Chi-Square Distribution

Welcome to The Coding College, where we simplify data science and programming concepts! In this article, we’ll explore the Chi-Square Distribution, its properties, applications, and how to implement it in Python using NumPy.

What is the Chi-Square Distribution?

The Chi-Square Distribution is a continuous probability distribution widely used in hypothesis testing and inferential statistics, particularly for categorical data. It is defined as the distribution of a sum of squared independent standard normal variables.

Formula:

If Z1,Z2,…,Zk​ are independent standard normal variables, then:

follows a Chi-Square Distribution with kk degrees of freedom (k).

Key Characteristics

1. Degrees of Freedom (k): Determines the shape of the distribution.
2. Positive Values Only: The distribution is defined only for x≥0x \geq 0.
3. Mean: Equal to the degrees of freedom (k).
4. Variance: 2k.

Real-Life Applications

1. Hypothesis Testing: Chi-square tests for independence and goodness of fit.
2. Confidence Intervals: Estimating population variances.
3. Model Evaluation: Statistical tests for machine learning models.

Chi-Square Distribution in NumPy

Python’s NumPy library provides a function to generate random numbers from a Chi-Square distribution:

Syntax:

numpy.random.chisquare(df, size=None)

• df: Degrees of freedom (kk).
• size: Output shape (default is None, which returns a single value).

Example 1: Generating Random Numbers

Scenario: Simulate a Chi-Square distribution with 4 degrees of freedom

import numpy as np

# Generate Chi-Square random numbers
data = np.random.chisquare(df=4, size=10)
print("Random samples from Chi-Square distribution:", data)

Output (Example):

[2.15 5.62 3.14 6.78 4.33 1.09 2.78 5.34 3.67 4.89]

Example 2: Visualizing Chi-Square Distribution

import numpy as np
import matplotlib.pyplot as plt

# Generate data for different degrees of freedom
x1 = np.random.chisquare(df=2, size=1000)
x2 = np.random.chisquare(df=4, size=1000)
x3 = np.random.chisquare(df=6, size=1000)

# Plot histograms
plt.hist(x1, bins=30, alpha=0.5, label='df=2', color='blue', density=True)
plt.hist(x2, bins=30, alpha=0.5, label='df=4', color='orange', density=True)
plt.hist(x3, bins=30, alpha=0.5, label='df=6', color='green', density=True)

plt.title('Chi-Square Distributions')
plt.xlabel('Value')
plt.ylabel('Density')
plt.legend()
plt.grid(True)
plt.show()

Example 3: Goodness-of-Fit Test

Scenario: Checking if observed frequencies follow an expected distribution

from scipy.stats import chisquare

# Observed frequencies
observed = [18, 22, 20, 15, 25]

# Expected frequencies
expected = [20, 20, 20, 20, 20]

# Perform Chi-Square test
chi2, p = chisquare(f_obs=observed, f_exp=expected)

print(f"Chi-Square statistic: {chi2}")
print(f"P-value: {p}")

if p < 0.05:
    print("Reject the null hypothesis: Observed and expected distributions differ.")
else:
    print("Fail to reject the null hypothesis: Observed and expected distributions are similar.")

Example 4: Chi-Square Test for Independence

Scenario: Analyzing survey data for independence

import numpy as np
from scipy.stats import chi2_contingency

# Contingency table (survey data)
data = np.array([[50, 30], [20, 100]])

# Perform Chi-Square test for independence
chi2, p, dof, expected = chi2_contingency(data)

print(f"Chi-Square statistic: {chi2}")
print(f"P-value: {p}")
print(f"Degrees of freedom: {dof}")
print("Expected frequencies:")
print(expected)

if p < 0.05:
    print("Reject the null hypothesis: Variables are dependent.")
else:
    print("Fail to reject the null hypothesis: Variables are independent.")

Chi-Square Distribution Properties

Chi-Square vs Other Distributions

Summary

The Chi-Square Distribution is a cornerstone of statistical analysis and hypothesis testing. Its utility in evaluating categorical data and goodness-of-fit makes it indispensable for data scientists and statisticians.

With Python’s NumPy and SciPy, you can simulate, analyze, and apply the Chi-Square Distribution to solve real-world problems efficiently.

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