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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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“20,21 In addition, for studies with missing data, all primary study authors were contacted; however, we did not receive any responses.”
Wang, F., Wei, W., & Liu, P. J. (2024) · Frontiers in Endocrinology · doi:10.3389/fendo.2024.1487998 · Cited by 36
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“13 The random-effects model was used to calculate the pooled risk ratio (RR) and mean difference (MD) with the corresponding 95% CI for proportional and continuous variables, respectively.”
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MethodWang, G., Chen, L., Luo, F., Luo, J., & Xu, J. (2024) · Knee Surgery Sports Traumatology Arthroscopy · doi:10.1002/ksa.12093 · Cited by 35
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“For studies where the sample size, median, and range were provided rather than the mean and standard deviation, the approaches used by Luo et al. [16] and Shi et al. [17] were considered to approximate the necessary measurements.”
da Cunha, G. P., Hoogteijling, T. J., Besselink, M. G., Alzoubi, M., Swijnenburg, R., & Hilal, M. A. (2025) · International Journal of Surgery · doi:10.1097/js9.0000000000002567 · Cited by 35
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“The method and its accuracy have been described elsewhere [21, 22].”
Uric acid levels in subjects with schizophrenia: A systematic review and meta-analysis
He, Q., You, Y., Yu, L., Yao, L., Lu, H., Zhou, X., Wu, S., Chen, L., Chen, Y., & Zhao, X. (2020) · Psychiatry Research · doi:10.1016/j.psychres.2020.113305 · Cited by 35
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“Data were calculated by online software based on the method used by McGrath et al, Wan et al, and Luo et al.(31,32,33) All meta-analysis was performed by comprehensive meta-analysis version 2 (CMA2).”
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“Mean SR was estimated from medians in studies where means were not reported.(30,31) Response rate was determined as SR of 50% compared to”
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“Data of continuous variables represented by median and interquartile range (or max-and-min) were converted to mean and standard deviation to perform statistical analysis and data synthesis [22, 23].”
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Systematic review of self-monitoring of blood glucose in patients with type 2 diabetes
Chircop, J., Sheffield, D., & Kotera, Y. (2021) · Nursing Research · doi:10.1097/nnr.0000000000000542 · Cited by 34
“If the median was available but not the mean, the latter was transformed from the former (Luo et al., 2018); SD was estimated from the interquartile range when only this was obtainable (Wan et al., 2014).”
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“Data expressed as order statistics (i.e., median, interquartile range and/or maximum–minimum values) were computed and transformed into means ± standard deviation (SD) using the methods proposed by Luo et al. (2018) and Wan et al. (2014) [18,19].”
MethodShi, H.-X., Li, C., Zhang, Y.-Q., Li, X., Liu, A.-F., Liu, Y.-E., Jiang, W., & Lv, J. (2023) · Journal of Integrative Neuroscience · doi:10.31083/j.jin2202052 · Cited by 34
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de Lange, G., Davies, J., Toso, C., Meurette, G., Ris, F., & Meyer, J. (2023) · Techniques in Coloproctology · doi:10.1007/s10151-023-02853-8 · Cited by 37
“For studies that only provided median and interquartile range, conversion into mean was done using the method proposed by Liu et al. [13] and standard deviation with the method from Wan et al. [14].”
MethodChen, T., Wang, J., Liu, Z., & Gao, F. (2024) · Frontiers in Endocrinology · doi:10.3389/fendo.2023.1282699 · Cited by 37
Liu, Y., Chuang, S., Chen, Y., & Shih, Y. (2024) · Archives of Dermatological Research · doi:10.1007/s00403-024-02994-2 · Cited by 34
“For studies providing NLR, PLR, MLR, SII, or PASI in the form of the median with interquartile or full range, mathematical conversion to the mean with standard deviation was performed [ 27 , 28 ].”
Nguyen, T. P. H., Patrick, C. J., Parry, L. J., & Familari, M. (2019) · PLoS ONE · doi:10.1371/journal.pone.0214671 · Cited by 34
“To pool results in a consistent format, information from these six studies were transformed to sample mean and standard deviation using a mathematical estimator optimised for meta-analyses [49].”
MethodMeta-analysis of vitamin D supplementation in the treatment of chronic heart failure
Wang, T., Liu, Z., Fu, J., & Min, Z. (2019) · Scandinavian Cardiovascular Journal · doi:10.1080/14017431.2019.1612084 · Cited by 34
“Some continuous variables could not be represented as mean±standard deviation; therefore, data were transformed as previously described [18,19].”
