Estimate mean and standard deviation from median, interquartile range, and/or range using the validated methods by Wan et al. (2014), Luo et al. (2018), and Hozo et al. (2005). Essential for including non-parametric study reports in your meta-analysis.
Enter the median, minimum, maximum, and sample size as reported in the paper.
Select the tab matching the data your paper reports: median + range, median + IQR, or all five summary statistics.
Input the median, quartiles or range bounds, and the sample size exactly as reported in the source paper.
The tool shows the estimated mean, estimated SD, the method used, and key assumptions for your reporting.
Copy the results to your extraction form and cite the appropriate method reference in your review.
The Wan, Luo, and Hozo methods assume the underlying data are roughly normally distributed. If the original authors chose to report medians because of severe skewness, the estimates may be biased. Always check whether the median is approximately centered between Q1 and Q3.
When you have median, IQR, and range, Scenario 3 (Luo method) gives the most accurate estimate. If only IQR is available, use Scenario 2 (Wan). If only the range is available, use Scenario 1. Never discard available information.
In the Methods section of your review, state clearly which conversion method was applied and for which studies. This transparency is essential for reproducibility and is required by PRISMA reporting guidelines.
After including converted estimates in your meta-analysis, run a sensitivity analysis excluding the converted studies. If results change meaningfully, report both analyses and discuss the implications for the certainty of evidence.
A median to mean converter solves one of the most common data extraction challenges in quantitative evidence synthesis. Many primary studies — particularly those measuring skewed outcomes such as hospital length of stay, biomarker concentrations, or cost data — report medians and interquartile ranges rather than means and standard deviations. Because standard meta-analytic pooling methods require means and SDs, the Cochrane Handbook (Higgins et al., 2023, Chapter 6) recommends using validated estimation techniques to convert summary statistics when direct computation is not possible. Hozo et al. (2005) provided the earliest widely adopted formulas for estimating the mean from the median and range, establishing the foundation upon which subsequent methods have built. These conversions enable reviewers to include studies that would otherwise be excluded from quantitative synthesis, strengthening the comprehensiveness demanded by PRISMA 2020 (Page et al., 2021). Clinical data frequently follow a log-normal distribution—common for biomarker concentrations, hospital costs, and length of stay—in which case a Box–Cox transformation can stabilize variance before estimation, and Cochrane RevMan requires the resulting mean and SD in its standard input format for inverse-variance pooling.
The IQR to standard deviation calculator in this tool implements three validated approaches that differ in their data requirements and statistical precision. The Wan method calculator (Wan et al., 2014) improved upon Hozo's formulas by incorporating sample size into the SD estimation through normal quantile functions, producing more accurate results especially for samples below 50. The Luo method estimator (Luo et al., 2018) further refined the mean estimation by applying an optimal weighted combination of range-based and IQR-based estimates, giving greater weight to the IQR component as sample sizes increase. McGrath et al. (2020) conducted a comprehensive simulation study comparing these methods and confirmed that the Luo-Wan combination performs reliably under approximate normality, with mean estimates typically within 5% of the true value. Beyond these three approaches, quantile estimation methods such as those proposed by Shi et al. (2020) extend the framework to accommodate additional percentile combinations that some primary studies report. After estimating the SD, you may need to interconvert between standard deviation and standard error for your pooling software — our SE/SD converter handles that step using the sample size.
All three estimation methods assume the underlying data follow an approximately symmetric distribution. When this assumption is violated — indicated by a median that sits much closer to Q1 than Q3, or by a range that is disproportionately wide relative to the IQR — the estimated mean may diverge meaningfully from the true population mean. The Cochrane Handbook advises reviewers to conduct sensitivity analyses excluding converted studies to assess the robustness of pooled results. Conducting a sensitivity analysis that compares estimates from different methods (Wan vs. Luo vs. Hozo) helps reviewers gauge how sensitive pooled results are to the choice of estimation approach. With your estimated means, SDs, and sample sizes in hand, you can compute standardized mean differences using our effect size calculator, which derives Cohen's d, Hedges' g, and their confidence intervals from group-level summary statistics. For studies that report p-values but omit confidence intervals, our p-value to confidence interval converter provides a complementary back-calculation pathway.
