The Chi-Square (χ²) distribution is widely used in statistical inference, especially for hypothesis testing and confidence intervals involving variances and categorical data.

With NumPy, generating and working with Chi-Square distributed values is simple and efficient for simulations and analysis.

What is the Chi-Square Distribution?

The Chi-Square distribution is a continuous probability distribution that describes the sum of the squares of independent standard normal variables.

Mathematically:

If Z1,Z2,...,Zk∼N(0,1)Z_1, Z_2, ..., Z_k \sim \mathcal{N}(0, 1) (standard normal), then:

χ2=Z12+Z22+⋯+Zk2\chi^2 = Z_1^2 + Z_2^2 + \dots + Z_k^2

This follows a Chi-Square distribution with k degrees of freedom (df).

Key Properties

NumPy's chisquare() Function

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

Parameters

✅ Returns

An array of random values from a Chi-Square distribution.

✅ Example: Generate Chi-Square Data

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns

rng = np.random.default_rng(seed=42)

# Generate 1000 chi-square values with 5 degrees of freedom
data = rng.chisquare(df=5, size=1000)

print(data[:5])  # First few samples

Visualizing the Distribution

sns.histplot(data, bins=40, kde=True, color='skyblue')
plt.title("Chi-Square Distribution (df=5)")
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.grid(True)
plt.show()

You’ll notice a right-skewed shape typical of the Chi-Square distribution.

Varying the Degrees of Freedom

Let’s compare Chi-Square distributions with different df values:

dfs = [1, 3, 5, 10, 20]

for df in dfs:
    data = rng.chisquare(df=df, size=1000)
    sns.kdeplot(data, label=f'df={df}')

plt.title("Chi-Square Distributions with Varying Degrees of Freedom")
plt.xlabel("Value")
plt.ylabel("Density")
plt.grid(True)
plt.legend()
plt.show()

Observation:

• Lower df: Highly skewed

• Higher df: Approaches a normal distribution

Practical Use Case: Goodness-of-Fit Test (Conceptual)

While NumPy simulates the distribution, statistical libraries like SciPy use it to perform hypothesis tests.

Chi-Square distribution is commonly used in:

• Goodness-of-fit tests (is data distributed as expected?)

• Test of independence in contingency tables

• Variance tests

Example (not using NumPy directly):

from scipy.stats import chisquare

observed = [18, 22, 20, 25, 15]
expected = [20, 20, 20, 20, 20]

stat, p = chisquare(f_obs=observed, f_exp=expected)
print(f"Chi-Square statistic: {stat:.2f}, p-value: {p:.3f}")

Full Simulation Example: Generate and Analyze Chi-Square Samples

# Simulate
samples = rng.chisquare(df=10, size=10000)

# Summary
print("Mean:", np.mean(samples))
print("Expected Mean:", 10)
print("Variance:", np.var(samples))
print("Expected Variance:", 2 * 10)

# Plot
sns.histplot(samples, bins=60, kde=True, color='orange')
plt.title("Chi-Square Distribution (df=10, 10k samples)")
plt.xlabel("Value")
plt.ylabel("Density")
plt.grid(True)
plt.show()

You’ll notice the empirical mean/variance aligns closely with theory.

Tips for Using Chi-Square in NumPy

⚠️ Common Pitfalls

Relation to Other Distributions

Conclusion

The Chi-Square distribution is vital for statistical modeling, especially when analyzing categorical data and variances. NumPy provides an easy and efficient way to generate data and perform simulations.

Summary