Question

How the heavily the tails and skewness of distribution and the normal distribution can be measured?

Answer 1

Kurtosis is a statistical measure that defines how heavily the tails of a distribution differ from the tails of a normal distribution. In other words, kurtosis identifies whether the tails of a given distribution contain extreme values.

For univariate data Y₁, Y₂, ..., Y_N, the formula for kurtosis is:

where Y¯ is the mean, s is the standard deviation, and N is the number of data points. Note that in computing the kurtosis, the standard deviation is computed using N in the denominator rather than N - 1.

An excess kurtosis is a metric that compares the kurtosis of a distribution against the kurtosis of a normal distribution. The kurtosis of a normal distribution equals 3. Therefore, the excess kurtosis is found using the formula below:

Excess Kurtosis = Kurtosis – 3

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.

Skewness essentially measures the symmetry of the distribution, while kurtosis determines the heaviness of the distribution tails.

There are several ways to measure skewness. Pearson’s first and second coefficients of skewness are two common ones. Pearson’s first coefficient of skewness, or Pearson mode skewness, subtracts the mode from the mean and divides the difference by the standard deviation. Pearson’s second coefficient of skewness, or Pearson median skewness, subtracts the median from the mean, multiplies the difference by three and divides the product by the standard deviation.

The formulae for Pearson's skewness are:

where:Sk_{​​​​​1}=Pearson’s first coefficient of skewness and S k_​​​2 the second

s=the standard deviation for the sample

Xˉ=is the mean value

Mo=the modal (mode) value

Md= is the median value.