Estimating the mean and variance from the median, range, and the size of a sample
Open accessAbstract
BACKGROUND: Usually the researchers performing meta-analysis of continuous outcomes from clinical trials need their mean value and the variance (or standard deviation) in order to pool data. However, sometimes the published reports of clinical trials only report the median, range and the size of the trial. METHODS: In this article we use simple and elementary inequalities and approximations in order to estimate the mean and the variance for such trials. Our estimation is distribution-free, i.e., it makes no assumption on the distribution of the underlying data. RESULTS: We found two simple formulas that estimate the mean using the values of the median (m), low and high end of the range (a and b, respectively), and n (the sample size). Using simulations, we show that median can be used to estimate mean when the sample size is larger than 25. For smaller samples our new formula, devised in this paper, should be used. We also estimated the variance of an unknown sample using the median, low and high end of the range, and the sample size. Our estimate is performing as the best estimate in our simulations for very small samples (n < or = 15). For moderately sized samples (15 < n < or = 70), our simulations show that the formula range/4 is the best estimator for the standard deviation (variance). For large samples (n > 70), the formula range/6 gives the best estimator for the standard deviation (variance). We also include an illustrative example of the potential value of our method using reports from the Cochrane review on the role of erythropoietin in anemia due to malignancy. CONCLUSION: Using these formulas, we hope to help meta-analysts use clinical trials in their analysis even when not all of the information is available and/or reported.
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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 [26]
The Background and Results describe a method for estimating the mean and variance from the median, range, and sample size. They do not establish what conversion Banach et al. used or identify the citing study’s method as an update; the cited method uses a range, not an interquartile range. (LLM, conf=0.82)
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- Authors
- Hozo, S. P., Djulbegovic, B., & Hozo, I.
- Affiliations
- Indiana University Northwest (US)
- Corresponding author
- Hozo, Iztok
- ORCID
- Hozo, Iztok
- Venue
- BMC Medical Research Methodology
- Published
- 2005-04-20
- Volume
- 5
- Issue
- 1
- Pages
- 13-13
- ISSN
- 1471-2288 (ISSN-L)
- PMC
- PMC1097734
- OpenAlex
- W1537972189
- Cited by
- 9,062
- References
- 14
- Citations by year
- 2026: 531 · 2025: 725 · 2024: 832 · 2023: 866 · 2022: 864 · 2021: 1046 · 2020: 940 · 2019: 638 · 2018: 559 · 2017: 478 · 2016: 405 · 2015: 398 · 2014: 276 · 2013: 181 · 2012: 127
- FWCI
- 13.40
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Locations
- BMC Medical Research Methodology (publishedVersion, cc-by)
- Europe PMC (PubMed Central) (publishedVersion, cc-by)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- 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
- mean estimation, variance estimation, median, range, sample size, distribution-free estimation, standard deviation estimation, small sample size, erythropoietin
- MeSH terms
- Analysis of Variance, Clinical Trials as Topic, Computer Simulation, Humans, Meta-Analysis as Topic, Models, Statistical, Sample Size
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.
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