Correlation coefficient and regression — the calculation mechanics

The regression line $\hat y = a + bx$ needed exactly two summary sums, $S_{xx}$ and $S_{xy}$ — both covered in depth already, including the exact documented trap of substituting $\bar x^2$ for $\frac{(\sum x)^2}{n}$ (see the regression-gradient-interpretation-extrapolation lesson, a prerequisite for this one). The product moment correlation coefficient, spec item 4.3, needs one more: $S_{yy} = \sum y^2 - \frac{(\sum y)^2}{n}$ — exactly the same computational shape as $S_{xx}$, with $y$ standing in for $x$ throughout. It's printed in full in the formula booklet's Statistics S1 section alongside $S_{xx}$ and $S_{xy}$, so nothing here needs memorising; what needs practising is not repeating, on a formula that looks almost identical, the one substitution error the archive already documents for $S_{xx}$ — confusing $\bar y^2$ for $\frac{(\sum y)^2}{n}$ inside it.

With all three sums available, the product moment correlation coefficient is $r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ — also given in full in the booklet, so this is a lookup, not a memorisation task. Where $b = S_{xy}/S_{xx}$ answers 'how much does $y$ change, on average, per unit of $x$', $r$ answers a genuinely different question: 'how tightly do the actual data points cluster around a straight line, in both directions at once?' $b$ carries real units — whatever $y$ is measured in, per whatever unit $x$ is measured in — and can be any size at all; $r$ is built to carry none, which is exactly why it always lands somewhere in the fixed range $-1 \le r \le 1$, whatever units $x$ and $y$ happen to be measured in.

The sign of $r$ always matches the sign of $b$ — both come from the same numerator, $S_{xy}$ — so a positive gradient and a positive $r$ aren't two separate facts to check on a scatter diagram, they're the same fact read off two different formulas. $r$ close to $+1$ or $-1$ means the data sits close to a straight line; $r$ close to $0$ means it doesn't — though, as the next section covers, 'doesn't sit close to a straight LINE' and 'has no relationship at all' are two different claims, and treating them as the same one is $r$'s single most consequential limitation.

Spec item 4.3 names 'its use, interpretation and limitations' as content in its own right, not just the calculation — and it's worth being honest about what backs the paragraphs below. Unlike the accuracy trap and the coding property covered later in this lesson, the reviewed archive doesn't contain a quoted examiner-report example testing the limitations of $r$ specifically. What follows is standard subject-matter content for this spec point, not a documented exam trap — flagged here plainly rather than dressed up as evidence it isn't.

$r$ specifically measures LINEAR association — how well a straight line fits the data — and nothing else. A dataset can show an extremely strong, entirely genuine relationship between $x$ and $y$ that curves rather than follows a straight line, and $r$ will still report a value close to $0$ for it, because a curve, however clean, is exactly the shape a straight-line measure is built to miss. A scatter diagram sitting next to a computed $r$ value isn't decoration — it's the one check that confirms $r$ is even measuring the right kind of pattern in the first place.

A high $|r|$ is evidence of association, not evidence of cause. Two variables can move together closely without one causing the other — both can instead be driven by some third factor that $r$ has no way to detect or rule out. Reading a strong correlation as proof of a causal mechanism is a step $r$ itself never licenses; it reports how closely two things happened to move together in this data, and nothing about why.

And like the regression line it's built from the same summary sums as, $r$ describes the data it was actually calculated from — a strong $r$ computed over one range of $x$ says nothing certain about whether the same tight linear pattern continues once $x$ moves well outside that range. That's the same caution the regression-gradient-interpretation-extrapolation lesson covers in depth for predictions made from the line itself; it applies with identical force to the correlation coefficient describing that line's fit.

Substituting all three into the PMCC formula for the coded data: $r_{uv} = \dfrac{S_{uv}}{\sqrt{S_{uu}S_{vv}}} = \dfrac{S_{xy}/(qt)}{\sqrt{(S_{xx}/q^2)(S_{yy}/t^2)}} = \dfrac{S_{xy}/(qt)}{\sqrt{S_{xx}S_{yy}}/|qt|} = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \times \dfrac{|qt|}{qt}$.

Everything before the final fraction is just $r_{xy}$ — the PMCC of the original, uncoded data. The final fraction, $\dfrac{|qt|}{qt}$, is $+1$ whenever $q$ and $t$ share a sign and $-1$ whenever they don't — it can never be anything else, since it's a magnitude divided by the exact same signed number. That's the whole result: $r_{uv} = \pm r_{xy}$, with the sign flipping if and only if exactly one of the two coding DIVISORS ($q$ or $t$ — not $p$ or $s$, the two shifts, which never appear in the final result at all) is negative.

Both coding examples this facts bank documents divide by a positive constant — the Jan 2024 Q2 coding (verified directly against the real mark scheme: a Fahrenheit-to-Celsius-shaped transform, dividing by 9 and multiplying by 5, both positive) and the Section 2.2 variance-coding example (Jun 2022 Q3(e), dividing by 2 — coding exists to shrink unwieldy numbers, not to flip their sign). Neither is a full audit of every coding question across all 14 series reviewed, so this is suggestive rather than exhaustive, but in both real examples this pass has checked, the credited phrase 'r not affected by (linear) coding' carries no qualification because it doesn't need one — sign included. The edge case here — one negative divisor, flipping the sign — is real mathematics, not a trick, but it's flagged as background understanding rather than exam content: nothing in the reviewed archive shows it being tested, and manufacturing a negative-divisor coding question to drill against would mean inventing a trap this pass's evidence doesn't actually show exists on this paper.