Deep Dive: Regression Gradient Interpretation Extrapolation

#CODEX Deep-Dive — Regression Gradient Interpretation and Extrapolation: the Statistical Case Behind the Exam Convention (WST01 spec 4.1/4.2)

Provenance, 2026-09-29: written after reading research/veridian/WST01-verified-facts.md in full (§1 spec transcription, §2a formula-booklet contents, §2b AO weighting, §4 "Section 4," §7–§8, and — in full, as the primary anchor for this document — the per-lesson audit entry for wst01-regression-gradient-interpretation-extrapolation.ts, which discloses the real Jun 2022 Q2 data line used in §3 below); the existing lesson file content/veridian/curriculum/wst01/regression-gradient-interpretation-extrapolation.ts in full, end to end, including its header note, every teach/mechanism/worked-chain block, its trap taxonomy, and its closing flag block; content/veridian/curriculum/wst01/correlation-coefficient-and-regression-calculation.ts, specifically its beyond-spec coding-invariance derivation (§2 below builds directly on it); the four non-deep-dive technique-hub documents in this folder (codex-formula-and-method-bank.md §1.3/§2.4, forge-exam-technique-guide.md §3.2/§4.2/its trap table, signal-frequency-and-boundaries.md §1, and signal-topic-brief-regression-gradient-interpretation-extrapolation.md in full — see the count correction immediately below); and content/veridian/question-bank/wst01.ts (grepped for "regression"/"gradient"/"extrapolat" — zero matches; both its entries are Jun 2019 questions on box plots and discrete random variables, not this sub-topic). All new arithmetic below was computed independently and is flagged as such at each point it appears; the two derivations in §2 and §3 are standard mathematical/statistical reasoning carried out from first principles in this document, not sourced from a Pearson document, and are labelled as such throughout.

Revision note (2026-09-29, same pass): the sibling signal-topic-brief-regression-gradient- interpretation-extrapolation.md in this folder independently re-checked how many examiner-report series actually carry a confirmed 4.1/4.2 citation and found 4, not the "5" this document originally borrowed from the facts bank's own §7 top-line synthesis — no fifth series or quote for this exact sub-topic (as distinct from 4.3/PMCC) is named anywhere in the facts bank's detailed §4 write-up; Jan 2021's confirmed examiner report is real but its content addresses Section 6 (the Normal distribution), not Section 4. Every "of 5" below has been corrected to "of 4" accordingly, and the denominator series list is now stated as Oct 2021, Jun 2022, Jan 2023, Jun 2024 only.

#Why this document exists, and what it does not re-teach

This is, by the facts bank's own top-line account, the single richest-evidenced sub-topic in the whole WST01 archive; re-verified for this document (see the revision note above), that holds as four examiner-report series with a distinct concrete slip each, not five, all mapping to the same skill gap — still the densest confirmed coverage of any WST01 sub-topic in the reviewed archive. The existing lesson file is, correspondingly, the most heavily audited lesson in this batch: a full bullet-by-bullet coverage audit against a real 14-mark anchor question (Jun 2022 Q2) already closed every gap that audit found. The lesson already teaches, in full, and this document does not repeat: fitting $S_{xx}$/$S_{xy}$ → $b$/$a$ and assembling the final line; reading the gradient as a rate at the correct scale of $x$ (with both real, quoted scale-slip examples — the $156/$1.56 case and the GDP/ population case); judging reliability by range-comparison when a range is given and by mean/$S_{xx}$- comparison when only summary statistics are given (with the real Jun 2022 Q2(e) marking note quoted verbatim); the basic reverse-gradient-solve ($\Delta y = b\Delta x$); the calculus derivation of why a least-squares line always passes through $(\bar x,\bar y)$; and a full nine-item trap taxonomy. Reading any of that again here would be restatement, not depth.

What follows is four things the reviewed archive and the existing material together support but that no existing WST01 document currently states: (1) the half of the coding-transformation derivation the PMCC lesson leaves as a bare assertion — how the gradient $b$, unlike $r$, actually rescales under a linear change of variable, derived rather than asserted; (2) a real statistical mechanism, not just an intuitive caution, for why extrapolation risk grows the way it does, applied to the real disclosed numbers from the Jun 2022 anchor question; (3) a proof that the mark scheme's two credited methods for a reverse-gradient solve are the same calculation reached two different ways, with the second method (quoted in the facts bank, taught nowhere in this repo) worked through on numbers this course already uses; (4) an honest audit of what spec 4.1/4.2 explicitly permits or requires that the reviewed archive does not show being tested at all.

