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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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Liu, Y., Chen, S., Zhou, P., Li, H., Wan, Q., Lu, Y., & Li, B. (2024) · The Science of The Total Environment · doi:10.1016/j.scitotenv.2024.174655 · Cited by 41
Infarct Growth despite Successful Endovascular Reperfusion in Acute Ischemic Stroke: A Meta-analysis
Bala, F., Ospel, J. M., Mulpur, B., Kim, B. J., Yoo, J., Menon, B. K., Goyal, M., Federau, C., Sohn, S., Hussain, M. S., et al. (2021) · American Journal of Neuroradiology · doi:10.3174/ajnr.a7177 · Cited by 42
Tang, R., Luo, R., Wu, B., Wang, F., Song, H., & Chen, X. (2021) · Journal of Critical Care · doi:10.1016/j.jcrc.2021.06.004 · Cited by 39
High levels of Von Willebrand factor markers in COVID-19: a systematic review and meta-analysis
Rostami, M., Mansouritorghabeh, H., & Parsa‐kondelaji, M. (2021) · Clinical and Experimental Medicine · doi:10.1007/s10238-021-00769-x · Cited by 39
“Data from articles reported as median and mean ± SD were estimated with online software using the methods described by Luo et al.[24] and Wan et al.”
MethodZhang, W., Bai, M., Yu, Y., Chen, X., Zhao, L., & Chen, X. (2021) · Seminars in Dialysis · doi:10.1111/sdi.12946 · Cited by 38
Chen, S., Zhang, W., Wang, D., & Chen, Z. (2024) · European Journal of Physical and Rehabilitation Medicine · doi:10.23736/s1973-9087.24.08354-0 · Cited by 38
Blood Astrocyte Biomarkers in Alzheimer Disease
Holper, S., Loveland, P. M., Churilov, L. P., Italiano, D., Watson, R., & Yassi, N. (2024) · Neurology · doi:10.1212/wnl.0000000000209537 · Cited by 43
do Nascimento, T. S., Pereira, R. O. L., Maia, E., Ohnuma, T., da Costa, M. G., Slawka, E., Galhardo, C., & Krishnamoorthy, V. K. (2024) · Perioperative Medicine · doi:10.1186/s13741-024-00439-y · Cited by 41
Xie, H.-M., Guo, T., Sun, X., Ge, H.-X., Chen, X.-D., Zhao, K.-J., & Zhang, L. (2021) · Archives of Physical Medicine and Rehabilitation · doi:10.1016/j.apmr.2020.12.010 · Cited by 38
Beran, A., Altorok, N., Srour, O., Malhas, S., Khokher, W., Mhanna, M., Ayesh, H., Aladamat, N., Abuhelwa, Z., Srour, K., et al. (2022) · Journal of Clinical Medicine · doi:10.3390/jcm11071971 · Cited by 43
The Use of Magnesium Sulfate and Peripheral Nerve Blocks
Zeng, J., Chen, Q., Yu, C., Zhou, J., & Yang, B. (2021) · Clinical Journal of Pain · doi:10.1097/ajp.0000000000000944 · Cited by 37
Robotic Living Donor Right Hepatectomy: A Systematic Review and Meta-Analysis
Lincango-Naranjo, E. P., Garces-Delgado, E., Siepmann, T., Mirow, L., Solís-Pazmiño, P., Alexander-León, H., Restrepo-Rodas, G., Mancero-Montalvo, R., Ponce, C. J., Semanate, R. C., et al. (2022) · Journal of Clinical Medicine · doi:10.3390/jcm11092603 · Cited by 37
Ma, Y., Wang, J., Guo, S., Meng, Z., Ren, Y. D., Xie, Y., & Wang, M. (2023) · Frontiers in Immunology · doi:10.3389/fimmu.2022.1064007 · Cited by 38
“If the mean or SD was not stated directly in the articles, we could estimate them from the sample size and the alternative measures, including median, range, and 95% confidence interval (CI) or interquartile range, utilizing Wan and Luo’s online data converter (26, 27).”
