Skewness and long-tailedness have relationship same as one way and two way tests. skewness is one-tailed long-tailedness and it has left long-tailedness for left skewed data and vise versa. Along with, Kurtosis is two-tailed long-tailedness. But you should be aware that the range in skewed data with no kurtosis is same as normal data. So, just the density function has long tail in one side, and short tail on the other.
OK. How to measure long tailedness? what is the best way to check whether a particular discrete distribution is long tailed or not.
in continuous variables, this can be measured using Z score standardization. In normally distributed data 63% of data are between ± 1, 95% between ±2 standard deviation. In long-tailed data you have data farther than this in one side (one tailed long-taildness) for skewed data and in both sides for data with negative kurtosis.
In discrete data longtailedness depends on "p" and sample size. The lower the "p" the longer the tails, and the higher the sample size, the shorter the tails.
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