MethodSekikawa, A. M., Cui, C., Sugiyama, D., Fabio, A., Harris, W. S., & Zhang, X. (2019) · Nutrients · doi:10.3390/nu11112599 · Cited by 38
PANTALOS, G., Patsouras, D., Spartalis, E. D., Dimitroulis, D., Tsourouflis, G. S., & Nikiteas, N. I. (2019) · In Vivo · doi:10.21873/invivo.11740 · Cited by 34
Li, J., Zhu, W., Gao, X., & Li, X. (2020) · BioMed Research International · doi:10.1155/2020/1709415 · Cited by 34
“When the SD was not available, it was calculated using the provided 95% confidential interval (95% CI) or estimated using the interquartile range [20].”
MethodShen, Q.-H., Li, H., Zhou, X., Yuan, X.-Z., & Lu, Y. (2020) · Journal of International Medical Research · doi:10.1177/0300060520913423 · Cited by 34
“For continuous data that were described as the median (range) in the studies, we converted it to the mean and standard deviation, based on the protocol.(18,19) The mean difference (MD) and 95% CI were calculated for continuous data with the same measure-evaluation methods and units.”
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“If a study reported the median [interquartile range], this was converted to the mean and standard deviation as follows: the mean was estimated according to the method of Luo and colleagues [11] and the standard deviation was calculated according to the method of Wan and colleagues [12].”
MethodLow-level laser therapy for oral mucositis in children with cancer
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“If mean and SD were not readily available, then mean and SD were calculated using median and interquartile range (IQR).(24) Heterogeneity was assessed using the I(2) and Cochran Q statistics.”
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“We extracted final value scores (means and standard deviations) for the meta-analysis and converted change scores into mean values [26, 27].”
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“Conversely, in cases where data were articulated as median interquartile ranges (IQR), the mean and SD were derived following the methodologies proposed by Wan et al. [31] and Luo et al. [32].”
New models for symbolic data analysis
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Kovalic, A. J., Khan, M. A., Malaver, D., Whitson, M. J., Teperman, L. W., Bernstein, D., Singal, A. K., & Satapathy, S. K. (2019) · European Journal of Gastroenterology & Hepatology · doi:10.1097/meg.0000000000001588 · Cited by 33
Xia, W., Luo, C., Li, Q., Hu, N., Xiao, Y., Liu, N., Lui, S., & Gong, Q. (2021) · Frontiers in Aging Neuroscience · doi:10.3389/fnagi.2020.610962 · Cited by 33
“For studies reported with interquartile range and median, we calculated the mean and SD (Wan et al., 2014; Luo et al., 2018; http://www.math.hkbu.edu.hk/tongt/ papers/median2mean.html).”
MethodShe, G., Teng, Q., Li, J., Zheng, X., Chen, L., & Hou, H. (2021) · Frontiers in Surgery · doi:10.3389/fsurg.2021.607743 · Cited by 33
Luo, R., Tong, X., Yan, W., Liu, H., Lei, Y., & Zuo, Y. (2021) · Pediatric Anesthesia · doi:10.1111/pan.14255 · Cited by 33
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“Some of the age data were mathematically processed (23, 24).”
MethodZhang, Q., Zheng, S., Li, S., Zeng, Y., Chen, L., Li, G., Li, S.-L., He, L.-L., Chen, S.-P., Zheng, X., et al. (2023) · Frontiers in Neurology · doi:10.3389/fneur.2023.1074922 · Cited by 24
Fan, Z., Liu, X., Liu, J., Chen, C., & Zhou, M. (2023) · Journal of Integrative Neuroscience · doi:10.31083/j.jin2204085 · Cited by 32
Self-citationRupprecht, M., Wagenpfeil, S., Schöpe, J., Vieth, R., VOGT, T. M., & Reichrath, J. (2023) · Nutrients · doi:10.3390/nu15183986 · Cited by 33
Chen, J.-W., Du, W.-Q., & Zhu, K. (2024) · European Child & Adolescent Psychiatry · doi:10.1007/s00787-024-02507-6 · Cited by 35
“When relevant statistics were not reported in the study, the means and standard deviations were estimated based on sample size, median, range (minimum and maximum), and interquartile range [40, 41].”
