How to Calculate Kurtosis?

Kurtosis can be calculated by dividing the fourth-order moment by the standard deviation of the population raised to the fourth power. Kurtosis is a measure of the fourth moment of a probability distribution of a random variable. It can be calculated as the ratio of the fourth moment to the square of the variance.

To calculate kurtosis in statistics, you can follow these steps:

1. Compute the Mean (μ): Calculate the arithmetic mean of the dataset.
2. Compute the Variance (σ2): Calculate the variance of the dataset, which is the average of the squared differences from the mean.
3. Compute the Standard Deviation (σ): Take the square root of the variance to find the standard deviation.
4. Compute the Fourth Moment (μ4): Calculate the fourth moment of the dataset, which is the average of the fourth power of the differences from the mean.
5. Compute Kurtosis: The formula for calculating kurtosis is:
   Kurtosis = μ4/σ4​
   ​
   Sometimes, you might also see a version of kurtosis that subtracts 3 from this calculation. This is called excess kurtosis, and it subtracts 3 because the kurtosis of a normal distribution is 3.
   So the formula becomes:
   Excess Kurtosis = (μ4/σ4​)​ − 3
6. This version is often used because it allows for easier comparison to the normal distribution, where excess kurtosis of 0 indicates normality.

Kurtosis can be classified as:

• Leptokurtic: Distributions with wide tails and positive kurtosis.
• Mesokurtic: When the excess kurtosis is zero or close to zero.
• Platykurtic: When the excess kurtosis is negative.

Kurtosis is a statistical measure used to describe the distribution of observed data around the mean. It is used to identify the tails and sharpness of a distribution. The kurtosis of a probability distribution for a random variable x is defined as the ratio of the fourth central moment (μ⁴​) to the fourth power of the standard deviation (σ⁴), expressed as: [Tex]κ= σ 4 μ 4 ​ ​ = (E[ σ x−E[x] ​ ]) 4 E[( σ x−E[x] ​ ) 4 ] ​ [/Tex]

In this article, we will explore how to calculate kurtosis in statistics.