Coaching Centre
Scatterplots and correlation
Bivariate data, explanatory and response variables, scatterplots, describing form, direction and strength, Pearson's correlation coefficient r, outliers, and why correlation does not imply causation.
Scatterplots
The explanatory (independent) variable goes on the horizontal axis, and the response (dependent) variable goes on the vertical axis. Each individual becomes one point.
Six students recorded hours of study $x$ and test score $y$.
| Hours $x$ | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Score $y$ | 52 | 55 | 61 | 60 | 68 | 70 |
Each dot is one student. The points rise from left to right, close to a straight line.
Which student did a little worse than the trend suggests?
Describe a scatterplot by its form (linear or not), direction (positive or negative) and strength (how closely the points follow the pattern). Also look for outliers, points far from the rest.
Pearson's correlation coefficient
The correlation coefficient $r$ measures the strength and direction of a linear relationship. It is found with a calculator, and always satisfies $-1 \le r \le 1$.
| $r = 1$ or $r = -1$ | Perfect: every point is on a line |
| $0.75 \le |r| \lt 1$ | Strong |
| $0.5 \le |r| \lt 0.75$ | Moderate |
| $0.25 \le |r| \lt 0.5$ | Weak |
| $|r| \lt 0.25$ | Little or no linear relationship |
The sign gives the direction. For Example 1, a calculator gives $r \approx 0.97$: a strong positive linear relationship.
Correlation is not causation
A strong correlation does not prove that one variable causes the other. Both may be driven by a third, lurking variable, or the link may be coincidence.
Ice cream sales and drownings at a beach have a strong positive correlation. Does ice cream cause drowning?
In Example 1, the explanatory variable is:
Study time explains the score.
$r = -0.88$ describes a relationship that is:
Negative sign, and $|r| \ge 0.75$.
Which value of $r$ shows the strongest linear relationship?
The largest $|r|$.
Which value is impossible for $r$?
$-1 \le r \le 1$.
As temperature rises, heating costs fall. The correlation is:
One goes up as the other goes down.
Points lie exactly on a U-shaped parabola. Which is true?
$r$ only measures linear relationships.
Shoe size and reading ability in primary school children are strongly correlated. The lurking variable is most likely:
Older children have bigger feet and read better.
A strong correlation between two variables shows that:
Correlation is not causation.
New numbers every time. Keep going until you get five in a row.