Topic Brief: Correlation Coefficient and Regression Calculation

#SIGNAL Topic Master Brief — Product Moment Correlation Coefficient Calculation (WST01, spec 4.3)

Provenance (2026-09-29): built from research/veridian/WST01-verified-facts.md (compiled 2026-09-12, spot-check addendum 2026-09-25), re-grepped for "correlation", "PMCC", "regression" and "Spp", plus the per-lesson audit trail at the end of that file for both wst01-correlation-coefficient-and-regression-calculation and its sibling wst01-regression-gradient-interpretation-extrapolation. Cross-checked against signal-frequency-and-boundaries.md (whole-Section-4 count) and codex-deep-dive-correlation-coefficient-and-regression-calculation.md (formal proofs and micro-examples) so this document doesn't restate either. It also corrects one figure: CODEX's own synthesis table puts spec 4.3 at only "2 of 6" examiner-report series — a count that missed the sibling lesson's audit entry, which independently re-read the real Jun 2022 question paper and states plainly that its parts (a)–(b) test spec 4.3. This document uses the corrected count throughout. No new PDF fetches were made; every claim cites a series and question from the facts bank or a lesson's own audit trail.

#1. Frequency — seen in 3 of 6 series checked

The facts bank has a fully-read Principal Examiner Feedback report for 6 of the 14 WST01 series with a verified mark scheme: Jun 2019, Jan 2021, Oct 2021, Jun 2022, Jan 2023, Jun 2024. Within that set, the product moment correlation coefficient itself (spec 4.3 — calculating $r$, or a property of it) is directly evidenced in 3 of those 6: Oct 2021, Jun 2022, Jun 2024.

Jan 2021 and Jun 2019 show no confirmed 4.3 citation anywhere in the facts bank. Jan 2023 Q6 is confirmed for this section only via its gradient-interpretation and extrapolation-reliability parts (spec 4.1/4.2, the sibling lesson's territory) — no part of Jan 2023 Q6 is confirmed testing $r$ itself. One further citation exists outside this 6-series denominator: Jan 2024, confirmed only via its mark scheme (no examiner report located for that series), covering a property of $r$ rather than its calculation — kept separate rather than folded into the "3 of 6" count.

#2. What was actually tested, by series

• Oct 2021 Q2(b). Computing $r$ required first finding $S_{pp}$ (the response variable's own sum of squared deviations). ER quote: "the most common error made was using $(273/12)^2$ in the calculation of $S_{pp}$ and giving their answer to an inappropriate degree of accuracy." Context: paper expenditure ($p$) against number of employees ($x$), $n=12$.
• Jun 2022 Q2(a)–(b), 4 of the question's 14 marks. Confirmed testing PMCC calculation and interpretation by direct re-reading of the real question paper during the sibling lesson's own audit — but no specific quoted error survives in the reviewed record for these two parts specifically; the facts bank's own Section 4 write-up quotes only Q2(c)–(e) (the regression-line and gradient parts of the same question) by name. Flagged here as confirmed-present-but-undetailed, not filled in with an invented mechanism. Context: GDP ($g$) against population ($t$).
• Jun 2024 Q4(b), 2 of the question's 13 marks. ER quote: "Part (b) was answered well with many students able to calculate a correct value of the product moment correlation coefficient. Common error included the omission of the square root in the denominator." Target: $r = S_{dg}/\sqrt{S_{dd}S_{gg}} = 0.9802\ldots \to$ awrt 0.98. Context: bear length ($d$) against girth ($g$).
• Jan 2024 Q2(d) (mark scheme only, no ER for this series). A credited general-knowledge point: "r not affected by (linear) coding" — PMCC's invariance under a linear change of variable, tested as a stated property rather than a fresh calculation.

#3. Mark-tariff pattern

The calculation step itself carries 2–4 marks wherever confirmed, and the two figures aren't directly comparable: Jun 2024's 2 marks isolate the calculation alone; Jun 2022's 4 marks bundle calculation and interpretation together, not broken out. Oct 2021 Q2(b)'s own mark allocation is not confirmed in the reviewed record.

