Skewness and Kurtosis

Overview

Skewness and kurtosis are numerical measures that quantify the shape of a distribution beyond what the mean and variance capture. Skewness measures asymmetry, while kurtosis measures the heaviness of the tails relative to a normal distribution.

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1. Symmetric and Skewed Distributions

Symmetric Distribution

A distribution is symmetric if its left and right sides mirror each other. The most common example is the normal distribution (bell curve), where mean, median, and mode are equal and located at the center.

Example: Heights of people often follow a symmetric distribution.

Symmetric Distribution: Mixture of Gaussians

A symmetric shape can also arise from a mixture of distributions, provided the components are centered at the same location.

import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats

def generate_and_plot_mixed_distribution(seed: int = 0):
    np.random.seed(seed)
    main_data = stats.norm().rvs(1_000)
    minor_1 = stats.norm(scale=2).rvs(200)
    minor_2 = stats.norm(scale=4).rvs(100)
    combined = np.concatenate((main_data, minor_1, minor_2))

    fig, ax = plt.subplots(figsize=(12, 3))
    ax.hist(combined, bins=30)
    plt.show()

if __name__ == "__main__":
    generate_and_plot_mixed_distribution()

Skewed Distributions

A skewed distribution has data that stretches more on one side than the other.

Right-Skewed (Positively Skewed): The tail extends to the right. Mean > Median > Mode. Example: income distribution.

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt

def generate_and_plot_right_skewed_distribution(seed: int = 0):
    np.random.seed(seed)
    main_data = stats.norm().rvs(1_000)
    right_1 = stats.norm(loc=2).rvs(200)
    right_2 = stats.norm(loc=4).rvs(100)
    combined = np.concatenate((main_data, right_1, right_2))

    fig, ax = plt.subplots(figsize=(12, 3))
    ax.hist(combined, bins=30)
    plt.show()

if __name__ == "__main__":
    generate_and_plot_right_skewed_distribution()

Left-Skewed (Negatively Skewed): The tail extends to the left. Mean < Median < Mode. Example: age at retirement.

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt

def generate_and_plot_left_skewed_distribution(seed: int = 0):
    np.random.seed(seed)
    main_data = stats.norm().rvs(1_000)
    left_1 = stats.norm(loc=-2).rvs(200)
    left_2 = stats.norm(loc=-4).rvs(100)
    combined = np.concatenate((main_data, left_1, left_2))

    fig, ax = plt.subplots(figsize=(12, 3))
    ax.hist(combined, bins=30)
    plt.show()

if __name__ == "__main__":
    generate_and_plot_left_skewed_distribution()

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2. Detecting Skewness via Box Plots

Box plots provide a quick visual diagnostic for skewness:

\[ \begin{array}{lll} \text{Left\_Box} > \text{Right\_Box} &\Rightarrow& \text{Skew to Left} \\ \text{Left\_Box} < \text{Right\_Box} &\Rightarrow& \text{Skew to Right} \\ \text{Left\_Box} = \text{Right\_Box},\; \text{Left\_Whisker} > \text{Right\_Whisker} &\Rightarrow& \text{Skew to Left} \\ \text{Left\_Box} = \text{Right\_Box},\; \text{Left\_Whisker} < \text{Right\_Whisker} &\Rightarrow& \text{Skew to Right} \\ \text{Left\_Box} = \text{Right\_Box},\; \text{Left\_Whisker} = \text{Right\_Whisker} &\Rightarrow& \text{Symmetric} \\ \end{array} \]

Box Plot: Symmetric Distribution

import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats

def generate_and_plot_histogram_and_box_plot_mixed_distribution(seed: int = 0):
    np.random.seed(seed)
    main_data = stats.norm().rvs(1_000)
    minor_1 = stats.norm(scale=2).rvs(200)
    minor_2 = stats.norm(scale=4).rvs(100)
    combined = np.concatenate((main_data, minor_1, minor_2))

    fig, (ax_hist, ax_box) = plt.subplots(2, 1, figsize=(12, 6))
    ax_hist.hist(combined, density=True, bins=30)
    ax_hist.set_title('Histogram of Combined Data (Density)')
    ax_box.boxplot(combined, vert=False)
    ax_box.set_title('Boxplot of Combined Data')
    plt.tight_layout()
    plt.show()

if __name__ == "__main__":
    generate_and_plot_histogram_and_box_plot_mixed_distribution()

Box Plot: Right-Skewed Distribution

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt

def generate_and_plot_histogram_and_box_plot_right_skewed(seed: int = 0):
    np.random.seed(seed)
    main_data = stats.norm().rvs(1_000)
    right_1 = stats.norm(loc=2).rvs(200)
    right_2 = stats.norm(loc=4).rvs(100)
    combined = np.concatenate((main_data, right_1, right_2))

    fig, (ax_hist, ax_box) = plt.subplots(2, 1, figsize=(12, 6))
    ax_hist.hist(combined, density=True, bins=30)
    ax_hist.set_title('Histogram of Combined Data (Density)')
    ax_box.boxplot(combined, vert=False)
    ax_box.set_title('Boxplot of Combined Data')
    plt.tight_layout()
    plt.show()

if __name__ == "__main__":
    generate_and_plot_histogram_and_box_plot_right_skewed()

Box Plot: Left-Skewed Distribution

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt

def generate_and_plot_histogram_and_box_plot_left_skewed(seed: int = 0):
    np.random.seed(seed)
    main_data = stats.norm().rvs(1_000)
    left_1 = stats.norm(loc=-2).rvs(200)
    left_2 = stats.norm(loc=-4).rvs(100)
    combined = np.concatenate((main_data, left_1, left_2))

    fig, (ax_hist, ax_box) = plt.subplots(2, 1, figsize=(12, 6))
    ax_hist.hist(combined, density=True, bins=30)
    ax_hist.set_title('Histogram of Combined Data (Density)')
    ax_box.boxplot(combined, vert=False)
    ax_box.set_title('Boxplot of Combined Data')
    plt.tight_layout()
    plt.show()

if __name__ == "__main__":
    generate_and_plot_histogram_and_box_plot_left_skewed()

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3. Skewness: Definition and Computation

Definition

\[ \text{Skewness}(X) = E\left(\frac{X - \mu}{\sigma}\right)^3 \approx \frac{1}{n}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^3 \]

• Skewness = 0: Symmetric distribution.
• Skewness > 0: Right-skewed (positive skew).
• Skewness < 0: Left-skewed (negative skew).

