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Cited by 11,735Meta-analysis and systematic reviewsStatistical Methods in EpidemiologyProbability and Statistical Research

The DOI matches its Crossref record; 15 of the 15 references it cites were found in a registry or an index.

Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range

Open access
Wan, Xiang, Wang, Wenqian, Liu, Jiming, Tong, Tiejun·2014·BMC Medical Research Methodology
DOI 10.1186/1471-2288-14-135PMC PMC4383202
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Abstract

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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Crossref · 10.1186/1471-2288-14-135

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Figures

Figure 1 Relative errors of the sample standard deviation estimation for normal data, where the red lines with solid circles represent Hozo et al.’s method, and the green lines with empty circles represent the new method.
Figure 1 Relative errors of the sample standard deviation estimation for normal data, where the red lines with solid circles represent Hozo et al.’s method, and the green lines with empty circles represent the new method.CC BY · PubMed Central
Figure 2 Relative errors of the sample standard deviation estimation for non-normal data (log-normal, beta, exponential and Weibull), where the red lines with solid circles represent Hozo et al.’s method, and the green lines with empty circles represent the new method.
Figure 2 Relative errors of the sample standard deviation estimation for non-normal data (log-normal, beta, exponential and Weibull), where the red lines with solid circles represent Hozo et al.’s method, and the green lines with empty circles represent the new method.CC BY · PubMed Central
Figure 3 Relative errors of the sample standard deviation estimation for normal data and log-normal data, where the red lines with solid circles represent Bland’s method, and the green lines with empty circles represent the new method.
Figure 3 Relative errors of the sample standard deviation estimation for normal data and log-normal data, where the red lines with solid circles represent Bland’s method, and the green lines with empty circles represent the new method.CC BY · PubMed Central
Figure 4 Relative errors of the sample mean and standard deviation estimations for normal data, where the black solid circles represent the method under scenario , the red solid triangles represent the method under scenario , and the green empty circles represent the method under scenario .
Figure 4 Relative errors of the sample mean and standard deviation estimations for normal data, where the black solid circles represent the method under scenario , the red solid triangles represent the method under scenario , and the green empty circles represent the method under scenario .CC BY · PubMed Central
Figure 5 Relative errors of the sample mean estimation for non-normal data (log-normal, beta, exponential and Weibull), where the black lines with solid circles represent the method under scenario , the red lines with solid triangles represent the method under scenario , and the green lines with empty circles represent the method under scenario .
Figure 5 Relative errors of the sample mean estimation for non-normal data (log-normal, beta, exponential and Weibull), where the black lines with solid circles represent the method under scenario , the red lines with solid triangles represent the method under scenario , and the green lines with empty circles represent the method under scenario .CC BY · PubMed Central
Figure 6 Relative errors of the sample standard deviation estimation for non-normal data (log-normal, beta, exponential and Weibull), where the black lines with solid circles represent the method under scenario , the red lines with solid triangles represent the method under scenario , and the green lines with empty circles represent the method under scenario .
Figure 6 Relative errors of the sample standard deviation estimation for non-normal data (log-normal, beta, exponential and Weibull), where the black lines with solid circles represent the method under scenario , the red lines with solid triangles represent the method under scenario , and the green lines with empty circles represent the method under scenario .CC BY · PubMed Central

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Research works printed but not linked: 1

  1. Triola M. F: Elementary Statistics, 11th Ed. 2009, Addison Wesley

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Metadata

Authors
Wan, X., Wang, W., Liu, J., & Tong, T.
Affiliations
  • Hong Kong Baptist University (HK)
  • Northwestern University (US)
Corresponding author
Tong, Tiejun
ORCID
Venue
BMC Medical Research Methodology
Publisher
Springer Science and Business Media LLC
Published
2014-12
Volume
14
Issue
1
Pages
135-135
ISSN
1471-2288 (ISSN-L)
DOI
10.1186/1471-2288-14-135
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
FWCI
83.87
Type
journal article
Language
en
Open access
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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