# Degrees of freedom for estimation

In the context of estimators, why is it that in general dividing by the degrees of freedom(instead of the sample size) leads to unbiasedness? I see the value in substituting degrees of freedom for sample size for an unbiased estimator in small samples but I don't get why it works.

• I am not sure the rule is general. I think it works for normally distributed variables, or at least variables that can be approximated to be normally distributed. Something similar may also work for Poisson variables (since a sum of Poisson variables is also Poisson distributed). In general, your estimator may not be a sum of underlying data points (or their squares). ...
– Cryo
Commented Apr 1 at 2:45
• Also, in general, a sum of random iid variables may not have the same distribution as the underlying variables (for example, sum of beta-distributed variables, I think, may not be beta-distributed). Perhaps it is a good idea to start with something simple & concrete. Are you confident with why it works for normally distributed variables?
– Cryo
Commented Apr 1 at 2:50
• No, I actually never thought about this in the context of distributions. By estimators, I mean things like the sample variance. With the sample variance, you divide by (n-1) since you lose a degree of freedom when calculating the mean. But does this hold for any estimator in general? Commented Apr 1 at 17:22
• for starters, variance is a natural parameter for normal and poisson distribution, but not, for example for beta or gamma distributions. You can define variance for all distributions, and it will follow the rule of dividing for $n-1$ for multiple samples, but i would only care about variance if the underlying distribution accepts it as a parameter, or at least something close to a parameter. For beta distribution, or Dirichlet, I wouldn't care
– Cryo
Commented Apr 2 at 20:13
• Coming back to your original question, I would suggest looking at the inference theory for normal distributions. You will get answers there. These would not be general, but you don’t even need such estimators in general - only in cases if suitable diatributions
– Cryo
Commented Apr 2 at 20:19