Statistics 1

Formula and Method Bank

#WST01 (Statistics 1) — Formula & Identity Masterlist and Worked-Method Reference

Provenance: written 2026-09-28. This is the CODEX-equivalent content layer for WST01, built on research/veridian/WST01-verified-facts.md (compiled 2026-09-12, spot-check addendum 2026-09-25) and cross-checked line-by-line against the reference-card and trap-taxonomy blocks already live in the 13 lessons at content/veridian/curriculum/wst01/, and against content/veridian/technique-hub/wst01/forge-exam-technique-guide.md (completed the same day). No new PDFs were fetched for this document — as that guide's own provenance note records, no local WST01 PDF archive exists anywhere on this machine, and every fact below traces to the facts bank's own already-verified, live-fetched record. Every formula's GIVEN/MEMORISE status is transcribed from the spec's own "Notation and formulae" box and the formula booklet's Statistics S1 section (both cited by page in the facts bank §2a), not inferred. Direct quotation from Pearson's copyrighted mark schemes, examiner reports and specification is kept to short fragments (under ~15 words) with series/question attribution; longer passages are paraphrased.

#What this document is, and isn't

The FORGE guide covers how the exam is built, timed and marked — the M/A/B system, notation conventions, and the four real "conclusion in context" question types. The 13 curriculum lessons teach each topic in full, with worked chains, MCQs and trap-taxonomies. This document sits between them: a single, fast lookup for every formula the spec actually requires (Part 1), and a short-form method reference for the recurring question shapes — distinct from the curriculum's full worked-chain exemplars, which stay the place to learn a topic from scratch. If you already know how to standardise a Normal variable or fit a regression line, this is the page to check which formula you're allowed to forget and how a credited answer is actually shaped, not to learn either skill for the first time.

#PART 1 — Formula & Identity Masterlist

Two primary sources, both cited precisely in the facts bank: the spec's own "Notation and formulae" box (Pearson Edexcel IAL Mathematics — Specification, Issue 3, April 2019, which states outright that formulae in that box "will not appear in the booklet"), and the Mathematical Formulae and Statistical Tables booklet (Issue 2, January 2021, pp. 13–17, "Statistics S1"). Every row below is tagged:

MEMORISE — spec-listed as a formula students must produce from memory; not printed in the booklet.
GIVEN — printed in the yellow booklet; look it up, but know where and how to use it under time pressure.
RULE GIVEN IN Q — no formula to memorise or look up, because the spec guidance explicitly states the exam question will supply it every time.
TECHNIQUE, not a formula — a method the spec expects (guidance: "simple interpolation may be required"), with no single algebraic form prescribed by Pearson in either source reviewed.

