Deep Dive: Reading and Comparing Distributions

#CODEX Deep-Dive — Reading Histograms and Stem-and-Leaf Diagrams, and Comparing Distributions in Words (WST01, spec 2.1, plus the shared 2.1/2.4 guidance point)

Provenance (2026-09-29): written after reading, in full, research/veridian/WST01-verified-facts.md (§1 spec transcription, §2/§2a assessment structure and formula-booklet contents, §5 Section 2, §7 topic-richness ranking, and the 2026-09-12 per-lesson audit entry for wst01-reading-and-comparing-distributions); the live lesson file content/veridian/curriculum/wst01/reading-and-comparing-distributions.ts in full, including its own header note and closing flag block; the sibling lesson content/veridian/curriculum/wst01/outliers-boxplots-comparing-distributions.ts (read in full, per the live lesson's own instruction to read it before writing anything that touches comparing distributions); content/veridian/technique-hub/wst01/codex-formula-and-method-bank.md, forge-exam-technique-guide.md and signal-frequency-and-boundaries.md (grepped for "histogram", "stem", "back-to-back" and "comparing distribution"); and content/veridian/question-bank/wst01.ts in full (two entries exist — a box-plot amendment question and an expectation-vs-probability question — neither on this topic). No new PDF fetches were made for this document. Every arithmetic figure in the micro-examples below was independently computed with Python (statistics/exact integer arithmetic) and is flagged as this document's own computation, not a transcribed exam answer, unless a specific past question and mark scheme line is named for it. Following this course's copyright discipline, Pearson's own examiner-report and mark-scheme wording is paraphrased throughout rather than quoted at length; where a phrase is quoted directly it is kept under fifteen words and attributed.

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

The live lesson already teaches, in full: the frequency-density mechanism for histograms with a worked 60-cyclist example and diagram; reading an ordinary and a back-to-back stem-and-leaf diagram, including a chain-drill on the Pine (Group B) data and a marked solution on the Oak (Group A) data; the three-part discipline for comparing two distributions in words (name a statistic, give both figures, state the direction); and five MCQs, one of which (rcd-mcq-5) already drills an even-$n$ median as the average of two ranked values. It explicitly defers the mean-vs-median resistance-to-outliers mechanism, and the outlier-fence/box-plot mechanics generally, to the sibling lesson outliers-boxplots-comparing-distributions.ts — a deferral this document also respects. None of that content is repeated here.

What follows instead is three things the reviewed archive supports but no existing WST01 document currently states in this form: (1) the equal-class-width case treated as a derived special case of the general area rule, rather than as a separate fact, which is what actually explains why the height-only reading error is invisible on some historical/simplified material and only decisively wrong once a real Edexcel question sets unequal widths; (2) a general position rule for either side of a back-to-back stem-and-leaf diagram, stated once as a formula rather than only demonstrated on one dataset, plus a fresh micro-example where the two groups sharing one diagram have different parities of $n$ and therefore need genuinely different median mechanics side by side; (3) a synthesis of every real exam angle this specific topic (as opposed to the sibling's outlier/box-plot angles) has actually been tested from, with the honest denominator stated for each one, plus a clearly-flagged (not overclaimed) decision procedure for which pair of statistics to quote in a comparison, built from a mechanism the sibling lesson proves but this document does not re-derive.

#1. The model, stated formally

#1.1 Histograms: why the equal-width case is a special case, not a separate rule

Spec 2.1 groups histograms with stem-and-leaf diagrams and box plots under "representation and summary of data," with an explicit guidance point that drawing them will not be the focus of a question — reading and using one will be. A histogram's vertical axis is frequency density, $fd = f/w$ for class frequency $f$ and class width $w$; rearranged, $f = fd \times w$, the bar's area.

The general rule already fully derives the special case that makes some simplified teaching describe a histogram as "bar height = frequency": if every class shares the same width $w = c$, then $f_i = fd_i \times c$ for every class $i$ — frequency is directly proportional to height, with the same constant of proportionality $c$ throughout. Under that condition, comparing two bars' heights directly does correctly rank their frequencies, because the shared factor of $c$ cancels in any height ratio: $f_i/f_j = fd_i/fd_j$. This is a genuinely true fact, not a misconception — it just only holds when $w$ is constant across every class being compared. The confirmed real error (Jan 2023, Q1(a), examiner report: most unsuccessful candidates read bar heights straight off the axis and reported them as the class frequencies, producing wrong totals) is therefore best understood as this special-case shortcut applied to a question that did not satisfy its own precondition. The reviewed record does not disclose the actual class widths used on that specific question, so this document does not claim to know them — but the fact that reading heights directly produced a genuinely wrong answer (rather than a right-ratio-wrong-scale one) is itself evidence the classes were not all the same width, since equal widths would have made the height-only method at least internally consistent.

