Sample variance¶
We define the sample variance as: $\( \hat{\sigma}^2_m = \frac{1}{m} \sum_{i=1}^m (x^{(i)} - \hat{\mu}_m)^2 \)$
Bias¶
The sample variance is a biased estimator
\[
\text{bias}(\hat{\delta}^2_m) = - \delta^2/m
\]
\(m\) is the number of samples
\(\delta^2\) is the true variance
If we want an unbiased estimator for variance we have:
\[
\tilde{\delta}^2 = \frac{1}{m-1} \sum_{i=1}^m (x^{(i)} - \hat{\mu}_m)^2
\]