# Degrees of freedom for standard deviation of sample

would someone please explain why the degrees of freedom for a random sample is n-1 instead of n ?

I'm looking for an explanation that is intuitive and easily understood by a high school student.

• I'd suggest avoiding talking about bias in the standard deviation as a motivation; the $n-1$ denominator is related to unbiasing the variance. Consequently - by the Jensen inequality - that guarantees bias in the standard deviation. – Glen_b May 6 '13 at 3:31
• Interesting though, dividing by $n$ gives a smaller mean square error $E [(s^2 - \sigma^2) ^2]$, and dividing by $n+1$ smaller still! Also, you don't get an unbiased estimate for $\sigma$ by dividing by $n-1$ (which is what $s^2$ is used for more likely) – probabilityislogic May 24 '16 at 10:30