and 95th percentiles of the population proportion, which we denote by $\theta_{(0.05)}$ and $\theta_{(0.95)}$ respectively, and these are described to the expert as approximate lower and upper limits.

It may help the expert to explicitly consider the two intervals $[\hat{m}, \hat{m}+c]$ and $(\hat{m}+c, U]$ and consider how the population is distributed between these two intervals. For example, for an appropriate choice of $c$, she might judge a split of 2%-48% across the two intervals highly unlikely, which can help prompt judgements of more plausible allocations (though the facilitator should remain cautious of anchoring effects). If the expert prefers to report judgements on the 0-1 scale rather than the 0-0.5 scale, she could instead be asked to consider the proportion in $[\hat{m}, \hat{m}+c]$, out of those known to be greater than $\hat{m}$.

The facilitator uses the expert’s judgements $\theta_{(0.05)}$ and $\theta_{(0.95)}$ to infer the corresponding quantiles of her distribution for the variance, ${\sigma}^2_{(0.95)}$ and ${\sigma}^2_{(0.05)}$. It is straightforward to show that $${\sigma}^2_{(\alpha)} = \left( \frac{c}{\Phi^{-1} \left( \theta_{(1-\alpha)} + \frac{1}{2} \right)}\right)^2,\label{equ:sigma-quantile}$$ for $\alpha\in(0,1)$.

The facilitator chooses $a$ and $b$ to be $$\label{equ:fitting-sigma}
    \hat{a}, \hat{b} = \underset{a, b}{\operatorname{arg-min}} \left[ \left\{F_{\sigma^2}\left(\hat{\sigma}^2_{(0.05)}; a , b \right) - 0.05\right\}^2 + \left\{F_{\sigma^2}\left(\hat{\sigma}^2_{(0.95)}; a , b\right) - 0.95\right\}^2 