#1.1 Spec Section 2 — Representation and summary of data

Formula / identity	Spec ref	Status	Notes
$\bar{x} = \dfrac{\Sigma x}{n}$ or $\dfrac{\Sigma fx}{\Sigma f}$	2.2	MEMORISE	Grouped data uses class midpoints for $x$ ; divide by total frequency $\Sigma f$ , never the number of classes — the exact denominator confusion documented at Jan 2023 Q1(c).
Median, mode	2.2	TECHNIQUE, not a formula	Positional (median) or highest-frequency (mode) identification — no summation formula exists for either. Spec guidance: calculating these "will not be the direct focus" of a question; interpreting them is.
$\sigma^2 = \dfrac{\Sigma f(x-\bar x)^2}{n}$ (deviation form) $= \dfrac{\Sigma fx^2}{n} - \bar x^2$ (computational form); $\sigma = \sqrt{\sigma^2}$	2.3	MEMORISE	The two forms are algebraically identical; the computational form is the one worth building a running table around. Divide by $n$ before subtracting $\bar x^2$ ; the square root wraps the whole result, taken once, last.
$/(n-1)$ "sample standard deviation"	2.3	Confirmed alternative, single series only	On one verified series (Jan 2023 Q1(d)) the mark scheme credited both the $/n$ population form (awrt 7.26) and the $/(n-1)$ form (awrt 7.3) for the same 2 marks. Checking June 2023 alongside it found no equivalent credit. This is not a general licence — the $/n$ form is what the spec expects reconstructed from memory, and is the safe default; treat $/(n-1)$ disagreeing with a model answer as a possible alternative, not automatically an error.
$IQR = Q_3 - Q_1$	2.3	MEMORISE	Explicitly grouped with mean/SD on the spec's "expected from memory" list.
Quartile position ( $n/4$ , $n/2$ , $3n/4$ or $(n{+}1)/4$ etc.) with linear interpolation inside the located class	2.3	TECHNIQUE, not a formula	Neither source reviewed prints a single interpolation formula — spec guidance only says "simple interpolation may be required." Both the $n$ -based and $(n{+}1)$ -based positional conventions are tolerated by real mark schemes; one verified series (Jan 2023 Q1(e)) found the $n$ -based convention empirically the stronger route for students that series, without ruling the other one out. The standard interpolation shape — locate the class containing the target position from a cumulative-frequency running total, then interpolate proportionally across that class's width — is domain-standard technique, not a form transcribed from a WST01 primary source in this pass; flagged here rather than presented as a verified quote.
Recovering $\Sigma x^2$ from a stated mean and SD: $\Sigma x^2 = n(\sigma^2 + \bar x^2)$	2.3	Derived, not separately sourced	A straightforward rearrangement of the computational SD formula above, not a separately-memorised identity — included because it is a genuinely common reverse-engineering step (e.g. "find $\Sigma x^2$ given the mean and SD") and easy to get backwards under pressure: square the SD first, then add $\bar x^2$ , then multiply by $n$ .
Coding $Y = \dfrac{X-a}{b}$ or $Y=b(X-a)$ : mean decodes by reversing the coding step; variance decodes by the square of the scaling constant, in the opposite direction	2.2	TECHNIQUE, not a formula	Not a separate formula — a direct consequence of $E(aX+b)$ / $\text{Var}(aX+b)$ (§1.3 below) applied to real data. The confirmed, repeatable trap: scaling variance by the constant instead of its square, or dividing instead of multiplying when decoding back (Jun 2022 Q3(e)).
Outlier fences: lower $= Q_1 - k \times IQR$ ; upper $= Q_3 + k \times IQR$	2.4	RULE GIVEN IN Q	Spec guidance states outright: "any rule to identify outliers will be specified in the question." There is no default $k=1.5$ to memorise for this paper — a confirmed real error is candidates substituting a remembered generic multiplier for the one the question actually gave (Jun 2022 Q1(b)). What's worth memorising is only the shape: fences point outward from the box (subtract for the lower fence, add for the upper), never the reverse — a confirmed, separately-documented direction error (Jan 2021 Q2).
Skewness (qualitative: comparing mean/median position, or a quartile-based measure)	2.4	TECHNIQUE, not a formula	Spec item 2.4 lists "skewness" with no computed skewness-coefficient formula surfaced anywhere in the facts bank's reviewed record; treated on this paper as a comparative/descriptive judgement (mean vs median position, or quartile spacing), not a numeric statistic to calculate from a memorised formula. Flagged as thin in the reviewed record — no standalone worked example was found to anchor a computed-skewness convention to.

