Converting a standard error to a standard deviation for meta-analysis

It’s 11pm, the trial you’re extracting reports “mean difference 0.7 (95% CI −0.4 to 1.8)”, your meta-analysis needs a standard deviation, and the paper never states one. This is one of the most common shapes in the RCT literature. It’s also one of the most common places extraction errors are born.

The recovery, step by step

The Cochrane Handbook (§6.5.2.3) gives the standard route. First, recover the standard error of the mean difference from the confidence interval:

SE = (CI_upper − CI_lower) / (2 × z)

where z is the normal quantile for the CI level (1.96 for a 95% CI). For our example:

SE = (1.8 − (−0.4)) / (2 × 1.96) = 2.2 / 3.92 = 0.5612

Then convert the SE of the difference to a pooled SD using the two arm sizes:

SD = SE / √(1/n₁ + 1/n₂)

With n₁ = 24 and n₂ = 24:

SD = 0.5612 / √(1/24 + 1/24) = 0.5612 / 0.2887 = 1.944

That SD is what your effect-size computation (and your software’s input sheet) actually wants.

The two mistakes that flip results

Mistake 1: treating the SE as the SD. They differ by a factor of √(1/n₁ + 1/n₂); at n=24 per arm that’s a ~3.5× error in the SD, which inflates your standardized effect by the same factor. This single confusion is a recurring finding in published meta-analysis error audits.

Mistake 2: using 1.96 for every CI. Trials report 90% CIs (z = 1.645) and 99% CIs (z = 2.576) more often than you’d think, especially in older cardiology and nutrition literature. Read the table header, not just the parentheses.

If the trial is a crossover

Per-condition means with SDs (rather than paired differences) need the paired route instead, from Handbook §23.2.7, because the within-person correlation changes the variance. Don’t apply the parallel-arm formula to crossover conditions; it overstates the variance and washes out the effect.

The audit-trail habit

Whatever tool or spreadsheet you use: record that the SD was derived, from what, and by which formula. A derived SD presented as a reported one is exactly the kind of silent step a statistical reviewer will ask about two years later. “We recovered it via Handbook §6.5.2.3 from the 95% CI with n₁=n₂=24” is a complete answer.