#1.2 Back-to-back stem-and-leaf diagrams: a general position rule, not just a worked instance

The live lesson correctly teaches, on one dataset, that a back-to-back diagram's two flanks share one stem column, both order their leaves ascending as you move away from the stem, and that this makes the left-hand flank print in the reverse of its own value order. Stated once as a rule rather than only demonstrated: for a row with $L$ leaves on the left, printed left to right on the page as $\ell_1,\ell_2,\ldots,\ell_L$, the leaf printed at page-position $k$ is the $(L-k+1)$-th smallest value in that row — the printed sequence is the reverse of the ranked sequence. For the right-hand flank, with $R$ leaves printed as $r_1,\ldots,r_R$, the leaf at page-position $k$ is simply the $k$-th smallest value directly — no reversal, because moving right is also moving away from the stem on that side. This single formula covers both the ordinary one-sided case (a diagram is just a right-hand flank with no partner) and any back-to-back row, of any $L$ and $R$, without needing a fresh verbal description each time. The confirmed real error this formula makes precise (Jun 2024, Q1(b), examiner report: a cross-series description of students reading the diagram in the wrong direction and misidentifying which value was $Q_1$ and which was $Q_3$) is exactly a failure to apply the $k \to L-k+1$ reversal on the left flank while correctly leaving the right flank unreversed.

Locating a quantile once a flank's values are correctly ranked uses simple rank-position counting — median at position $\frac{n+1}{2}$, $Q_1$ at $\frac{n+1}{4}$, $Q_3$ at $\frac{3(n+1)}{4}$ — which the live lesson already uses on data chosen so every position lands on a whole number. This document is explicit about a genuine limitation that choice conceals: for even $n$, $n+1$ is odd, so $\frac{n+1}{4}$ is never a whole number, meaning $Q_1$/$Q_3$ from an even-sized ranked list generically falls between two ranks. Only the median (position $\frac{n+1}{2}$, a whole-or-half number) has a settled convention for that case in the material read for this document — average the two middle ranks, which is exactly what the live lesson's own even-$n$ MCQ (rcd-mcq-5) does. No confirmed convention for interpolating an even-$n$ quartile position from a ranked list (as opposed to interpolating within a grouped-frequency class, which is a different, already-taught spec 2.3 skill) was found in the reviewed archive, so §2.2 below deliberately restricts its even-$n$ flank to the median only, rather than inventing an interpolation rule the source material doesn't establish.

#1.3 Comparing two distributions: a decision procedure, clearly separated from confirmed exam content

The live lesson and its sibling both already establish, correctly, that a scoring comparison must name a statistic, state both groups' figures for it, and give the direction of the difference — and that treating the mean as inherently "more accurate" than the median is the single most-recorded way to lose that mark (confirmed in three series: Jun 2022 Q1(e), Jun 2024 Q1(e), Oct 2021 Q3(e), examiner reports). The sibling lesson proves, in full, why an outlier moves a mean but not a median; that proof is not repeated here.

What is added here, flagged explicitly as this document's own synthesis rather than a confirmed past-paper instruction, is the practical decision it implies: when nothing in a question suggests either group has outliers or marked skew, mean and standard deviation are a complete, efficient pair to quote, because they use every value; when either group's data shows an extreme value, or skew is visible in a stem-and-leaf diagram's shape, median and IQR are the safer pair, because neither statistic is moved by a value in the tail — the median because it is a rank, not a sum, and the IQR because $Q_1$ and $Q_3$ are themselves interior ranks that a single extreme value cannot shift. No citation in the reviewed archive states this as an explicit rule for choosing which pair to quote; it is this document's own generalisation of a mechanism the sibling lesson already proves, offered as reasoning a student can apply, not as a memorised exam pattern.

#2. Real, distinct sub-cases — short worked micro-examples

Every figure below is independently computed for this document and verified with Python; none reuses a dataset from either lesson.

#2.1 Equal-width classes make height ranking correct; unequal widths break it

Three classes: Class 1 (width 5, density 3), Class 2 (width 5, density 6), Class 3 (width 10, density 3). Frequencies: $3\times5=15$, $6\times5=30$, $3\times10=30$. Comparing Class 1 against Class 2 (same width): the height ratio $6/3=2$ matches the frequency ratio $30/15=2$ exactly — confirming §1.1's special case. The more striking pair is Class 2 and Class 3: they have different heights (6 and 3) but the same frequency (30 each), because Class 3's doubled width exactly offsets its halved density. A student ranking these three classes by height alone would place Class 2 above Class 3; ranking by frequency, they are tied — a concrete demonstration that height alone stops being a reliable ranking the moment widths differ.

#2.2 A back-to-back diagram where the two flanks need genuinely different median mechanics

Two fresh, VERIDIAN-original groups sharing one diagram. Group L, $n=8$ (even), ranked ascending: 12, 14, 15, 17, 19, 21, 22, 25. Group R, $n=7$ (odd), ranked ascending: 13, 16, 18, 20, 24, 26, 29.

