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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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“When articles reported median and interquartile range, we estimated means and standard deviations (SDs) according to the methodology described by Luo et al. [18] and Wan et al. [19].”
Zhu, N., Weldegiorgis, M., Carter, E., Brown, C. S., Holmes, A. H., & Aylin, P. P. (2024) · JMIR Public Health and Surveillance · doi:10.2196/53828 · Cited by 24
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“For studies reporting only ranges or 95% CIs, we calculated the means [31,32,35].”
Papazisi, O., van der Schoot, M. M., Berendsen, R. R., Arbous, M. S., le Cessie, S., Dekkers, O. M., Klautz, R. J. M., Marczin, N., Palmen, M., & de Waal, E. E. C. (2025) · Journal of Cardiothoracic and Vascular Anesthesia · doi:10.1053/j.jvca.2025.02.027 · Cited by 20
Music Interventions in Percutaneous Coronary Procedures: A Meta-Analysis
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“Where median and interquartile ranges were reported, mean and standard deviation were estimated as per special formulas((12,13)).”
BackgroundSomagutta, M. K. R., Pormento, M. K. L., Khan, M. A., Hamdan, A. H. Y., Hange, N., Manish, K. C., Pagad, S., Jain, M. S., Lingarajah, S., Sharma, V., et al. (2021) · Acute and Critical Care · doi:10.4266/acc.2021.00108 · Cited by 21
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“Where median and interquartile range was presented for an outcome interest, normal distribution of cohort was assumed and these parameters were converted to mean and standard deviation by method outlined by Luo et al.7 The random‐effects model was used to assess effect estimates.”
MethodWang, L., Yang, L., Pei, S.-F., Chong, Y., Guo, Y., Gao, X. L., Tang, Q. Y., Li, Y., & Feng, F. M. (2021) · Scandinavian Journal of Clinical and Laboratory Investigation · doi:10.1080/00365513.2021.2000635 · Cited by 20
“According to literature, median and the interquartile range (IQR) were converted to the mean value and standard deviation for meta-analysis [11,12] The Newcastle–Ottawa Scale (NOS) was used for assessing the quality of the included studies.”
MethodDebi, H., Itu, Z. T., Amin, M. T., Hussain, F., & Hossain, M. S. (2021) · Expert Review of Endocrinology & Metabolism · doi:10.1080/17446651.2022.2002146 · Cited by 22
“[34], the median and interquartile range (IQR) were used to calculate the mean and SD.”
MethodOropharyngeal colostrum immunotherapy and nutrition in preterm newborns: meta-analysis
Ramos, M. D. S. X., Martins, C. D. C., Souza, E. S., Vieira, G. O., Gomes‐Filho, I. S., Figueiredo, A. C. M. G., Pereira, M. G., & da Cruz, S. S. (2021) · Revista de Saúde Pública · doi:10.11606/s1518-8787.2021055003051 · Cited by 20
Behera, B. K., Misra, S., Sarkar, S., & Mishra, N. (2021) · Pain Medicine · doi:10.1093/pm/pnab353 · Cited by 20
Tey, K. Y., Tan, S. Y., Ting, D. S. J., Mehta, J. S., & Ang, M. (2022) · Frontiers in Medicine · doi:10.3389/fmed.2022.857200 · Cited by 20
Chen, X., Guan, X.-X., Tang, Y., Deng, J., & Zhang, X. (2022) · International Journal of Food Sciences and Nutrition · doi:10.1080/09637486.2022.2046711 · Cited by 20
“If the study only reported the mean and upper and lower quartiles, we would use some methods to estimate (Luo et al. 2018).”
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Jiang, Z., Zhao, Z., Feng, T., Cheng, Y., Zhang, G., Zhong, M., & Hu, S. (2023) · International Journal of Surgery · doi:10.1097/js9.0000000000000402 · Cited by 20
“If the values of the obtained continuous variables were represented as meancross-checked and median-cross-checked with maximum and minimum values, the mean and SD were obtained based on the previous description([10,11]).”
