In a normal distribution, the 68-95-99.7 rule imparts standard deviation a lot of meaning, but what would standard deviation mean in a non-normal distribution (multimodal or skewed)? Would all data values still fall within 3 standard deviations? Do we have rules like the 68-95-99.7 one for non-normal distributions?
Solved – What does standard deviation tell us in non-normal distribution
normal distributionskewnessstandard deviation
Best Answer
It's the square root of the second central moment, the variance. The moments are related to characteristic functions(CF), which are called characteristic for a reason that they define the probability distribution. So, if you know all moments, you know CF, hence you know the entire probability distribution.
Normal distribution's characteristic function is defined by just two moments: mean and the variance (or standard deviation). Therefore, for normal distribution the standard deviation is especially important, it's 50% of its definition in a way.
For other distributions the standard deviation is in some ways less important because they have other moments. However, for many distributions used in practice the first few moments are the largest, so they are the most important ones to know.
Now, intuitively, the mean tell you where the center of your distribution is, while the standard deviation tell you how close to this center your data is.
Since the standard deviation is in the units of the variable it's also used to scale other moments to obtain measures such as kurtosis. Kurtosis is a dimensionless metric which tells you how fat are the tails of your distribution compared to normal