Group R's positions are all whole numbers: median at position $\frac{7+1}{2}=4$ (the value 20), $Q_1$ at position $\frac{7+1}{4}=2$ (the value 16), $Q_3$ at position $\frac{3\times8}{4}=6$ (the value 26), $IQR=26-16=10$. Group L's median position is $\frac{8+1}{2}=4.5$, so — following §1.2's stated convention — the median is the average of the 4th and 5th ranked values, $\frac{17+19}{2}=18$; its quartile positions ($\frac{9}{4}=2.25$, $\frac{27}{4}=6.75$) are not whole numbers, and no confirmed convention for that case was found, so this micro-example stops at the median for Group L rather than inventing one. The point this makes concrete: two groups printed on the very same diagram can each require a different rule for the same statistic, purely because one group's count is even and the other's is odd — a reader who has only ever practised on odd-$n$ data (as the live lesson's own core dataset happens to be, $n=11$ and $n=15$) has not yet met this case.

#2.3 Why median/IQR and mean/SD can tell two different stories about the same single change

Seven values, symmetric: 10, 12, 14, 16, 18, 20, 22. Mean $=16$, median $=16$ (identical, as expected for symmetric data); $Q_1=12$, $Q_3=20$, $IQR=8$; population variance $=16$, so $SD=4$ exactly.

Now change only the largest value, from 22 to 82, leaving the other six untouched: 10, 12, 14, 16, 18, 20, 82. The median is still the 4th ranked value, still 16 — completely unchanged, because replacing the maximum doesn't move which value sits in rank position 4. $Q_1$ (2nd ranked value, 12) and $Q_3$ (6th ranked value, 20) are also both unchanged, so $IQR=8$ is identical to before — neither quartile's rank position was touched by a change confined to the single largest value. The mean, by contrast, is pulled from 16 to $172/7\approx24.6$ (3 s.f., matching the paper's own stated rounding convention for inexact answers); the standard deviation rises from 4 to $\approx23.7$ (3 s.f.) — nearly a six-fold increase from one changed value. A "compare the two distributions" answer built on mean and SD alone would report a large shift that a median-and-IQR answer would correctly show as no shift at all in three of the four statistics — exactly the reasoning behind §1.3's decision procedure, made concrete with one clean, fully-verified change rather than asserted in the abstract.

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

Stated against the facts bank's own honest denominator: 6 of the unit's 14 mark-scheme series have a fully-read examiner report (Jan 2021, Oct 2021, Jun 2022, Jan 2023, Jun 2024, plus the 2026-09-25 Jun 2019 spot-check); the other 8 remain unread as of this writing.

Honest frequency statement. The comparing-distributions-in-words guidance point is the one angle in this table confirmed across multiple series (3 of the 6 fully-read: Jun 2022, Jun 2024, Oct 2021); the histogram and stem-and-leaf-direction traps are each confirmed exactly once in the reviewed record. That is a real difference in evidence weight, and this document states it rather than rounding both up to "recurring."

#4. Adjacent territory this document deliberately leaves alone

Spec 2.1 shares its guidance point with spec 2.4 (skewness, outliers), and several real, confirmed exam angles sit on that side of the boundary rather than this one: the outlier-fence formula and its substitution errors (Jan 2021 Q2; Jun 2022 Q1(b)), box-plot whisker placement against a fence versus a real data value (Jan 2021 Q2), "show that $N$ outliers exist" completeness (Oct 2021 Q3(c)), and a verified full worked box-plot amendment after two data values change (June 2019 Q2(c), already present as wst01-jun19-q2c-box-plot-outliers in the question bank). All of these are owned in full by outliers-boxplots-comparing-distributions.ts and, for the June 2019 question specifically, by the question bank — repeating them here would be exactly the restatement this document's own scope section disclaims.

#5. Open flags — where the real evidence runs out

• No confirmed past-paper convention for interpolating a quartile position from an even-sized ranked list (as opposed to a grouped-frequency class) was found; §1.2/§2.2 state this limitation plainly rather than inventing a rule to fill it.
• §1.3/§2.3's statistic-pair decision procedure is this document's own generalisation of an already-proven mechanism, not itself a confirmed exam instruction — repeated here because it is easy to misread the table in §3 as claiming otherwise.
• 8 of the unit's 14 mark-scheme series have no independently-verified examiner report read yet. The one spot-check outside the original five (June 2019) found two real gaps, neither on this topic — a lower bound on what a full re-read might still surface here, not evidence this topic's record is exhausted.
• The live lesson's own scale-calibration marked-solution question (inferring a histogram's frequency axis from one class's known frequency) is stated there as VERIDIAN-original; this document neither newly confirms nor denies a real past-paper instance of that specific mechanic, since none was located in the material read for this pass.

#Sources

research/veridian/WST01-verified-facts.md — §1 (spec and guidance text), §2/§2a (assessment structure, rounding convention, formula-booklet contents), §5 Section 2 (all examiner-report findings cited above), §7 (topic-richness ranking), and the 2026-09-12 audit entry for wst01-reading-and-comparing-distributions. content/veridian/curriculum/wst01/reading-and-comparing-distributions.ts and outliers-boxplots-comparing-distributions.ts (both read in full). content/veridian/technique-hub/wst01/codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md (checked, no overlapping depth found). content/veridian/question-bank/wst01.ts (checked in full — two entries, neither on this topic).