Transparent reporting of statistical conversions is essential for reproducibility. PRISMA 2020 requires reviewers to document how effect estimates and their precision measures were obtained, including any derivations from non-standard summary statistics. Reviewers should cite the specific estimation method used (Wan et al., 2014; Luo et al., 2018; or Hozo et al., 2005) and identify which studies required conversion in both the Methods section and a supplementary table. To organize all extracted and converted data systematically, our extraction template builder generates structured forms with dedicated fields for original summary statistics, estimated means and SDs, and the conversion method applied. Once your synthesis is complete, present the results visually with our forest plot generator, which creates publication-ready diagrams showing each study's weight, confidence interval, and the pooled diamond summary estimate.
In meta-analysis, most pooling methods require means and standard deviations. However, some primary studies -- especially those with skewed outcomes -- report medians, interquartile ranges, or ranges instead. To include these studies in your quantitative synthesis, you need to estimate the mean and SD from the reported summary statistics. This is common in medical and clinical research where outcomes like length of stay or biomarker levels are often right-skewed.
Hozo et al. (2005) provided simple formulas for estimating mean and SD from median and range, but these can be inaccurate for small samples. Wan et al. (2014) improved on Hozo by incorporating sample size into the SD estimation using normal quantile functions. Luo et al. (2018) further refined the mean estimation when both IQR and range are available, using a weighted combination that gives more weight to the IQR-based estimate for larger samples.
Under the assumption of normality, Wan and Luo methods perform well, especially for sample sizes above 15. Simulation studies show mean estimates are typically within 5% of the true mean. SD estimates are less precise but generally within 10-15% for moderate sample sizes. Accuracy decreases substantially when the underlying distribution is heavily skewed. Always note in your review that these are estimated values.
All methods assume the underlying data follow an approximately normal (symmetric) distribution. This is a critical assumption: if the data are heavily skewed, the median and mean diverge substantially, and using the estimated mean may be misleading. Some reviewers recommend sensitivity analyses excluding studies where data appear highly skewed (e.g., when the median is much closer to Q1 than Q3, or when the range is extremely wide relative to the IQR).
It depends on which scenario you use. Scenario 1 requires the median, minimum, maximum, and sample size. Scenario 2 requires the median, Q1 (25th percentile), Q3 (75th percentile), and sample size. Scenario 3 (the most accurate) uses all five summary statistics plus the sample size. Use whichever scenario matches the data reported in your source paper.
Cite the specific method used: Wan X et al. BMC Med Res Methodol. 2014;14:135 for the Wan method; Luo D et al. Stat Methods Med Res. 2018;27(6):1785-1805 for the Luo method; or Hozo SP et al. BMC Med Res Methodol. 2005;5:13 for the Hozo method. State in your Methods section that means and SDs were estimated from medians and IQR/ranges, and identify which studies required this conversion.
Yes, but you must first estimate the mean and SD. Methods by Wan et al. (2014), Luo et al. (2018), and Hozo et al. (2005) provide validated formulas for this conversion. The Cochrane Handbook (Chapter 6) endorses these approaches. However, median reporting often signals skewed data, so consider a sensitivity analysis excluding these converted estimates to test robustness.
Wan et al. (2014) generally produces the most accurate estimates, especially for the SD, because it incorporates sample size into the formula. Luo et al. (2018) refines the mean estimation further. Hozo et al. (2005) was the earliest method and tends to overestimate SD for small samples. McGrath et al. (2020) recommend Wan/Luo for most scenarios. If possible, compare all three methods as a sensitivity check.
The conversion formulas assume approximate normality. When data are substantially skewed (indicated by the median being far from the midpoint of Q1 and Q3), the estimated mean may be inaccurate. Options include: analyzing on a transformed scale (e.g., log), using the median directly in a non-parametric meta-analysis, or conducting a sensitivity analysis excluding skewed studies.
Our team handles complex data extraction, statistical conversions, and quantitative synthesis so you can focus on interpreting the clinical evidence.