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Xu, M., Zhang, G., Tang, Y., Wang, R., & Yang, J. (2023) · Journal of Cardiothoracic and Vascular Anesthesia · doi:10.1053/j.jvca.2023.05.003 · Cited by 20
Hemostatic biomarkers associated with postpartum hemorrhage: a systematic review and meta-analysis
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“6 Missing means and SDs were calculated from quartiles, if reported, using the method developed by Luo et al. 7 Results from the RE models were presented graphically with forest plots.”
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“[14] .”
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“To convert the median and quartile to mean and standard deviation (SD), we first tested the skewness and then applied a new piecewise function based on the sample size [35,36].”
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MethodWang, F., Dou, P., Wei, W., & Liu, P. J. (2024) · Nutrition & Diabetes · doi:10.1038/s41387-024-00263-9 · Cited by 20
“The SD of each difference was calculated as follows: If the target data in an included study were expressed as medians (quantile interval), the mean value of the target data was determined using the method of Luo et al. [25] or Hozo et al. [26] when appropriate.”
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“The Newcastle–Ottawa Scale (NOS) was used to assess the quality of the included studies [24–26], as shown in Table S3.”
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RetractedHosseini, K., Soleimani, H., Maleki, S., Nasrollahizadeh, A., Tayebi, S., Nelson, J. R., & Heffron, S. P. (2024) · BMC Cardiovascular Disorders · doi:10.1186/s12872-024-04057-w · Cited by 20
“In studies that reported median and interquartile (IQR) ranges, we used the method developed by Lou et al. [13] and Wan et al. [14] to calculate means and standard deviations.”
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Sun, H., Ren, Q., Zhao, X., Tian, Y., Pan, J., Wei, Q., Li, Y., Chen, Y., Zhang, H., Zhang, W., et al. (2020) · Prostaglandins Leukotrienes and Essential Fatty Acids · doi:10.1016/j.plefa.2020.102184 · Cited by 21
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“The means and medians were extracted, and the medians were converted to means as previously described (Hozo et al., 2005; Luo et al., 2015).”
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Gabbai‐Armelin, P. R., de Oliveira, A. B., Ferrisse, T. M., Sales, L. S., Barbosa, E. R. O., Miranda, M. L., Salomão, K. B., & Brighenti, F. L. (2021) · European Journal of Clinical Investigation · doi:10.1111/eci.13559 · Cited by 20
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“The methods developed by Wan et al. [18] and Luo et al. [19] were employed to convert median and interquartile range values into mean and standard deviation.”
Higa, K. C., Mayer, K. P., Quinn, C., Jubina, L., Suarez‐Pierre, A., Colborn, K. L., Jolley, S. E., Enfield, K., Zwischenberger, J. B., Sevin, C. M., et al. (2023) · Critical Care Medicine · doi:10.1097/ccm.0000000000005900 · Cited by 22
McHugh, F. T., Ryan, É. J., Ryan, O. K., Tan, J., Boland, P. A., Whelan, M. C., Kelly, M. E., McNamara, D. A., Neary, P. C., O’Riordan, J. M., et al. (2024) · Diseases of the Colon & Rectum · doi:10.1097/dcr.0000000000003256 · Cited by 19
Acute effects of outdoor versus indoor exercise: a systematic review and meta-analysis
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“Median and interquartile range were used to estimate mean ± SD using an online calculator found in the literature (Luo et al., 2018; Shi,2020a , 2020b).”
Mandarino, F. V., Barchi, A., Salmeri, N., Azzolini, F., Fasulo, E., Dell’Anna, G., Vespa, E., Sinagra, E., Jacques, J., & Danese, S. (2024) · DEN Open · doi:10.1002/deo2.70021 · Cited by 20
“…using the method by Shi et al., 14 if the data did not show signifi-cant skewness from normality,we extrapolated the mean from median values using the method by Luo et al. 15 and calculated the SD from the IQR or range using the equation by Wan et al., when either of these measures was available.”
