Formula for a skewed bell curve given mean, standard deviation, and median

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I have a large dataset of the income of various counties. I am given the median, mean, and standard deviation. I would like to do some analysis, but I would like to model the income these individual counties as skewed bell curves (feel free to offer a better idea). Thing is, I can't seem to find any good ways to derive the formula for a skewed bell curve given a mean, standard deviation, and median. I would be fine with even a decent approximation as I have been having trouble with deriving such a formula due to the error function.