I have several datasets, which data can be represented by a random variable $X$.
Some of the datasets follow an approximately symmetric distribution, while other ones follow a skewed distribution.
For all the datasets I can calculate the traditional statistics, like median $med$, mean $\mu$, standard deviation $\sigma$, quartiles $Q_1$, $Q_2$, $Q_3$, etc..
For a generic distribution, and given a certain value $a$ (please see the Figure), is there a formula/inequality to calculate the approximate area (or a bound) of a distribution tail, by using the summary statistics?
In other words, for any distribution, and by using the summary statistics, is there a way to get an approximate probability (or a bound) that the random variable $X$ is smaller than $a$?
$P\left(X\leq a\right) \approx \;?$
Note 1: If possible, I would like to avoid to use the cumulative distribution function, unless it is the only way to get that probability.
Note 2: As far as I know the Chebyshev's inequality should apply to any type of distribution. Is there anything similar that I could apply to my case?