#1. What spec 4.1/4.2 and the formula booklet require, restated only as far as this document needs it

Spec 4.1: "Scatter diagrams. Linear regression. (Guidance: calculation of the least-squares regression line; students may be required to draw it on a scatter diagram.)" Spec 4.2: "Explanatory (independent) and response (dependent) variables. Applications and interpretations. (Guidance: use to make predictions within the range of the explanatory variable and the dangers of extrapolation; derivations not required; variables other than x/y may be used; linear change of variable may be required.)" Three of those five guidance clauses — drawing the line, variables other than $x$/$y$, and linear change of variable — are not the existing lesson's focus and are addressed directly in §5. The formula booklet prints $S_{xx}$, $S_{xy}$ (both forms) and $b=S_{xy}/S_{xx}$, $a=\bar y-b\bar x$ in full (facts bank §2a); nothing below needs memorising for that reason, only for the two derivations that follow.

#2. Completing the coding derivation: what happens to $b$ that never happens to $r$

Suppose $x$ and $y$ are linearly coded as $u=\dfrac{x-p}{q}$ and $v=\dfrac{y-s}{t}$ for constants $p,q,s,t$ ($q,t\neq0$) — exactly the transform spec 4.2's own guidance names. The sibling lesson on the PMCC already derives, in its own beyond-spec block, the three sums this coding produces: $S_{uu}=S_{xx}/q^2$, $S_{vv}=S_{yy}/t^2$, $S_{uv}=S_{xy}/(qt)$ — and uses them to prove $r$ survives coding unchanged up to sign. That lesson leaves the gradient's own behaviour under coding as a bare assertion elsewhere ("the gradient genuinely does need converting back through the coding constants"), with no derivation supplied anywhere in this repo. Completing that half of the same calculation:

$$b_{uv} = \frac{S_{uv}}{S_{uu}} = \frac{S_{xy}/(qt)}{S_{xx}/q^2} = \frac{q}{t}\cdot\frac{S_{xy}}{S_{xx}} = \frac{q}{t}\,b_{xy} \quad\Longrightarrow\quad b_{xy} = \frac{t}{q}\,b_{uv}.$$

The reason $r$ and $b$ behave differently under the identical transform is visible directly in this algebra: $r$ divides $S_{uv}$ by $\sqrt{S_{uu}S_{vv}}$, and every factor of $q$ and $t$ cancels between numerator and denominator (up to the sign question the PMCC lesson already resolves), because $r$ is comparing two matched, same-kind spreads. $b$ divides $S_{uv}$ by $S_{uu}$ alone — a $y$-shaped quantity over an $x$-shaped one — so the $t$ and $q$ do not cancel; they combine into the single conversion factor $t/q$ that carries the coded gradient back to the real one. This is exactly why $b$ has units at all and $r$ does not, now shown as a formula rather than argued from unit-cancellation alone.

Micro-worked check. A dataset codes cost in thousands of dollars, $u=x/1000$ ($p=0$, $q=1000$), against demand in single units, left uncoded ($v=y$, so $s=0$, $t=1$). The fitted coded line is $\hat v = 40-0.6u$, i.e. $b_{uv}=-0.6$. Applying $b_{xy}=(t/q)\,b_{uv}$: $b_{xy}=(1/1000)\times(-0.6)= -0.0006$ — demand falls by 0.0006 units for every extra dollar spent, or equivalently by 0.6 units for every extra $1000, the number already visible in $b_{uv}$ once its scale is correctly named. This is the same "name what one unit of the variable actually is" discipline the existing lesson already teaches case-by-case for the $156/$1.56 and GDP/population examples, now derived as a general rule rather than checked instance by instance. No examiner report or mark scheme reviewed for this unit applies a coding transform to a regression-gradient question specifically (as opposed to the PMCC or mean/ variance coding questions the archive does confirm — facts bank §4, "Coding (2.2)" and the Jan 2024 MS PMCC-coding citation) — this section is spec-grounded theory (4.2's own guidance permits it), not a documented exam pattern, and is presented as such.

#3. Why extrapolation risk is a quantifiable statistical fact, not only a caution

Spec 4.2 names "the dangers of extrapolation" and waives derivations; the existing lesson's mechanism block argues the danger correctly but qualitatively — no data on the far side of the range to borrow strength from. Here is the quantitative version, standard simple-linear-regression theory (beyond spec, exactly in the spirit of the existing lesson's own least-squares beyond-spec block), carried out from first principles because no WST01 document states it.

