Inferential Statistics • Lesson 17
Chi-Square Tests
Learn how chi-square tests help analyze categorical data, compare observed and expected frequencies, and determine whether categorical variables are associated.
What You Will Learn
1. What Is a Chi-Square Test?
A chi-square test is a statistical test commonly used with categorical data.
It compares observed frequencies with frequencies that would be expected under a particular hypothesis.
Observed
What happened
Expected
What H₀ predicts
The test asks whether the difference between observed and expected counts is larger than we would reasonably expect from random variation under the null hypothesis.
2. Categorical Data
Categorical variables place observations into groups or categories.
Payment Method
Cash, Card, UPI
Customer Type
New, Returning
Product Category
Laptop, Phone, Tablet
Chi-square methods work with counts or frequenciesrather than directly comparing numerical means.
3. Observed vs Expected Frequency
The observed frequency is the actual number of observations in a category.
The expected frequency is the number we would expect if the null hypothesis were true.
| Category | Observed | Expected |
|---|---|---|
| A | 40 | 50 |
| B | 55 | 50 |
| C | 45 | 50 |
| D | 60 | 50 |
Key idea:
A chi-square statistic becomes larger when the observed frequencies differ substantially from the expected frequencies, relative to the expected counts.
4. Chi-Square Statistic
χ² = Σ (O − E)² / E
O
Observed frequency
E
Expected frequency
Σ
Add the contribution from every category
Worked Example: Customer Preference
Suppose a company expects customers to choose four payment methods equally. From 200 customers, the observed counts are:
| Payment Method | Observed (O) | Expected (E) |
|---|---|---|
| Cash | 40 | 50 |
| Card | 60 | 50 |
| UPI | 70 | 50 |
| Wallet | 30 | 50 |
Expected frequency:
200 ÷ 4 = 50 per category
Cash: (40 − 50)² / 50 = 2
Card: (60 − 50)² / 50 = 2
UPI: (70 − 50)² / 50 = 8
Wallet: (30 − 50)² / 50 = 8
χ² = 2 + 2 + 8 + 8 = 20
5. Chi-Square Goodness-of-Fit Test
A goodness-of-fit test examines whether observed categorical frequencies are consistent with a specified distribution or set of category proportions.
H₀
The population category proportions follow the specified distribution.
Hₐ
The population category proportions do not follow the specified distribution.
Example questions
- • Are four payment methods equally popular?
- • Does a product category distribution match the company's target?
- • Do customer choices follow an expected market distribution?
6. Chi-Square Test of Independence
A chi-square test of independence examines whether two categorical variables are associated in a population.
Variable 1
Device Type
Mobile, Desktop, Tablet
Variable 2
Purchase
Yes, No
Business question:
Is purchase behavior associated with the customer's device type?
H₀: Independent
The two categorical variables are independent in the population.
Hₐ: Associated
The two categorical variables are associated in the population.
7. Contingency Tables
Independence tests commonly use a contingency table showing counts for combinations of two categorical variables.
| Device | Purchased | Did Not Purchase | Total |
|---|---|---|---|
| Mobile | 80 | 120 | 200 |
| Desktop | 60 | 90 | 150 |
| Tablet | 20 | 30 | 50 |
8. Expected Frequency in an Independence Test
For a contingency table, the expected count for a cell is calculated using the corresponding row and column totals.
Expected Count = (Row Total × Column Total) / Grand Total
Example:
If a row total is 200, a column total is 140, and the grand total is 400:
Expected = (200 × 140) / 400 = 70
9. Degrees of Freedom
Goodness-of-Fit
df = k − 1
k is the number of categories.
Independence
df = (r − 1)(c − 1)
r is the number of rows and c is the number of columns.
Example:
A 3 × 2 contingency table has:
df = (3 − 1)(2 − 1) = 2
10. Interpreting the Result
Like other hypothesis tests, the chi-square test can be interpreted using a p-value and a chosen significance level α.
If p ≤ α
Reject H₀ because the observed data provide sufficient evidence against the null hypothesis under the chosen procedure.
If p > α
Fail to reject H₀ because the evidence is insufficient to reject the null hypothesis at that significance level.
11. Important Conditions
The data should be categorical counts or frequencies.
Observations should be appropriately independent for the study design.
Expected cell counts should generally be sufficiently large for the usual chi-square approximation.
Very small expected counts can make the chi-square approximation unreliable.
The test should match the research question: goodness-of-fit or independence.
12. Which Chi-Square Test?
| Feature | Goodness-of-Fit | Independence |
|---|---|---|
| Main question | Does one categorical distribution match a specified pattern? | Are two categorical variables associated? |
| Data structure | One categorical variable | Two categorical variables |
| Example | Do four products have equal preference? | Is purchase associated with device type? |
Interactive Practice
Test your understanding of chi-square tests.
CHECK YOUR UNDERSTANDING
Chi-square tests are commonly used with which type of data?
CHECK YOUR UNDERSTANDING
What does the observed frequency represent?
CHECK YOUR UNDERSTANDING
Which formula represents the chi-square statistic?
CHECK YOUR UNDERSTANDING
Which test checks whether observed category frequencies match a specified distribution?
CHECK YOUR UNDERSTANDING
A chi-square test of independence is used to investigate:
CHECK YOUR UNDERSTANDING
For a 4 × 3 contingency table, what are the degrees of freedom?
Fill in the Blanks
A chi-square test commonly works with categorical ______.
The expected frequency represents what the null hypothesis ______.
For a goodness-of-fit test with k categories, df = k − ______.
For an r × c contingency table, df = (r − 1)(c − ______).
Data Analytics Challenge
Customer Support Channel
A company expects four support channels to be used equally. It records 400 support requests:
Phone
80
120
Chat
140
Social
60
Expected frequency:
400 ÷ 4 = 100
Chi-square statistic:
(80 − 100)² / 100 = 4
(120 − 100)² / 100 = 4
(140 − 100)² / 100 = 16
(60 − 100)² / 100 = 16
χ² = 40
Degrees of freedom:
df = 4 − 1 = 3
Final Challenge
Device Type and Purchase
An online store wants to know whether device type is associated with whether a visitor makes a purchase.
| Device | Purchased | No Purchase |
|---|---|---|
| Mobile | 80 | 120 |
| Desktop | 60 | 90 |
| Tablet | 20 | 30 |
Your task:
- Identify the appropriate chi-square test.
- Write H₀ and Hₐ.
- Calculate the expected count for Mobile + Purchased.
- Calculate the degrees of freedom.
Test: Chi-square test of independence
H₀: Device type and purchase are independent.
Hₐ: Device type and purchase are associated.
Mobile row total = 200
Purchased column total = 160
Grand total = 400
Expected Mobile + Purchased = (200 × 160) / 400
Expected count = 80
df = (3 − 1)(2 − 1)
df = 2
Lesson Summary
✓ Chi-square tests are commonly used with categorical counts.
✓ Observed counts come from the actual data.
✓ Expected counts are calculated under H₀.
✓ χ² = Σ(O − E)² / E.
✓ Goodness-of-fit tests compare one distribution with a specified pattern.
✓ Independence tests examine association between two categorical variables.
✓ For independence, df = (r − 1)(c − 1).
✓ Use the p-value and chosen α to make the statistical decision.