How this calculator works
For independent, identically distributed observations with finite variance, the variance of their average is the population variance divided by sample size. Taking the square root gives sigma divided by sqrt(n). This spread relation does not require normally distributed observations; normal approximations for probabilities need additional conditions. The sample-size mode rounds upward to achieve a target spread at most as large as requested. No finite-population correction or correlation adjustment is applied.
Formula
SD(mean) = sigma/sqrt(n); sigma = SD(mean) x sqrt(n); n_required = [sigma/target]². Planned n = max(1, ceiling(n_required)).
Worked example
A population SD of 12 with 36 observations gives SD(mean) = 12/6 = 2. For sigma 10 and a target of 3, n_required is 100/9, so use 12 observations; achieved SD is about 2.88675.
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Frequently asked questions
Is this the same as an estimated standard error?
It uses a known population SD. If you estimate SD from the observed sample, use the standard error page.
Does quadrupling the sample halve the spread?
Yes, under independence and the same population variance. The square root of four is two.
What if observations are correlated?
The simple expression can understate or overstate spread. Use an effective sample size or a model of the covariance.