Yu, Z., Chen, L., Xu, Q., Yuan, Z., Chen, M., Li, R., Zhou, S., Li, P., Wei, B., & Zhao, X. (2024) · International Journal of Surgery · doi:10.1097/js9.0000000000002056 · Cited by 37
“The method reported by Luo and Wan was used to convert the median and numerical range to mean and SD [20,21] .”
Santo, P. A. D. E., Meine, G. C., Nau, A. L., Barbosa, E. C., Baraldo, S., Lenz, L., & Maluf‐Filho, F. (2024) · Endoscopy · doi:10.1055/a-2368-4608 · Cited by 33
Samavarchitehrani, A., Cannavò, A., Behnoush, A. H., Abadi, A. K., Varniab, Z. S., & Khalaji, A. (2024) · Nutrition and Diabetes · doi:10.1038/s41387-024-00341-y · Cited by 32
“Table 1 illustrates the TyG levels in patients with PAD vs control Three studies assessed TyG levels in patients with PAD and compared them with healthy controls [19, 30, 31].”
ResultSoh, N. H., Yau, C. R. Z., Low, X. Z., Kadir, H. B. A., Fong, W. J., Ramalingam, M. B., Tan, P. L., Ng, K. Y. Y., Hsing, Y. T., Cai, M., et al. (2024) · Annals of Surgical Oncology · doi:10.1245/s10434-024-16527-8 · Cited by 32
“19,20 To explore the impact of uni-versus mutimodal prehabili-tation, an a priori stratified analysis based on a fixed-effects (plural) model, incorporating both random effects within strata and fixed effects were utilized.”
Esposito, G., Mullineris, B., Colli, G., Curia, S., & Piccoli, M. (2025) · Cancers · doi:10.3390/cancers17010150 · Cited by 32
Li, Y.-K., Li, W., Ren, H., Xiao, C., Guo, Z., & Luo, J. (2025) · Inflammopharmacology · doi:10.1007/s10787-025-02044-x · Cited by 32
Human–large language model collaboration in clinical medicine: a systematic review and meta-analysis
Wang, G., Zhang, K., Jiang, J., Wang, C., Bi, H., Liang, H., Qi, Z., Huang, Y., Li, Y., & Yang, X. (2026) · npj Digital Medicine · doi:10.1038/s41746-026-02382-2 · Cited by 48
Carvedilol for prevention of variceal bleeding: a systematic review and meta analysis
Malandris, K. (2019) · Annals of Gastroenterology · doi:10.20524/aog.2019.0368 · Cited by 32
“In studies where the duration of follow up was provided as median (range or interquartile range) rather than mean and standard deviation the latter was calculated as described previously [16,17].”
MethodGonzález‐Moles, M. Á., Warnakulasuriya, S., & Ramos‐García, P. (2021) · Cancers · doi:10.3390/cancers13194843 · Cited by 32
“Data expressed as order statistics (i.e., medians, interquartile, and/or minimum-maximum ranges) were computed and transformed into means ± standard deviations (SD) using the methods proposed by Luo et al. (2018) and Wan et al. (2014) [20,21].”
MethodAmer, H., Kartikasari, A. E. R., & Plebanski, M. (2021) · Journal of Personalized Medicine · doi:10.3390/jpm11121335 · Cited by 34
“When median values were recorded, we estimated the mean values following Cochrane recommendations, with established optimal formulation [32].”
MethodA systematic review and network meta-analysis comparing treatments for anal fissure
Jin, J. Z., Bhat, S., Park, B., Hardy, M.-O., Unasa, H., Mauiliu‐Wallis, M., & Hill, A. G. (2022) · Surgery · doi:10.1016/j.surg.2021.11.030 · Cited by 35
Wang, F., Ma, Y., Xu, S., Liu, H., Chen, Y., Yang, H. J., Shao, M., Xu, W., Kong, J., Chen, L., et al. (2022) · Clinical Rheumatology · doi:10.1007/s10067-022-06087-1 · Cited by 32
“Characteristics about the patient’s age and the duration of the underlying disease are uniformly adjusted by showing in mean with standard deviation (SD) [26, 27].”
