Often the estimation is coupled to a so-called confidence interval (cf. Cumming 2012).  For ease of exposition, assume that \(\Theta\) consists of the real numbers and that every sample \(s\) is labelled with a unique \(\hat{\theta}(s)\). We define the set \(R_{\tau} = \{ s:\: \hat{\theta}(s) = \tau \}\), the set of samples for which the estimator function has the value \(\tau\). We can now collate a region in sample space within which the estimator function \(\hat{\theta}\) is not too far off the mark, i.e., not too far from the true value \(\theta^{\star}\) of the parameter. For example,  \[ C^{\star}_{\Delta} = \{ R_{\tau} :\: \tau \in [ \theta^{\star} - \Delta , \theta^{\star} + \Delta ] \} . \]  So this set is the union of all \(R_{\tau}\) for which \(\tau \in [ \theta^{\star} - \Delta , \theta^{\star} + \Delta ]\). Now we might set this region in such a way that it covers a large portion of the sample space, say \(1 - \alpha\), as measured by the true distribution \(P_{\theta^{\star}}\). We choose \(\Delta\) such that  \[ P_{\theta^{\star}}(C^{\star}_{\Delta}) = \int_{\theta^{\star} - \Delta}^{\theta^{\star} + \Delta} P_{\theta^{\star}}(R_{\tau}) d\tau = 1 - \alpha .\]  Statistical folk lore typically sets \(\alpha\) at a value 5%. Relative to this number, the size of \(\Delta\) says something about the quality of the estimate. If we were to repeat the collection of the sample over and over, we would find the estimator \(\hat{\theta}\) within a range \(\Delta\) of the true value \(\theta^{\star}\) in 95% of all samples. This leads us to define the symmetric 95% confidence interval:  \[ CI_{95} = [ \hat{\theta} - \Delta , \hat{\theta} + \Delta ] \]  The interpretation is the same as in the foregoing: with repeated sampling we find the true value within \(\Delta\) of the estimate in 95% of all samples. 