r=\frac{S{xy}}{\sqrt{S{xx}S{yy}}}

What type of relationship does r measure?

The strength and direction of linear correlation only.

Perfect positive linear correlation: all points lie on a straight line with positive gradient.

What does r near 0 mean?

Little or no linear correlation (there may still be a non-linear relationship).

S{yy}=\sum y^2-\frac{(\sum y)^2}{n}

How should you interpret r in an exam answer?

Describe strength, direction and the variables in context, using 'tend to'.

Does correlation imply causation?

No. A third variable may affect both, or the link may be coincidence.

Effect of an outlier on r?

It can change r a lot; r is sensitive to outliers.

Can r=1.2 be correct?

No; r must lie between -1 and 1.

How are the signs of r and the regression gradient b related?

They are always the same sign, because S{xx}0.

Why can data on a curve give r near 0?

r only detects straight-line patterns.

Product moment correlation coefficientEdexcel International A Level Maths: Flashcards

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Range of values of $r$?

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Range of values of $r$ ?: $-1\le r\le1$
Formula for $r$ ?: $r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$
What type of relationship does $r$ measure?: The strength and direction of linear correlation only.
What does $r=1$ mean?: Perfect positive linear correlation: all points lie on a straight line with positive gradient.
What does $r$ near 0 mean?: Little or no linear correlation (there may still be a non-linear relationship).
Formula for $S_{yy}$ ?: $S_{yy}=\sum y^2-\frac{(\sum y)^2}{n}$
How should you interpret $r$ in an exam answer?: Describe strength, direction and the variables in context, using 'tend to'.
Does correlation imply causation?: No. A third variable may affect both, or the link may be coincidence.
Effect of a linear change of units on $r$ ?: None; $r$ is unchanged.
Effect of an outlier on $r$ ?: It can change $r$ a lot; $r$ is sensitive to outliers.
Can $r=1.2$ be correct?: No; $r$ must lie between $-1$ and $1$ .
How are the signs of $r$ and the regression gradient $b$ related?: They are always the same sign, because $S_{xx}>0$ .
Why can data on a curve give $r$ near 0?: $r$ only detects straight-line patterns.

Exam questions on Product moment correlation coefficient

A dealer records the age, $x$ years, and the value, $y$ thousand pounds, of 10 used cars of the same model. The summary statistics are $S_{xx}=40$ , $S_{yy}=90$ and $S_{xy}=-48$ .
The dealer converts the ages into months and the values into pounds. State the new value of $r$ and justify your answer.2 marks
A seaside town records, for each of 12 months, the ice cream sales, $x$ hundred cones, and the number of sunburn cases, $y$ , treated at the local clinic. The product moment correlation coefficient for these data is $r=0.91$ .
Suggest a reason why the two variables are strongly correlated, even though neither is likely to cause the other.2 marks
A teacher records the number of practice tests, $x$ , completed by five students and their scores, $y$ , out of 5 in a final quiz. The values of $x$ are 1, 2, 3, 4, 5 and the scores are 2, 4, 5, 4, 5. For these data $\sum x=15$ , $\sum y=20$ , $\sum x^2=55$ , $\sum y^2=86$ and $\sum xy=66$ .
Find $S_{xx}$ , $S_{yy}$ and $S_{xy}$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).