Kurtosis

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Understanding Kurtosis

Definition

Kurtosis is a measure of the "tailedness" of a probability distribution. It quantifies how heavy the tails of a distribution are compared to a normal distribution.

Formula

Sample Kurtosis:

\text{Kurtosis} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^4 / n}{s^4} - 3

Where:

$x_i$ is each value in the sample
$\bar x$ is the mean of the sample
$n$ is the number of values
$s$ is the sample standard deviation

Interpretation Guidelines

Kurtosis = 0: Normal distribution (mesokurtic)

Kurtosis > 0: Heavy-tailed distribution (leptokurtic)

Kurtosis < 0: Light-tailed distribution (platykurtic)

Visual Examples of Kurtosis

The following examples illustrate how kurtosis affects the shape of a distribution.

Mesokurtic Distribution

Kurtosis ≈ 0

Characteristics: Moderate peak height and tail weight, typical of normal distribution

Similar to normal distribution with balanced tails.

Leptokurtic Distribution

Kurtosis > 0

Characteristics: Taller peak with more concentration of data, thicker tails indicating more extreme values

Higher peak and heavier tails than normal distribution.

Platykurtic Distribution

Kurtosis < 0

Characteristics: Flatter peak with more even spread of data, thinner tails indicating fewer extreme values

Lower peak and lighter tails than normal distribution.

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