Chi-Square Goodness-of-Fit Test

Chi-Square Goodness-of-Fit Test

Tests whether observed frequencies match a set of expected (theoretical) frequencies.

$\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}, \quad df = k-1$

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

# Example: Are die rolls fair?
# Observed 60 rolls: each face should appear 10 times
observed = np.array([8, 12, 9, 11, 13, 7])  # Observed counts
expected = np.array([10, 10, 10, 10, 10, 10])  # Equal probability

chi2, p_value = stats.chisquare(observed, expected)
df = len(observed) - 1

print("=== Chi-Square Goodness-of-Fit Test ===")
print("H₀: Die is fair (each face equally likely)")
print("H₁: Die is NOT fair")
print()
print(f"{'Face':<8} {'Observed':>10} {'Expected':>10} {'(O-E)²/E':>10}")
print("-" * 40)
for i, (o, e) in enumerate(zip(observed, expected), 1):
    component = (o-e)**2/e
    print(f"Face {i:<3} {o:>10} {e:>10.1f} {component:>10.4f}")
print("-" * 40)
print(f"{'Total':<8} {observed.sum():>10} {expected.sum():>10.1f} {chi2:>10.4f}")
print(f"
χ²({df}) = {chi2:.4f}, p = {p_value:.4f}")
print(f"Decision: {'Reject H₀ — die is biased' if p_value < 0.05 else 'Fail to reject H₀ — die appears fair'}")

# Visualize
fig, axes = plt.subplots(1, 2, figsize=(12, 4))
faces = [f'Face {i}' for i in range(1, 7)]
x = np.arange(6)
axes[0].bar(x - 0.2, observed, 0.4, label='Observed', color='steelblue', alpha=0.8)
axes[0].bar(x + 0.2, expected, 0.4, label='Expected', color='coral', alpha=0.8)
axes[0].set_xticks(x)
axes[0].set_xticklabels(faces)
axes[0].set_title('Observed vs Expected Frequencies')
axes[0].legend()

# Chi-square distribution
x_chi = np.linspace(0, 20, 500)
axes[1].plot(x_chi, stats.chi2.pdf(x_chi, df=df), 'b-', linewidth=2)
axes[1].fill_between(x_chi, stats.chi2.pdf(x_chi, df=df), where=x_chi >= chi2, alpha=0.4, color='red')
axes[1].axvline(chi2, color='red', linewidth=2, linestyle='--', label=f'χ²={chi2:.3f}')
axes[1].axvline(stats.chi2.ppf(0.95, df=df), color='black', linewidth=1.5, linestyle=':',
               label=f'Critical value={stats.chi2.ppf(0.95,df=df):.3f}')
axes[1].set_title(f'χ²({df}) Distribution (p={p_value:.3f})')
axes[1].legend()
plt.tight_layout()
plt.savefig('chi_square_gof.png', dpi=150)
plt.show()

# Real example: Testing normality of data
np.random.seed(42)
data = np.random.normal(50, 10, 200)
# Bin into 5 intervals and compare to expected normal proportions
bins = [-np.inf, 35, 45, 55, 65, np.inf]
obs, _ = np.histogram(data, bins=bins)
# Expected proportions under N(50,10)
probs = np.diff(stats.norm.cdf(bins, 50, 10))
exp = probs * len(data)
chi2_norm, p_norm = stats.chisquare(obs, exp)
print(f"
Normality test via chi-square: χ²={chi2_norm:.3f}, p={p_norm:.4f}")

Assumptions

• Each expected frequency Eᵢ ≥ 5 (merge small categories if needed)
• Observations are independent
• Categories are mutually exclusive and exhaustive

Key Takeaways

1. Goodness-of-fit tests whether data matches a theoretical distribution
2. df = k − 1 (k = number of categories)
3. Expected frequencies must be ≥ 5 — merge bins if violated
4. Any theoretical distribution can be tested: normal, Poisson, binomial, uniform
5. Cramér's V = √(χ²/n) measures effect size