Lesson 13 ~30 min Unit 4 · Data & Probability +85 XP

Misleading Graphs

The data can be correct but the graph still lies. Learn to spot truncated axes, 3D effects, inconsistent scales and cherry-picked time periods.

Today's hook: In 2014, a major news network showed a bar chart where two candidates' bars looked almost equal — but the y-axis started at 1,999,000. In reality, one had three times as many votes. Misleading graphs are everywhere. Your job: spot them.

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Think First

warm-up

Before you read on — a bar chart shows two companies' profits. Company A's bar reaches the top of the chart; Company B's bar is about half the height. But the y-axis starts at $980,000. Company A earned $1,010,000 and Company B earned $995,000. Are these companies actually very different in profit? What went wrong?

Record your answer in your workbook.

The Big Idea

+5 XP

Graphs can mislead through truncated axes (not starting at zero), inconsistent scales, 3D effects, omitted data, or cherry-picked time periods. The data itself can be correct — the presentation creates the deception.

A graph showing unemployment at 5.2% vs 5.1% — a tiny 0.1% difference — can look like a dramatic change if the y-axis starts at 5.0%. The bars appear completely different in height even though the actual values are nearly identical. This is the truncated axis trick, and it is used all the time in media and politics.

Same data, different axis start → completely different impression

Always check where the axis starts

If a bar chart's y-axis doesn't start at zero, be suspicious.

Data correct ≠ graph honest

The numbers can be real but the visual impression can still mislead.

Read the numbers directly

Don't trust the bar heights — read the actual values from the scale.

What You'll Master

objectives

Know

The main techniques used to create misleading graphs
The STALC checklist for evaluating any graph
What a truncated axis is and why it distorts perception

Understand

Why correct data can still produce a misleading graph
How 3D effects and inconsistent scales distort visual comparisons
Why cherry-picking time periods can misrepresent trends

Can Do

Identify multiple problems in a given graph and explain each
Redraw a misleading graph in an honest way
Apply the STALC checklist to evaluate any statistical graph

Words You Need

vocabulary

Truncated axisAn axis that does not start at zero, making differences look larger than they are.

Misleading graphA graph that creates a false or exaggerated impression through its design choices.

3D effectA visual technique that distorts perception of size and proportion in charts.

Inconsistent scaleA scale where the intervals between values are not equal, distorting the graph shape.

Cherry-pickingChoosing only the time period or data points that support your argument, ignoring others.

Visual distortionWhen a graph's visual appearance misrepresents the actual proportional differences in the data.

Spot the Trap

heads-up

Wrong: "The data is correct, so the graph must be fine." A graph can use real numbers and still create a completely false visual impression.

Right: Always check: where does the axis start? Are the scale intervals equal? Is there a 3D effect? Was the time period cherry-picked?

Wrong: Only looking at the bar heights without reading the y-axis values. A bar twice as tall doesn't always represent twice the value.

Right: Read the actual numbers off the axis scale. Calculate the real proportional difference: (larger − smaller) ÷ smaller × 100%.

The Truncated Axis Trick

+5 XP

A truncated axis starts at a value above zero. This makes the differences between bars look much larger than they really are. It is the most common misleading technique in news media, advertising and politics.

Party A receives 2,100,000 votes and Party B receives 2,000,000 votes. Real difference: 100,000 ÷ 2,000,000 = 5%. But if the y-axis starts at 1,950,000, Party A's bar is three times taller than Party B's — giving the impression of a landslide when it was barely a win.

Fix: always start y-axis at zero for bar charts

Check the axis start

The very first thing to check on any bar chart is where the y-axis starts.

Calculate the real % difference

Use numbers, not bar heights: (bigger−smaller) ÷ smaller × 100%.

The zigzag break symbol

A zigzag on the axis warns you the scale is not continuous from zero.

3D Effects and Visual Distortions

+5 XP

3D effects on bar charts and pie charts distort the visual proportions. Slices at the front of a 3D pie chart appear larger than they really are due to perspective. Inconsistent scale intervals (e.g., 0, 10, 20, 50, 100) make trends look different from reality.

A 3D pie chart shows three slices: Slice A = 30%, Slice B = 35%, Slice C = 35%. If Slice A is placed at the front of the 3D pie, it appears larger than Slices B and C, even though it is actually the smallest. A flat 2D pie chart with the same data shows the truth correctly.

3D effects distort proportions · flat charts are always more accurate

Beware 3D pie charts

Front slices always look bigger. Read the percentage labels instead.

Check scale intervals

Are the gaps between tick marks all equal? If not, the graph is distorted.

Prefer flat, simple charts

Simple 2D charts with zero-based axes are always the most honest.

