Two measurements per individual
Two-variable data records two characteristics for each person or object, such as height and arm span, or study time and test score. The goal is to see whether there’s an association between them.
Explanatory and response variables
| Role | Description | Axis |
|---|---|---|
| Explanatory variable | the one that may explain or predict | x |
| Response variable | the outcome | y |
Example: Temperature (explanatory) and ice cream sales (response).
Numerical data: scatter plots
For two numerical variables, use a scatter plot. Describe:
| Feature | Questions |
|---|---|
| Direction | positive, negative or none? |
| Form | linear or curved? |
| Strength | strong, moderate or weak? |
| Outliers | any unusual points? |
Then quantify with the correlation coefficient r and a line or curve of best fit.
Categorical data: two-way tables
For two categorical variables, use a two-way table.
Example: Survey of 200 students: grade level vs preferred lunch.
| Cafeteria | Packed | Total | |
|---|---|---|---|
| Grade 9–10 | 70 | 30 | 100 |
| Grade 11–12 | 45 | 55 | 100 |
| Total | 115 | 85 | 200 |
Conditional percentages:
- Grade 9–10: 70% cafeteria
- Grade 11–12: 45% cafeteria
The percentages differ, so there is an association between grade and lunch choice.
Mixed data
One categorical and one numerical variable (e.g., gender and height): compare groups with side-by-side box plots.
From description to conclusion
- Identify the variables and their types.
- Choose a display (scatter plot, two-way table, side-by-side box plots).
- Describe the pattern.
- Quantify it (r, regression line, conditional percentages).
- Draw a cautious conclusion; association ≠ causation.
Worked examples
Example 1: Study time and test score: which is explanatory? Study time.
Example 2: In a two-way table, 30 of 50 adults and 10 of 50 teens prefer news on TV. Association? Yes: 60% vs 20%.
Example 3: A scatter plot of age and reaction time rises in a curve. Form? Non-linear (curved).
Common mistakes
- Swapping explanatory and response variables.
- Comparing raw counts in two-way tables when group sizes differ (use percentages).
- Concluding causation from association.
Try it yourself
- Explanatory or response: fertilizer amount in a study of crop yield?
- Which display for "favourite music genre" vs "age group"?
- In the lunch table, what percent of Grade 11–12 students pack lunch?
- Name the four features to describe in a scatter plot.
Answers: 1) explanatory 2) a two-way table 3) 55% 4) direction, form, strength, outliers
Quick recap
- Two-variable data looks for an association.
- Numerical: scatter plot; categorical: two-way table.
- Identify the explanatory variable and response variable, then describe and quantify.