Skewness Simulation

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

def generate_samples(main_size, right_size, left_size):
    main_sample = np.random.normal(0, 1, main_size)
    right_sample = np.random.normal(2, 1, right_size)
    left_sample = np.random.normal(-2, 1, left_size)
    return np.concatenate([main_sample, right_sample, left_sample])

def calculate_statistics(data):
    n = data.shape[0]
    mean = data.sum() / n
    std_dev = np.sqrt(np.sum((data - mean) ** 2) / n)
    skewness = stats.describe(data).skewness
    return mean, std_dev, skewness

def plot_distribution_with_normal_fit(data, mean, std_dev, skewness, title):
    fig, ax = plt.subplots(figsize=(12, 3))
    _, bins, _ = ax.hist(data, density=True, bins=100, label="Samples")
    normal_pdf = stats.norm(loc=mean, scale=std_dev).pdf(bins)
    ax.plot(bins, normal_pdf, "--r", label="Normal PDF")
    ax.set_title(f"{title}\nSkewness = {skewness:.4f}")
    ax.legend()
    ax.spines["left"].set_visible(False)
    ax.spines["right"].set_visible(False)
    ax.spines["top"].set_visible(False)
    plt.show()

def main():
    np.random.seed(0)
    main_size = 10_000
    right_size = 3_000
    left_size = 3_000

    samples = generate_samples(main_size, right_size, left_size)
    mean, std_dev, skewness = calculate_statistics(samples)

    if right_size > left_size:
        title = "Right-Skewed Distribution"
    elif right_size < left_size:
        title = "Left-Skewed Distribution"
    else:
        title = "Symmetric Distribution"

    plot_distribution_with_normal_fit(samples, mean, std_dev, skewness, title)

if __name__ == "__main__":
    main()

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4. Kurtosis

Definition

Kurtosis measures the "tailedness" of a distribution—how much probability mass is in the tails relative to the center.

\[ \text{Kurtosis}(X) = E\left(\frac{X - \mu}{\sigma}\right)^4 \approx \frac{1}{n}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^4 \]

Excess Kurtosis subtracts the kurtosis of the normal distribution (which equals 3):

\[ \text{Excess Kurtosis}(X) = \text{Kurtosis}(X) - 3 \]

• Excess Kurtosis = 0 (Mesokurtic): Normal-like tails.
• Excess Kurtosis > 0 (Leptokurtic): Heavier tails than normal; more extreme outliers.
• Excess Kurtosis < 0 (Platykurtic): Lighter tails than normal; fewer extreme values.

Kurtosis Simulation

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

def generate_samples(main_size, peak_size):
    main_sample = np.random.normal(0, 1, main_size)
    peak_sample = np.random.normal(0, 0.2, peak_size)
    return np.concatenate([main_sample, peak_sample])

def calculate_statistics(data):
    mean = np.mean(data)
    std_dev = np.std(data)
    skewness = stats.describe(data).skewness
    kurtosis = np.mean(((data - mean) / std_dev) ** 4)
    excess_kurtosis = kurtosis - 3
    return mean, std_dev, skewness, kurtosis, excess_kurtosis

def plot_distribution_with_normal_fit(data, mean, std_dev, excess_kurtosis, title):
    fig, ax = plt.subplots(figsize=(12, 3))
    _, bins, _ = ax.hist(data, density=True, bins=100, label="Sample Data")
    normal_pdf = stats.norm(loc=mean, scale=std_dev).pdf(bins)
    ax.plot(bins, normal_pdf, "--r", label="Normal PDF")
    ax.set_title(f"{title}\nExcess Kurtosis = {excess_kurtosis:.4f}")
    ax.legend()
    ax.spines["left"].set_visible(False)
    ax.spines["right"].set_visible(False)
    ax.spines["top"].set_visible(False)
    plt.show()

def main():
    np.random.seed(0)
    main_size = 10_000
    peak_size = 500

    data = generate_samples(main_size, peak_size)
    mean, std_dev, skewness, kurtosis, excess_kurtosis = calculate_statistics(data)

    if excess_kurtosis > 0:
        title = "Leptokurtic Distribution"
    elif excess_kurtosis < 0:
        title = "Platykurtic Distribution"
    else:
        title = "Mesokurtic Distribution"

    plot_distribution_with_normal_fit(data, mean, std_dev, excess_kurtosis, title)

if __name__ == "__main__":
    main()

Computing Kurtosis in Python

SciPy provides convenient functions that compute excess kurtosis directly:

from scipy import stats
import numpy as np

data = np.random.normal(0, 1, 10000)

# All three return excess kurtosis (kurtosis - 3)
print(stats.kurtosis(data))
print(stats.describe(data).kurtosis)

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Summary

Skewness and kurtosis extend the description of a distribution beyond its center and spread. Skewness reveals directional asymmetry, guiding the choice between mean and median as a representative center. Kurtosis quantifies tail behavior, which is critical in risk management and finance where extreme events (heavy tails) have outsized consequences.