Kurtosis Calculator

What is Kurtosis?

Kurtosis is a statistical measure that describes the shape of a probability distribution. Specifically, it quantifies the "tailedness" of the distribution of data. Kurtosis indicates whether the tails of a given distribution contain extreme values.

Formula and Its Meaning

The formula for kurtosis is:

\[ Kurt = \frac{\frac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^4}{s^4} - 3 \]

Where:

• \(x_i\) are the individual values in a dataset
• \(\bar{x}\) is the mean of the dataset
• \(n\) is the number of data points
• \(s\) is the standard deviation of the dataset

The "-3" in the formula is used to make the kurtosis of a normal distribution equal to zero.

Interpretation of Kurtosis Values

• Kurtosis = 0: Indicates a normal distribution (mesokurtic)
• Kurtosis > 0: Indicates a heavy-tailed distribution (leptokurtic)
• Kurtosis < 0: Indicates a light-tailed distribution (platykurtic)

Calculation Steps

1. Calculate the mean of the dataset.
2. Calculate the standard deviation of the dataset.
3. For each data point, subtract the mean and raise the result to the fourth power.
4. Calculate the average of these values.
5. Divide by the fourth power of the standard deviation.
6. Subtract 3 from the result.

Example Calculation

Let's calculate the kurtosis for the dataset: 1, 2, 3, 4, 5

1. Mean: \(\bar{x} = \frac{1 + 2 + 3 + 4 + 5}{5} = 3\)
2. Variance: \(s^2 = \frac{(1-3)^2 + (2-3)^2 + (3-3)^2 + (4-3)^2 + (5-3)^2}{5} = 2\)
3. Standard Deviation: \(s = \sqrt{2} = 1.414\)
4. Fourth moment: \(m_4 = \frac{(1-3)^4 + (2-3)^4 + (3-3)^4 + (4-3)^4 + (5-3)^4}{5} = 4.8\)
5. Kurtosis: \(Kurt = \frac{4.8}{2^2} - 3 = -1.8\)

Visual Representation

This graph illustrates different kurtosis shapes. The blue curve represents a leptokurtic distribution (high kurtosis), the green curve a mesokurtic distribution (normal kurtosis), and the red curve a platykurtic distribution (low kurtosis).