BackgroundHang, K., Zhou, L., Liu, H., Huang, Y., Zhang, H., Tan, C., Xiong, J., & Li, K. (2022) · International Journal of Surgery · doi:10.1016/j.ijsu.2022.106686 · Cited by 32
Zhang, K., Wang, L., Zhang, Z.-H., Zhang, D., Lin, X., Meng, T., & Qi, F. (2022) · Journal of Pain Research · doi:10.2147/jpr.s384215 · Cited by 32
Cordray, H., Mahendran, G. N., Tey, C. S., Nemeth, J. K., Sutcliffe, A. G., Ingram, J. C., & Raol, N. (2022) · Acta Paediatrica · doi:10.1111/apa.16609 · Cited by 39
Yang, X., Zhang, Y., Chen, Y., Xu, M., Lei, X., & Fu, Q. (2023) · BMC Anesthesiology · doi:10.1186/s12871-023-01969-6 · Cited by 32
“If continuous variables were expressed as median and range (minimum to maximum or interquartile range), Luo and Wan’s formula was used to estimate the mean and standard deviation [18, 19].”
MethodKhalaji, A., Behnoush, A. H., Saeedian, B., Khanmohammadi, S., Varniab, Z. S., & Peiman, S. (2023) · Diabetology & Metabolic Syndrome · doi:10.1186/s13098-023-01076-z · Cited by 33
Sugai, S., Yamawaki, K., Sekizuka, T., Haino, K., Yoshihara, K., & Nishijima, K. (2023) · American Journal of Obstetrics & Gynecology MFM · doi:10.1016/j.ajogmf.2023.101027 · Cited by 38
Rayan, M., Gruehn, D., & Khayyam, U. (2022) · Sustainability · doi:10.3390/su14137966 · Cited by 35
Tao, Y., Sun, M., Miao, M., Han, Y., Yang, Y., Cong, X., & Zhang, J. (2022) · Frontiers in Surgery · doi:10.3389/fsurg.2022.949614 · Cited by 31
“In some trials, mean and standard deviations (SD) were estimated from the provided median and interquartile range (IQR) (19, 20).”
MethodAlterations in levels of cytokine following treatment to predict outcome of sepsis: A meta-analysis
Li, X., Liu, M., Fu, Y., Jiang, Y., & Zhang, Z. (2022) · Cytokine · doi:10.1016/j.cyto.2022.156056 · Cited by 31
Yang, H., Huang, D., Luo, J. F., Liang, Z., & Li, J. (2023) · Heart & Lung · doi:10.1016/j.hrtlng.2023.02.016 · Cited by 31
Lu, C., Yang, C., Li, X., Du, G., Zhou, X., Luo, W., Du, Q., & Tang, G. (2023) · BMC Oral Health · doi:10.1186/s12903-023-03441-w · Cited by 31
“Data, such as the mean differences in burning pain, quality of life, and anxiety before and after interventions, were converted to the mean ± SD [34].”
MethodHussain, K., Nso, N., Tsourdinis, G. E., Haider, S., Mian, R., Sanagala, T., Erwin, J. P., & Pursnani, A. K. (2023) · Current Problems in Cardiology · doi:10.1016/j.cpcardiol.2023.102146 · Cited by 31
Wang, T., Hong, J., Huang, J.-W., Liao, C.-Y., Lu, C., & Wu, Y. (2024) · Radiotherapy and Oncology · doi:10.1016/j.radonc.2024.110344 · Cited by 35
Yang, R., Zhang, L., Guo, J., Wang, N., Zhang, Q., Qi, Z., Wu, L., Qin, L., & Liu, T. (2025) · Journal of Translational Medicine · doi:10.1186/s12967-025-06302-y · Cited by 35
March, D. S., Jones, A. W., BISHOP, N. C., & Burton, J. O. (2019) · Journal of Renal Nutrition · doi:10.1053/j.jrn.2019.07.006 · Cited by 31
Sanfilippo, F., Rosa, V. L., Grasso, C., Santonocito, C., Minardi, C., Oliveri, F., Iacobelli, R., & Astuto, M. (2020) · Pediatric Critical Care Medicine · doi:10.1097/pcc.0000000000002622 · Cited by 31
“Mean values and sd of echocardiography data were collected; if data were reported as median and interquartile range (or CI), we followed recent evidence to approximate mean and sd (18, 19).”
MethodZawadzka, K., Więckowski, K., Małczak, P., Wysocki, M., Major, P., Pędziwiatr, M., & Pisarska, M. (2021) · European Journal of Endocrinology · doi:10.1530/eje-20-1301 · Cited by 31
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.
What counts as a claim ›How we measure confidence ›How we check this ›
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