Does using multiple imputations for missing data cause the standard deviation to drop?

In a recent analysis, I used 20 sets of multiple imputations to account for missing data. I decided to use the following suggestion to calculate the standard deviations of my variables: https://stats.stackexchange.com/questions/229542/multiple-imputation-pooling-mean-and-standard-deviation.

The standard deviations were really low (lower than 0.00 on a scale of 1 - 7!)

My code format is as follows:

#sadism variable

QW.sad <- array(dim = m)
UW.sad <- array(dim = m)

for (i in 1:m){
  QW.sad[i] <- mean(complete(Imputed.W.add.mids, i)$sad_tot)
  UW.sad[i] <- var(complete(Imputed.W.add.mids, i)$sad_tot)/nrow(DataW.regress.test)
}

B <-  var(QW.sad)
W.sad.mean <- mean(QW.sad) #mean
W.sad.SD <- mean(UW.sad) + (1 + 1/m) * B #standard deviation

Is it simply the case that using multiple imputations causes the standard deviation to drop? (I guess it makes intuitive heuristic sense due to each imputation being similar to each other).

EDIT: I used MICE to impute data

Imputed.W <- mice(DataW.regress.test[1:53], maxit=50, seed = 513, m = 20)

There were 0.2822456 % and 0.3577555 % missing data in my datasets (I show code only for the 0.3577555 % dataset here).

The standard deviation tested (in the sample item provided was for mean values taken from 9 inputs for individual cases.

I attempted to export a dput file but the code is too long to share in this post.