One structural pattern holds without exception across all three confirmed series: the PMCC calculation has never appeared as its own standalone question. It is always one part of a larger, multi-part Section 4 question that also fits a regression line and/or interprets it — Oct 2021 Q2(a)–(e), Jun 2022 Q2(a)–(f), Jun 2024 Q4(a)–(d). Its question number tracks the paper's total question count, not a fixed slot: Q2 on the two six-question papers (Oct 2021, Jun 2022), Q4 on Jun 2024's eight-question paper. Within that question, the PMCC part always comes early — before the "show that" or gradient-interpretation parts that close it out.

#4. Recurring context/angle across the confirmed series

No subject-matter domain repeats: business expenditure, national economic/demographic data, and animal biometrics are three genuinely different real-world settings. What does repeat is structural, not thematic: in every confirmed instance, the calculation starts from at least one sum or $S$-value already given or already found in an earlier part — $S_{xx}$ is handed directly in Oct 2021; $S_{dg}$ and $S_{dd}$ are carried forward from Jun 2024's own part (a) — never from a full list of raw data pairs the student must reduce to sums unaided. The two confirmed calculation-error types are also structurally related rather than coincidental: omitting the square root (Jun 2024) and substituting a mean-squared term inside an $S$-formula (Oct 2021) both produce a wrong value for $r$ that lands safely inside $[-1,1]$ — neither trips the naive "check $r$ is between $-1$ and $1$" self-check a student might rely on instead of checking their method.

#5. If this topic appears, here's what to expect

Expect the PMCC calculation to arrive as one middle part — not the opening or closing part — of a larger correlation-and-regression question that also asks for the regression line and/or its interpretation, worth roughly 2–4 marks on its own within that question. At least one of the three $S$-values will likely already be available from an earlier part or the question's own data line, so the actual new work is usually the formula substitution itself: $r = S_{xy}/\sqrt{S_{xx}S_{yy}}$. Two specific slips are confirmed real on this exact step — dividing by $S_{xx}S_{yy}$ instead of its square root, and substituting $\bar x^2$ (or $\bar y^2$) for $(\Sigma x)^2/n$ inside whichever $S$-formula feeds the calculation — and both produce an answer that still looks plausible, so checking the method line-by-line matters more than sanity-checking the final number's range. A stated property of $r$ (most confirmed: invariance under linear coding) is also fair game as its own separate mark, tested as bookwork rather than as a fresh calculation.

#Sources and limitations

• All citations trace to research/veridian/WST01-verified-facts.md §5 (Section 4) and its two per-lesson audit entries (wst01-correlation-coefficient-and-regression-calculation and wst01-regression-gradient-interpretation-extrapolation, both dated 2026-09-12).
• Frequency is stated only as "N of 6 series checked" (or noted separately when mark-scheme-only) — no percentage or likelihood claim is made anywhere in this document, and no grade-boundary figure is used or implied (none exists in the source material; see signal-frequency-and-boundaries.md §2).
• The 8 series without a located examiner report (Jun 2021, Jan 2022, Oct 2022, Jun 2023, Oct 2023, Jan 2024 [ER specifically], Oct 2024, Jan 2025) were not searched question-by-question for this topic and are excluded from both the numerator and denominator in §1, except where a mark-scheme-only citation is separately flagged as such (Jan 2024, §2).
• This document does not repeat the formal boundedness proof, the coding-invariance derivation, or the quantitative $S_{pp}$-error reconstruction already in codex-deep-dive-correlation-coefficient-and-regression-calculation.md, nor the whole-Section-4 frequency row in signal-frequency-and-boundaries.md.
• Oct 2021 Q2(b)'s exact mark allocation, and any specific quoted detail for Jun 2022 Q2(a)–(b) beyond its confirmed presence and 4-mark share, were not located in the reviewed record — flagged as open gaps rather than estimated.