Mathematics • Year 8 • Unit 4 • Lesson 13
Misleading Graphs
Build fluency with spotting the four major misleading-graph tricks: truncated axis, inconsistent intervals, 3D/pictograph distortion, and cherry-picking. One worked example, one guided fill-in, then eight independent identifications and percentage-change calculations.
1. I do — fully worked example
Read every line. Each step has a short reason.
Problem. A company shows a bar chart of quarterly sales (in thousands of units): Q1 = 98, Q2 = 99, Q3 = 100, Q4 = 101. Their y-axis starts at 97 (not 0). On the graph, Q4's bar looks roughly 4 times taller than Q1's bar. Identify the trick and calculate the actual percentage change from Q1 to Q4.
Step 1 — Identify the misleading technique.
The y-axis does not start at 0. This is a truncated y-axis.
Reason: bar/column charts MUST start at 0. Starting elsewhere exaggerates differences visually.
Step 2 — Calculate the actual percentage change.
% change = (new − old) ÷ old × 100
= (101 − 98) ÷ 98 × 100 = 3 ÷ 98 × 100 ≈ 3.06%
Reason: this is the real growth. About 3% — not 4× larger as the graph implies.
Step 3 — Explain how the graph misleads.
Because the y-axis starts at 97, the visible bar heights are 1, 2, 3, 4 units tall instead of 98, 99, 100, 101. The ratio of visible heights (4 ÷ 1 = 4) makes Q4 look 4 times Q1, when actual sales grew by only ~3%.
Answer: Truncated y-axis. Actual change ≈ 3.06%, not 4× growth.
2. We do — fill in the missing steps
Same shape as Section 1, but the working is faded. Fill in each blank. 4 marks
Problem. A school newsletter shows enrolment numbers as: 2022 = 480 students, 2023 = 492 students. Their bar chart has a y-axis starting at 475. On the graph, the 2023 bar looks twice as tall as the 2022 bar.
Step 1 — Identify the technique:
The y-axis starts at ______, not 0. This is a ______________________ axis.
Step 2 — Calculate the actual % change:
% change = (492 − ______) ÷ ______ × 100 = ______ ÷ ______ × 100 ≈ ______ %
Step 3 — Compare visual vs actual:
The graph suggests enrolment ______________ (e.g. doubled), but the real change is only about ______ %.
3. You do — independent practice
Show your working in the space under each problem. Foundation → Standard → Extension.
Foundation — name the trick
3.1 A bar chart of share prices starts its y-axis at $950 instead of $0. Name the misleading technique. 1 mark
3.2 A 3D pie chart shows slices for four products. The slice closest to the front appears much larger than the slice at the back, although both represent 25%. Name the misleading technique. 1 mark
3.3 A graph of unemployment shows ONLY the years 2010-2014 (when it was falling) and hides 2008-2010 and 2014-2018. Name the misleading technique. 1 mark
3.4 What does "truncated y-axis" mean? Write a one-sentence definition. 1 mark
Standard — calculate the real change
3.5 A truncated graph shows the price of milk rose from $1.40 to $1.55 with the y-axis starting at $1.30. Calculate the actual percentage change to 1 d.p. 2 marks
3.6 In a pictograph, a company doubles BOTH the height AND width of a coin icon to show sales doubled. What does the icon's area actually look like (relative to the original), and why is this misleading? 2 marks
Extension — redesign honestly
3.7 A bar chart of charity donations is misleading because its y-axis starts at $9 800 (showing donations of $9 850, $9 900, $9 950, $10 050). Describe TWO specific changes you would make to redraw the chart honestly. 2 marks
3.8 A media report shows: "Sales rose by 50%!" with a bar chart whose new bar is FIVE times taller than the old. Identify (a) what kind of graph trick is at play, and (b) what the bar height ratio SHOULD be if the 50% rise is genuine. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (school enrolment)
Step 1: starts at 475, not 0. This is a truncated axis.
Step 2: % change = (492 − 480) ÷ 480 × 100 = 12 ÷ 480 × 100 ≈ 2.5 %.
Step 3: The graph suggests enrolment roughly doubled / tripled, but the real change is only about 2.5 %.
3.1 — Y-axis at $950
Truncated y-axis — the axis does not start at zero.
3.2 — 3D pie chart
3D effect — tilting the pie distorts area perception, making front slices look larger than equal back slices.
3.3 — Selected years only
Cherry-picking — showing only the data range that supports the desired narrative, hiding contradicting data.
3.4 — Define truncated y-axis
A truncated y-axis is a y-axis that does not start at zero — instead it starts at a higher value, which exaggerates the visual difference between bars.
3.5 — Milk price rise
% change = (1.55 − 1.40) ÷ 1.40 × 100 = 0.15 ÷ 1.40 × 100 ≈ 10.7%. (On the truncated graph, the visible heights are 0.10 and 0.25, making the new bar look 2.5× taller — but the real rise is only about 10.7%.)
3.6 — Pictograph area trick
If both height AND width double, the icon's area = 2 × 2 = 4 times the original. Our eyes read AREA, so doubling both dimensions makes the icon look like a 4× increase — but sales only doubled (a 2× increase). The icon overstates the change by 100%.
3.7 — Redraw honestly
(1) Start the y-axis at $0 (not $9 800) so the bar heights reflect the true scale. (2) Choose a sensible interval (e.g. $0, $2 000, $4 000, …, $10 000) so the small differences ($150 between bars) are visible but not exaggerated. Optional: add the exact donation values as data labels above each bar so readers don't need to interpret height alone.
3.8 — Sales claim
(a) Truncated y-axis — the chart exaggerates the visual size of the change. (b) A 50% rise means new = 1.5 × old, so the new bar should be exactly 1.5 times the old bar's height — not 5 times. The graph misrepresents a modest growth as massive.