confused about unbiasedness of sample mean Recently, in a different thread, I was convinced by others ( after claiming that they were wrong ) that the sample mean is unbiased for the mean of its underlying distribution. But the case of the chi-squared confuses me. Suppose one has a sample of $n$ chi-squared random variables.
Then $E(X_{i}) = 1$. So, $E(\bar{X}) = \frac{n}{n} = 1$.
Now the sample has a chi-squared($n$) distribution so it's expectation is $n$. So, how can the sample mean whose expectation equals $1$ be unbiased for $n$ ?
Should the statement be re-stated and stated as the "the sample mean is always unbiased for the mean of one observation of the random variable ".

Hold on here
If $X \sim \chi^2_{k}$  then $E(X) = k$.  This means that
$$ E(\bar{X}) = E \left( \dfrac{1}{n} \sum_i^n X_i \right) = \dfrac{1}{n} \sum_i^n E (X_i) = \dfrac{1}{n} nk = k $$
Here, I've used linearity of the expectation.  I think you've just confused $n$ as the degrees of freedom and $n$ as the sample size.