#1.2 Spec Section 3 — Probability

Formula / identity	Spec ref	Status	Notes
$P(A) = \dfrac{n(A)}{n(\Omega)}$ for equally likely outcomes	3.1	TECHNIQUE, not a formula	The classical-probability definition; not separately listed in either source as memorise-vs-given because it's foundational notation, not a named result.
$P(A') = 1 - P(A)$	3.2	MEMORISE	Explicitly on the spec's "will not appear in the booklet" list.
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$	3.2	GIVEN	Printed in the booklet.
$P(A \cap B) = P(A)P(B \mid A)$ (general form)	3.2	GIVEN	Printed in the booklet — this is the general, non-independent form. Do not confuse with the independence-specific simplification below, which is memorise-only.
$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ , and the full two-event Bayes'-ratio form of $P(A\mid B)$	3.2	GIVEN	The facts bank confirms the booklet prints "the full Bayes'-theorem form of $P(A\mid B)$ as a ratio of two products" but does not transcribe its exact typeset layout. The standard two-event partition form, $P(A\mid B) = \dfrac{P(B\mid A)P(A)}{P(B\mid A)P(A) + P(B\mid A')P(A')}$ , is presented here as the conventional reconstruction of that description — flagged as a reconstruction from standard convention, not a verified transcription of the booklet's own typesetting.
$P(B\mid A) = P(B)$ , $P(A\mid B) = P(A)$ , $P(A \cap B) = P(A)P(B)$ (independence-specific forms)	3.3	MEMORISE	The spec box lists these as the "independence-specific forms only" students must produce unprompted; any one of the three implies the others. Confusing these with the general conditional forms above — assuming independence rather than testing for it — is Pearson's own stated general diagnosis of a paper-wide weak point in at least one series (Jun 2022).
Mutually exclusive: $A \cap B = \varnothing \Rightarrow P(A\cap B)=0$ , so $P(A\cup B)=P(A)+P(B)$	3.2	Derived, not separately listed	A special case of the $P(A\cup B)$ addition rule above with $P(A\cap B)=0$ substituted in — not a separately memorised identity. If $P(A)>0$ and $P(B)>0$ , mutually exclusive and independent can never both hold simultaneously (overlap is necessary but not sufficient for independence).
Tree-diagram product/sum laws: multiply along a path, add across complete mutually-exclusive paths; branches from one node always sum to 1	3.4	TECHNIQUE, not a formula	Direct application of $P(A\cap B)=P(A)P(B\mid A)$ along successive branches — not a new formula. With replacement: composition and totals are unchanged between draws. Without replacement: both the running total and the count just removed change on every draw — a confirmed, repeatable error is holding the denominator fixed across draws (Jun 2022 Q3(d)) or re-using an earlier branch's probability on a later one after removal (Jan 2023 Q2).

#1.3 Spec Section 4 — Correlation and regression

Formula / identity	Spec ref	Status	Notes
$S_{xx} = \Sigma(x-\bar x)^2 = \Sigma x^2 - \dfrac{(\Sigma x)^2}{n}$ ; $S_{yy}$ , the same shape with $y$	4.1 / 4.3	GIVEN	Both the deviation and computational forms are printed. The confirmed, repeatable substitution trap: using $\bar x^2$ in place of $\dfrac{(\Sigma x)^2}{n}$ — algebraically different numbers, not a rounding variant, and verified in the record for the equivalent $S_{pp}$ quantity in a PMCC calculation (Oct 2021 Q2(b)); relabelled variables ( $S_{pp}$ , $S_{tt}$ , etc.) are the same formula under a different letter, not a different identity.
$S_{xy} = \Sigma xy - \dfrac{(\Sigma x)(\Sigma y)}{n}$	4.1	GIVEN	$\Sigma xy$ alone is not $S_{xy}$ — the correction term must be subtracted first.
$b = \dfrac{S_{xy}}{S_{xx}}$ ; $a = \bar y - b\bar x$ ; $\hat y = a + bx$	4.1	GIVEN	Confirmed real errors at this exact step: the fraction inverted ( $S_{xx}/S_{xy}$ instead of $S_{xy}/S_{xx}$ ), and raw totals substituted for means inside the $a$ formula (both Oct 2021 Q2(c)). Finding $a$ and $b$ is not the same task as answering "find the equation of the line" — the line itself, values substituted in, is the deliverable (same question). Final line must be in decimal form — a fraction left in a final regression equation is explicitly not accepted, verified verbatim: fractions are "not... appropriate" for $a$ and $b$ as final estimates (Jun 2022 Q2(c)).
$r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$	4.3	GIVEN	$-1 \le r \le 1$ always; $\lvert r\rvert>1$ from a calculation signals an arithmetic error, not an unusual dataset. Sign of $r$ always matches the sign of $b$ (same numerator, $S_{xy}$ ). $r$ is invariant under a linear coding of the data — confirmed verbatim in mark-scheme use: "r not affected by (linear) coding" (Jan 2024 MS Q2(d)) — recomputing $r$ from original units after already finding it on coded data is wasted work, not extra rigour.
Derivations of $a$ , $b$ , or $r$ ; significance test of $r$	4.1–4.3	Explicitly NOT required	Spec guidance states plainly: derivations are not required for 4.2, and "derivations and tests of significance will not be required" for 4.3. Do not build a lesson expectation that a derivation could be examined.

