When summarizing a one dimensional continuous distribution (e.g. a posterior distribution) it is common to use either an equal tailed interval (aka quantile-based) or a highest density interval. The 95% equal tailed interval conceptually corresponds to the median in that when the coverage of the interval $\rightarrow 0\%$ the interval converges to the median. In the same way the highest density interval corresponds to the mode as the mode is the point of the highest density. But another popular point summary of a continuous distribution is the mean and my question is:
What is the interval that corresponds to the mean?
- What is that interval called?
- How is it defined/calculated?
If someone has a comment on the conundrum why it is the case that the mean is a really popular way of summarizing a distribution while the corresponding interval is not as popular (as is my impression).