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?

Who told you that? First, there's no particular reason why those percentiles are more useful than other percentiles. As noticed by other answers, mean and standard deviation easily translate to percentiles for normal distribution, but not necessarily for other distributions. Depending on what characteristics of the distribution are important for your particular use case, you can use any descriptive statistics that are useful for that case. In some cases, percentiles may be easier to interpret (e.g. think of exponential distribution that has a mean close to zero and is skewed).
