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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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“Where the median and interquartile range were shown, these were transformed into mean and standard deviation using the appropriate formulas [26, 34].”
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“The medians were converted to mean values through the method by Luo et al. [12].”
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“For continuous variables, the available data on medians and IQRs (interquartile range) have been converted into means and standard deviations applying the methods proposed by Luo et al. and Wan et al. [19, 20].”
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“[21, 22] to count the means and standard deviations.”
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“[26] 2020 Wuhan, China 24 121 41–72 Retrospective study 7 1”
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“When continuous variables were reported as median and range in the main literature, we calculated the mean and SD (11).”
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“Missing baseline and final mean and SD data were calculated by approximation according to Luo et al. and Shi et al., respectively, as recommended by Cochrane [20, 21].”
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“If median and interquartile range were reported, data were checked for normality [47] and converted to mean [48] and standard deviation [49].”
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“Wherein median and range of FVC and DLCO were provided, the mean and standard deviation values were calculated using Luo et al. 9 and Wan et al.’s 10 methods, respectively.”
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“Where the mean and standard deviation were not provided, the mean was estimated using the Luo method, 36 and the standard deviation was estimated using the Shi method 37 based on the median and quartiles or minimum and maximum values.”
Mousavi, S. M., Younesian, S., & Ejtahed, H. (2024) · Microbial Pathogenesis · doi:10.1016/j.micpath.2024.107266 · Cited by 18
Dewi, D. T. K., Nguyen, C. T. T., Chen, S., Lee, G. T., Tsai, S., Huda, M. H., & Kuo, S. (2025) · International Journal of Nursing Studies · doi:10.1016/j.ijnurstu.2025.105128 · Cited by 18
Gędek, A., Modrzejewski, S., Materna, M., Iwański, M., Wichniak, A., & Dominiak, M. (2025) · International Journal of Molecular Sciences · doi:10.3390/ijms262110362 · Cited by 18
Varikasuvu, S. R., Panga, J. R., & Satyanarayana, M. V. (2019) · Gynecological Endocrinology · doi:10.1080/09513590.2018.1536736 · Cited by 17
“For studies [12,13] where median and ranges were reported, the mean and SD values were obtained as described previously for suitable use in meta-analysis [18,19].”
MethodCharoenkwan, K., & Nantasupha, C. (2019) · Journal of obstetrics and gynaecology research · doi:10.1111/jog.14152 · Cited by 17
“The I2 statistic of ≥50% was considered as indication of significant heterogeneity.34 For meta-analysis of pain scores, which were continuous variable, weighted mean difference (WMD) with 95% confidence interval (CI) were estimated from mean and SD reported in original articles by employing the RevMan program.33 If no…”
MethodDing, R., Zhang, Q., Duan, Y., Wang, D.-J., Sun, Q.-B., & Shan, R. (2020) · European Journal of Pediatrics · doi:10.1007/s00431-020-03802-5 · Cited by 17
“When the data of gestational age and birth weight were given as the median (interquartile space), we calculated the mean using the statistical methods provided by Luo et al [23].”
MethodLateral Thoracotomy for Ventricular Assist Device Implantation: A Meta-Analysis of Literature
Mariani, S., Li, T., Boethig, D., Napp, L. C., Chatterjee, A., Homann, K., Bounader, K., Hanke, J. S., Dogan, G., Lorusso, R., et al. (2021) · ASAIO Journal · doi:10.1097/mat.0000000000001359 · Cited by 17
Anagnostis, P. G., Florou, P., Bosdou, J. Κ., Grimbizis, G. F., Iakovou, I. P., Kolibianakis, E. M., & Goulis, D. G. (2021) · Maturitas · doi:10.1016/j.maturitas.2021.04.002 · Cited by 17
Sheehan, D. M., Trepka, M. J., Dawit, R., Jebai, R., & Gbadamosi, S. O. (2022) · Aids Reviews · doi:10.24875/aidsrev.21000007 · Cited by 17
Bakaloudi, D. R., Chrysoula, L., Leonida, I., Kotzakioulafi, E., Theodoridis, X., & Chourdakis, M. (2021) · Clinical Nutrition · doi:10.1016/j.clnu.2021.10.002 · Cited by 17
Makker, P. G. S., Koh, C., Solomon, M. J., & Steffens, D. (2022) · ANZ Journal of Surgery · doi:10.1111/ans.17577 · Cited by 17
Chen, Y., Zhu, G., Liu, X., Wu, W., Chai, H., Tao, M., Kong, D., Li, Y., & Wang, L. (2022) · Frontiers in Cardiovascular Medicine · doi:10.3389/fcvm.2022.927642 · Cited by 17
“Data presented as median (interquartile range) in the included studies were converted to mean ± SD, as described in previous studies (11, 12).”
MethodRayan, M., Gruehn, D., & Khayyam, U. (2022) · International Journal of Environmental Research and Public Health · doi:10.3390/ijerph191911844 · Cited by 18
Jaiswal, V., Khan, N., Jaiswal, A., Dagar, M., Joshi, A., Huang, H. Y. R., Naz, H., Attia, A. M., Ghanim, M., Baburaj, A., et al. (2022) · IJC Heart & Vasculature · doi:10.1016/j.ijcha.2022.101125 · Cited by 17
“According to the recommendations, we converted the median and interquartile ranges into mean and standard deviation according to the recommendations [19].”