#1.4 Spec Section 5 — Discrete random variables

Formula / identity	Spec ref	Status	Notes
$p(x) = P(X=x)$ , with $0 \le p(x) \le 1$ and $\Sigma p(x) = 1$ over the whole domain	5.2	TECHNIQUE, not a formula	Definitional notation, not a computed result — but the $\Sigma p(x)=1$ constraint is the single most useful self-check on this paper: a confirmed, repeatable real error is a missing domain value, most often $x=0$ , only caught by summing the stated probabilities and finding they fall short of 1 (Oct 2021 Q4(d)); the same idea from the construction side — forgetting to write the $\Sigma p(x)=1$ equation itself when solving for an unknown constant — is separately confirmed (Jun 2022 Q5(d)).
$F(x_0) = P(X \le x_0) = \Sigma p(x)$ for $x \le x_0$	5.2	MEMORISE	Explicitly on the spec's memorise list. Note the $\le$ , not $<$ — a boundary value belongs to $F(x_0)$ , not just values strictly below it.
$E(X) = \mu = \Sigma x_i P(X=x_i)$ ; $\text{Var}(X) = \sigma^2 = \Sigma(x_i-\mu)^2P(X=x_i) = \Sigma x_i^2 P(X=x_i) - \mu^2$ ; $E(g(X)) = \Sigma g(x_i)P(X=x_i)$	5.3	GIVEN	The booklet prints the base definitions, not the $aX+b$ shortcut versions below — those are a separate, memorise-only pair. $E(X)>0$ is a long-run average across every outcome; it does not imply $P(\text{a single favourable outcome}) > 0.5$ — a confirmed real conflation, one series (Jun 2019 Q5(e)).
$E(aX+b) = aE(X)+b$ ; $\text{Var}(aX+b) = a^2\text{Var}(X)$	5.3	MEMORISE	Explicitly on the spec's memorise list — the shortcut forms, distinct from the base definitions above. The single most repeatedly-confirmed error on this whole paper is forgetting to square the constant $a$ in the variance version: independently confirmed across a three-year span, three separate series (Jan 2021 Q4, Jun 2024 Q2(c), Jun 2024 Q3(d)) — the identical mechanism each time. Note also: no $b$ term survives into $\text{Var}(aX+b)$ at all, whatever sign $b$ has — a pure translation changes no variance.
Discrete uniform: $p(x) = \dfrac{1}{n}$ for $n$ equally likely values	5.4	TECHNIQUE, not a formula	Values need not be consecutive integers, so no single closed-form mean/variance shortcut is prescribed by the spec or printed in the booklet for this distribution specifically — both must be computed from the general $E(X)$ / $\text{Var}(X)$ definitions above. Genuine, flagged gap in this reviewed record: for a symmetric, evenly-spaced discrete uniform variable, $E(X) = \frac{1}{2}(\min+\max)$ by symmetry is a valid shortcut (used directly in the one fully-worked real example found, Jan 2025 Q1(c)) — but no equivalent variance shortcut is confirmed anywhere in the facts bank; every variance calculation on a discrete uniform variable in the reviewed record sums $x_i^2 p(x_i)$ term by term. Do not reach for a remembered $\frac{n^2-1}{12}$ -style closed form for this paper — it is not confirmed as examined content here, and using it without deriving it would not show the method a mark scheme needs to see.

#1.5 Spec Section 6 — The Normal distribution

(printed under spec heading "6." but with its one content row numbered "5.1" — a confirmed genuine typo in the official Issue 3 spec, not a transcription artefact; see the facts bank §1 for the page-image verification. "6.1" is used here as the sensible reference, matching the curriculum's own convention.)

