How to evaluate different type of kurtosis in statistics

In statistics, kurtosis is a measure of the shape of a probability distribution. It measures the degree of peakedness of the distribution and the thickness of the tails compared to the normal distribution. There are different types of kurtosis that are used to describe different shapes of distributions. Here's how to evaluate different types of kurtosis with an example:

Consider the following dataset of exam scores: 85, 90, 75, 80, 95.

Step 1: Calculate the mean

Add up all the values in the dataset and divide by the total number of values to find the mean:

85 + 90 + 75 + 80 + 95 / 5 = 85

The mean is 85.

Step 2: Calculate the deviations from the mean

Calculate the deviation of each value from the mean by subtracting the mean from each value:

85 - 85 = 0
90 - 85 = 5
75 - 85 = -10
80 - 85 = -5
95 - 85 = 10

Step 3: Calculate the squared deviations

Square each deviation to eliminate the negative values:

0^2 = 0
5^2 = 25
(-10)^2 = 100
(-5)^2 = 25
10^2 = 100

Step 4: Calculate the variance

Calculate the variance by summing the squared deviations and dividing by the number of observations:

(0 + 25 + 100 + 25 + 100) / 5 = 50

The variance is 50.

Step 5: Calculate the fourth moment

Calculate the fourth moment of the dataset around the mean:

((0 - 85)^4 + (5 - 85)^4 + (-10 - 85)^4 + (-5 - 85)^4 + (10 - 85)^4) / 5 = 5085000

Step 6: Calculate the excess kurtosis

The excess kurtosis is the fourth moment divided by the variance squared, minus 3:

5085000 / 50^2 - 3 = -0.34

The excess kurtosis is -0.34, which indicates that the distribution is slightly platykurtic (i.e., flatter and more spread out than the normal distribution). A negative excess kurtosis indicates a flatter distribution with thinner tails compared to the normal distribution, while a positive excess kurtosis indicates a more peaked distribution with fatter tails compared to the normal distribution.

In summary, evaluating different types of kurtosis involves calculating the fourth moment and excess kurtosis of a distribution. The excess kurtosis can be used to compare the shape of a distribution to the normal distribution and to identify departures from normality.