How is kurtosis of a distribution calculated in R

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While skew measures if the distribution is left skewed or right skewed, kurtosis measures if the tail is thin or thick.

A tail is thick if the standard distribution is large. For example,

> p_kurtosis = data.frame(v=rnorm(1000,mean=100,sd=50))

Similarly, if the sd is small ( more tightly packed around the center ), the tail is thin.

> n_kurtosis = data.frame(v=rnorm(1000,mean=100,sd=10))

Let’s visualize this.

ggplot() + 
    geom_density(data = p_kurtosis,aes(p_kurtosis,fill="red",alpha=0.1)) + 
    geom_density(data = n_kurtosis, aes(n_kurtosis,fill="green",alpha=0.1))

The distribution in green has a thick tail , hence a positive kurtosis. The distribution in red has a relatively think tail and hence a negative kurtosis.

> kurtosis(p_kurtosis$v)
[1] 0.1529404
> kurtosis(n_kurtosis$v)
[1] -0.01435051

skewness and kurtosis are a bit related. Learn more about how to measure skewness of a distribution in R here.

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