Deep Dive: Outliers Boxplots Comparing Distributions

#CODEX Deep-Dive — Outliers, Box Plots, and Comparing Distributions (WST01, spec 2.1/2.4)

Provenance (2026-09-29): Read/checked for this document — research/veridian/WST01-verified-facts.md in full, grepped specifically for "outlier", "box", "whisker", "quartile", "IQR", "fence" and "skew" (§1 spec transcription; §2 memorise-vs-booklet list; §2b AO weighting; §3 source-archive counts; §5 Section-2 examiner-report findings; §6 the "show that" cross-cutting pattern; §7 topic-richness ranking; and the two per-lesson audit entries for outliers-boxplots-comparing-distributions.ts dated 2026-09-12 and 2026-09-25); the live lesson file content/veridian/curriculum/wst01/outliers-boxplots-comparing-distributions.ts in full; the sibling lesson content/veridian/curriculum/wst01/reading-and-comparing-distributions.ts (grepped for overlap/scope boundary — it explicitly delegates all outlier/box-plot content to the lesson this document is about); content/veridian/technique-hub/wst01/codex-formula-and-method-bank.md, forge-exam-technique-guide.md and signal-frequency-and-boundaries.md (grepped for this topic); content/veridian/question-bank/wst01.ts (grepped — one real past-paper item exists on this topic, wst01-jun19-q2c-box-plot-outliers, already a full reproduction with marked solution — not duplicated here). No new PDF fetches were made; every specific claim below traces to one of those sources, named inline. All arithmetic in the micro-examples was independently checked by hand before being written in.

#Scope — what this adds, and what it deliberately does not repeat

The live lesson already teaches the five-number summary, derives the fence-direction mechanism from first principles, works two full worked-chains (a 15-runner race, and the real June 2019 Q2(c) amended-value question), a chain-drill, a marked solution, seven MCQs, and a formal concept-ladder proof of why the mean moves and the median doesn't. The formula bank tabulates the fence formula's memorise/given status and the skewness gap. The forge guide's trap table already cross-references this lesson's confirmed traps by name and series. None of that is reproduced here.

This document does three things the existing material doesn't: (1) formalises, as an explicit general rule with an exact numeric check, the "a fence can move even though its own anchor quartile never changed" mechanism the live lesson only argues in prose for one specific dataset, plus a matching formalisation of the skew-direction test as a checkable inequality rather than a descriptive comparison; (2) assembles every real exam angle this specific sub-topic has actually been tested from into one synthesis table, organised by angle and citation rather than by lesson stage, including one adjacent upstream angle (a misread stem-and-leaf quartile) that the sibling lesson owns but that directly poisons this topic's fence calculations if it goes uncaught; (3) states, more sharply than the formula bank's existing "thin" flag, that skewness specifically has zero confirmed past-paper or examiner-report citations anywhere in the 14-series record reviewed for this course — not "thin evidence," but no evidence at all beyond the spec's own two-sentence listing.

#1. The formal model

The five-number summary and the box. A box plot draws exactly five numbers, in rank order: minimum, $Q_1$, median, $Q_3$, maximum. The box runs from $Q_1$ to $Q_3$ (the IQR, the middle 50% of the data) with a line at the median; whiskers extend outward toward the extremes. Spec 2.1's own guidance states plainly that "drawing of histograms/stem-and-leaf/box plots will not be the direct focus of exam questions" — reading and using a given plot is the tested skill, not constructing one from scratch. $IQR = Q_3 - Q_1$ sits on the spec's own list of formulae students must know from memory rather than look up — spec §S1.2 states outright that "formulae that students are expected to know" will "not appear in the booklet" — alongside mean and standard deviation.

The outlier rule — general form, and why it is never a fixed constant on this paper. Spec 2.4's guidance states outright: "any rule to identify outliers will be specified in the question." The general form is

$$\text{lower fence} = Q_1 - k\times IQR, \qquad \text{upper fence} = Q_3 + k\times IQR,$$

with $k$ given by the question — commonly, though not necessarily, $1.5$ in the reviewed record. The direction is not a convention to memorise: the lower fence must sit below $Q_1$ (subtract) and the upper fence above $Q_3$ (add), because a fence built the other way moves into the box rather than away from it.

The whisker rule. If neither extreme lies beyond its fence, both whiskers run to the true minimum and maximum. The moment a value is flagged as beyond a fence, its own whisker stops short: it runs only to the most extreme value that is NOT itself flagged, and the flagged value is marked separately. The safe, taught default is that a whisker's end is always a real data value, never the fence itself — an empirically confirmed but (per §2.2 below) not perfectly symmetric convention.

The recompute-cascade rule, formalised. Suppose a correction to the data changes $Q_1$ from $Q_{1,\text{old}}$ to $Q_{1,\text{new}}$ while $Q_3$ is untouched (the argument is symmetric if $Q_3$ moves instead). The new $IQR' = Q_3 - Q_{1,\text{new}}$, and comparing the new and old upper fences:

$$F_{\text{upper,new}} - F_{\text{upper,old}} = k\big(Q_3 - Q_{1,\text{new}}\big) - k\big(Q_3 - Q_{1,\text{old}}\big) = k\big(Q_{1,\text{old}} - Q_{1,\text{new}}\big).$$

The upper fence — anchored to $Q_3$, which never moved — shifts anyway, by exactly $k$ times the change in the OTHER quartile, because the width term feeding it (the IQR) changed. This is the general statement of what the live lesson's mechanism block argues in prose for one dataset; it is checked against that real dataset in §2.4 below.