Formula / identity	Spec ref	Status	Notes
$Z = \dfrac{X-\mu}{\sigma}$	6.1	MEMORISE	Explicitly on the spec's memorise list — standardise first, always, before any table lookup.
$\Phi(z) = P(Z<z)$ , read from the Normal Distribution Function table ( $z=0.00$ to $4.00$ )	6.1	GIVEN (table)	$P(Z>z) = 1-\Phi(z)$ is a direct consequence, not a separate memorised formula — but deciding when to subtract from 1 is the paper's single most repeated Normal-distribution error, confirmed independently in both directions across three series (under-subtracting: Jan 2021 Q3(a), Jan 2023 Q5(a); over-subtracting: Jun 2024 Q5(a)). Pearson's own examiner-report fix, twice stated: draw a diagram first.
Percentage Points of the Normal Distribution table: the inverse lookup, $z$ such that $P(Z>z)=p$ , to 4 d.p.	6.1	GIVEN (table)	Confirmed real, repeatable accuracy trap across three independent series: a rounded 2-3 d.p. table value substituted for the required 4 d.p. one (e.g. 1.64 for 1.6449 — Oct 2021 Q6(b); 1.03/1.04 for 1.0364 — Jan 2021 Q3(b); 0.25 for a more precise equivalent — Jun 2024 Q5(b)). Quote table values in full, per the front-page rounding instruction — see FORGE guide §3.3 for the complete convention.
Conditional probability with a Normal model: $P(A\mid B) = P(A\cap B)/P(B)$ ; if $A$ sits entirely inside $B$ , $P(A\cap B)=P(A)$ ; if $A$ and $B$ merely overlap, $P(A\cap B)$ is the interval probability between the two boundaries (subtract two $\Phi$ values)	6.1 / 3.2	GIVEN (general formula) + TECHNIQUE (Normal application)	Not a new formula — the ordinary conditional-probability identity applied to Normal-derived probabilities. The single most consistently "most discriminating" sub-skill in the whole facts bank: recognising a conditional probability is even required, confirmed across every one of the 5 fully-reviewed examiner reports (e.g. Jan 2021 Q3(c), Oct 2021 Q6(c), Jan 2023 Q5(e)) — see Part 2 §2.6 below for the short method template.
p.d.f. of the Normal distribution; derivation of its mean, variance, or CDF; interpolation within the Normal tables	6.1	Explicitly NOT required	Spec guidance states all three outright: knowledge of the p.d.f. is not required, derivation of mean/variance/CDF is not required, and interpolation is not necessary. Only the shape/symmetry and correct table use are examined.
Hypothesis testing of any kind ( $H_0$ / $H_1$ , test statistic, critical value)	—	Does not exist on this paper	Confirmed twice directly in spec guidance (items 2.2 and 4.3) and by a full-text search of the facts bank returning nothing under "hypothesis," "H0," "H1," "critical value," or "test statistic" beyond that guidance line. See FORGE guide §3.2 for the full correction — restated here only so this masterlist doesn't imply a test-statistic formula belongs somewhere in Part 1 by its absence being unexplained.

#PART 2 — Worked-Method Reference

Organised by spec topic. Each entry is a short method template, grounded in a real, cited mark-scheme or examiner-report convention — not a full worked chain (those live in the curriculum lessons cited alongside each entry). Four entries — marked [AO4] — cover the specific question shapes the FORGE guide identifies as this paper's real equivalent of "conclusion in context" (§3.2 of that guide); the others are shorter method-convention notes for the more mechanical question shapes that recur across the paper.

#2.1 Spec Section 1 — Mathematical models (1.1)

[AO4] Modelling assumptions and refinement. No standalone past-paper question tests this spec point in isolation anywhere in the reviewed 14-series record — its AO3 marks (15–20 of 75, unusually high for this qualification) arrive embedded as sub-parts riding on other questions, most often a probability question built on a stated assumption ("the die is fair," "growth follows a discrete uniform model").

Method template:

Name the assumption explicitly — a statement like "the spinner is fair" is something the question asks you to accept for the calculation, not an established fact; treating it as provable is the confirmed, VERIDIAN-named failure mode this template exists to guard against (no examiner-report misconception is quoted for this exact point, since no standalone question exists to generate one — flagged as inference from the general pattern, not a sourced quote).
State the distributional family the assumption implies (fair spinner with $n$ sectors → discrete uniform, $p(x)=1/n$ ).
If a later part of the same question supplies evidence against the assumption (an observed frequency table, say), a credited refinement answer names the specific evidence reported — which value was over- or under-represented — and its specific consequence for the model (which probability moves, and the constraint that all probabilities must still sum to 1). A generic, true-sounding statement ("real spinners are never perfectly fair") is confirmed to earn nothing on this paper's real "critique this model" parts, however reasonable it sounds, because it isn't tied to the evidence actually given.

Full treatment: mathematical-modelling-in-probability-and-statistics.ts.

