Independent random samples

The formula for the standard deviation of a sample mean,

 = 

is only accurate if the sample values are independent.

Dependent random samples

When sample values are correlated with each other, they are said to be dependent and the formula

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

Always check that a random sample is independently selected from the whole population before using the formula for the standard deviation of the sample mean.