SEM to SD converter (and back)

What this does

Trial tables report a group's spread as a standard error of the mean (SEM) about as often as a standard deviation (SD), but meta-analysis software (RevMan, metafor, the standardized-mean-difference formulas) wants the SD. The two differ by a single factor of the square root of the sample size, so the conversion is exact once you know n:

SD  = SEM × √n
SEM = SD  ÷ √n

Why the √n

The SEM is not the spread of the observations. It is the spread of the sample mean, the standard deviation of the sampling distribution. By definition SEM = SD / √n: it shrinks as the group grows, because a mean of many people is estimated more precisely than any one person's value. To get back to the observation-level SD that effect-size formulas assume, you multiply the reported SEM by √n. This is the identity given in the Cochrane Handbook §6.5.2.2.

When it's valid

Use it whenever n is the number of participants the statistic summarizes and the value really is that group's SEM (or SD). The same n applies on both sides: the SEM of a 50-person arm converts with n = 50, not with the pooled total across arms. For continuous outcomes measured on each participant, the conversion is exact, not an approximation.

When NOT to use it

Don't apply it to a confidence-interval half-width: a 95% CI is mean ± t × SEM, so recovering the SD from a CI needs the t (or z) multiplier first, then this √n step. That is a different calculator. Don't reuse one group's SD across a subgroup with a different n. And watch the header trap: a table that reads mean (SEM) versus mean (SD) is the single most common extraction mixup, and it is invisible in the numbers. Mislabeling one as the other puts every downstream effect size off by exactly √n. The values can't tell you which statistic you have; only the header (or the paper's methods) can. When in doubt, check the header, don't guess.