Kurtosis

Created:September 24, 2024

Last Updated:January 23, 2025

The Kurtosis Calculator helps you measure the "tailedness" of your data distribution - how heavy or light the tails are compared to a normal distribution. This can reveal important patterns in your data, such as whether extreme values occur more frequently than expected. For example, in financial analysis, high kurtosis could indicate a higher risk of extreme market movements.

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Note: Column names will be converted to snake_case (e.g., "Product ID" → "product_id") for processing.

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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}

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
The excess kurtosis is calculated as: kurtosis - 3, if you are interested

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.

How to Calculate Kurtosis in R

Use the kurtosis() function from the moments package to calculate kurtosis.

library(tidyverse)
library(moments)

tips <- read.csv("https://raw.githubusercontent.com/plotly/datasets/master/tips.csv")

kurtosis(tips$tip) # 6.549552

# histogram with density 
ggplot(tips, aes(x = tip)) +
  geom_histogram(aes(y = after_stat(density)), binwidth = 0.5, fill = "darkblue", alpha = 0.7) +
  geom_density(adjust=1.5, color = "red", linewidth = 1) + # the adjust parameter controls the smoothness of the density curve
  geom_vline(aes(xintercept = mean(tip)), 
         linetype = "dashed", 
         color = "blue", 
         size = 1) +
  geom_vline(aes(xintercept = median(tip)), 
         linetype = "dotted", 
         color = "green", 
         size = 1) +
  labs(title = "Histogram and Density of Tip Amounts",
       x = "Tip Amount ($)",
       y = "Density") +
  theme_minimal()