As is well known for the normal distribution, 68% of the probability mass is within one standard deviation of the mean, 95% within two standard deviations and 99.7% within 3 standard deviations.
However, I have some empirical distributions that are leptokurtic and negatively skewed. In such circumstances is there a formula based on their higher-order moments to calculate how much of the probability mass is within so many standard deviations of the mean?
I have a measurement and would like to give some sense of how far it is from the midpoint (mean or some other measure of central tendency).
Can this be done?
Best Answer
You can always calculate how many SDs values are from the mean by just plugging in sample values, (value $-$ mean)/SD, and then binning and counting.
Precise numerical facts such as you cite for the normal (Gaussian) in general depend on knowing one or more of the density, distribution or quantile functions, numerically if not analytically.
However, there aren't general relations available on just knowing the skewness or kurtosis. Skewness and kurtosis don't pin down the form of the distribution in general, as higher moments can vary too.