Why should we use 16-50-84 percentile instead of $\mu$ and $\sigma$ to characterize sampling results such as MCMC Can someone explain why we should use the 16-50-84 percentile rather than mean and standard deviation to characterize the average and uncertainties of the sampling results, such as MCMC (the Markov Chain Monte Carlo sampling)?
I look up the source code of a python package corner and I find they are using the 16-50-84 percentile. Can someone explain the difference between the 16-50-84 percentile and mean/standard deviation and when we should use which?

When you have a normal distribution, the following are true.

*

*The mean $\mu$ is at the $50^{\text{th}}$ percentile.


*Within one standard deviation $\sigma$ of the mean, $68\%$ of the data are contained.


*The distribution is symmetric about the mean.
This means that $34\%$ of the density is above $\mu$ but below $\mu+\sigma$, and $34\%$ of the density is below $\mu$ but above $\mu-\sigma$.
$$
50+34 = 84\\
50-34 = 16
$$
Those numbers$-$$16$, $50$, and $84$$-$look familiar, don't they?
In the normal distribution, $\mu-\sigma$ corresponds to the $16^{\text{th}}$ percentile, and $\mu+\sigma$ corresponds to the $84^{\text{th}}$ percentile.
In other words, when you have a normal distribution from your MCMC, you get $\mu-\sigma$, $\mu$, and $\mu+\sigma$ by looking at those those three percentiles, $16$, $50$, and $84$, respectively. When you don't have a normal distribution, you still get the same quantiles with which you are familiar from the normal distribution.
If you do not have a normal distribution, then $\mu-\sigma$, $\mu$, and $\mu+\sigma$ might correspond to percentiles other than $16$, $50$, and $84$, respectively.
