Chi-Square Test

 

What is the Chi-Square Test?

The Chi-Square (χ²) test is a non-parametric statistical test used to determine if there is a significant association between two categorical variables or if a categorical variable follows a specific distribution.

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✅ Types of Chi-Square Tests

1. Chi-Square Test for Independence

Used to determine if two categorical variables are independent or related.

Example: Is there a relationship between gender and preference for a product?

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2. Chi-Square Goodness-of-Fit Test

Used to determine whether a single categorical variable follows a hypothesized distribution.

Example: Does a die roll show numbers uniformly?

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Assumptions of the Chi-Square Test

• Data are frequencies or counts (not percentages or means).

• Categories are mutually exclusive.

• Observations are independent.

• Expected frequency in each cell should be ≥ 5 (or use Fisher's exact test if < 5).

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General Formula

$$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$

Where:

• $O_i$ = Observed frequency

• $E_i$ = Expected frequency

• The sum is over all categories or cells

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📊 1. Chi-Square Test for Independence – Example

Question: Is there an association between gender and ice cream preference?

Ice Cream	Male	Female	Total
Vanilla	20	30	50
Chocolate	25	25	50
Total	45	55	100

Step 1: Hypotheses

• H₀: Gender and ice cream preference are independent.

• H₁: Gender and ice cream preference are associated.

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Step 2: Calculate Expected Frequencies

$$E_{ij} = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}$$

Ice Cream	Male (Expected)	Female (Expected)
Vanilla	$\frac{50×45}{100} = 22.5$	$\frac{50\times 55}{100}=27.5$
Chocolate	$\frac{50×45}{100} = 22.5$	$\frac{50×55}{100} = 27.5$

Step 3: Apply Chi-Square Formula

$$\chi^2 = \frac{(20 - 22.5)^2}{22.5} + \frac{(30 - 27.5)^2}{27.5} + \frac{(25 - 22.5)^2}{22.5} + \frac{(25 - 27.5)^2}{27.5}$$ $$= \frac{6.25}{22.5} + \frac{6.25}{27.5} + \frac{6.25}{22.5} + \frac{6.25}{27.5}$$ $$= 0.278 + 0.227 + 0.278 + 0.227 = 1.01$$

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Step 4: Degrees of Freedom

$$df = (r - 1)(c - 1) = (2 - 1)(2 - 1) = 1$$

Critical value at α = 0.05 and df = 1 ≈ 3.841

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Step 5: Conclusion

Since χ² = 1.01 < 3.841 → Fail to reject H₀

✅ Conclusion: No significant association between gender and ice cream preference.

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📊 2. Chi-Square Goodness-of-Fit – Example

Question: A die is rolled 60 times. Observed frequencies:
[10, 11, 8, 9, 12, 10]. Is the die fair?

Step 1: Hypotheses

• H₀: Die is fair → all faces have equal probability.

• H₁: Die is not fair.

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Step 2: Expected Frequency

If fair:

$$E_i = \frac{60}{6} = 10 \text{ for each face}$$

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Step 3: Calculate χ²

$$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} = \frac{(10-10)^2}{10} + \frac{(11-10)^2}{10} + \frac{(8-10)^2}{10} + \frac{(9-10)^2}{10} + \frac{(12-10)^2}{10} + \frac{(10-10)^2}{10}$$ $$= 0 + 0.1 + 0.4 + 0.1 + 0.4 + 0 = 1.0$$

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Step 4: Degrees of Freedom

$$df = k - 1 = 6 - 1 = 5$$

Critical value at α = 0.05, df = 5 ≈ 11.07

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Step 5: Conclusion

Since χ² = 1.0 < 11.07 → Fail to reject H₀

✅ Conclusion: The die appears to be fair.

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📌 Summary Table

Test Type	Use Case	df
Goodness-of-Fit	One categorical variable vs. expected values	$k - 1$
Test for Independence	Relationship between 2 categorical variables	$(r - 1)(c - 1)$