AI Basics with AK
Season 03 - Introduction to Statistics
Arun Koundinya Parasa
Episode 13 - Chi-Square Tests
Recap: Episode 10 till 12
- What is Hypothesis Testing?
- The Null and Alternative Hypotheses
- Type I and Type II Errors
- p-values and Significance Levels
- t-tests for comparing means
This is all great for numerical data, how can we compare when there is categorical data?
Our Earlier Hypothesis questions was like: “Is the average height different between groups?”
A Different Kind of Data
Numeric Data (What We’ve Tested)
- Heights, weights, test scores
- Blood pressure, wait times
- We compute means, standard deviations
- We ask: “Is the average different?”
These are continuous or discrete numbers.
Categorical Data (What Chi-Square Tests)
- Colours, genders, political parties
- Yes/No responses, product preferences
- We count how many fall in each category
- We ask: “Are the counts what we expected?”
These are frequencies and proportions.
You can’t take the mean of “Red, Blue, Green.” But you can count them.
Two Questions Chi-Square Answers
Test 1 — Goodness of Fit
“Does the distribution of my data match an expected pattern?”
Example: Is a die fair? Do customers prefer all flavours equally?
You have one categorical variable. You compare observed counts to expected counts.
Test 2 — Test of Independence
“Are two categorical variables related to each other?”
Example: Is gender related to product preference? Is smoking related to disease status?
You have two categorical variables. You test whether knowing one tells you anything about the other.
The Chi-Square Distribution
The chi-square distribution is right-skewed and always positive. As df increases, it shifts right and becomes more symmetric. Our test statistic must exceed the critical value to reject H₀.
The Core Idea: Observed vs Expected
The Question We Always Ask
If the null hypothesis were true —
what counts would we expect to see?
Then we compare those expected counts to what we actually observed.
If the gap is small → data is consistent with H₀
If the gap is large → something is going on
The Chi-Square Statistic
\[\chi^2 = \sum \frac{(O - E)^2}{E}\]
- O = Observed count in each cell
- E = Expected count under H₀
Each cell contributes to the total.
Large \(\chi^2\) → observed and expected are far apart → evidence against H₀
Small \(\chi^2\) → data fits the expected pattern → no evidence against H₀
Test 1 — Goodness of Fit
Setup: You roll a die 60 times. If the die is fair, you expect each face 10 times.
| Face | Observed (O) | Expected (E) | (O−E)²/E |
|---|---|---|---|
| 1 | 8 | 10 | 0.40 |
| 2 | 12 | 10 | 0.40 |
| 3 | 7 | 10 | 0.90 |
| 4 | 14 | 10 | 1.60 |
| 5 | 11 | 10 | 0.10 |
| 6 | 8 | 10 | 0.40 |
\[\chi^2 = 0.40+0.40+0.90+1.60+0.10+0.40 = 3.80\]
df = k − 1 = 5 | Critical value at α=0.05 → 11.07
3.80 < 11.07 → Fail to Reject H₀ ✅
No evidence the die is unfair.
Test 2 — Chi-Square Test of Independence
The Question: Is there a relationship between two categorical variables?
Setup: A survey of 200 customers asks: “Do you prefer Product A, B, or C?” — split by gender.
| Product A | Product B | Product C | Row Total | |
|---|---|---|---|---|
| Male | 40 | 35 | 25 | 100 |
| Female | 30 | 45 | 25 | 100 |
| Col Total | 70 | 80 | 50 | 200 |
H₀: Gender and product preference are independent
H₁: Gender and product preference are associated
Computing Expected Counts
For each cell, the expected count under independence is:
\[E_{ij} = \frac{\text{Row Total}_i \times \text{Column Total}_j}{\text{Grand Total}}\]
| Product A | Product B | Product C | |
|---|---|---|---|
| Male | (100×70)/200 = 35 | (100×80)/200 = 40 | (100×50)/200 = 25 |
| Female | (100×70)/200 = 35 | (100×80)/200 = 40 | (100×50)/200 = 25 |
\[\chi^2 = \frac{(40-35)^2}{35}+\frac{(35-40)^2}{40}+\frac{(25-25)^2}{25}+\frac{(30-35)^2}{35}+\frac{(45-40)^2}{40}+\frac{(25-25)^2}{25}\]
\[= 0.714 + 0.625 + 0 + 0.714 + 0.625 + 0 = \mathbf{2.678}\]
df = (rows−1)(cols−1) = 1×2 = 2 | Critical value = 5.99
2.678 < 5.99 → Fail to Reject H₀ ✅ No significant association.
Interactive: Test of Independence
Switch between weak, moderate, and strong associations. Watch how χ² and the p-value respond as the gap between groups widens.
Real World Applications
Goodness of Fit In Practice
Marketing: Do customers buy all products equally? Or is one product dominating?
Genetics: Do offspring ratios match Mendel’s predicted 3:1 ratio?
Operations: Are defects distributed equally across shifts? Or is one shift producing more errors?
Independence Test In Practice
Healthcare: Is smoking status associated with lung disease?
HR analytics: Is employee attrition related to department?
Retail: Is purchase behaviour related to customer age group?
A/B Testing: Is click-through rate independent of the ad version shown?
