Statistics - Chi-square Test

The Chi-square test (also written as χ² test) is a statistical hypothesis test used to determine whether there is a significant association between two categorical variables. It compares the observed frequencies in a contingency table to the expected frequencies that would occur if the variables were independent.

---

🧪 Types of Chi-Square Tests

1. Chi-square test of independence
   ➤ Tests whether two categorical variables are independent.
   (Most commonly used in contingency tables.)

2. Chi-square goodness-of-fit test
   ➤ Tests whether an observed frequency distribution matches an expected distribution.

---

📊 Example (Chi-square test of independence)

Suppose you want to know whether gender and voting preference are related:

	Voted A	Voted B	Total
Male	30	20	50
Female	10	40	50
Total	40	60	100

 

We want to test if voting preference is independent of gender.

---

🧮 Chi-square Test Formula

 

import numpy as np
from scipy.stats import chi2_contingency

# Contingency table
table = np.array([[30, 20],
                  [10, 40]])

# Perform chi-square test of independence
chi2, p, dof, expected = chi2_contingency(table)

print("Chi-square statistic:", chi2)
print("Degrees of freedom:", dof)
print("Expected frequencies:\n", expected)
print("P-value:", p)

 

Chi-square statistic: 15.041666666666668
Degrees of freedom: 1
Expected frequencies:
 [[20. 30.]
 [20. 30.]]
P-value: 0.00010516355403363114

 

📌 Output (interpreted)

• Chi-square statistic: How different the observed frequencies are from expected ones.

• Degrees of freedom: ({#rows} - 1) * ({#cols} - 1)

• P-value: Probability of observing such a difference if the variables were truly independent.

---

✅ Interpretation

• If p-value < 0.05 → Reject the null hypothesis → Variables are likely associated

• If p-value ≥ 0.05 → Do not reject the null → No significant association