimum and maximum values, or sometimes the first and third quartiles to report the results. Thus, to pool results in a consistent format, researchers need to transform those information back to the sample mean and standard deviation. In this article, we investigate the optimal estimation of the sample mean for meta-analysis from both theoretical and empirical perspectives. A major drawback in the literature is that the sample size, needless to say its importance, is either ignored or used in a stepwise but somewhat arbitrary manner, e.g. the famous method proposed by Hozo et al. We solve this issue by incorporating the sample size in a smoothly changing weight in the estimators to reach the optimal estimation. Our proposed estimators not only improve the existing ones significantly but also share the same virtue of the simplicity. The real data application indicates that our proposed estimators are capable to serve as "rules of thumb" and will be widely applied in evidence-based medicine.
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“When studies reported a median and interquartile range, we estimated the mean and standard deviation (SD) using conventional formulae (Luo et al., 2018;Wan et al., 2014).When both of these were absent, we estimated the standard deviation using prognostic imputation (Ma et al., 2008).”
From Decision making capacity for treatment in psychiatric inpatients: a systematic review and meta-analysis (2024) · cites this paper as Luo et al., 2018
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“Means and standard deviations were the only accepted data entry form and medians with interquartile ranges were converted to means and standard deviations using methods outlined by Wan et al. [17] and Luo et al. [18].”
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“Where data were expressed as a median (IQR), we calculated the mean and SD by the statistical formula described previously (42).”
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Global cost of postoperative ileus following abdominal surgery: meta-analysis
Traeger, L., Koullouros, M., Bedrikovetski, S., Kroon, H. M., Moore, J. W., & Sammour, T. (2023) · BJS Open · doi:10.1093/bjsopen/zrad054 · Cited by 48
Jawanda, H. S., Khan, Z. A., Warrier, A. A., Acuña, A. J., Allahabadi, S., Kaplan, D. J., Ritz, E. M., Jackson, G. R., Mameri, E. S., Batra, A. K., et al. (2024) · Arthroscopy The Journal of Arthroscopic and Related Surgery · doi:10.1016/j.arthro.2024.01.037 · Cited by 52
Deprato, A., Ruchat, S.-M., Ali, M. U., Cai, C., Forte, M., Gierc, M. S. H., Meyer, S., Sjwed, T. N., Shirazi, S., Matenchuk, B. A., et al. (2024) · British Journal of Sports Medicine · doi:10.1136/bjsports-2024-108478 · Cited by 55
Kask, G., Barner-Rasmussen, I., Repo, J. P., Kjäldman, M. E., Kilk, K., BLOMQVIST, C. P., & Tukiainen, E. J. (2019) · Annals of Surgical Oncology · doi:10.1245/s10434-019-07698-w · Cited by 51
“were calculated using median and IQR results.(65,66) The SDs for studies by Friedmann et al.”
MethodSun, R., Zhao, N., Zhao, K., Su, Z., Zhang, Y., Diao, M., & Li, L. (2020) · PLoS ONE · doi:10.1371/journal.pone.0239857 · Cited by 48
“[15], medians with ranges and midquartile with ranges were converted into means with standard deviations.”
MethodGómez-Chiguano, G. F., Navarro‐Santana, M. J., Cleland, J. A., Arias‐Buría, J. L., Fernández‐de‐las‐Peñas, C., Ortega‐Santiago, R., & Plaza‐Manzano, G. (2020) · Pain Medicine · doi:10.1093/pm/pnaa342 · Cited by 48
“[22], the results were converted to the mean [22] and standard deviation [21].”
MethodLi, Z., Zhu, G., Li, C., Lai, H., Liu, X., & Zhang, L. (2021) · Nutrients · doi:10.3390/nu13124319 · Cited by 55
Ramos‐García, P., González‐Moles, M. Á., & Warnakulasuriya, S. (2022) · Oral Oncology · doi:10.1016/j.oraloncology.2022.105734 · Cited by 48
Zawadka, M., Via, L. L., Wong, A., Olusanya, O., Muscarà, L., Continella, C., Andruszkiewicz, P., & Sanfilippo, F. (2023) · Critical Care Medicine · doi:10.1097/ccm.0000000000005819 · Cited by 48
Standard error estimation in meta-analysis of studies reporting medians
McGrath, S., Katzenschlager, S., Zimmer, A., Seitel, A., Steele, R. J., & Benedetti, A. (2022) · Statistical Methods in Medical Research · doi:10.1177/09622802221139233 · Cited by 47
“, see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]) have almost entirely focused on the performance of these methods at the study-level (i.”