MethodWang, L., Hua, Y., Wang, L., Xia, Z., Zhang, Y., & Ou, X. (2023) · Frontiers in Medicine · doi:10.3389/fmed.2023.1202754 · Cited by 37
Ou, W., Chen, Y., Ju, Y., Ma, M., Qin, Y., Bi, Y., Liao, M., Liu, B., Liu, J., Zhang, Y., et al. (2023) · Journal of Affective Disorders · doi:10.1016/j.jad.2023.07.078 · Cited by 44
Self-citationZhang, Y., Jia, R., Zhang, Y., Sun, X., Mei, Y., Zou, R., Niu, L., & Dong, S. (2023) · BMC Oral Health · doi:10.1186/s12903-023-03383-3 · Cited by 39
“For studies that reported median and interquartile ranges, we estimated means and standard deviations through the given series estimation formulas through simulation studies provided by Wan X, McGrath, S. et al. [48, 49, 51].”
MethodSoderberg, L., Ergun, O., Ding, M., Parker, R. M. N., Borofsky, M. S., Pais, V. M., & Dahm, P. (2023) · Cochrane Database of Systematic Reviews · doi:10.1002/14651858.cd013445.pub2 · Cited by 37
Ma, Y., Jian, Z., Chi, Y., Li, H., & Wang, K. (2020) · Surgical Infections · doi:10.1089/sur.2020.013 · Cited by 37
“When only the median interquartile range (IQR) age information was provided in the involved studies, the mean age was estimated by methods described previously [5].”
Li, Z., Wei, A., Xia, N., Zheng, L., Yang, D., Ye, J., Xiong, J., & Hu, W. (2020) · Scientific Reports · doi:10.1038/s41598-020-74812-4 · Cited by 37
Navarro‐Santana, M. J., Sánchez-Infante, J., Gómez-Chiguano, G. F., Cleland, J. A., Fernández‐de‐las‐Peñas, C., Martín‐Casas, P., & Plaza‐Manzano, G. (2021) · Pain Medicine · doi:10.1093/pm/pnab188 · Cited by 37
Pan, H., Xian, P., Yang, D., Zhang, C., Tang, H., He, X., Lin, H., Wen, X., Ma, H., & Lai, M. (2021) · Endocrine · doi:10.1007/s12020-021-02886-9 · Cited by 37
Li, J., Tang, J., Liu, F., & Fang, J. (2022) · Frontiers in Surgery · doi:10.3389/fsurg.2022.811005 · Cited by 37
Mandigers, T. J., Bissacco, D., Domanin, M., D’Alessio, I., Tolva, V., Piffaretti, G., van Herwaarden, J. A., & Trimarchi, S. (2022) · European Journal of Vascular and Endovascular Surgery · doi:10.1016/j.ejvs.2022.05.004 · Cited by 37
Sanchez, S., Hickerson, M. M., Patel, R. R., Ghazaleh, D., Tarchand, R., Paranjape, G. S., Pope, H., Ortega‐Gutiérrez, S., Pederson, J. M., Hasan, D. M., et al. (2023) · Stroke Vascular and Interventional Neurology · doi:10.1161/svin.122.000707 · Cited by 39
“Most studies reported continuous outcome data as means and SDs; however, in cases where medians and interquartile ranges were reported, we employed the statistical methods described by Luo et al 11 and Wan et al 12 to estimate means and SDs, respectively.”
Rakhmat, I. I., Putra, I. C. S., Wibowo, A., Henrina, J., Nugraha, G. I., Ghozali, M., Syamsunarno, M. R. A. A., Pranata, R., Akbar, M. R., & Achmad, T. H. (2022) · Clinical Obesity · doi:10.1111/cob.12523 · Cited by 39
“range) were transformed into mean ± SD.(25,26) Data extraction was performed independently by five authors (AW, JH, GIN, MG and MRAAS).”
MethodLi, X., Liu, Z., Li, L., Wang, H., & Wang, H. (2022) · Frontiers in Endocrinology · doi:10.3389/fendo.2022.1026901 · Cited by 36
Chiavaroli, L., Cheung, A., Ayoub‐Charette, S., Ahmed, A., Lee, D., Au‐Yeung, F., Qi, X., Back, S., McGlynn, N. D., Ha, V., et al. (2023) · American Journal of Clinical Nutrition · doi:10.1016/j.ajcnut.2023.01.023 · Cited by 39
Downing, J. V., Yardi, I., Ren, C., Cardona, S., Zahid, M., Tang, K., Bzhilyanskaya, V., Patel, P., Pourmand, A., & Tran, Q. K. (2023) · The American Journal of Emergency Medicine · doi:10.1016/j.ajem.2023.06.046 · Cited by 36
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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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