Correlation Coefficient Summarizes Linear Relationship
When you collect bivariate data (paired values $(x,y)$), a scatter plot can suggest whether the variables are related.
A more precise way to describe this relationship is the correlation coefficient, usually written as $r$.
It measures how closely the points lie to a straight line.
Definition
Bivariate data
Data consisting of pairs of values $(x, y)$ where two quantitative variables are measured for each case.
Definition
Correlation coefficient (Pearson’s r)
A number between $-1$ and $1$ that measures the strength and direction of a linear relationship between two variables.
Interpreting The Value Of r: Direction And Strength
Two features are encoded in $r$:
Direction (sign): whether $y$ tends to increase or decrease as $x$ increases.
Strength (magnitude): how closely the points follow a straight line.
Always: $$-1\le r\le 1$$
Direction: The Sign Of r
$r>0$ means a positive correlation: larger $x$ values tend to come with larger $y$ values.
$r<0$ means a negative correlation: larger $x$ values tend to come with smaller $y$ values.
$r=0$ means no linear correlation (but other relationships may still exist).
Strength: The Size Of |r|
$|r|\approx 1$: points lie very close to a straight line.
$|r|\approx 0$: points form a loose cloud with little linear pattern.
Note
"Strong" correlation does not mean "steep" slope.
A steep line can have $r\approx 1$ or $r\approx 0$ depending on how tightly points cluster around the line.
Common Mistake
A value of $r=0$ does not mean "no relationship."
It means "no linear relationship."
For example, data on a U-shaped curve can have $r\approx 0$ even though $x$ clearly influences $y$.
Scatter Diagrams Show What r Cannot
A single number cannot show everything.
A scatter diagram lets you check whether it is reasonable to summarize the relationship with a correlation coefficient.
You should look for:
Linearity: do points follow a roughly straight trend?
Outliers: are there points far from the pattern?
Clusters: are there separate groups (which can distort $r$)?
Tip
Before quoting $r$, always sketch or generatea scatter plot.
Correlation is a summary of the plot, not a replacement for it.
Outliers Can Change $r$ Dramatically
Because $r$ uses deviations from the mean, a single unusual point can pull the correlation strongly up or down.
Example
If most points form a weak cloud, but one extreme point lies far to the right and high up, it can create a moderate positive $r$ even though the main cluster shows little relationship.
Example
Attitudes To News Media
Ten students were scored for their attitude to social media as a news source ($x$) and television as a news source ($y$).
Higher scores mean higher satisfaction.
Student
1
2
3
4
5
6
7
8
9
10
Social media $x$
5
0
3
1
2
2
5
3
5
4
Television $y$
1
2
1
3
3
4
3
1
0
2
A scatter plot of these data is shown below.
For this data set, the correlation coefficient is approximately $r\approx -0.19$.
This suggests a very weak negative linear relationship: students who score slightly higher for social media tend (very slightly) to score lower for television, but the points are quite scattered.
Exam technique
When you "comment on your findings," include both direction and strength, and refer back to the scatter plot.
A good sentence structure is: "There is a [weak/moderate/strong] [positive/negative] linear correlation because the points are [tightly clustered/quite scattered] around a line that slopes [up/down]."
Interpretation Should Match The Context
Even if you can compute $r$ perfectly, the meaning still depends on the situation:
Is a linear relationship reasonable?
Are the data reliable and measured consistently?
Does it make sense to generalizebeyond the sample?
Common Mistake
Never write "$x$ causes $y$" based only on a correlation coefficient. Y
ou need a study design or additional evidence for causation.
Hint
If the scatter plot looks curved, consider transforming variables or using a different model instead of relying on $r$.
Active recall
If $r=0.82$, what can you say about direction and strength?
If $r=-0.05$, is it fair to say there is "no relationship"? Why or why not?
Give one reason why a scatter plot is essential even when you know $r$.