BackgroundWang, J., Liu, Q., Jiang, S., Zhang, J., He, J., Li, Y., & Wang, D. (2023) · International Journal of Surgery · doi:10.1097/js9.0000000000000390 · Cited by 47
“In studies where outcomes were reported as medians and quartiles, means and SDs were calculated using methods proposed by Luo et al.([20]) andWan et al([21]).”
MethodZhang, Y., Zhao, W., Liu, K., Chen, Z., Fei, Q., Ahmad, N., & Yi, M. (2023) · SLEEP · doi:10.1093/sleep/zsad207 · Cited by 52
Wolk, D. M., Parrott, J. S., Babady, N. E., Mochon, A. B., Tom, R. J., Diel, C., Bard, J. D., Harrington, A., Hata, D. J., Roberts, A. L., et al. (2025) · Clinical Microbiology Reviews · doi:10.1128/cmr.00137-24 · Cited by 47
“…were not available, the following procedures were used to estimate these measures from median and associated dispersion measures: the method proposed by Cai et al. (36) to adjust the estimates of those proposed by Luo et al. 2018 (37) for the mean and by Wan et al. (38) for the standard deviation.”
Thillainadesan, J., Yumol, M. F., Hilmer, S. N., Aitken, S., & Naganathan, V. (2020) · Journal of the American Medical Directors Association · doi:10.1016/j.jamda.2020.03.023 · Cited by 47
Roorda, D., Königs, M., Schattenkerk, L. D. E., van der Steeg, L., van Heurn, E. L. W., & Oosterlaan, J. (2021) · Archives of Disease in Childhood Fetal & Neonatal · doi:10.1136/archdischild-2021-322158 · Cited by 46
Beamish, N., Davenport, M. H., Ali, M. U., Gervais, M. J., Sjwed, T. N., Bains, G., Sivak, A., Deering, R. E., & Ruchat, S.-M. (2024) · British Journal of Sports Medicine · doi:10.1136/bjsports-2024-108619 · Cited by 46
The inflammatory cytokine profile of myelodysplastic syndromes
Shi, X., Zheng, Y., Xu, L., Cao, C., Dong, B., & Chen, X. (2019) · Medicine · doi:10.1097/md.0000000000015844 · Cited by 46
“The data present by median and range were transferred to mean and SD by the methods proposed byWen et al[13] and Luo et al[14] (http://www. comp.hkbu.edu.hk/∼xwan/median2mean.html).”
MethodNi, X., Jia, D., Guo, Y., Sun, X., & Suo, J. (2019) · International Journal of Surgery · doi:10.1016/j.ijsu.2019.07.034 · Cited by 45
Wang, Z., Zhu, Y., Cui, L., & Qu, B. (2019) · JMIR mhealth and uhealth · doi:10.2196/14404 · Cited by 48
“For studies reporting a median and interquartile range for adherence outcomes, we converted the outcomes into the mean (SD) as previously reported [42,43].”
Zhou, Y., Chen, M., Zhang, Y., Zhou, H., Yu, X., & Chen, G. (2019) · Medicine · doi:10.1097/md.0000000000017545 · Cited by 46
“When only the median and range were offered, the mean and standard deviation (SD) were estimated by the calculator with a compiled formula recommended by Luo et al.([15])”
Huang, Z.-D., Zhao, Y.-F., Li, S., Gu, H.-Y., Lin, L., Yang, Z.-Y., Niu, Y., Zhang, C., & Luo, J. (2020) · Frontiers in Pharmacology · doi:10.3389/fphar.2020.00559 · Cited by 50
Fernández‐de‐las‐Peñas, C., Plaza‐Manzano, G., Navarro‐Santana, M. J., Olesen, J., Jensen, R., & Bendtsen, L. (2020) · Cephalalgia · doi:10.1177/0333102420958384 · Cited by 46
“If the study presented non-parametric values, they were converted to means and standard deviations (25,26).”
