Calculate The Kurtosis of The Following Distribution Using The Formula

Kurtosis is a statistical measure that describes the shape of a distribution, particularly the "tailedness" of the distribution. This guide explains how to calculate kurtosis using the formula, interpret the results, and use our calculator.

What is kurtosis?

Kurtosis is a statistical measure that describes the shape of a probability distribution, particularly the "tailedness" of the distribution. It quantifies whether the data is heavy-tailed or light-tailed relative to a normal distribution.

There are two main types of kurtosis:

Excess kurtosis: Measures how much the tails of the distribution are heavier or lighter than those of a normal distribution. A positive excess kurtosis indicates heavy tails, while a negative excess kurtosis indicates light tails.
Pearson's kurtosis: Measures the kurtosis relative to a uniform distribution. It is less commonly used than excess kurtosis.

Kurtosis is important in finance, risk management, and quality control, where understanding the shape of distributions helps in making informed decisions.

Kurtosis formula

The formula for calculating kurtosis depends on whether you're calculating excess kurtosis or Pearson's kurtosis. Here are the formulas:

Excess kurtosis formula

Excess kurtosis is calculated using the following formula:

K = (n(n+1)(n-1)(n-2)(n-3)Σ(xi - μ)⁴) / (n-1)(n-2)(n-3)(σ⁴) - 3

Where:

K = Excess kurtosis
n = Number of observations
xi = Each individual observation
μ = Mean of the distribution
σ = Standard deviation of the distribution

Pearson's kurtosis formula

Pearson's kurtosis is calculated using the following formula:

K = (Σ(xi - μ)⁴) / (nσ⁴)

Where:

K = Pearson's kurtosis
n = Number of observations
xi = Each individual observation
μ = Mean of the distribution
σ = Standard deviation of the distribution

In practice, excess kurtosis is more commonly used because it provides a direct comparison to the normal distribution (which has an excess kurtosis of 0).

How to calculate kurtosis

Calculating kurtosis involves several steps:

Collect the data: Gather the observations or data points for which you want to calculate kurtosis.
Calculate the mean (μ): Compute the average of the data points.
Calculate the standard deviation (σ): Compute the standard deviation of the data points.
Compute the fourth moment: Calculate the sum of the fourth powers of the deviations from the mean.
Apply the kurtosis formula: Use the appropriate formula (excess kurtosis or Pearson's kurtosis) to calculate the kurtosis.

Important notes

Kurtosis is sensitive to outliers, so it's important to ensure your data is clean and free of extreme values.
For small sample sizes, kurtosis estimates can be unreliable. Larger sample sizes provide more stable estimates.
Kurtosis is not a measure of central tendency or dispersion. It only describes the shape of the distribution.

Interpreting kurtosis

Interpreting kurtosis involves understanding the shape of the distribution:

Excess kurtosis = 0: The distribution has the same kurtosis as a normal distribution (mesokurtic).
Excess kurtosis > 0: The distribution is leptokurtic, meaning it has heavier tails and a sharper peak than a normal distribution.
Excess kurtosis < 0: The distribution is platykurtic, meaning it has lighter tails and a flatter peak than a normal distribution.

Here's a table summarizing the interpretation of kurtosis:

Excess kurtosis	Distribution shape	Example
0	Mesokurtic	Normal distribution
Positive	Leptokurtic	Stock returns, financial data
Negative	Platykurtic	Uniform distribution, dice rolls

Understanding the kurtosis of a distribution helps in making informed decisions, especially in fields like finance, risk management, and quality control.

Example calculation

Let's calculate the kurtosis of the following distribution: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29].

We'll use the excess kurtosis formula.

Calculate the mean (μ):
μ = (2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29) / 10 = 120 / 10 = 12
Calculate the standard deviation (σ):
First, calculate the variance:

Variance = Σ(xi - μ)² / n = [(2-12)² + (3-12)² + ... + (29-12)²] / 10

Variance ≈ 62.8

σ = √Variance ≈ 7.92
Compute the fourth moment:
Fourth moment = Σ(xi - μ)⁴ / n ≈ [(2-12)⁴ + (3-12)⁴ + ... + (29-12)⁴] / 10

Fourth moment ≈ 12,000
Apply the excess kurtosis formula:
K = (Fourth moment / σ⁴) - 3 ≈ (12,000 / (7.92)⁴) - 3 ≈ 1.5

The excess kurtosis of this distribution is approximately 1.5, indicating that the distribution is leptokurtic (heavier tails than a normal distribution).

FAQ

What is the difference between kurtosis and skewness?: Kurtosis measures the "tailedness" of a distribution, while skewness measures the asymmetry of the distribution. Kurtosis describes the shape of the tails, while skewness describes the shape of the peak.
Can kurtosis be negative?: Yes, kurtosis can be negative. A negative kurtosis indicates a platykurtic distribution, which has lighter tails and a flatter peak than a normal distribution.
How does sample size affect kurtosis?: Kurtosis estimates are less reliable for small sample sizes. Larger sample sizes provide more stable and accurate estimates of kurtosis.
What is the difference between excess kurtosis and Pearson's kurtosis?: Excess kurtosis is calculated by subtracting 3 from Pearson's kurtosis. Excess kurtosis provides a direct comparison to the normal distribution (which has an excess kurtosis of 0), while Pearson's kurtosis is relative to a uniform distribution.
How is kurtosis used in finance?: In finance, kurtosis is used to assess the risk of investments. Higher kurtosis (heavier tails) indicates a higher probability of extreme returns, which can be both positive and negative. Financial analysts use kurtosis to manage risk and make informed investment decisions.