Graphing pairs of data
A scatter plot shows two numerical variables for each item as a point. It’s used to look for a relationship.
Example: Hours studied (x) and test score (y) for 8 students:
| Hours | 1 | 2 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Score | 55 | 60 | 58 | 68 | 72 | 75 | 84 | 88 |
Plot each pair as a point. Don’t join the points.
Types of correlation
Correlation describes the pattern:
| Pattern | Name | Example |
|---|---|---|
| points rise left to right | positive correlation | study time and test scores |
| points fall left to right | negative correlation | car age and resale value |
| no clear pattern | no correlation | shoe size and test score |
Strength:
- Strong: points cluster tightly around a line.
- Weak: points are scattered but show a general trend.
Line of best fit
A line of best fit runs through the middle of the points, with roughly equal numbers above and below. It summarizes the trend.
From the study data, a line might be: score ≈ 5.5 × hours + 50.
Making predictions
- Interpolation: predicting within the range of data (e.g., 4.5 hours → about 75%). Usually reliable.
- Extrapolation: predicting beyond it (e.g., 15 hours → 132%?!). Risky and sometimes impossible.
Outliers
An outlier is a point far from the pattern. It might be an error, or something unusual worth investigating. Outliers can pull a line of best fit off course.
Correlation is not causation
Two variables can move together without one causing the other.
- Ice cream sales and sunburns both rise in summer. Ice cream doesn’t cause sunburn: hot sunny weather affects both. This hidden third factor is called a lurking or confounding variable.
Worked examples
Example 1: As temperature rises, heating bills fall. Type? Negative correlation.
Example 2: A scatter plot of height vs. favourite number shows random points. Type? No correlation.
Example 3: Using score ≈ 5.5h + 50, predict the score for 3.5 h. ≈ 69%.
Common mistakes
- Joining points like a line graph.
- Forcing the line through the first and last points instead of the middle of all points.
- Claiming causation from correlation.
- Extrapolating too far.
Try it yourself
- Describe the likely correlation between hours of exercise per week and resting heart rate.
- Describe the correlation between a person’s height and arm span.
- What is a point far from the pattern called?
- Why is predicting a score for 20 hours of study risky?
Answers: 1) negative 2) strong positive 3) an outlier 4) It’s extrapolation far beyond the data, and scores can’t exceed 100%.
Quick recap
- A scatter plot shows paired data as points.
- Describe correlation: positive, negative or none; strong or weak.
- Use a line of best fit for predictions, carefully.