Bootstrapping, recipe
Let us assume we have done this \( K \) times and thus have \( K \) sets of \( N \)
data values each.
Of course some values will enter more than once in the new sets. For each of these sets one computes the observable \( \theta \) resulting in values \( \theta_k \) with \( k = 1,...,K \). Then one determines
$$
\tilde{\theta} = \frac{1}{K} \sum_{k=1}^K \theta_k,
$$
and
$$
sigma^2_{\tilde{\theta}} = \frac{1}{K} \sum_{k=1}^K \left(\theta_k-\tilde{\theta}\right)^2.
$$
These are estimators for \( \angle\theta\rangle \) and its variance. They are not unbiased and therefore
\( \tilde{\theta}\neq\hat{\theta} \) for finite K.
The difference is called bias and gives an idea on how far away the result may be from
the true \( \angle\theta\rangle \). As final result for the observable one quotes \( \angle\theta\rangle = \tilde{\theta} \pm \sigma_{\tilde{\theta}} \) .