Model $y_i=\alpha+\beta x_i+\varepsilon_i$, with the $x_i$ fixed and the errors $\varepsilon_i$ independent, mean 0, constant variance $\sigma^2$ — the standard assumption behind least squares, never stated explicitly in spec 4.1/4.2 itself. Writing the fitted value at a new point $x_0$ as $\hat y_0=a+bx_0=\bar y+b(x_0-\bar x)$ and substituting $b=\sum(x_i-\bar x)(y_i-\bar y)/S_{xx}$, the terms involving $\bar y$ cancel (since $\sum(x_i-\bar x)=0$), leaving $\hat y_0$ as a weighted sum of the original $y_i$ alone:

$$\hat y_0 = \sum_{i=1}^n y_i\left[\frac1n + \frac{(x_0-\bar x)(x_i-\bar x)}{S_{xx}}\right].$$

Since the $y_i$ are independent with variance $\sigma^2$, $\operatorname{Var}(\hat y_0)=\sigma^2$ times the sum of the squared weights. The cross term in that expansion vanishes because $\sum(x_i-\bar x)=0$ by definition of $\bar x$; the remaining squared term collapses because $\sum(x_i-\bar x)^2=S_{xx}$ exactly. What survives is

$$\operatorname{Var}(\hat y_0) = \sigma^2\left[\frac1n + \frac{(x_0-\bar x)^2}{S_{xx}}\right].$$

This is the real mechanism behind "the dangers of extrapolation": prediction uncertainty has a fixed floor of $\sigma^2/n$ that never disappears (no fitted line, however good, predicts with zero uncertainty, even exactly at $\bar x$), plus a second term that grows with the square of the distance from the mean — not gently, quadratically. It also explains, rather than merely describes, why the real mark scheme's credited fallback compares a predicted $x_0$ against $\bar x$ using $S_{xx}$ as the reference spread (Jun 2022 Q2(e)'s own marking note, already quoted in the existing lesson) instead of inventing a range that question never supplied: $\bar x$ and $S_{xx}$ are exactly the two quantities this formula says actually drive the uncertainty.

Applying it to the real disclosed numbers. The facts bank's audit of the Jun 2022 anchor question discloses $n=9$, $\sum t=7.87$, $S_{tt}=1.29$, and the prediction point $t_0=7$ directly. From these: $\bar t = 7.87/9 = 0.8744$ (4 d.p.), matching the "$[=0.874]$" the facts bank itself quotes from the marking note. $(t_0-\bar t)^2/S_{tt} = (7-0.8744)^2/1.29 = 37.52/1.29 \approx 29.1$ — roughly 260 times the fixed floor $1/n=1/9\approx0.111$, even before either is multiplied by $\sigma^2$. Framed the way this course already teaches "typical spread" (via $\sqrt{S_{xx}/n}$): a typical single point sits about $\sqrt{1.29/9}\approx0.379$ (million people) from $\bar t$; $t_0=7$ sits $7-0.874\approx6.13$ from it — about 16.2 of those typical spreads away. Both figures are this document's own computation from the four disclosed numbers, not a transcription of a printed mark-scheme target (no variance/$\sigma$ figure for this question is disclosed anywhere in the facts bank), but they quantify, for the first time in this course's material, exactly how far "much greater than the mean" (the mark scheme's own words) actually is: not a marginal, arguable distance, but sixteen-plus multiples of the data's own typical spread.

#4. The mark scheme's two reverse-gradient-solve methods are one method in disguise

The facts bank's audit of the same anchor question quotes its Q2(f) marking note in full: "M1: Equating 0.1 with 'their b' × x. Or substituting two values for g with a difference of 0.1 in their equation leading to a value of x... Do not allow fractions." The existing lesson teaches, and its reg-mcq-5 tests, only the first of these two credited methods. The second — checked by grep across the existing lesson, the formula-and-method bank, and the FORGE guide, with zero matches for its wording — appears nowhere else in this repo, despite being explicitly credited by the real mark scheme.

