Correlation Between Variables: Contingency Tables and Scatter Plots
To understand the relationship between two variables, we use correlation.
But how we analyze that relationship depends on the type of data we’re working with — categorical or quantitative.
📚 This post is part of the "Intro to Statistics" series
🔙 Previously: A Real-World Statistics Example
🔜 Next: Understanding Pearson's R
🎓 Real-Life Case: Study Habits and Exam Performance
Imagine a high school counselor wants to investigate the relationship between how often students study and whether they pass or fail a weekly quiz.
She surveys 30 students and records two things:
- 📚 Study Time Category: Rarely, Sometimes, Often
- ✅ Quiz Result: Pass or Fail
🧮 Step 1: The Contingency Table
This type of table is used for categorical variables. It shows how often combinations of categories occur.
| Study Frequency \ Quiz Result | Pass | Fail | Total |
|---|---|---|---|
| Rarely | 3 | 7 | 10 |
| Sometimes | 6 | 4 | 10 |
| Often | 9 | 1 | 10 |
| Total | 18 | 12 | 30 |
🔁 Step 2: Conditional Proportions
The raw counts don’t tell the full story. So we calculate the percentage of each outcome within each group.
For example:
- Among students who study Rarely, 3/10 passed = 30%
- Among those who study Often, 9/10 passed = 90%
| Study Frequency | % Passed | % Failed |
|---|---|---|
| Rarely | 30% | 70% |
| Sometimes | 60% | 40% |
| Often | 90% | 10% |
✅ These are conditional proportions — percentages within each row.
📊 Step 3: Understanding Proportions — Quick Summary
We use conditional proportions to look within groups, and marginal proportions to summarize a variable on its own.
- Conditional example:
Among those who study Rarely → 3/10 passed = 30% - Marginal example:
Overall pass rate → 18/30 = 60%
📚 Want a full breakdown with examples, visual tables, and when to use each?
👉 Read: Conditional vs. Marginal Proportions →
🔍 Step 4: Interpreting the Categorical Correlation
The more a student studies, the more likely they are to pass.
We can see a positive association in the conditional proportions:
- Rarely study → low pass rate
- Often study → high pass rate
➡️ But contingency tables don’t quantify correlation — they only show patterns.
🔄 Step 5: Let’s Make It Quantitative
Now let’s change the scenario:
The counselor asks students for their exact number of study hours per week and records their quiz scores out of 100.
Here’s a sample:
| Hours Studied | Quiz Score |
|---|---|
| 2 | 50 |
| 3 | 55 |
| 5 | 65 |
| 7 | 70 |
| 8 | 76 |
| 10 | 85 |
| 12 | 92 |
📈 Step 6: Scatter Plot
This type of plot is perfect for quantitative variables.
It helps us visually assess correlation:
- Each point = one student
- X-axis: Hours studied
- Y-axis: Quiz score
You’ll notice: the more hours students study, the higher their scores.
This is a strong positive relationship.
🧠 Level Up: Choosing the Right Correlation Approach Based on Data Types
Correlation analysis isn’t one-size-fits-all — the type of variables determines the best method:
- 📊 For two quantitative variables, measures like Pearson's r capture linear relationships.
- 📋 For two categorical variables, contingency tables and tests like Chi-square help assess association.
- 🔄 For mixed variable types, specialized methods like point-biserial correlation or ANOVA are used.
Understanding your data types ensures you pick the most powerful and appropriate analysis technique.
📌 Try It Yourself
Q: If your data has outliers that raise the mean, which measure of center is more reliable: mean or median?
💡 Show Answer
✅ Median — because it's resistant to outliers, unlike the mean which gets skewed.
✅ Conclusion
| Type of Data | Tool to Use | Example |
|---|---|---|
| Categorical (Nominal/Ordinal) | Contingency Table | Study Frequency vs Pass/Fail |
| Quantitative | Scatter Plot | Hours Studied vs Quiz Score |
🧠 Choose the right tool based on your variable types.
🔜 Up Next
Next, we’ll calculate the Pearson correlation coefficient (r) — a number that tells us how strong a linear relationship really is.