#2.2 Spec Section 2 — Representation and summary of data (2.1–2.4)

Reading a histogram. Method: frequency = frequency density × class width — area, not bar height, is frequency. A confirmed real failure is reading heights straight off the axis as frequencies (Jan 2023 Q1(a)). Self-check: do your recovered frequencies sum to the total the question states?

Interpolating a quartile from grouped/continuous data. Method: locate the class containing the target cumulative position (using either the $n$ - or $(n{+}1)$ -based convention — both are tolerated; see Part 1 §1.1), then interpolate proportionally across that class's width. A confirmed "show that" failure on this exact method: reverse-engineering two plausible-looking values that happen to produce the given answer, rather than genuinely interpolating from the data (Oct 2021 Q3(b)).

Outlier fences and box plots. Method: take the $k$ and the rule exactly as the question states them (never a remembered default), apply the fence formula from Part 1 §1.1, then draw whiskers to the most extreme real data value that is not itself an outlier — never to the fence itself. A confirmed, specifically named "show that N outliers exist" failure: finding the correct fences but never listing the $N$ values that actually fall outside them (Oct 2021 Q3(c)). If a data value is later corrected, the IQR and both fences must be recomputed from scratch — a confirmed real question where the majority failure mode was redrawing the plot with only the corrected point moved, fences left as they were (Jun 2019 Q2(c)).

[AO4] Comparing two distributions in words. The richest single "worded answer" trap in this section, confirmed with near-identical wording across three series.

Method template:

Identify which comparison the question actually asks for — a difference, a similarity, or a named statistic specifically — since answering the wrong one of these scores nothing even when the observation made is true (a confirmed real failure: discussing similarities when the question asked for differences, Oct 2021 Q3(e)).
Name a specific statistic (mean, median, IQR, range — not "the average," unqualified).
State its value for both groups being compared — a bare qualitative claim ("Branch A was higher") with no supporting figures is confirmed to lose the mark even when the direction stated is correct (Jun 2024 Q1(e)).
Never claim one measure of location is "more accurate" than another — a confirmed, named, chronic misconception on this exact question type is treating the mean as inherently more accurate than the median; the credited contrast is structural ("the mean uses every value; the median does not"), not a claim that one is a better estimate of some true value (Jun 2022 Q1(e)).

Full treatment: outliers-boxplots-comparing-distributions.ts, reading-and-comparing-distributions.ts.

#2.3 Spec Section 3 — Probability (3.1–3.4)

Setting up a conditional probability. Method: write $P(A\mid B) = P(A\cap B)/P(B)$ explicitly before substituting numbers — a confirmed, repeatable real error across every one of the 5 fully-reviewed examiner reports is treating a conditional probability as a plain ratio of the two given raw values (e.g. $0.35/0.4$ with no intersection term at all, Jan 2021 Q1(c)), or assuming independence instead of testing it (a confirmed paper-wide diagnosis, Jun 2022). On a multi-branch tree, count how many branches genuinely contribute to the numerator event before multiplying — a confirmed real error used only 2 of 3 relevant branches in a numerator product (Oct 2021 Q4(c)).

Sampling without replacement. Method: after each draw, reduce both the running total and the count of whatever was just removed — a confirmed real error holds the denominator fixed across all draws (e.g. $(62/88)^4$ instead of decreasing denominators, Jun 2022 Q3(d)), and a separate confirmed error reuses an earlier branch's probability on a later branch after the item it represents has already been removed (Jan 2023 Q2).

Venn diagrams. Method: every region gets a value written in, including a region that is genuinely zero — a confirmed real error leaves the zero region blank, and blank is never read as 0 on this paper (Jun 2022 Q4(c)).

Full treatment: elementary-probability-and-conditional-notation.ts, conditional-probability-independence-vs-mutually-exclusive.ts, sampling-tree-diagrams-venn-diagrams.ts.

#2.4 Spec Section 4 — Correlation and regression (4.1–4.3)

Computing $S_{xx}$ , $S_{xy}$ , and the regression line. Method: build the four running sums ( $\Sigma x$ , $\Sigma y$ , $\Sigma x^2$ , $\Sigma xy$ — or $\Sigma x^2$ / $\Sigma xy$ /etc. under whatever variable letters the question uses), form $S_{xx}$ and $S_{xy}$ from the computational formulas in Part 1 §1.3 (never substituting $\bar x^2$ for $(\Sigma x)^2/n$ ), then $b=S_{xy}/S_{xx}$ , $a=\bar y - b\bar x$ , and write the full line $\hat y = a+bx$ as the actual deliverable — not just the two numbers. Keep the final line in decimal form, 3 s.f., no fraction.