The STALC Checklist

+5 XP

Use the STALC checklist to critically evaluate any graph: Scale (does the y-axis start at zero? are intervals equal?), Title (is it objective or biased?), Axes (are both axes labelled with units?), Legend (is a key provided?), Context (what time period? is data cherry-picked?).

A graph is titled "Sales SOAR as competitors FAIL" — the title itself is biased and emotional. The y-axis starts at 80% rather than 0. The axes have no units. The data covers only the best 3 months of the year. All five STALC checks reveal problems.

STALC: Scale · Title · Axes · Legend · Context

Use STALC every time

Apply all five checks to every graph you see in an exam or in real life.

Title bias is subtle

Emotional or loaded words in a title hint at bias in the graph.

Context = the full picture

Ask: what happened before and after the period shown in the graph?

Watch Me Solve It · 3 examples

Watch Me Solve It · Identify why a graph is misleading

+15 XP per step

PROBLEM

A bar chart compares two phone brands' customer satisfaction: Brand X = 87%, Brand Y = 84%. The y-axis starts at 80%. Brand X's bar appears about four times taller than Brand Y's. Identify why this is misleading and explain how to fix it.

1

Identify the misleading technique

The y-axis starts at 80% instead of 0%. This is a truncated axis.

Starting at 80% means only the top 7% (Brand X) vs 4% (Brand Y) above the baseline is shown. This makes a 3% difference look enormous.
2

Calculate the actual difference

Real difference: 87% − 84% = 3%.
Percentage difference: 3 ÷ 84 × 100% ≈ 3.6%

Brand X is only 3.6% better than Brand Y — a very small difference, not the dramatic gap the graph suggests.
3

Explain the fix

Redraw the bar chart with the y-axis starting at 0% and ending at 100%. Both bars will appear nearly the same height, accurately reflecting the small difference.

A zero-based axis ensures the visual impression matches the actual proportional difference in the data.

AnswerTruncated axis (starts at 80%). Real difference = 3%. Fix: start y-axis at 0%.

Watch Me Solve It · Spot 3 problems in a bar chart

+15 XP per step

PROBLEM

A 3D bar chart titled "Our Product Dominates the Market" has a y-axis starting at 50 with unequal intervals (50, 60, 75, 100). No axis labels or units are shown. The time period shown is the 3 best months of the year. Identify 3 problems.

1

Problem 1: Biased title and 3D effect

"Our Product Dominates" is biased and promotional. The 3D effect makes bars appear taller and differences look larger.

An objective title should describe what the graph shows, not make a claim.
2

Problem 2: Truncated axis and unequal intervals

The y-axis starts at 50 (not zero) AND the intervals are unequal (50, 60, 75, 100). This doubly distorts the visual impression.

Unequal intervals compress some differences and stretch others, making trends look completely different from reality.
3

Problem 3: No labels and cherry-picked period

Without axis labels or units, we don't know what is being measured. Showing only the 3 best months hides the full-year picture.

Cherry-picking creates a falsely positive impression. Missing labels make the graph impossible to verify.

AnswerBiased title + 3D effect; truncated axis + unequal intervals; no labels + cherry-picked period.

Watch Me Solve It · Explain how to fix a truncated-axis graph

+15 XP per step

PROBLEM

A line graph shows a company's share price over 6 months with the y-axis starting at $48. The price moves from $50 to $52 — but the graph looks like a dramatic spike. Explain step by step how to correctly redraw this graph.

1

Identify the problem

The y-axis starts at $48, not $0. A $2 increase ($50 to $52) looks like a huge rise because it fills most of the chart's vertical space.

The actual increase is only 2 ÷ 50 × 100% = 4% — modest, not dramatic.
2

Choose an appropriate scale

For a line graph showing share price changes, starting at $0 may make the changes too small to see. A better approach: start at $0 but use a clear zigzag break symbol to show the scale is not continuous, OR use a context-appropriate range (e.g., $0–$60).

A zigzag break is the honest way to truncate a line graph axis when needed. It warns the reader.
3

Redraw with honest labels

Add a title that describes (not interprets): "Company XYZ Share Price, Jan–Jun". Label both axes with units ($). Use equal intervals. Show the full 6-month period.

An honest graph lets the reader form their own interpretation without being pushed toward a conclusion.

AnswerStart at $0 with zigzag break, or use $0–$60 range. Label axes. Neutral title. Equal intervals.

Common Pitfalls

heads-up

Thinking "correct data = honest graph"

Students often assume that if the numbers are real, the graph is honest. But a truncated axis, 3D effect, or cherry-picked period can make real data tell a completely false story.

Fix: Always evaluate the design choices, not just the data. Ask: "Does this graph's visual impression match the actual proportional difference?"

Not checking if scale increments are equal

A scale that goes 0, 10, 20, 50, 100 has unequal intervals. The gap from 20 to 50 looks the same size as 10 to 20, but represents 30 units instead of 10.

