How is S{xy} calculated from sums?

S{xy}=\sum xy-\frac{\sum x\sum y}{n}.

How are S{xx} and S{yy} calculated?

S{xx}=\sum x^2-\frac{(\sum x)^2}{n} and S{yy}=\sum y^2-\frac{(\sum y)^2}{n}.

What does the PMCC measure?

The strength of linear correlation, between -1 and 1.

State the condition for using the PMCC in a test.

The data come from a bivariate normal population (roughly elliptical scatter).

What is the effect of linear coding on r?

The size is unchanged; the sign reverses only if exactly one multiplier is negative.

State the formula for Spearman's rank correlation coefficient.

rs=1-\frac{6\sum d^2}{n(n^2-1)}.

What does Spearman's coefficient measure?

The strength of monotonic association, using ranks.

How do you rank tied values?

Give each the mean of the ranks they would have occupied, e.g. 3.5 for joint 3rd and 4th.

Why is the Spearman formula approximate when there are ties?

It assumes there are no tied ranks.

When is Spearman's coefficient preferred to the PMCC?

For ranked data, or a monotonic non-linear relationship, or when outliers are present.

The two rankings agree perfectly.

Product moment and Spearman's rank correlationEdexcel A-Level Further Maths: Flashcards

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State the formula for the product moment correlation coefficient.

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State the formula for the product moment correlation coefficient.: $r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ .
How is $S_{xy}$ calculated from sums?: $S_{xy}=\sum xy-\frac{\sum x\sum y}{n}$ .
How are $S_{xx}$ and $S_{yy}$ calculated?: $S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}$ and $S_{yy}=\sum y^2-\frac{(\sum y)^2}{n}$ .
What does the PMCC measure?: The strength of linear correlation, between $-1$ and $1$ .
State the condition for using the PMCC in a test.: The data come from a bivariate normal population (roughly elliptical scatter).
What is the effect of linear coding on $r$ ?: The size is unchanged; the sign reverses only if exactly one multiplier is negative.
State the formula for Spearman's rank correlation coefficient.: $r_s=1-\frac{6\sum d^2}{n(n^2-1)}$ .
What does Spearman's coefficient measure?: The strength of monotonic association, using ranks.
How do you rank tied values?: Give each the mean of the ranks they would have occupied, e.g. $3.5$ for joint 3rd and 4th.
Why is the Spearman formula approximate when there are ties?: It assumes there are no tied ranks.
When is Spearman's coefficient preferred to the PMCC?: For ranked data, or a monotonic non-linear relationship, or when outliers are present.
What does $r_s=1$ mean?: The two rankings agree perfectly.

Exam questions on Product moment and Spearman's rank correlation

A researcher records the hours of sleep, $x$ , and the reaction time, $y$ ms, of each of 8 randomly chosen students. The summary statistics are $S_{xx}=62.5$ , $S_{yy}=40$ and $S_{xy}=-45$ .
Interpret the value of $r$ between $x$ and $y$ in context, and state an assumption about the population needed to use $r$ in a hypothesis test.2 marks
Two judges each rank six paintings, $A$ to $F$ , from 1 (best) to 6 (worst). Judge 1 ranks $A$ to $F$ as $3, 1, 2, 5, 4, 6$ respectively. Judge 2 ranks them as $2, 1, 4, 5, 3, 6$ respectively.
State, with a reason, whether Spearman's rank correlation coefficient or the product moment correlation coefficient is more appropriate for these data.2 marks
Seven students sit a mathematics test and a physics test. Their scores, in the order of students $1$ to $7$ , are: mathematics $56, 61, 61, 70, 48, 75, 66$ and physics $50, 58, 49, 64, 44, 70, 60$ .
Rank the mathematics scores and the physics scores from lowest (rank 1) to highest, and hence show that $\sum d^2=3.5$ .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).