Median and IQR to mean and SD converter

What this computes

Trials often report a continuous outcome as a median with an interquartile range (the first and third quartiles, Q1 and Q3) rather than a mean and standard deviation. Meta-analysis needs the mean and SD. This calculator applies the Wan et al. (2014) Scenario C3 estimators to recover both from the median, Q1, Q3, and the sample size n.

The formulas

The mean is the simple average of the three quartile-order statistics:

mean ≈ (Q1 + median + Q3) / 3

The SD scales the interquartile range by a sample-size-corrected normal factor. Under an assumed normal distribution the IQR spans 2·Φ⁻¹(0.75) ≈ 1.35 standard deviations; Wan's correction replaces the fixed 0.75 with a term that accounts for the expected position of the sample quartiles at size n:

SD ≈ (Q3 − Q1) / ( 2 · Φ⁻¹( (0.75n − 0.125) / (n + 0.25) ) )

Φ⁻¹ is the inverse standard-normal CDF, evaluated here with the Acklam / Beasley–Springer–Moro rational approximation. As n grows the correction factor converges to the classic IQR / 1.35 rule; at small n it widens the denominator so the SD is not overstated.

When it is valid (and when it is not)

These estimators assume the underlying data are approximately normal. That is the load-bearing assumption, and it is exactly what a median-plus-IQR summary tends to hide. This tool runs a live check: if the median sits asymmetrically inside the IQR (when the distance from Q1 to the median differs from the median to Q3 by more than a quarter of the IQR), it flags likely skew and warns that the normal-based conversion is unreliable. For skewed data, prefer methods that pool medians directly (e.g. the metamedian approach, or Cochrane Handbook §10.5.3) rather than forcing a mean and SD.

A second check guards non-negative outcomes: if you mark the outcome as one that cannot go below zero (a count, a duration, a concentration) and the estimated mean − 2·SD falls below zero, the implied normal distribution would produce impossible values, another sign the data are not near-normal and the SD should be treated with caution.

Citations

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;14:135.