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:
- Compute the Mean (μ): Calculate the arithmetic mean of the dataset.
- Compute the Variance (σ2): Calculate the variance of the dataset, which is the average of the squared differences from the mean.
- Compute the Standard Deviation (σ): Take the square root of the variance to find the standard deviation.
- 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.
- 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 - 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.
How to Calculate Kurtosis in Statistics?
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.
Table of Content
- What is Kurtosis in Statistics?
- Types of Kurtosis
- How to Calculate Kurtosis?