Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range
Open accessAbstract
BACKGROUND: In systematic reviews and meta-analysis, researchers often pool the results of the sample mean and standard deviation from a set of similar clinical trials. A number of the trials, however, reported the study using the median, the minimum and maximum values, and/or the first and third quartiles. Hence, in order to combine results, one may have to estimate the sample mean and standard deviation for such trials. METHODS: In this paper, we propose to improve the existing literature in several directions. First, we show that the sample standard deviation estimation in Hozo et al.'s method (BMC Med Res Methodol 5:13, 2005) has some serious limitations and is always less satisfactory in practice. Inspired by this, we propose a new estimation method by incorporating the sample size. Second, we systematically study the sample mean and standard deviation estimation problem under several other interesting settings where the interquartile range is also available for the trials. RESULTS: We demonstrate the performance of the proposed methods through simulation studies for the three frequently encountered scenarios, respectively. For the first two scenarios, our method greatly improves existing methods and provides a nearly unbiased estimate of the true sample standard deviation for normal data and a slightly biased estimate for skewed data. For the third scenario, our method still performs very well for both normal data and skewed data. Furthermore, we compare the estimators of the sample mean and standard deviation under all three scenarios and present some suggestions on which scenario is preferred in real-world applications. CONCLUSIONS: In this paper, we discuss different approximation methods in the estimation of the sample mean and standard deviation and propose some new estimation methods to improve the existing literature. We conclude our work with a summary table (an Excel spread sheet including all formulas) that serves as a comprehensive guidance for performing meta-analysis in different situations.
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Banach et al. used a conversion for median and interquartile range but this study used an updated method.
“Banach et al. used a conversion for median and interquartile range but our study used an updated method [20,26].”
From Effect of Coenzyme Q10 on statin-associated myalgia and adherence to statin therapy: A systematic review and meta-analysis (2020) · cites this paper as [20]
Missing data were handled by estimating mean and standard deviation from median and interquartile range using conventional formulae, and by prognostic imputation when both were absent.
“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 Wan et al., 2014
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Cites
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“can also estimate the sample standard deviation by S ≈ q3 −q1 2Φ−1 0.75n− 125 n+0.25 . (16) A similar estimator for estimating the standard deviation from IQR is provided in the Cochrane Handbook [13], which is defined as S ≈ q3 −q1 1.35 . (17) Note that the estimator (17) is also independent of the sample size n and t…”
MethodInfluentialEstimating the mean and variance from the median, range, and the size of a sample
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“In contrast, the following procedure is commonly adopted in the recent literature including [11,12]: “If the study provided medians and IQR, we imputed the means and standard deviations as described by Hozo et al.”
MethodEstimating Mean and Standard Deviation from the Sample Size, Three Quartiles, Minimum, and Maximum
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“In this setting, Bland’s method [10] extended Hozo et al.”
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“−∞ ≤ X¯ ≤ ∞. This shows that the approaches based on the inequalities do not apply to Scenario C3. In contrast, the following procedure is commonly adopted in the recent literature including [11] and [12]: “If the study provided medians and IQR, we imputed the means and standard deviations as described by Hozo et al.…”
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“For instance, Antman et al. (1992) pointed out that systematic reviews and meta-analysis of randomized control trials would have led to earlier recognition of the benefits of thrombolytic therapy for myocardial infarction and may save a large number of patients.”
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- Authors
- Wan, X., Wang, W., Liu, J., & Tong, T.
- Affiliations
- Hong Kong Baptist University (HK)
- Northwestern University (US)
- Corresponding author
- Tong, Tiejun
- Venue
- BMC Medical Research Methodology
- Published
- 2014-12
- Volume
- 14
- Issue
- 1
- Pages
- 135-135
- ISSN
- 1471-2288 (ISSN-L)
- PMC
- PMC4383202
- OpenAlex
- W2170892587
- Cited by
- 11,735
- References
- 15
- Citations by year
- 2026: 1931 · 2025: 1918 · 2024: 1806 · 2023: 1600 · 2022: 1448 · 2021: 1273 · 2020: 803 · 2019: 438 · 2018: 248 · 2017: 156 · 2016: 90 · 2015: 12 · 2013: 5
- FWCI
- 83.87
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- BMC Medical Research Methodology (publishedVersion, cc-by)
- Europe PMC (PubMed Central) (publishedVersion, cc-by)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HKBU Institutional Repository (Hong Kong Baptist University) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- Hong Kong Baptist University (FRG2/13-14/062)
- 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 › Probability and Statistical Research
- Keywords
- sample mean estimation, median, range, interquartile range, meta-analysis, skewed data, normal data
- MeSH terms
- Algorithms, Computer Simulation, Humans, Review Literature as Topic, Statistics as Topic, Meta-Analysis as Topic, Reproducibility of Results, Sample Size, Biomedical Research
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