Sample mean variability

01For n IID observations with finite individual variance v, the average has variance v/n#

For $n$ IID observations with finite individual variance $v$, the average has variance $v/n$. Here $n$ is a positive integer. The formula concerns variation across fresh samples.

The sum of $n$ independent observations has variance $nv$. Dividing that sum by $n$ multiplies its variance by $1/n^2$.

With four observations of variance $4$, the sum variance is $16$. Averaging multiplies it by $1/16$, leaving variance $1$.

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02If an individual standard deviation is supplied, square it to obtain v#

If an individual standard deviation is supplied, square it to obtain $v$. For standard deviation $5$ and $n=5$, mean variance is $25/5=5$. Variance uses squared measurement units.

03At n=1, the variance stays v#

At $n=1$, the variance stays $v$. At $v=0$, it stays zero for every positive $n$. Copies of a nonconstant random variable are dependent; the IID shortcut is unjustified.