MethodChen, C., Zhu, C., Yan, D., Liu, H., Li, D., Zhou, Y., Fu, X., Wu, J., Ding, C., Tian, G., et al. (2021) · International Journal of Infectious Diseases · doi:10.1016/j.ijid.2021.01.017 · Cited by 46
A Comprehensive Review of Continuous Glucose Monitoring Accuracy during Exercise Periods
Fabra, E. M., Díez, J.-L., Bondía, J., & Sanz, A. J. L. (2021) · Sensors · doi:10.3390/s21020479 · Cited by 48
“[21] to estimate missing sample mean values from the reported sample size, median, range, and/or interquartile range.”
BackgroundPederson, J. M., Reierson, N. L., Hardy, N. J., Touchette, J. C., Medam, S., Barrett, A., Schmidt, M., Brinjikji, W., Kallmes, D. F., & Kallmes, K. M. (2021) · World Neurosurgery · doi:10.1016/j.wneu.2021.07.034 · Cited by 46
Nordgård, R., & Låg, T. (2021) · Frontiers in Virtual Reality · doi:10.3389/frvir.2021.699383 · Cited by 53
Notbohm, H. L., Moser, F., Goh, J., Feuerbacher, J. F., Bloch, W., & Schumann, M. (2023) · Acta Physiologica · doi:10.1111/apha.14013 · Cited by 54
Osborne-Grinter, M., Ali, A., & Williams, M. C. (2023) · European Radiology · doi:10.1007/s00330-023-10439-z · Cited by 47
“If the skew of the data was significant, it was excluded from the analysis and if it was not, mean and standard deviation was calculated using the method reported by Shi et al [16] and Luo et al [17].”
Zhang, Y., Bai, Y. S., Chen, M., Zhou, Y., Yu, X., Zhou, H., & Chen, G. (2019) · BMC Anesthesiology · doi:10.1186/s12871-019-0761-3 · Cited by 45
Hassanin, A. I., Aherne, T. M., Greene, G., Boyle, E., Egan, B. M., Tierney, S., Walsh, S. R., McHugh, S. M., & Aly, S. (2019) · Journal of Vascular Surgery Venous and Lymphatic Disorders · doi:10.1016/j.jvsv.2019.06.009 · Cited by 45
Laka, M., Milazzo, A., & Merlin, T. (2019) · Journal of Antimicrobial Chemotherapy · doi:10.1093/jac/dkz543 · Cited by 46
Diagnostic role of postmortem CK-MB in cardiac death: a systematic review and meta-analysis
Xu, C., Zhang, T., Zhu, B.-L., & Cao, Z. (2020) · Forensic Science Medicine and Pathology · doi:10.1007/s12024-020-00232-5 · Cited by 45
“In addition, the two researchers estimated the means and SDs based on the sample sizes, medians, and ranges of the studies that did not provide a mean or SD directly according to the methods of previous studies [23, 24].”
MethodKovalic, A. J., Majeed, C. N., Samji, N. S., Thuluvath, P. J., & Satapathy, S. K. (2020) · Alimentary Pharmacology & Therapeutics · doi:10.1111/apt.16078 · Cited by 45
Metadata
- Authors
- Luo, D., Wan, X., Liu, J., & Tong, T.
- Affiliations
- Hong Kong Baptist University (HK)
- Corresponding author
- Wan, Xiang
- Venue
- Statistical Methods in Medical Research
- Publisher
- SAGE Publications
- Published
- 2016-09-27
- Volume
- 27
- Issue
- 6
- Pages
- 1785-1805
- ISSN
- 0962-2802 (ISSN-L)
- 1477-0334
- OpenAlex
- W1534181333
- Cited by
- 4,066
- References
- 21
- Citations by year
- 2026: 789 · 2025: 833 · 2024: 700 · 2023: 601 · 2022: 462 · 2021: 379 · 2020: 188 · 2019: 91 · 2018: 20 · 2017: 1
- FWCI
- 34.66
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Statistical Methods in Medical Research (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Hong Kong Baptist University (FRG2/14-15/077, FRG1/14-15/084, FRG2/15-16/019)
- Topics
- Social Sciences › Decision Sciences › Statistics, Probability and Uncertainty › Meta-analysis and systematic reviews
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Epidemiology
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Clinical Trials
- Keywords
- sample mean estimation, meta-analysis, median, sample size, standard deviation estimation, optimal estimators, evidence-based medicine
- MeSH terms
- Algorithms, Sampling Studies, Meta-Analysis as Topic, Sample Size, Evidence-Based Medicine
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