Standard deviation of mean from independent random samples

The formula for the standard deviation of a sample mean,

 = 

is only accurate if the sample is collected in such a way that each successive value is unaffected by other values that have already been collected. This is called an independent random sample and is implied when the term random sample is used.

It can badly underestimate the variability (and hence accuracy) of the sample mean of dependent random samples.

Sampling incomes