Asymmetric bar errors Generally, when gaussian or normal distribution is assumed, we can calculate $\sigma$, the standard deviation of the population and $s$, the SD of the sample. When we have a non-symmetric probability distribution, how are asymmetric $\sigma_L$ and $\sigma_R$ calculated? In what cases are preferred and how can we understand precision with asymmetric errors? And accuracy?
 A: Given the probability density $p(x)$, with average value $\overline{x} = \int_{X} x p(x) dx$ (that can be interpreted as the center of mass, with a mechanical analogy), you can split the variance
$\sigma^2 = \displaystyle \int_X (x - \overline{x})^2 p(x) dx$
as the sum of two integrals over two domains $X^-:x \le \overline{x}$, and $X^+:x \ge \overline{x}$
$\sigma^2 = \displaystyle \int_X (x - \overline{x})^2 p(x) dx = \displaystyle \int_{x_1}^{\overline{x}} (x - \overline{x})^2 p(x) dx + \displaystyle \int_{\overline{x}}^{x_2} (x - \overline{x})^2 p(x) dx = {\sigma_-}^2 + {\sigma_+}^2$.
Choosing the probability of the confidence value $f$ (the probability of finding the variable in the desired interval), you can find the lower and upper limits of the confidence variable distributing the probability in a symmetric way around $\overline{x}$,
$\dfrac{f}{2} = \displaystyle \int_{x_L}^{\overline{x}} p(x) dx$$\quad , \qquad$$\dfrac{f}{2} = \displaystyle \int_{\overline{x}}^{x_U} p(x) dx$
providing you the values of $x_L$ and $x_U$, that you can use to find the contributions to the variance
$\sigma_L^2 = \int_{x_L}^{\overline{x}} (x-\overline{x})^2 p(x) dx$$\quad , \qquad$
$\sigma_U^2 = \int_{\overline{x}}^{x_U} (x-\overline{x})^2 p(x) dx$.
For non-symmetric probability density, I'd use the average value $\overline{x}$, indicating the limits $x_L$, $x_U$ of the desired interval of confidence centered in the average value, stating the probability $f$ of finding measurements in that interval, without using $\sigma_L$, $\sigma_U$ if not explicitly required, since the information $x_L$, $x_U$ provides information that is easier to understand
A: Asymmetric error bars are common in experimental high energy physics, where the Particle Data Group  convention is that uncertainties are defined by 68.3% Confidence Intervals (CI), so the error bars represent the 15.9% and 84.1% CI points.  As noted in @basics answer, this or similar CI conventions are probably easier to understand whenever uncertainties are asymmetric or otherwise non-Gaussian.