Methodvan Boven, M. R., Hutten, G. J., Richardson, R., Königs, M., Leemhuis, A. G., Onland, W., Terheggen‐Lagro, S. W. J., Oosterlaan, J., & van Kaam, A. H. (2024) · European Respiratory Review · doi:10.1183/16000617.0114-2024 · Cited by 19
“If only the median and (interquartile) range were reported, normality of the data was checked using the method described by S HI et al. [21] before estimating the sample mean± SD by the method of L UO et al. [22] and W AN et al .”
Patton, A. G., Davey, M. G., Quinn, E., Reinhardt, C., Robb, W. B., & Donlon, N. E. (2024) · Diseases of the Esophagus · doi:10.1093/dote/doae086 · Cited by 22
Deriglazov, D., Halamová, J., & Kernová, L. (2025) · Worldviews on Evidence-Based Nursing · doi:10.1111/wvn.70033 · Cited by 34
“We evaluated the skewness of the data and estimated sample means using the method suggested by Luo et al. (2016) and the standard deviations proposed by Wan et al. (2014).”
Gao, J., Yu, S., Zhu, X., Yan, Y., Zhang, Y., & Pei, D. (2020) · Journal of Evidence Based Dental Practice · doi:10.1016/j.jebdp.2020.101398 · Cited by 20
Yuan, X., Xu, B.-R., Wong, M., Chen, Y., Tang, Y., Deng, L., & Tang, D. (2020) · The Surgeon · doi:10.1016/j.surge.2020.06.014 · Cited by 19
Bernal-Soriano, M. C., Lumbreras, B., Hernández‐Aguado, I., Pastor‐Valero, M., López‐Garrigós, M., & Parker, L. A. (2020) · Clinical Chemistry and Laboratory Medicine (CCLM) · doi:10.1515/cclm-2020-0145 · Cited by 19
“In studies that reported medians and ranges, we estimated the means and standard deviations [15, 16].”
MethodCovic, A. C., Çöpür, S., Tapoi, L. C., Afşar, B., Ureche, C., Siriopol, D. C., Nistor, I., & Kanbay, M. (2020) · American Journal of Cardiovascular Drugs · doi:10.1007/s40256-020-00453-7 · Cited by 19
Navarro‐Santana, M. J., Sánchez-Infante, J., Gómez-Chiguano, G. F., Cummings, M., Fernández‐de‐las‐Peñas, C., & Plaza‐Manzano, G. (2020) · Acupuncture in Medicine · doi:10.1177/0964528420967364 · Cited by 19
Lin, X., Yang, Y., Sun, B., Liu, B., Zeng, Q., & Xiong, C. (2021) · Journal of the American Heart Association · doi:10.1161/jaha.120.018374 · Cited by 19
The epidemiological and clinical characteristics of the hospital-acquired influenza infections
Li, Y., Wang, L., Xie, L., Hou, W.-L., Liu, X., & Yin, S. W. (2021) · Medicine · doi:10.1097/md.0000000000025142 · Cited by 19
Predictive Performance of Serum β-hCG MoM Levels for Preeclampsia Screening: A Meta-Analysis
Zhang, X., Huangfu, Z., Shi, F., & Xiao, Z. (2021) · Frontiers in Endocrinology · doi:10.3389/fendo.2021.619530 · Cited by 19
“003 7 11-13 large 4063 64/3999 29(17-44) 30(16-47) 0.”
Formica, F., Maestri, F., D’Alessandro, S., Mauro, M. D., Singh, G., Gallingani, A., & Nicolini, F. (2021) · Journal of Thoracic and Cardiovascular Surgery · doi:10.1016/j.jtcvs.2021.06.062 · Cited by 19
“Median and interquartile range were converted into mean and standard deviation following the recommendations of Luo and colleagues.(19) Each study included in the meta-analysis was identified by study design, study period, first authors, affiliation, and year of publication.”
