Scatter Graphs and Correlation
A scatter graph investigates the relationship between two numerical variables measured on the same individuals. Use the pattern to make cautious estimates, while keeping association separate from cause.
Foundation & Higher, including lines of best fit, interpolation and the limitations of extrapolation. You need to plot coordinate pairs accurately and read graph scales. Review Linear Graphs if needed.
Plot paired measurements
Each point represents one pair of measurements. Keep the pairing intact: sorting the two columns separately would create different data. Label both axes and choose consistent scales that show the observations clearly.
| Hours | Score / 100 |
|---|---|
| 1 | 42 |
| 2 | 43 |
| 3 | 52 |
| 4 | 53 |
| 5 | 62 |
| 6 | 63 |
| 7 | 72 |
| 8 | 73 |
Worked example — plot one student
The student who revised for 3 hours scored 52. Plot (3, 52): move to 3 on the hours axis, then to 52 on the score axis. Do not join the observations in the order they were recorded.
Describe direction and strength
Positive correlation: larger values of one variable tend to go with larger values of the other. Negative correlation: larger values tend to go with smaller values. No correlation: no clear relationship is visible.
A pattern is stronger when points lie closer to a clear trend; it is weaker when they are more scattered. A steep line is not automatically a strong correlation. A perfect straight-line pattern has every point on one line.
The heating example shows negative correlation: warmer days tend to use less heating energy. Describe the tendency, not a rule that every individual pair must follow.
In this small sample, travel distance does not usefully predict puzzle time. Both variables are measurements, but there is no clear upward or downward pattern. Do not force a line of best fit when there is no useful relationship.
Draw and use a line of best fit
For an approximately linear pattern, use a ruler to draw a line through the middle of the point cloud, with points reasonably balanced above and below it across the graph. It need not pass through a particular point or the origin. Do not simply join the first and last points.
Worked example — interpolate within the observed range
The dashed line passes through (1,40) and (8,75). At 4.5 hours, move vertically to the line and then horizontally to the score axis: about 57.5, or about 58 marks. This is interpolation because 4.5 lies within the observed 1–8 hours.
To estimate the hours associated with a score of 60, read horizontally from 60 to the line, then down: about 5 hours. These are typical estimates, not promised results for an individual.
For this drawn line, the rise is 35 marks over 7 hours, giving 5 marks per hour and the convenient equation score ≈ 5 × hours + 35. You can read the graph directly; different sensible hand-drawn fits give slightly different estimates.
Treat extrapolation and outliers carefully
Extrapolation predicts outside the range of observed data. It is less reliable because the relationship may change. Distance beyond the observed range and the context both matter.
Worked example — a model has limits
Continuing the revision line to 10 hours gives 5 × 10 + 35 = 85 marks. This is an extrapolation beyond 8 hours. At 14 hours the same rule gives 105, impossible for a test out of 100, so the line cannot describe unlimited increases.
An isolated point far from the pattern may be an outlier. For example, a further student at (5,30) would be far below this trend. Check a possible recording error or a real difference in circumstances. Do not automatically erase an inconvenient observation or drag the fit towards one unusual point.
Correlation does not establish cause
Two variables may vary together because another factor affects both, because the relationship runs in the other direction, or by chance. A scatter graph alone does not show which explanation is right.
Worked example — consider a third factor
Ice-cream sales and swimming attendance may rise together because warm weather affects both. The graph would not show that buying ice cream causes swimming.
Similarly, revision hours and scores may also depend on prior understanding, teaching and study quality. The observed association is useful, but it does not prove that one extra hour will cause exactly five extra marks for every student.
Support a conclusion with the direction, strength, range and context of the observed data. Avoid the word “proves” when the evidence only suggests an association.
Think Like a Mathematician
A useful prediction needs both a visible relationship and a sensible context. A line drawn perfectly with a ruler can still give a poor prediction if it extends too far beyond the data or confuses correlation with causation.
Common Mistakes
Broken pairs: each plotted point must use measurements from the same individual.
Joining every point: use an overall fit for a suitable pattern.
Forcing the origin: the fitted line need not pass through (0,0).
Certainty from correlation: a trend gives estimates and does not by itself prove cause.
Practice
1. A student revised for 6 hours and scored 63. Give the coordinate pair when hours is horizontal.
(6,63).
2. As a car gets older, its resale value generally decreases. Describe the correlation.
Negative correlation: greater age tends to accompany lower value.
3. Does a steeper fitted line necessarily mean stronger correlation?
No. Strength concerns how closely the points follow the trend, not its steepness.
4. Use the example fitted line to estimate the score at 6.5 hours.
5 × 6.5 + 35 = 67.5, about 68 marks. This is interpolation within 1–8 hours.
5. Use that line to estimate the hours associated with 70 marks.
70 = 5 × hours + 35, so hours ≈ (70 − 35)/5 = 7 hours. Reading the graph gives the same estimate.
6. Is a prediction at 9 hours interpolation or extrapolation? State one caution.
Extrapolation, because observations stop at 8 hours. The trend may not continue beyond the observed range.
7. Why is a point at (5,30) worth investigating?
It is far below the main trend: the fitted value at 5 hours is 60. It may be an error or a genuine unusual result; investigate rather than automatically deleting it.
8. Umbrella sales and traffic delays rise on rainy days. Does this prove umbrellas cause delays?
No. Rain may affect both. Correlation alone does not establish causation.
9. A scatter graph has no clear pattern. Should you draw a line and make a precise prediction?
No. There is no useful observed relationship to support that prediction.
Frequently Asked Questions
Can several lines of best fit be acceptable?
Yes. A hand-drawn line is an estimate. It should follow the main linear pattern with a reasonable balance of points, and predictions should be consistent with the line you drew.
Is interpolation always accurate?
No. It usually has more support than extrapolation, but scatter, sample limitations and the context still create uncertainty.