The correlation coefficient r: meaning and properties

The correlation coefficient $\displaystyle r$ is a number that measures how close the points of a scatter diagram are to a straight line – the strength of the linear relation.

• The sign of $\displaystyle r$ is the direction: $\displaystyle r>0$ – a positive relation, $\displaystyle r<0$ – a negative relation.
• The closer $\displaystyle r$ is to 1 or to $\displaystyle -1$, the closer the points are to a straight line.

The heights of the fathers and sons, from the last lesson, lie close to a rising straight line:

Here the relation is weak: there is a slight tendency to rise, but the points are scattered far from any straight line:

In the next lesson you will work out $\displaystyle r$ yourself. For now – how to read it.

Likewise for a negative relation: $\displaystyle -1<r\le-0.7$ – a strong negative linear relation, and so on.

• $\displaystyle {r}={\pm1}$: all the points lie on one straight line – a perfect linear relation.
• The strength of the relation depends on $\displaystyle |r|$: a relation with $\displaystyle {r}={-0.9}$ is stronger than one with $\displaystyle {r}={0.6}$.

$\displaystyle {r}={0}$ means there is no linear relation – not necessarily that there is no relation at all. For the points on the parabola $\displaystyle {y}={x^2}$, $\displaystyle y$ is fixed entirely by $\displaystyle x$, but the relation is not linear:

On the left the points fall and on the right they rise, and $\displaystyle r$ “balances” the two sides to 0. So before concluding anything from $\displaystyle r$, look at the scatter diagram.

These properties follow from the formula for $\displaystyle r$, which you will learn in the next lesson:

• Always $\displaystyle -1\le r\le1$.
• $\displaystyle r$ is a number with no units: it is the same number whether the height is in centimetres or in metres.
• Adding a number to all the values of $\displaystyle x$ (or $\displaystyle y$) does not change $\displaystyle r$: the points move together, and their shape stays.
• Multiplying all the values of $\displaystyle x$ (or $\displaystyle y$) by a positive number – a change of units – does not change $\displaystyle r$; multiplying by a negative number reverses its sign.
• $\displaystyle r$ is symmetric: the correlation coefficient between $\displaystyle x$ and $\displaystyle y$ equals the correlation coefficient between $\displaystyle y$ and $\displaystyle x$.

Here are six scatter diagrams, and a list of their correlation coefficients:
$\displaystyle 0.63$, $\displaystyle 1$, $\displaystyle -0.20$, $\displaystyle 0.27$, $\displaystyle -0.93$, $\displaystyle -0.75$.
Match each diagram with its correlation coefficient. Explain your choice.

A head teacher found that the correlation coefficient between a class’s marks in a final maths test and their marks in a final physics test is 0.8.
The maths teacher took 4% off every pupil’s mark, and the physics teacher took 3% off every pupil’s mark.
What is the correlation coefficient between the marks after the cuts? Explain.

Describe the relation – its direction and strength – for each of these correlation coefficients: $\displaystyle {r}={0.85}$, $\displaystyle {r}={-0.5}$, $\displaystyle {r}={0.2}$, $\displaystyle {r}={-0.95}$.

Danny measured the height of a ball thrown upwards every half second, from the moment it was thrown until it fell to the ground. He got a correlation coefficient very close to 0 between the time and the height.
Is it right to conclude that there is no relation between the time and the height? Explain.