Product moment correlation coefficientEdexcel International A Level Maths: Revision notes

Section 1

What the PMCC measures

The product moment correlation coefficient (PMCC), $r$ , measures the strength and direction of linear correlation between two variables. It always satisfies $-1\le r\le1$ .

$r=1$ : perfect positive linear correlation (all points on a line with positive gradient).
$r=-1$ : perfect negative linear correlation.
$r=0$ (or close): no linear correlation. Values close to $\pm1$ show strong correlation; values near 0 show weak correlation. A value outside $-1$ to $1$ must be a calculation error.

Key termsproduct moment correlation coefficientlinear correlationpositive correlationnegative correlation

Common mistake

Giving an $r$ value greater than 1 or less than $-1$ . Recheck the arithmetic.

Section 2

Calculating r

Using the sums of squares $S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}$ , $S_{yy}=\sum y^2-\frac{(\sum y)^2}{n}$ and $S_{xy}=\sum xy-\frac{\sum x\sum y}{n}$ : $r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}.$ The sign of $r$ is the sign of $S_{xy}$ . Example: $S_{xx}=10$ , $S_{yy}=6$ , $S_{xy}=6$ gives $r=\frac{6}{\sqrt{60}}=0.775$ . If $S_{xx}=40$ , $S_{yy}=90$ , $S_{xy}=-48$ then $r=\frac{-48}{\sqrt{3600}}=-0.8$ . A calculator in statistics mode can also give $r$ directly from the data.

Key terms$S_{xx}$, $S_{yy}$, $S_{xy}$

Exam tip

Take one square root of the whole product $S_{xx}S_{yy}$ in the denominator.

Common mistake

Dividing $S_{xy}$ by $S_{xx}$ (that is the regression gradient, not $r$ ).

Section 3

Interpreting r in context

Describe three things: strength, direction and the variables in context. Example: $r=-0.8$ for age and value of cars: 'There is strong negative linear correlation between age and value; older cars tend to have lower values.' $r$ is not a percentage, and it does not say how many items follow the pattern. It is also unchanged by a linear change of units or coding of either variable, such as converting pounds to euros or years to months.

Key termsstrong correlationweak correlation

Exam tip

Always name both variables in context and use the word 'tend': older cars tend to have lower values.

Section 4

Limitations of the PMCC

Correlation is not causation. A strong $r$ shows an association only. Another variable may affect both. For example, hot weather raises both ice cream sales and sunburn cases.
Only linear correlation. Data on a curve, such as $y=\frac{1000}{x}$ or a U-shape, can give $r$ close to 0 or with $|r|<1$ even though there is a strong relationship.
Outliers. One unusual point can change $r$ dramatically. Check the data for outliers.
Sample size. With very few points, a high $r$ can occur by chance. Significance tests for $r$ are not required.

Key termscausationoutlierthird variable

Common mistake

Saying 'x causes y' because r is close to 1.

Section 5

Using r with regression

The PMCC tells you whether a linear model is sensible. If $|r|$ is close to 1, a regression line gives reasonable predictions inside the range of the data. If $r$ is near 0, the regression line is of little use for prediction, but a non-linear relationship may still exist. $r$ and the regression gradient $b=\frac{S_{xy}}{S_{xx}}$ always have the same sign, because $S_{xx}$ is positive.

Key termslinear model

Exam tip

If a question gives $r$ near 0 and a curved pattern, mention that PMCC cannot detect non-linear relationships.

That's the notes covered.

Carry on to the next subtopic.

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).