Correlation
A scatterplot shows a pattern. Correlation gives a number for its linear direction and strength. Learn what that number tells you, what it leaves out, and how to explain it clearly.
By the end of this lesson, you should be able to:
- Interpret the sign and magnitude of a correlation in context.
- Compare linear strength without confusing negative direction with weak association.
- Use a scatterplot to recognize curvature, unusual points and limits on causal claims.
Before you start: Review Topic 5.1: scatterplots. Be comfortable with paired quantitative variables and describing form, direction, strength and unusual features.
First time learning this? Start with the student example and visual gallery. The formula explanation is optional support.
Here to revise? Use the correlation checklist, then try the practice questions before opening solutions.
The concept in 60 seconds
Correlation, written , measures the direction and strength of a linear association between two quantitative variables. It summarizes how closely the points follow a straight pattern.
Read it in two steps: the sign gives direction; the distance from gives linear strength. A value nearer or indicates a stronger linear association.
For example, describes a stronger linear association than . The first is negative and the second is positive. A minus sign does not make a relationship weak.
Always look at the scatterplot as well. Correlation does not describe curvature, explain why a relationship exists, or establish that one variable causes the other to change.
Quick check: what does a positive sign mean?
Larger values of one variable tend to occur with larger values of the other. It does not mean all observations are positive or that every point follows the tendency.
A real situation: practice time and quiz scores
Continue the invented teaching data from Topic 5.1. A teacher records weekly practice time and quiz score for volunteers in one class. The same student supplies both measurements in each row; scores are on a -point scale.
| Student ID | Practice time (hours) | Quiz score (points) |
|---|---|---|
| A | ||
| B | ||
| C | ||
| D | ||
| E | ||
| F | ||
| G | ||
| H | ||
| I | ||
| J | ||
| K |
Invented teaching data. Student K uses a diamond marker and a text label; the distinction does not rely on colour alone.
Using all recorded students:
Interpretation: In this volunteer group, students who report more practice time generally tend to have higher quiz scores. The correlation indicates a moderate positive linear association. Most points follow an increasing, approximately straight pattern closely, but student K at departs from it.
The description “moderate” is a qualitative judgment, not a universal numerical boundary. State the actual value and describe the graph so the reader can see the evidence.
How much does student K affect the number?
For the first students alone, ; with student K included, . This comparison shows that the unusual pair weakens this dataset’s linear correlation. It is not a reason to discard the observation. Investigate recording errors or different circumstances, and report any justified exclusion transparently.
An unusual point can increase or decrease correlation depending on its position. Do not assume the effect from its label alone.
The study is observational and based on volunteers. These results describe the observed group; the correlation alone does not show that extra practice caused higher scores or justify generalizing to all students.
Key ideas and notation
Correlation coefficient:
A numerical summary of linear direction and strength in paired quantitative data. It has no measurement units.
Possible values
The endpoints represent perfect straight-line association, provided both variables vary.Sign
indicates positive direction; indicates negative direction. indicates no linear association in the data.
Magnitude:
The absolute value measures distance from . A magnitude closer to indicates stronger linear association.
and .
Linear versus nonlinear
A straight pattern can be summarized by correlation. A clear curved pattern may have a small correlation or a surprisingly large one.
Association versus causation
Correlation describes measurements that occur together. Study design and other evidence determine whether a causal conclusion is justified.
Keep three ideas separate: correlation describes linear strength; slope describes change in the response per unit of the explanatory variable; a causal claim describes what would happen if an intervention changed a variable.
Read sign and linear strength
- Check the variables. They should be quantitative measurements paired from the same individuals.
- Read the sign. State whether the direction is positive or negative.
- Read the magnitude. Compare absolute values when judging linear strength.
- Return to the graph. Check for a straight form, unusual points and groups.
- Write in context. Name the variables and observed group, then explain the tendency.