MethodSystematic review of the incidence and risk factors for chyle leak after pancreatic surgery
Varghese, C., Wells, C. I., Lee, S., Pathak, S., Siriwardena, A. K., & Pandanaboyana, S. (2021) · Surgery · doi:10.1016/j.surg.2021.07.021 · Cited by 19
Cai, J., & Lin, Z. (2021) · Heart & Lung · doi:10.1016/j.hrtlng.2021.07.010 · Cited by 19
Huang, Y. C., Cao, D., Chen, Z., Chen, B., Li, J., Guo, J., Dong, Q., Wei, Q., & Liu, L. (2021) · Frontiers in Oncology · doi:10.3389/fonc.2021.730662 · Cited by 19
“When continuous variables in the study were reported as median with range or interquartile range, we calculated the mean ± standard deviation through the validated mathematical method (33, 34).”
MethodBlood Neurofilament Light Chain in Genetic Ataxia: A Meta‐Analysis
Peng, L., Wang, S., Chen, Z., Peng, Y., Wang, C., Long, Z., Peng, H., Shi, Y., Hou, X., Lei, L., et al. (2021) · Movement Disorders · doi:10.1002/mds.28783 · Cited by 19
“We performed log transformation to get the log (IQR), and then obtained the log (mean SD) using Luo et al’s formula.(41,42) For studies lacking original NfL concentrations, data were extracted from the related graphs by Getdata Graph Digitizer software (version 2.”
MethodDeng, J., Zhou, F., Hou, W., Heybati, K., Ali, S. H., Chang, O., Silver, Z. A., Dhivagaran, T., Ramaraju, H. B., Wong, C. Y., et al. (2022) · Future Virology · doi:10.2217/fvl-2021-0066 · Cited by 20
Zhang, F., Ren, Y., Wang, H., Bai, Y., & Huang, L. (2022) · Frontiers in Medicine · doi:10.3389/fmed.2022.842423 · Cited by 19
Alfalasi, R. B., Downing, J. V., Cardona, S., Lowie, B., Fairchild, M., Chan, C., Powell, E. M., Pourmand, A., Grazioli, A., & Tran, Q. K. (2022) · Healthcare · doi:10.3390/healthcare10030591 · Cited by 19
Bijlenga, D., Overeem, S., Fronczek, R., & Lammers, G. J. (2022) · SLEEP · doi:10.1093/sleep/zsac091 · Cited by 19
Jiang, T., Huang, B., Huo, S., Mageta, L., Guo, J., Lv, J., & Li, L. (2022) · Frontiers in Surgery · doi:10.3389/fsurg.2022.859205 · Cited by 19
Adusumilli, G., Pederson, J. M., Hardy, N. J., Kallmes, K. M., Hutchison, K., Kobeissi, H., Heiferman, D. M., & Heit, J. J. (2022) · Interventional Neuroradiology · doi:10.1177/15910199221100796 · Cited by 19
Zhou, F., Deng, J., Heybati, K., Zuo, Q., Ali, S. H., Hou, W., Wong, C. Y., Ramaraju, H. B., Chang, O., Dhivagaran, T., et al. (2022) · Future Virology · doi:10.2217/fvl-2021-0244 · Cited by 21
Zhang, L., Xie, X., Sun, Y., & Zhou, F. (2022) · Neuropsychiatric Disease and Treatment · doi:10.2147/ndt.s383654 · Cited by 19
Statins Have an Anti‐Inflammation in CKD Patients: A Meta‐Analysis of Randomized Trials
Wang, J., Chen, Z., Qiu, Y., Wu, L., Wang, H., Wu, L., Zhao, L., & Xie, D. (2022) · BioMed Research International · doi:10.1155/2022/4842699 · Cited by 19
“If data were reported as median, interquartile range, 95% CI, or standard error (SE), the data were converted to mean and standard deviation (SD) by the formula [37, 38].”