Skewness as a checkable inequality, not a formula. No skewness coefficient appears in the WST01 formula booklet or on the "expected to know" list; spec 2.4 groups "skewness" with outliers under a single guidance-free bullet. The two comparisons the live lesson's teach block describes in prose reduce to two inequalities:

$$\text{positive (right) skew signal:}\ (Q_3-\text{median}) > (\text{median}-Q_1), \qquad \text{negative (left) skew signal:}\ (\text{median}-Q_1) > (Q_3-\text{median}),$$

corroborated, where neither whisker is cut short by an outlier, by the matching comparison of whisker lengths, $(\max - Q_3)$ against $(Q_1-\min)$.

Why the mean moves and the median doesn't. $\bar{x}=\frac{1}{n}\sum x_i$ is a linear function of every value, so a change of $\delta$ in one value changes $\bar{x}$ by exactly $\delta/n$ — small but never zero. The median is an order statistic, fixed by rank alone; a change that doesn't alter a value's rank leaves the median completely unchanged, however large that change is. This is exactly what the credited real answers name — "the mean uses all the data," "the mean includes the outliers" (Jun 2022, Q1(e)) — against the uncredited claim that the mean is simply "more accurate."

#2. Real, distinct sub-cases, each with a short micro-example

#2.1 Applying a fence rule that is stated, not remembered

A data set has $Q_1=30$, $Q_3=54$ ($IQR=24$). The question states its own rule: an outlier lies more than $1.25\times IQR$ beyond the nearer quartile — a multiplier the live lesson doesn't otherwise use (it drills $1.5$ and $2$). $1.25\times24=30$, so the upper fence is $54+30=84$. A value of $87$ IS an outlier under this stated rule ($87>84$). Substitute the more familiar $1.5\times IQR$ instead — the exact error the confirmed record names — and $1.5\times24=36$ gives an upper fence of $54+36=90$, under which the same value $87$ is NOT an outlier ($87<90$). Same data, same value, opposite conclusion, purely from using the remembered multiplier instead of the one the question actually gave: "a small number of candidates opted to apply the more commonly used outlier formula" (Jun 2022, Q1(b)).

#2.2 Whisker termination, and the mark scheme's own confirmed asymmetry

Real anchor, reconstructed from the facts bank's independently-verified figures (Jan 2021, Q2): $Q_1=68$, $Q_3=80$, $IQR=12$, $k=1.5$, giving fences at $50$ (lower) and $98$ (upper). The real data's lowest non-outlier value is $54$ and one outlier sits at $43$; the real maximum non-outlier value is $97$, with no outlier above it. The taught safe default — draw every whisker to the most extreme real, non-outlier value — puts the lower whisker at $54$ and the upper at $97$. The real mark scheme, verified against the primary PDF, instead reads (truncated to stay under the copying limit): "for upper whisker ending at 97 and lower whisker ending at 54 or 50." It credits the lower whisker at EITHER the real value (54) or the fence itself (50), but the upper whisker only ever at the real value (97), never at its fence (98). The taught rule (always draw to the real value) remains the correct, universally-safe choice; the asymmetry is evidence that the fence is an unreliable shortcut that happens to work on one side of one question, not a second valid method to rely on.

#2.3 "Show that N outliers exist" — completeness, not just the fences

A data set has $Q_1=22$, $Q_3=41$ ($IQR=19$), rule: outlier if more than $1.5\times IQR$ beyond the nearer quartile. Fences: lower $=22-28.5=-6.5$, upper $=41+28.5=69.5$. Suppose the data's largest three values are $71,\,74,\,80$. A response that stops at "the upper fence is 69.5, so there are outliers" earns the method mark but not the accuracy mark for a "show that there are 3 outliers" instruction: the values themselves — $71,\,74,\,80$, each individually checked against $69.5$ — have to be listed. This is a confirmed, named failure mode on a real question: "some did not list the 3 outliers in this part" (Oct 2021, Q3(c)), despite having found the correct limits — the same paper-wide "show that" discipline flagged as Pearson's single most repeated meta-level complaint across every examiner report read for this course.