It is worth teaching because it looks like a different calculation and provably isn't. Suppose the fitted line is $\hat y=a+bx$, and two response values $y_1,y_2$ differ by a known amount $\Delta y=y_2-y_1$. Method 2 solves $y_1=a+bx_1$ and $y_2=a+bx_2$ separately for $x_1$ and $x_2$, then reports $\Delta x=x_2-x_1$. Subtracting the two equations directly: $y_2-y_1=b(x_2-x_1)$, i.e. $\Delta y=b\,\Delta x$ — Method 1's equation exactly, with the intercept $a$ cancelling out of the subtraction regardless of which two $y$-values were chosen. This is why the mark scheme is indifferent between them: they are algebraically the same statement, and Method 2 is the more useful route for a candidate who has correctly fitted the line but hasn't spotted that only $b$ matters for a rate-of-change question.

Micro-worked check, on numbers this course already uses. The existing lesson's reg-mcq-5 already establishes $\hat y=10.1+4.25x$, a stated rise of 21.25 in $y$, and the correct answer $\Delta x=5$ units of $x$ ($500) via Method 1. Applying Method 2 to the identical line, picking an arbitrary pair of $y$-values 21.25 apart — say $y_1=35$, $y_2=56.25$ — and solving each separately: $x_1=(35-10.1)/4.25= 5.859$ (3 d.p.), $x_2=(56.25-10.1)/4.25=10.859$ (3 d.p.). $\Delta x=10.859-5.859=5.000$ exactly, the identical answer, for a genuinely arbitrary choice of $y_1$ — choosing a different starting value moves $x_1$ and $x_2$ individually but never their difference, exactly as the general proof above predicts.

#5. What spec 4.1/4.2 explicitly permits or requires that the reviewed archive does not confirm tested

Named honestly, since the hard rule against manufacturing exam evidence applies here as much as anywhere else in this document:

• "Students may be required to draw it on a scatter diagram" (spec 4.1 guidance) — a genuine, spec-named skill with no confirmed worked past-paper instance found across the six examiner-report series or fourteen mark-scheme series the facts bank reviewed for this unit.
• "Variables other than x/y may be used" (spec 4.2 guidance) — true in substance in the real anchor question itself, which uses $g$ (GDP) and $t$ (population) throughout and never $x$/$y$ — but this is a labelling fact about how the real question is dressed, not a distinct skill tested separately from the interpretation and extrapolation content the existing lesson already covers in full.
• The coding transform derived in §2 — spec-permitted (4.2's own "linear change of variable may be required"), and the archive does confirm coding is tested in this unit generally (the PMCC lesson's own Jan 2024 mark-scheme citation, and the separate mean/variance-coding citation at spec 2.2), but no reviewed examiner report or mark scheme applies a coding transform to a regression-gradient question specifically. §2 is genuine spec-grounded theory extending an assertion this repo's own material already makes elsewhere; it is not evidence of a tested pattern, and is not presented as one.

#6. Synthesis: every real exam angle this specific sub-topic has actually been tested from

Denominators throughout use the four examiner-report series that actually carry a confirmed 4.1/4.2 citation in the facts bank's own §4 write-up — Oct 2021, Jun 2022, Jan 2023, Jun 2024. Jan 2021 and Jun 2019 are excluded from the denominator: Jan 2021's confirmed examiner report addresses Section 6 (the Normal distribution) in the facts bank, with no cited content for Section 4, and the 2026-09-25 Jun 2019 spot-check added findings for two other lessons (outliers/box plots; discrete random variables), not this one. This corrects an unsupported "5" carried in this document's own §7 flags before this revision — see the revision note at the top of this document.

#7. What remains flagged, not resolved, by this document

• §3's "≈29" and "≈16.2" figures are this document's own arithmetic from the four numbers the facts bank discloses for Jun 2022 Q2 ($n$, $\sum t$, $S_{tt}$, $t_0$); no $\sigma$ or residual-variance figure for this question is disclosed anywhere in the reviewed material, so no absolute prediction interval is claimed here, only the relative comparison the disclosed numbers actually support.
• §2 and §3 are self-contained mathematical derivations, correct by construction, but neither is claimed as content this course's spec requires or the archive shows being examined — both are labelled beyond-spec throughout, consistent with spec 4.1/4.2's own explicit waiver of derivations.
• §5's three gaps (scatter-diagram drawing, variables-other-than-x/y as a distinct skill, coding applied to a gradient specifically) reflect the facts bank's own reviewed archive (14 mark-scheme series, 6 of them with a read examiner report) at the time of this reading — none is asserted as absent from the paper generally, only as unconfirmed in the material this document and its sources have actually read.
• This document performed no new PDF fetches; every cited exam fact traces to research/veridian/WST01-verified-facts.md or the two existing lesson files' own headers, and every arithmetic result is independently computed from numbers those sources already disclose.