# Is it possible to manually calculate standard deviation for a multiply-imputed survey variable based on the standard error (SE)?

I am analyzing a multiply-imputed complex sample survey data using Stata. For normally distributed numerical variables I want to report the mean and standard deviation. However, the Stata command for estimating mean of a multiply-imputed survey data mi estimate: svy: mean [varlist] does give the standard error of the mean, not the standard deviation. I tried to search for valuable help using Google but in vain. My question is this: Under such circumstances, is it possible to obtain an unbiased estimate of the standard deviation using the formula $\sigma$ $=$ $SE$ $\Huge.$ $\sqrt{n}$?

• i know. the answer to your question depends on the microdata. is it one of the ones covered on asdfree.com? – Anthony Damico Oct 15 '14 at 13:24
• Not covered on asdfree.com. I collected my own data from a population of university students using a two-stage stratified cluster sampling technique. – Ayalew A. Oct 15 '14 at 13:35

## 2 Answers

If you really want to get to the standard deviation of the population distribution, you should mi xeq : generate y2 = y*y the squares of the variables, and then

mi estimate (sd : sqrt( _b[y2] - _b[y]*_b[y] ) ) : svy : mean y y2
mi testtransform sd


Note that the interface of multiple imputation and inference with complex survey data is extremely poorly researched into given the ubiquity of the issue. I outlined the literature and the steps elsewhere on statalist.

• Should I do the mi xeq: generate y2 = y*y on M=0 and then follow it by mi updateor just do it on one of the imputations M=1 or M=2 or ... or M=20? – Ayalew A. Oct 16 '14 at 5:37
• mi xeq should do it on all of them at once. – StasK Oct 17 '14 at 3:46

Short answer: NO, you can't do that, because the relationship between the standard error and standard deviation is only valid for simple random samples. For a longer answer look at Stata's FAQ. It does not include the issue with multiple imputation, but it shows how the standard deviation could be computed from survey data.