🧪 Chi-Square Test (χ² Test)

Last Updated: Jan 2026

The Chi-Square Test (χ² Test) is a non-parametric hypothesis test used to determine whether there is a significant relationship between categorical variables.

It compares observed frequencies with expected frequencies.

🗣 Hinglish Tip: Chi-Square test = actual data vs expected data ka comparison

When to Use Chi-Square Test?

Use Chi-Square Test when:

Data is categorical
Values are in frequency/count
Sample size is sufficiently large
Observations are independent

✘ Not used for:

Mean comparison
Numerical data

Types of Chi-Square Test

Chi-Square Test of Independence
Chi-Square Test of Goodness of Fit

👉 In this tutorial, we cover Test of Independence (most common)

Chi-Square Notation (Math Standard)

Observed frequency → $(O)$
Expected frequency → $(E)$
Chi-square statistic → $(\chi^2)$
Degrees of freedom → $(df)$
Significance level → $(\alpha)$

Chi-Square Formula

 $\chi^2 = \sum \frac{(O - E)^2}{E}$

Example

Problem Statement

A survey was conducted to see whether Gender and Preference for Online Course are independent.

Gender	Like Course	Dislike Course
Male	30	10
Female	20	40

Test at 5% significance level.

Step 1: State the Hypotheses

 $H_0: Gender and course preference are independent \\[0.5em] H_1: Gender and course preference are dependent$

Step 2: Create Observed Frequency Table (O)

Gender	Like	Dislike	Total
Male	30	10	40
Female	20	40	60
Total	50	50	100

Step 3: Calculate Expected Frequencies (E)

Formula:

 $E = (Row Total x Column Total) / Grand Total$

Cell	Calculation	E
Male-Like	(40 x 50) / 100	20
Male-Dislike	(40 x 50) / 100	20
Female-Like	(60 x 50) / 100	30
Female-Dislike	(60 x 50) / 100	30

Step 4: Compute $X^2$ Value

$O$	$E$	$(O-E)^2 / E$
30	20	5
10	20	5
20	30	3.33
40	30	3.33

 $\chi^2 = 5 + 5 + 3.33 + 3.33 = 16.66$

Step 5: Degrees of Freedom

 $df = (\text{rows} - 1)(\text{columns} - 1) \\[0.5em] df = (2 - 1)(2 - 1) = 1$

Step 6: Critical Value

At:

$\alpha$ = 0.05
df = 1

From Chi-Square table:

 $\chi^2_{0.05,1} = 3.84$

Step 7: Decision

Calculate χ² Value= 16.66
Compare with Critical Value= 3.84

Since:

 $16.66 > 3.84$

👉 Reject $H_0$

Step 8: Conclusion

There is significant evidence to conclude that Gender and course preference are dependent.

🗣 Hinglish Tip: Chi-square zyada aaya → relation exist karta hai

Report An Issue

← Previous Next →