8.1 Data Representation & Analysis

Why data representation matters on QL

The CEA Quantitative Literacy construct treats data representation and analysis as a core subdomain — it is QL2 in the six-subdomain list scored in the NBT National Report. On the AQL you will not be asked to invent a fancy statistical model. You will be asked to read a table or chart the way a first-year student must read a lab report, a fee schedule, or a municipal water notice: find the right cell, combine cells, compare series, and say what the pattern means in context.

Calculators are not allowed. That constraint rewards clean reading and short arithmetic, not button-mashing. The habit that scores marks is: label → units → structure → derive → interpret.

Chance, probability trees, and uncertainty language belong in Section 8.2. Here the focus stays on tables and graphs.

The pre-read checklist (do this every time)

1. Title / caption — What population, place, and time window is shown?
2. Axes and column headers — What is measured on each dimension?
3. Units — Rand, litres, percentages, thousands of students, hours?
4. Legend / key — Which shade or marker is which series?
5. Scale quirks — Broken axes, dual axes, stacked totals that must sum to 100%, or a note that values are rounded.

If two options differ only by a unit mix-up (for example treating “thousands of litres” as litres), the wrong option is usually there on purpose.

Multi-row, multi-column tables

Tables on QL often hide the answer in a derived quantity: a row total, a column total, a difference between two cells, or a share of a grand total. Do not stop at the first number that matches an option.

Worked read: campus library visits

Step 1 — structure. Rows are campuses; columns are weekdays; each cell is visit counts for one campus-day.

Step 2 — combine columns for a daily total.

Wednesday is the busiest combined day (95 visits).

Step 3 — row totals and shares.

• North week total: 42 + 38 + 55 + 47 + 30 = 212
• South week total: 28 + 35 + 40 + 33 + 22 = 158
• Grand total: 212 + 158 = 370
• North’s share: 212 / 370. Simplify by dividing by 2 → 106 / 185. As a percent: 106 ÷ 185 ≈ 57% (a little over half).

Step 4 — interpret in context. If a stem asks whether North “always” outranks South, check day by day: North is higher every listed day here, so the claim holds for this week. If even one day flipped, “always” would be false even if North’s week total were larger.

Representation skill: rebuild a missing cell

Suppose a footnote says the Friday combined total was 52 (as above) but South Friday is blank and North Friday is 30. Then South Friday = 52 − 30 = 22. QL loves this reverse use of totals.

Pie charts: parts of one whole

A pie chart partitions a single total into mutually exclusive slices. Slice percents should sum to 100% (allowing for rounding). Absolute counts = percent × total.

Worked read: faculty enrolment shares

Imagine a pie for 2 500 registered first-years:

Check: 30 + 24 + 18 + 16 + 12 = 100%.

• Commerce headcount: 0.30 × 2 500 = 750
• Arts + Education: 24% + 16% = 40% → 0.40 × 2 500 = 1 000
• Science exceeds Health by 18% − 12% = 6% → 0.06 × 2 500 = 150 students

Meaning in context: a pie answers “of this whole, what fraction is X?” It does not by itself show change over time. If a stem compares this year’s pie with last year’s pie, you need both displays (or a table of both years).

Bar charts: simple, grouped (compound), and stacked

Simple bar

One bar per category; height/length encodes a single measure. Compare heights directly; watch whether the vertical axis starts at zero (a truncated axis can exaggerate differences).

Grouped / compound bars

Side-by-side bars for the same categories across two or more series (for example 2024 vs 2025 fees for each residence). Read the legend first, then compare within a category (paired bars) or across categories (same series).

Stacked bars

Segments stacked in one bar show composition of a total. The full bar height is the sum of segments. To find one segment’s share, divide that segment by the bar total — not by a neighbouring bar.

Worked read: stacked monthly water use (kilolitres)

• Residential fell from 120 to 100 → change = −20 kL → percent change = 20/120 = 1/6 ≈ 16.7% decrease.
• Academic’s share of January = 80/240 = 1/3 ≈ 33%.
• Academic’s share of February = 90/225 = 0.4 = 40%. Academic rose as a share even though the campus total fell — composition and total can move in opposite directions.

Broken-line (time-series) charts

A broken-line (line) chart plots a measure against an ordered time axis (days, weeks, months). Focus on:

• Direction — rising, falling, or flat stretches
• Steepness — larger vertical change per time step means a faster rate of change
• Peaks and troughs — which period is highest/lowest
• Crossing series — when two lines swap rank

Worked read: tutorial attendance (percent present)

Stream A falls each week (88 → 75). Stream B rises (80 → 87). They cross between Week 2 and Week 3: at Week 2, A (84) still leads B (82); at Week 3, B (85) leads A (79). A stem asking “in which week does B first exceed A?” is answered by Week 3, not by looking only at Week 4’s larger gap.

Scatter plots: association, not automatic causation

A scatter plot places paired observations (x, y). QL typically asks whether points trend upward (positive association), downward (negative), or show little pattern — and whether a labelled point is an outlier.

Example cloud: hours of self-study last week (x) versus quiz score out of 20 (y). Points climbing to the right suggest that more study hours associate with higher scores in this sample. That does not prove that forcing every student to study longer would raise every score by the same amount; exam language usually stays with “associated with” or “tends to.”

If one student studied 2 hours and scored 19 while peers who studied 2 hours scored near 8–10, that point is an outlier relative to the cloud — mention it only if the stem asks.

Two-way tables as a display cousin

A compact two-way table (campus × preference, faculty × residence type) is still a data representation. Totals along margins support the same derive-then-interpret habit as multi-column grids. Prefer a two-way table when the stem needs joint counts; prefer a grouped bar when the visual comparison of heights is the main task.

Simple representation skills (producing or choosing a display)

QL sometimes asks which display fits a claim:

Choosing a pie to show “how fees changed from 2024 to 2025” is usually weak unless you show two pies or a better time-aware chart. Choosing a scatter when the stem only lists category totals wastes the association idea.

Exam traps for data items

• Reading the wrong series from a legend (solid vs dashed line).
• Comparing raw counts when the stem asks for rates or percentages (or the reverse).
• Using a segment as a percent of the wrong total in a stacked bar.
• Ignoring a footnote such as “figures in thousands.”
• Treating a scatter association as proof of cause.
• Stopping at one cell when the question needs a row–column combination.

Practise saying the answer in a short sentence: “Wednesday’s combined visits are 95, the week maximum.” If you cannot name the quantity in words, you may have matched a distractor number.

Data representation on AQL rewards deliberate reading more than advanced formulae. Master the checklist, derive across the grid, and interpret the story the display is telling about a real situation.