Ever wondered how researchers check whether a claim about categories, say customer preferences or product choices, actually holds true? That’s where the Chi-Square Test steps in. It’s the most widely used tool for categorical data.
Let’s walk step by step, starting from the basics, building all the way to practical applications.
Purpose of Chi-Square Test
The chi-square test answers one simple but powerful question: Is the difference we see in categorical data just by chance, or does it reflect something meaningful?
- Goodness of Fit: Checks whether an observed distribution matches an expected one.
- Test of Independence: Examines whether two categorical variables are related or independent.
When to Use Chi-Square
Use this test when:
- Data is categorical (e.g., gender, product preference, type of service used).
- You want to compare observed frequencies (what actually happened) with expected frequencies (what should have happened theoretically).
- Sample size is reasonably large, since the test relies on approximation.
Formula and Degrees of Freedom
The heart of the chi-square test is its formula:
χ² = Σ ( (Oi − Ei)² / Ei )
Where:
- Oi = Observed frequency
- Ei = Expected frequency
Degrees of freedom (df):
- Goodness of fit: df = (n − 1), where n = number of categories.
- Test of independence: df = (r − 1)(c − 1), where r = rows, c = columns in a contingency table.
Worked-Out Examples
a. Goodness of Fit Example
Problem: A dice is rolled 60 times. The observed frequencies of outcomes are:
1: 8, 2: 9, 3: 10, 4: 11, 5: 12, 6: 10.
Is the dice fair?
Step 1: Expected frequency
If fair, each outcome = 60/6 = 10.
Step 2: Apply formula
χ² = ( (8−10)²/10 ) + ( (9−10)²/10 ) + … + ( (10−10)²/10 )
= (4/10) + (1/10) + (0/10) + (1/10) + (4/10) + (0/10)
= 1.0
Step 3: Compare with table
df = 5. At 5% level, χ²(0.05,5) ≈ 11.07. Since 1.0 < 11.07, the dice can be considered fair.
b. Test of Independence Example
Problem: A survey is conducted on 100 people about their preference for tea or coffee, classified by gender.
| Preference | Male | Female | Total |
|---|---|---|---|
| Tea | 30 | 10 | 40 |
| Coffee | 20 | 40 | 60 |
| Total | 50 | 50 | 100 |
Step 1: Expected frequencies
E = (Row total × Column total) / Grand total.
- For Male-Tea: (40×50)/100 = 20
- For Female-Tea: (40×50)/100 = 20
- For Male-Coffee: (60×50)/100 = 30
- For Female-Coffee: (60×50)/100 = 30
Step 2: Apply formula
χ² = Σ (O−E)²/E = (30−20)²/20 + (10−20)²/20 + (20−30)²/30 + (40−30)²/30
= (100/20) + (100/20) + (100/30) + (100/30)
= 5 + 5 + 3.33 + 3.33 = 16.66
Step 3: Compare with table
df = (2−1)(2−1) = 1. At 5% level, χ²(0.05,1) ≈ 3.84. Since 16.66 > 3.84, gender and drink preference are not independent.
Assumptions & Limitations
Assumptions
- Data must be in frequency form, not percentages.
- Observations must be independent.
- Expected frequency in each cell should ideally be at least 5.
- Sample should be random and large enough for valid approximation.
Limitations
- It’s sensitive to sample size: large samples may exaggerate small differences.
- Not suitable for small expected frequencies (< 5).
- Only applicable to categorical data, not continuous data without grouping.
- Doesn’t measure strength of association—only tells if association exists.
Applications in Business Research
In business statistics, chi-square is especially valuable:
- Studying consumer preference patterns (brand loyalty, product choice).
- Testing independence between demographic factors and buying behavior.
- Checking fairness of random sampling in surveys.
- Verifying claims in market research (e.g., whether advertising influences preference).
The chi-square test gives us a sharp lens to examine categorical data. It helps identify whether patterns we observe are real or just chance accidents. It’s simple, formula-based, and extremely exam-relevant. But remember:
While it can flag relationships, it doesn’t measure their strength.
Next time you see a survey result in newspapers, ask yourself, would a chi-square test support that claim? That’s the kind of reflective thinking UGC NET demands.