IQR to standard deviation converter

What this computes

Many trials report a median with an interquartile range (IQR) instead of a mean with a standard deviation (SD). To pool such a study in a meta-analysis you usually need the SD. This converter recovers it from the quartiles under the assumption that the underlying data are approximately normally distributed.

The formula

The IQR spans the middle 50% of the data. For a normal distribution the distance between the first and third quartiles is a fixed multiple of the SD, so the SD is the IQR divided by that multiple:

SD ≈ IQR / ξ(n)

ξ(∞) = 2·Φ⁻¹(0.75) = 1.349        (large-n rule)
ξ(n) = 2·Φ⁻¹((0.75n − 0.125)/(n + 0.25))   (Wan 2014, finite n)

Here IQR = Q3 − Q1 and Φ⁻¹ is the inverse standard-normal CDF (the probit). With no sample size the tool uses the classic IQR / 1.349; supply n and it uses the Wan et al. 2014 small-sample denominator, which is more accurate for the modest sample sizes typical of clinical trials.

When it is valid

The conversion is only as good as the normality assumption. It works well for symmetric, roughly bell-shaped outcomes (many continuous biomarkers, scores and lab values). The Wan correction additionally assumes the reported quartiles are the ordinary sample quartiles.

When NOT to use it

Skewed outcomes (lengths of stay, costs, counts, biomarkers with a floor at zero) violate the normal assumption, and the recovered SD will be wrong (usually understated). Authors report a median precisely because the data are skewed, so treat any converted SD with suspicion and check the median/IQR against the range for symmetry first. For heavily skewed data, prefer analysing on a transformed scale or extracting the SD directly. This page is a fast recovery, not a licence to ignore the distribution.