[AO4] Interpreting the regression gradient in context. The single most consistently-flagged "worded answer" weak point in the whole facts bank — confirmed in every one of the 5 fully-reviewed examiner reports, a different concrete slip each time (a hundred-fold scale misread, $vs$ /100 — Oct 2021 Q2(d); stopping at "positive correlation" instead of stating a rate — Jun 2022 Q2(d); mismatched units — Jan 2023 Q6(a); reversed direction — Jun 2024 Q4(d)).

Method template — one sentence, three components, all required:

Name both variables, in the words the question uses for them.
State the rate: "for each extra [one real unit of $x$ ], [ $y$ ] changes by $b$ [correct unit of $y$ ], on average" — not a bare direction statement.
Check the scale: what does "1 unit of $x$ " actually mean in this question's own units (one person, one hundred people, one million people)? The gradient's numeric value is only meaningful attached to that unit.

Worked snippet (short, illustrative — not a full chain): a regression line fitted to $\hat c = 2.1 + 0.35m$ , where $m$ is miles driven and $c$ is fuel used, in litres. Credited interpretation: "for each extra mile driven, fuel used increases, on average, by 0.35 litres." Not credited: "there is a positive correlation between distance and fuel" (states direction, no rate — the exact confirmed failure mode above) or "fuel increases by 0.35" (rate stated, unit dropped — also confirmed, mixed-unit case, Jan 2023 Q6(a)).

[AO4] Judging the reliability of an extrapolated prediction. Confirmed directly: it was rare to see a response that correctly identified which value fell outside the data — many stated "unreliable" with no value or boundary named at all (Jun 2022 Q2(e)).

Method template:

State whether the $x$ -value used for the prediction falls inside or outside the range the data actually covers.
If it falls outside: name the specific predicted value and the specific reference point the question's own data supplies — the stated minimum/maximum of $x$ if individual data or an explicit range is given (credited real form: naming that a value "outside of the range" makes it unreliable, Jan 2023 Q6(c)); if only summary statistics are given with no explicit range, compare instead to $\bar x$ (and the spread $S_{xx}$ implies) — confirmed as the credited fallback on a real question whose data line never gave a min/max at all, where the mark scheme explicitly zero-credits "reference to [a value being] out of range" when no range was ever supplied (Jun 2022 Q2(e) marking note).
Never submit "extrapolation so unreliable" alone — confirmed explicitly B0 on its own on a real mark scheme (Jun 2022 Q2(e)).

PMCC "show that" calculations. Method: when a question builds to $r$ via a "show that" step, the standardisation/ $S$ -formula working must appear on the page at one extra decimal place beyond the printed target — a confirmed real failure loses the final accuracy mark purely on insufficient shown precision, not a wrong method (Jun 2024 Q4(c)).

Full treatment: regression-gradient-interpretation-extrapolation.ts, correlation-coefficient-and-regression-calculation.ts.

#2.5 Spec Section 5 — Discrete random variables (5.1–5.4)

Verifying or solving for an unknown in a probability function. Method: before anything else, write $\Sigma p(x) = 1$ as its own line — both as a self-check on a completed table (catches a missing domain value, the single most-quoted error in this category, Oct 2021 Q4(d)) and as the actual equation to solve when a constant is unknown (a confirmed failure scores a method mark for everything else but skips writing this equation down at all, Jun 2022 Q5(d)).

$E(aX+b)$ / $\text{Var}(aX+b)$ . Method: apply the shortcut formulas from Part 1 §1.4 directly — never re-derive from $E(X^2)$ for a simple linear transformation. The one habit worth over-checking: is the constant squared in the variance line? Confirmed independently three times across a three-year span (Jan 2021 Q4, Jun 2024 Q2(c), Jun 2024 Q3(d)) as the single most repeated numeric error on this whole paper.

Full treatment: discrete-random-variables-probability-function.ts, expectation-and-variance-of-linear-transformations.ts.

