📐 NumPy Chi-Square Distribution – Analyze Variability with Python

🧲 Introduction – Why Learn the Chi-Square Distribution in NumPy?

The Chi-Square (χ²) distribution is used in statistics to measure variance, test hypotheses, and perform goodness-of-fit tests. It’s especially common in statistical inference, regression diagnostics, and machine learning model evaluation (like feature selection using Chi-Square scores).

With NumPy’s np.random.chisquare(), you can easily generate samples for simulations or statistical modeling.

🎯 By the end of this guide, you’ll:

• Generate χ²-distributed values using NumPy
• Understand the df (degrees of freedom) parameter
• Visualize how the shape changes with df
• Use chi-square values in real-world statistical simulations

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🔢 Step 1: Generate Chi-Square Samples with NumPy

import numpy as np

data = np.random.chisquare(df=2, size=10)
print(data)

🔍 Explanation:

• df=2: Degrees of freedom
• size=10: Generate 10 values from the χ² distribution
  ✅ Output: Array of positive float values, typically right-skewed

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📊 Step 2: Visualize the Chi-Square Distribution

import matplotlib.pyplot as plt
import seaborn as sns

samples = np.random.chisquare(df=4, size=1000)
sns.histplot(samples, bins=30, kde=True, color="skyblue", edgecolor="black")
plt.title("Chi-Square Distribution (df=4)")
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.show()

🔍 Explanation:

• Right-skewed histogram
• As df increases, the distribution becomes more symmetric
  ✅ Visual confirmation of Chi-Square’s shape

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📈 Step 3: Compare Distributions with Different Degrees of Freedom

for df in [2, 4, 8, 16]:
    sns.kdeplot(np.random.chisquare(df, 1000), label=f'df={df}', fill=True)

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

🔍 Explanation:

• Smaller df = sharper skew
• Larger df = smoother, more bell-shaped curve
  ✅ Helps understand how degrees of freedom control variability

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🧪 Step 4: Real-World Use Case – Goodness-of-Fit Simulation

observed = np.array([18, 22, 20])
expected = np.array([20, 20, 20])
chi_square_stat = ((observed - expected) ** 2 / expected).sum()
print("Chi-Square Statistic:", chi_square_stat)

🔍 Explanation:

• Manual Chi-Square test formula: χ2=∑(O−E)2E\chi^2 = \sum \frac{(O – E)^2}{E}

✅ Used to test if observed frequencies differ significantly from expected

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📐 Step 5: Create 2D Chi-Square Data

chi_matrix = np.random.chisquare(df=5, size=(3, 4))
print(chi_matrix)

🔍 Explanation:

• Generates a 3×4 matrix of chi-square samples
  ✅ Good for simulating grouped experimental results or parallel simulations

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🧠 Real-World Applications of Chi-Square Distribution

Use Case	Description
Goodness-of-Fit Testing	Compare observed vs expected distributions (e.g., dice rolls)
Independence Testing	Chi-Square tests for contingency tables (e.g., gender vs major)
Feature Selection in ML	Select best features for classification using χ² scores
Simulation of Variance Models	Test variability in simulated systems or outcomes
Residual Analysis	Assess model accuracy in regression diagnostics

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⚠️ Common Mistakes to Avoid

Mistake	Correction
Using non-positive degrees of freedom	df must be positive (usually ≥ 1)
Expecting symmetric distribution	χ² is right-skewed, especially with small df
Confusing χ² with normal distribution	Only becomes symmetric as df gets large
Forgetting that outputs are positive	Chi-Square values are always ≥ 0

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📌 Summary – Recap & Next Steps

The Chi-Square distribution is vital for statistical hypothesis testing, modeling sample variance, and evaluating categorical relationships. NumPy’s np.random.chisquare() makes it simple to simulate and visualize this distribution.

🔍 Key Takeaways:

• Use np.random.chisquare(df, size) to generate χ² samples
• df controls the shape — higher df = smoother distribution
• Output is continuous and always non-negative
• Perfect for simulations, hypothesis testing, and machine learning analysis

⚙️ Real-world relevance: Core tool in statistical analysis, ML feature evaluation, and experimental modeling.

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❓ FAQs – NumPy Chi-Square Distribution

❓ What does df mean in np.random.chisquare()?
✅ It stands for degrees of freedom and controls the shape of the distribution.

❓ Can Chi-Square values be negative?
❌ No. Values are always non-negative floats.

❓ What’s the relationship between Chi-Square and Normal distributions?
✅ A χ² distribution with k degrees of freedom is the sum of the squares of k standard normal variables.

❓ How do I use Chi-Square for feature selection?
✅ Use sklearn.feature_selection.chi2() to score features against the target.

❓ When should I use Chi-Square in modeling?
✅ For comparing frequencies, independence tests, or measuring variance.

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