Kurtosis (Fourth Central Moment)
- is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution
- data sets with high kurtosis tend to have heavy tails, or outliers
- data sets with low kurtosis tend to have light tails, or lack of outliers
Formula
for univariate dataset {𝑋1, …, 𝑋𝑛}:
- sample kurtosis = 𝐄[(𝑋𝑖- 𝑋̅)4] = [𝛴1≤𝑖≤𝑛(𝑋𝑖- 𝑋̅)4] / [𝑠4𝑛]
where:
- 𝑋̅ - mean
- 𝑠 - sample standard deviation computed with 𝑛 as the denominator instead of (𝑛 - 1)
Excess Kurtosis
kurtosis for a standard normal distribution is 3
- excess kurtosis = [ [𝛴1≤𝑖≤𝑛(𝑋𝑖- 𝑋̅)4] / [𝑠4𝑛] ] - 3
Kurtosis Types