#2.6 Spec Section 6 — The Normal distribution (6.1)

Basic table use: standardise, sketch, decide. Method: standardise with $Z=(X-\mu)/\sigma$ first, sketch a Normal curve and shade the required region before touching the table — Pearson's own examiner-report advice, stated after both an under-subtraction and a sign-reversal error, in each case attributing the mistake to skipping the sketch (Jan 2023 Q5(a); Jan 2021 Q3(b)) — then read $\Phi(z)$ and decide, from the sketch, whether the required probability is $\Phi(z)$ directly or $1-\Phi(z)$ .

Inverse-Normal (finding an unknown parameter from a stated probability). Method: use the Percentage Points table to find $z$ to 4 d.p. for the stated tail probability, quote it in full, then solve $Z=(X-\mu)/\sigma$ for whichever parameter is unknown. Confirmed as the hardest single Normal-distribution skill type in the reviewed record by a clear margin — a multi-step version of this method is where Pearson's own examiner report notes candidates abandoning the question partway through (Jun 2024 Q5(c)).

Conditional probability under a Normal model. Method: identify whether the conditioning event $B$ fully contains the target event $A$ (in which case $P(A\cap B)=P(A)$ and the answer is a straight ratio) or the two events merely overlap (in which case $P(A\cap B)$ is the probability of the interval between the two boundaries, found by subtracting two $\Phi$ -values, not by taking $P(A)$ alone). Confirmed as the single most consistently "most discriminating" sub-skill on this paper: recognising a conditional probability is even required at all, not just computing one correctly once recognised (Jan 2021 Q3(c), Oct 2021 Q6(c), Jan 2023 Q5(e)).

"Show that" a Normal-distribution probability. Method: the standardisation itself must be visible on the page — a confirmed real failure writes the standardisation followed immediately by the target value with the intermediate table-lookup step missing, and a further confirmed failure writes only the decimal target with no working shown at all (both Oct 2021 Q6(a)).

Full treatment: normal-distribution-standardisation-and-conditional-probability.ts.

#Genuine gaps in this document — stated plainly

The general interpolation formula (Part 1 §1.1) is presented in its standard textbook form; neither primary source reviewed for this pass (the spec's own guidance, or the formula booklet) prints a single prescribed algebraic form for it, only the guidance that "simple interpolation may be required." This is domain-standard technique, not a claim traceable to a specific WST01 document — flagged rather than presented as sourced.
The Bayes'-ratio form of $P(A\mid B)$ (Part 1 §1.2) is reconstructed in its conventional two-event partition form from the facts bank's own general description ("a ratio of two products"); the exact typeset layout as printed in the booklet was not independently re-transcribed for this pass.
The $/(n-1)$ sample-standard-deviation alternative credit (Part 1 §1.1) is confirmed on one series only (Jan 2023) and explicitly not found on a second series checked alongside it (June 2023) — presented as a one-series confirmed fact, not a general convention, matching the same caveat already carried in the FORGE guide and the curriculum's own reference card.
No confirmed variance shortcut for the discrete uniform distribution (Part 1 §1.4) — every real worked example in the reviewed record computes it by direct summation; this document does not supply a closed-form alternative because none is confirmed as examined content on this paper.
Skewness (Part 1 §1.1) has no standalone worked example in the reviewed 14-series record to confirm whether a numeric skewness coefficient is ever actually computed on this paper, versus purely qualitative comparison — flagged as thin, not resolved.
Modelling assumptions/refinement (Part 2 §2.1) has no examiner-report-sourced misconception to cite, because no standalone past-paper question exists in the reviewed record to generate one — the method template there is built from spec guidance and the curriculum lesson's own worked example (Jan 2025 Q1), not from a quoted examiner-report pattern, and is labelled as such rather than dressed as one.
Examiner-report coverage remains 6 of 14 mark-scheme series (the same lower bound the FORGE guide and facts bank both state) — every method template above sourced from an examiner report should be read as evidence from that 6-series record, not as exhaustive coverage of every way each question type can be answered incorrectly on the remaining 8 unread series.
This document performed no new PDF fetches or independent primary-source verification of its own — every claim traces either to WST01-verified-facts.md's own already-verified record or to the already-audited curriculum .ts reference cards and trap-taxonomies, both cited by file, series, and question number throughout.