Topic Brief: Regression Gradient Interpretation Extrapolation

#SIGNAL Topic Master Brief — Regression: Gradient Interpretation and Extrapolation/Reliability (WST01, spec 4.1/4.2)

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 "regression", "gradient" and "extrapolat", plus that file's per-lesson audit entry for wst01-regression-gradient-interpretation-extrapolation.ts. Also checked against the existing lesson file (to avoid restating its worked exemplars), codex-deep-dive-regression-gradient-interpretation-extrapolation.md (to avoid restating its derivations), and content/veridian/question-bank/wst01.ts (no items for this sub-topic). No new PDF fetches were made; every claim below cites a series and question from the facts bank.

A correction, stated plainly. The facts bank's §7 synthesis, this lesson's own header comment, and the CODEX deep-dive's synthesis table each describe this topic as tested in "5" examiner-report series, but none names a fifth series or quote beyond the four cited below. signal-frequency-and-boundaries.md already carries the corrected figure for spec 4.1–4.3 combined ("4 of 6"); this document independently re-verifies that count for 4.1/4.2 specifically (excluding PMCC calculation, the sibling signal-topic-brief-correlation-coefficient-and-regression-calculation.md's territory) and uses it throughout.

#1. Frequency — seen in 4 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. Gradient interpretation and/or extrapolation-reliability (spec 4.1/4.2) is directly evidenced, by a specific quoted examiner-report finding, in 4 of those 6: Oct 2021, Jun 2022, Jan 2023, Jun 2024. Jan 2021 and Jun 2019 show no confirmed 4.1/4.2 citation anywhere in the facts bank — the Jun 2019 spot-check specifically surfaced gaps in two other lessons (outliers/box plots and discrete random variables), not this one.

#2. What was actually tested, by series

• Oct 2021 Q2(d). Context: number of employees against paper expenditure. The confirmed error is reading the gradient at the wrong scale of the explanatory variable — one report records candidates giving "$1.56" instead of the correct "$156" for the effect of one extra employee.
• Jun 2022 Q2(d)–(f). Context: national population ($t$) against GDP ($g$), $n=9$. (d): most credited answers had to state the rate in real terms — "GDP increases by 31.2 billion dollars for every 1 million increase in the population" — with most attempts stopping at bare direction instead. (e)(ii), worth a single B1 mark: judge reliability of a prediction at $t=7$ million with no stated minimum or maximum for $t$ anywhere in the question — the credited line compares the value to the mean ("much greater than the mean population/t-bar for the 9 years"), and the marking note states plainly that citing range, or "extrapolation," alone earns nothing ("reference to g out of range is also B0"). (f): given a stated real-world rate of change, solve backward through the gradient alone for the corresponding change in $t$ — the examiner report notes it was rarely identified that "the rate of change... was the relevant part of the regression equation."
• Jan 2023 Q6(a),(c). Context: a dataset in grams and °C. (a): the same two failure modes at once — answers giving bare direction/correlation only, and answers that "mixed up the units (grams and °C)." (c): reliability judged against a stated range this time, not a mean — credited answers "made reference to 90 being outside of the range and therefore unreliable."
• Jun 2024 Q4(d). Context: bear body length against girth (the same anchor question as the PMCC sub-topic's Q4(b)). Confirmed failure: context dropped, or the interpretation "gave the interpretation the wrong way round."

#3. Mark-tariff pattern

This topic has never been confirmed as a standalone question — in all four series it is the back half of one larger, multi-part correlation-and-regression question. Whole-question totals, where confirmed: Jun 2022 Q2 = 14 marks (parts (c)–(f), fitting through reverse-solve, carry 10 of those 14); Jan 2023 Q6 = 14 marks; Jun 2024 Q4 = 13 marks. Oct 2021 Q2's own total is not confirmed anywhere in the reviewed record. Individual gradient-interpretation/reliability parts are small — Jun 2022's reliability judgement is a single B1 mark — rewarding one precisely-worded sentence, not extended working. The question number tracks each paper's own question count, not a fixed slot: Q2 on the two six-question papers reviewed (Oct 2021, Jun 2022), Q6 or Q4 on the longer ones — revise for a fixed late position inside a regression question, not a fixed question number.

#4. Recurring context/angle across the confirmed series

No real-world subject repeats — employee numbers, national population/GDP, a grams-and-°C dataset, and animal biometrics are four genuinely different settings. What repeats is structural. Two distinct failure families recur: reading the gradient at the wrong scale of the explanatory variable (confirmed in Oct 2021, and implicated in Jan 2023's unit mix-up), and a bare direction-only or reversed-direction sentence with no rate of change attached (confirmed in Jun 2022, Jan 2023, Jun 2024). Separately, the reliability judgement always needs a named, specific reference point — the stated data range where one is given (Jan 2023), or the mean when the question supplies only summary statistics and no range at all (Jun 2022) — and a bare "unreliable" or "extrapolation" verdict with nothing attached is confirmed, in both of those series, to earn zero marks. The reverse-gradient-solve step (given a real-world $\Delta y$, solve for $\Delta x$) is the thinnest-evidenced angle here — confirmed just once, Jun 2022 Q2(f) — so treat it as a real but occasional extra step, not a fixed fourth part.

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

Expect it as the back half of a larger regression question, not its own question, worth roughly a handful of that question's marks across two or three short parts. First, interpret the gradient in one full sentence naming both variables in context, the correct real-world size of "one unit" of the explanatory variable, and the direction — bare direction or a generic "as x increases, y increases" scores nothing on this skill in the confirmed record. Second, judge reliability by naming a specific value and a specific reference point — the stated range if the question gives one, otherwise the mean — and connecting the two; "extrapolation" alone is confirmed worthless. A less common third step, seen once, asks for the reverse: given a stated rate of change in the response variable, solve back through the gradient (not the whole fitted line) for the corresponding change in the explanatory variable. Read the stated units on the explanatory variable before writing anything — every confirmed slip at the interpretation step across all four series traces to units or scale, not arithmetic.

#Sources and limitations

• All citations trace to research/veridian/WST01-verified-facts.md's Section 4 write-up and its per-lesson audit entry for wst01-regression-gradient-interpretation-extrapolation.ts (2026-09-12), cross-checked against signal-frequency-and-boundaries.md's whole-Section-4 row.
• Frequency is stated only as "N of 6 series checked" — no percentage, likelihood, or grade-boundary figure is used or implied anywhere in this document (none exists in the source material; see signal-frequency-and-boundaries.md §2).
• The 8 series without a located examiner report were not searched question-by-question for this topic and are excluded from both the numerator and denominator in §1.
• Oct 2021 Q2's whole-question mark total, and any exact mark split for Jun 2022 Q2(d), Jan 2023 Q6(a)/(c), and Jun 2024 Q4(d) individually (beyond the single confirmed B1 at Jun 2022 Q2(e)(ii)), were not located in the reviewed record — flagged as open gaps, not estimated.
• This document does not repeat the formal derivations, the reverse-gradient-solve proof, or the quantitative worked comparison already in codex-deep-dive-regression-gradient-interpretation-extrapolation.md, nor the lesson's own worked exemplars, marked solutions, or MCQs.