# Questions about unbiased sample variance

I have three related questions:

1. Is it right to say that an unbiased estimate is close to the true value of population parameter?
2. If 1) is true, then does the sample variance after correction become an unbiased estimator?
3. If the corrected sample variance is an unbiased estimator, the shouldn't the corrected sample variance follow the normal distribution? I don't understand why a correct sample variance follow t-distribution even after its bias is removed.
• 1. No; it can have large variance. ... (the premise of 2 is gone) 3. the corrected sample variance is neither normal nor t-distributed. Did you mean to ask something else in 3? Commented Nov 30, 2015 at 0:33

2) I don't see the connection with 1): this is an unrelated question. But yes, if "correction" means using the denominator $n-1$ in place of $n$, then the sample variance becomes an unbiased estimator. But if $n$ is not large, it could be very far off!
3) There is a fog of misunderstandings here. You seem to think that an unbiased estimator must be normally distributed. But there is NO such connection. And, the variance is never $t$-distributed. Appropriately scaled, it might be chi squared-distributed, in the normal case. As the sample variance never can be negative, it cannot have any distribution which assigns positive probability to negative values, and that includes all normal distributions and all $t$-distributions.