Estimate mean and SD from median and quartiles or range
For a meta-analysis: estimate the mean and standard deviation of a study that reported only the median with quartiles or with the range.
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- Estimated mean
- Estimated standard deviation
How it is calculated
The formulas of Wan and colleagues (standard deviation) and Luo and colleagues (mean) are used. Both were derived for roughly normal data.
The result is an estimate for a meta-analysis, not a replacement for the raw data; say so in your methods.
Mean ≈ (0.7 + 0.39/n) × (Q1 + Q3)/2 + (0.3 − 0.39/n) × medianSD ≈ (Q3 − Q1) / [2 × Φ⁻¹((0.75n − 0.125)/(n + 0.25))]Range: mean ≈ 4/(4 + n^0.75) × (min + max)/2 + n^0.75/(4 + n^0.75) × median; SD ≈ (max − min) / [2 × Φ⁻¹((n − 0.375)/(n + 0.25))]
Worked example: Q1 10, median 15, Q3 20, n = 100: estimated mean 15.0, estimated SD 7.52.
Sources
- Wan X, Wang W, Liu J, Tong T. Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC Med Res Methodol. 2014.
- Luo D, Wan X, Liu J, Tong T. Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range. Stat Methods Med Res. 2018.
These calculators support research; they do not replace the advice of a statistician or a clinician.
Questions
How reliable is the estimate?
It is an estimate, and it assumes the data are roughly normal. For clearly skewed data, which is why medians were reported, it can be misleading; the tool warns when the quartiles are uneven.