MethodBusinge, C. B., Musarurwa, H. T., Longo‐Mbenza, B., & Kengne, A. P. (2022) · Systematic Reviews · doi:10.1186/s13643-022-02072-6 · Cited by 17
“2) In order to check for heterogeneity and publication bias, the mean UIC and standard deviation were derived from the median UIC using the methods described elsewhere [63, 64] (Table 3).”
MethodHe, J., Zhang, J., & Ren, X. (2022) · Frontiers in Immunology · doi:10.3389/fimmu.2022.1041098 · Cited by 17
“The values reported in terms of median and quartile in several articles were altered to mean and standard deviation (17, 18).”
BackgroundZhong, G., Cheng, S., Zhou, M., Xie, J., Xu, Z., Lai, H., Yan, Y. C., Xie, Z., Zhou, J., Xie, X., et al. (2023) · The Spine Journal · doi:10.1016/j.spinee.2023.01.008 · Cited by 18
Zhou, X., Cao, J., Topatana, W., Xie, T., Chen, T., Hu, J., Li, S., Juengpanich, S., Lu, Z., Zhang, B., et al. (2023) · Immunotherapy · doi:10.2217/imt-2022-0168 · Cited by 17
Lin, H., Liu, Q., Zhao, L., Liu, Z., Cui, H., Li, P., Fan, H., & Guo, L. (2023) · International Journal of Molecular Sciences · doi:10.3390/ijms24076090 · Cited by 16
HUANG, X., Lin, F., Chen, Q., & Hu, X. (2023) · Minerva Anestesiologica · doi:10.23736/s0375-9393.23.17100-8 · Cited by 22
Davey, M. G., Ryan, É. J., Donlon, N. E., Ryan, O. K., Azzawi, M. A., Boland, M. R., Kerin, M. J., & Lowery, A. (2023) · Langenbeck s Archives of Surgery · doi:10.1007/s00423-023-02911-7 · Cited by 21
“Estimates of mean and SDs were calculated using standard statistical methods, where applicable [36, 37].”
MethodOkoli, G. N., Caeseele, P. G. V., Askin, N., & Abou-Setta, A. M. (2023) · Scandinavian Journal of Infectious Diseases · doi:10.1080/23744235.2023.2214609 · Cited by 17
Liu, W., Bai, D., & Kou, L. (2023) · BMC Ophthalmology · doi:10.1186/s12886-023-02987-1 · Cited by 17
“Most studies reported the daily corticosteroid dosage with median and interquartile range, so we converted the data to mean ± SD according to the aforementioned literature [13, 25].”
MethodDong, F., Zheng, L., An, W., Xue, T., & Zhong, X. (2023) · PLoS ONE · doi:10.1371/journal.pone.0286956 · Cited by 17
“We calculated the mean ± SD value in the study by Luo (2018) and Wan (2014) [23, 24], as the researchers provided only the number of participants and median and interquartile range of NLR.”
MethodMa, Y., Yang, M., Chen, X., Qu, W., Qu, X., & He, P. (2023) · PLoS ONE · doi:10.1371/journal.pone.0287833 · Cited by 17
Gu, S., Du, X., Wang, D., Yu, Y., & Guo, S. (2023) · PLoS ONE · doi:10.1371/journal.pone.0290362 · Cited by 20
Wang, L., Deng, J.-Y., Liang, C., & Zhu, P.-Y. (2023) · Frontiers in Oncology · doi:10.3389/fonc.2023.1195910 · Cited by 17
“(12) to transform raw data from the median and interquartile range or maximum and minimum values into normal distribution data, as well as McGrath et al.”
BackgroundCavaleri, D., Capogrosso, C. A., Guzzi, P., Bernasconi, G., Re, M., Misiak, B., Crocamo, C., Bartoli, F., & Carrà, G. (2023) · Psychoneuroendocrinology · doi:10.1016/j.psyneuen.2023.106392 · Cited by 19
“If raw means and SDs were not reported, they were estimated using conventional transformation methods, when feasible D. Cavaleri et al. (Drevon et al., 2017; Higgins et al., 2022; Luo et al., 2018).”
MethodSugai, S., Yamawaki, K., Sekizuka, T., Haino, K., Yoshihara, K., & Nishijima, K. (2023) · American Journal of Obstetrics & Gynecology MFM · doi:10.1016/j.ajogmf.2023.101197 · Cited by 22
de Albuquerque Pessoa dos Santos, Y., Tomazini, B. M., dos Santos, M. H. C., de Queiroz, E. L., Pastore Júnior, L., Costa, E. L. V., & da Silva Ramos, F. J. (2025) · Critical Care · doi:10.1186/s13054-025-05623-0 · Cited by 20
“…reported as medians and interquartile ranges, we estimated means and standard deviations using the validated method proposed by Wan et al. and Luo et al. [18, 19] A random-effects model (DerSimonian and Laird) was applied to account for anticipated clinical and methodological heterogeneity.”
Schmidt, P. H. S., Pasqualotto, E., dos Santos, H. V., de Souza, L. S. N., dos Santos, B. E., Chavez, M. P., Ferreira, R. O. M., Hohl, A., Ronsoni, M. F., & van de Sande-Lee, S. (2025) · Obesity Research & Clinical Practice · doi:10.1016/j.orcp.2025.01.009 · Cited by 18
Grewal, S., Zaman, N., Borgatta, L., Nudy, M., Foy, A. J., & Peterson, B. (2021) · The American Journal of Cardiology · doi:10.1016/j.amjcard.2021.05.043 · Cited by 14
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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