Direct parameter finding of the split normal distribution

## Abstract

A common standard for reporting results in astrophysics (and likely much of science) is flawed. When reporting a distribution we often assume it is distributed as a Gaussian and report the position of the \(16^{th},50^{th}\) and \(84^{th}\) percentile of the data. This is a useful summary statistic - it tells us concrete information about the general form of the distribution. However, it can be misleading if read as a result - something that gives a representative reporting of the distribution.

If a distribution is asymmetric then the three stated percentiles have no direct relevance to the data, and it is not possible to recreate an approximation to the distribution from them. This is of particular concern when one author quotes results from another (particularly if they are actually repurposing summary statistics) to use for meta-data analysis or as priors for their own work.

Asymmetric extensions of the Gaussian normal exist, but are rarely used, because their parameters need to be estimated via costly methods (whilst computing the mean or the median is a direct and efficient calculation). In this paper I show a method for calculating the 3 parameters that define the split normal directly and robustly.

I suggest that this could become the new minimal standard for pragmatic result reporting - encoding a more representative approximate distribution without any extra overhead and allowing authors to be more definite in separating summary statistics and the reporting of usable results.