He, F., Chen, W., Xu, W., Liu, D., Xiao, Z., Tang, Y., Lin, Z., Liao, Y., Bin, J., Chen, G., et al. (2022) · Diabetes Obesity and Metabolism · doi:10.1111/dom.14908 · Cited by 20
Red Blood Cell Transfusion in Patients With Placenta Accreta Spectrum
Miller, S. E., Leonard, S. A., Meza, P. K., Ku, S., Ren, L. Y., Lyell, D. J., Sultan, P., & Butwick, A. J. (2022) · Obstetrics and Gynecology · doi:10.1097/aog.0000000000004976 · Cited by 19
Şener, H., Evereklioğlu, C., Horozoğlu, F., & Sener, A. B. G. (2023) · Ocular Immunology and Inflammation · doi:10.1080/09273948.2022.2157288 · Cited by 19
“25 was used to estimate the sample mean and standard deviation from sample size, median, range, mid-range and/or interquartile range and mid-quartile range.”
MethodYin, Y., Xie, H., Ren, J., Jiang, N.-J., & Dai, L. (2023) · Frontiers in Oncology · doi:10.3389/fonc.2022.1059924 · Cited by 19
“(34) to ensure that all of the re levant data could be inc luded in the final quantitative synthesis.”
BackgroundAssociation of arterial stiffness and neuropathy in diabetes: a systematic review and meta-analysis
Beros, A. L., Sluyter, J., & Scragg, R. K. R. (2023) · BMJ Open Diabetes Research & Care · doi:10.1136/bmjdrc-2022-003140 · Cited by 21
Asadi, B., Cuenca‐Zaldívar, J. N., Ansari, N. N., Ibáñez, J., Herrero, P., & Calvo, S. (2023) · Healthcare · doi:10.3390/healthcare11050666 · Cited by 19
“In the articles showing the results with median and interquartile range, these were transformed into mean and standard deviation using the appropriate formulas [38,39].”
Chen, X., Li, H., Li, S., Wang, Y., Ma, R., Qian, W., Chen, G., & Li, J. (2023) · International Journal of Surgery · doi:10.1097/js9.0000000000000291 · Cited by 19
“[32] and Wan et al .”
Wibawa, I. D. N., Mariadi, I. K., Shalim, C. P., & Sindhughosa, D. A. (2023) · Clinical and Experimental Hepatology · doi:10.5114/ceh.2023.128768 · Cited by 19
“to calculate the mean if data were given as median with Q1 and Q3 or range [13, 14].”
BackgroundCastellana, R., Natrella, M., Fanelli, G., Lancellotta, V., Cornacchione, P., Castellana, M., Filippiadis, D. K., Tagliaferri, L., & Iezzi, R. (2023) · European Journal of Radiology · doi:10.1016/j.ejrad.2023.110943 · Cited by 19
Ebrahimi, N. D., Sadeghi, A., Ala, M., Ebrahimi, F., Pakbaz, S., & Azarpira, N. (2023) · Frontiers in Endocrinology · doi:10.3389/fendo.2023.1202560 · Cited by 22
“When minimum, fi rst quartile, median, third quartile, and maximum were the only available statistics, mean and standard deviation was estimated utilizing methods outlined in prior research (26, 27).”
Vignarajah, M., Berg, A., Abdallah, Z., Arora, N., Javidan, A. P., Pitre, T., Fernando, S. M., Spence, J. D., Centofanti, J., & Rochwerg, B. N. (2023) · British Journal of Anaesthesia · doi:10.1016/j.bja.2023.05.029 · Cited by 19
Olid, M. N. R., Bucataru, E., Ramos‐García, P., & González‐Moles, M. Á. (2023) · Antibiotics · doi:10.3390/antibiotics12101552 · Cited by 19
Khaboushan, A. S., Moeinafshar, A., Ersi, M. H., Teixeira, A. L., Zolbin, M. M., & Kajbafzadeh, A. M. (2023) · Journal of Neuroimmunology · doi:10.1016/j.jneuroim.2023.578243 · Cited by 19
Santi, D., Debbi, V., Costantino, F., Spaggiari, G., Simoni, M., Greco, C., & Casarini, L. (2023) · Clocks & Sleep · doi:10.3390/clockssleep5040050 · Cited by 20
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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