#2.4 The recompute cascade — checking the general rule from §1 against the real question it comes from

The real anchor (June 2019, Q2(c)): original $Q_1=33$, $Q_3=47$ ($IQR=14$, $k=1.5$). One correction moves $Q_1$ to $35$ (Q3 untouched). The general rule from §1 predicts the upper fence shifts by $k(Q_{1,\text{old}}-Q_{1,\text{new}}) = 1.5\times(33-35) = -3$. Direct check: old upper fence $=47+1.5(14)=68$; new upper fence $=47+1.5(12)=65$; $65-68=-3$ — the formula matches exactly. A second, fresh check with different numbers (not the real question's data): $Q_1=40\to46$, $Q_3=64$ fixed, $k=1.5$. Old $IQR=24$, old upper fence $=64+36=100$. New $IQR=18$, new upper fence $=64+27=91$. Predicted shift: $1.5\times(40-46)=-9$; actual shift $91-100=-9$ — matches again. The practical consequence in both cases is identical: a value sitting just below the OLD upper fence (in the fresh example, a value like $95$) is now beyond the new one, a newly-created outlier that was never itself corrected — exactly the mechanism the real examiner report names as the reason most candidates lost marks: "The major problem was that few candidates realised that the lower quartile would change" — and, once it does, that the untouched-side fence has to be rebuilt from scratch, not patched.

#2.5 Skewness read from box-plot geometry — and the honest limit of the evidence

A box plot has $Q_1=12$, median $=18$, $Q_3=22$, whiskers to $4$ and $26$ (neither an outlier). Lower half of the box: $18-12=6$. Upper half: $22-18=4$. Since $6>4$, this signals negative (left) skew by the box-half test in §1 — the middle 50% of the data is more spread out below the median. Checking the whiskers: lower whisker length $12-4=8$; upper whisker length $26-22=4$; the longer lower whisker corroborates the same conclusion. This is a domain-standard reading skill, correctly derivable from the definitions above, but it is not a confirmed WST01 exam angle in the record reviewed for this course: a full-text search of WST01-verified-facts.md for "skew"/"skewed"/"skewness" returns exactly one hit, the spec's own guidance line — no examiner-report quote, no mark-scheme line, and no past-paper question part citing skewness was found anywhere in the 14 downloaded series. The formula bank already flags this as "thin"; the honest statement is that it is currently unevidenced beyond the spec text, not merely under-evidenced.

#2.6 Comparing two distributions — naming a statistic and showing the figures

Class A: median $=52$, $IQR=8$. Class B: median $=61$, $IQR=15$. A credited-shape answer names a statistic and both figures: "Class B's median (61) is higher than Class A's (52), so Class B's scores are typically higher; Class B's IQR (15) is larger than Class A's (8), so Class B's scores are more variable." An uncredited answer of the kind the confirmed record flags — "the mean is more accurate than the median" (Jun 2022, Q1(e)) — asserts a ranking between two different measures of the same idea rather than comparing either group's actual figures. Two further confirmed failure modes, independent of this specific pair of numbers: giving no figures at all — "surprisingly too many students failed to give supporting figures" (Jun 2024, Q1(e)) — and answering the wrong half of the question, such as describing similarities when asked for differences — "few comments referring to the distribution... were seen" (Oct 2021, Q3(e)).

#3. Synthesis — every real exam angle this topic has actually been tested from

Honest frequency statement, no invented likelihoods. Six examiner-report series have been fully read for this course (Jan 2021, Oct 2021, Jun 2022, Jan 2023, Jun 2024, Jun 2019 — the 2026-09-25 addition). This specific sub-topic — outliers, box plots, and comparing distributions as their own tested skill — is directly evidenced in 5 of those 6: Jan 2021, Jun 2022, Oct 2021, Jun 2024 and Jun 2019. Jan 2023's read report contributes confirmed content to this unit's Section 2 (histogram reading, grouped mean, SD/interpolation), but nothing specific to outliers, box plots, or skewness — so it is correctly absent from the table above, not an oversight.

#4. Open flags — where the real evidence currently runs out

• Skewness has no confirmed past-paper or examiner-report citation anywhere in the reviewed record (§2.5) — taught here and in the live lesson as a correct, spec-grounded reading skill, but it should not be presented to a student as a confirmed, frequently-tested angle, because the evidence for that specific claim does not exist yet in this course's research.
• 8 of the 14 downloaded mark-scheme series have no examiner report read yet (Jun 2021, Jan 2022, Oct 2022, Jun 2023, Oct 2023, Jan 2024, Oct 2024, Jan 2025, per the facts bank's own count) — the synthesis table above is a lower bound, not a claim that this topic's exam-angle catalogue is exhausted.
• The §1 fence-cascade formula is a derivation from definitions already established elsewhere in this course's material, checked against one real dataset and one fresh one — it is sound mathematics, not a new empirical finding, and is presented as such.

#Sources

research/veridian/WST01-verified-facts.md — §1 (spec text, both 2.1 and 2.4 guidance), §2 (memorise-vs-booklet list), §3 (source-archive counts, the 6-of-14 examiner-report denominator), §5 Section 2 (all examiner-report quotes cited above), §6 (the cross-cutting "show that" pattern), §7 (topic-richness ranking), and the 2026-09-12 and 2026-09-25 per-lesson audit entries for outliers-boxplots-comparing-distributions.ts. content/veridian/curriculum/wst01/outliers-boxplots-comparing-distributions.ts (live lesson, read in full). content/veridian/curriculum/wst01/reading-and-comparing-distributions.ts (grepped for scope boundary and the Jun 2024 Q1(b) stem-and-leaf misread citation). content/veridian/technique-hub/wst01/codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md. content/veridian/question-bank/wst01.ts (checked — one existing item, not duplicated).