and calibration studies tend to find performance worsens as more extreme quantiles are elicited [@Alpert:1969; @Lichtenstein:1981]. However, in our experience, experts may find moderate quantiles harder to judge, and so there can be a trade-off between aiming for the best calibration performance, and asking the experts questions that they are comfortable answering. We tend towards the latter, given that a population mean is a more abstract quantity than an individual observable population member. In our example, we elicited the 5th and 95th percentiles of the expert’s distribution of $\mu$.

If we denote the cumulative distribution function of the standard normal distribution by $\Phi$, we have $$  \Phi^{-1} (\alpha_i) \sqrt{v} + m = \mu_{(\alpha_i)}, \\$$ and so we choose values for $m$ and $v$ as follows: $$\begin{aligned}
  \hat{m} &=& \frac{\mu_{(\alpha_1)} \Phi^{-1}(\alpha_2) - \mu_{(\alpha_2)}\Phi^{-1}(\alpha_1)} {\Phi^{-1}(\alpha_2)-\Phi^{-1}(\alpha_1)}, \\
  \hat{v} &=& \left( \frac{\mu_{(\alpha_2)}-\mu_{(\alpha_1)}}{\Phi^{-1}(\alpha_2)-\Phi^{-1}(\alpha_1)} \right)^2.\end{aligned}$$ The facilitator then feeds back different summaries of the fitted distribution, including showing the density function of N($\hat{m}, \hat{v}$). The expert may make adjustments until she is satisfied that the fitted distribution is a reasonable representation of her opinion about $\mu$.

Eliciting judgements about the population variance
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To elicit the expert’s judgements about the population variance, she is asked to 