Fix: Check every gap on the axis. Subtract consecutive tick marks: they should all be the same number.

Missing the cherry-picking problem

A graph showing only a subset of the time period can hide inconvenient parts of the data. A business might show only their best 3 months, hiding losses in the other 9.

Fix: Always ask "what time period is shown?" and "what might have happened before or after this period?"

Copy Into Your Books

4 Misleading Techniques

Truncated axis (not starting at zero)
3D effects (distort proportions)
Inconsistent scale intervals
Cherry-picking (showing best period only)

STALC Checklist

S: Scale — starts at zero? equal intervals?
T: Title — objective or biased?
A: Axes — labelled with units?
L: Legend — key provided?
C: Context — full time period?

How to Fix a Misleading Graph

Start y-axis at zero (or use zigzag break)
Use equal scale intervals
Use flat 2D chart, not 3D
Show the full time period
Write a neutral, descriptive title
Label both axes with units

Real Difference Formula

Read actual values from the axis scale
% difference = (bigger−smaller) ÷ smaller × 100%
Don't trust bar heights — read the numbers

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Misleading Graphs

4 problems

Four problems to sharpen your critical graph-reading skills. Work each, then reveal the answer.

1 A bar chart's y-axis goes 95, 96, 97, 98, 99. Name the technique being used and explain its effect.

This is a truncated axis — the y-axis starts at 95, not 0. The effect is that any differences between bars are greatly exaggerated. A 1-unit difference (e.g., 97 vs 98) takes up 20% of the chart's height, making it look much larger than the real 1% difference.Truncated axis: small differences look enormous
2 Why can a 3D pie chart make one slice look larger than it actually is?

In a 3D pie chart, the chart is tilted toward the viewer. Slices at the front appear taller and wider due to the perspective effect, even if they represent a smaller percentage. The eye interprets 3D depth as size, creating a false impression of proportion.3D perspective → front slices appear larger than their actual percentage
3 A graph shows only the last 3 months of a company's sales trend. What might be misleading about this?

Showing only the last 3 months is cherry-picking. If those months are the company's best (e.g., Christmas season), the graph creates a false impression of strong, sustained performance. The full-year picture might show declining sales for 9 months before a seasonal peak.Cherry-picking hides the full picture and may misrepresent long-term trends
4 List 4 questions to ask when evaluating any statistical graph.

Any four of: (1) Does the y-axis start at zero? (2) Are the scale intervals equal? (3) Is the title neutral and descriptive? (4) Are both axes labelled with units? (5) Is a legend/key provided? (6) Does the graph show the full time period, or is data cherry-picked? (7) Are any 3D effects distorting the proportions?STALC: Scale, Title, Axes, Legend, Context

Complete in your workbook.

Quick Check · 5 questions

What is a truncated axis?

+10 XP

How do 3D effects mislead in a pie chart?

+10 XP

A company shows only its 4 best months of sales data. This is an example of:

+10 XP

In the STALC checklist, what does the S stand for?

+10 XP

The most important fix for a misleading bar chart with a truncated axis is to:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. A bar chart shows two smartphone brands' screen sizes: Brand A = 6.1 inches, Brand B = 6.4 inches. The y-axis starts at 6.0 and Brand A's bar is tiny while Brand B's reaches the top. Identify the misleading technique, calculate the real percentage difference, and explain how to fix the graph.

Answer in your workbook.

Understand Easy 2 MARKS

Q7. Apply the STALC checklist to evaluate this graph description: "A 3D pie chart titled 'Why Our Party Wins' showing voting percentages from only the 2 most favourable districts. No axis labels. Percentage labels are shown on slices." Identify 3 specific STALC problems.

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A health food company claims: "Our product increased customer energy levels by 400%!" They show a bar chart with a y-axis starting at 90 (out of 100), with one bar at 92 and another at 96. Explain fully why this claim and graph are misleading. Calculate the true percentage increase and describe what an honest graph would look like.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — A truncated axis starts above zero, exaggerating differences.

2. D — 3D effects make front slices appear larger due to perspective.

3. B — Showing only the best months is cherry-picking.

4. A — S = Scale: starts at zero? equal intervals?

5. B — Start the y-axis at zero.

Show Your Working Model Answers

Q6 (3 marks): Technique: truncated axis (y-axis starts at 6.0 not 0) [1]. Real difference: 6.4−6.1 = 0.3 inches. Percentage difference = 0.3÷6.1×100% ≈ 4.9% [1]. Fix: redraw with y-axis from 0 to 7.0, using equal intervals. Both bars will appear nearly the same height, accurately reflecting the small size difference [1].

Q7 (2 marks): Any 3 of: T — title "Why Our Party Wins" is biased and promotional [1]; C — only 2 most favourable districts shown = cherry-picking [1]; the 3D effect on the pie chart distorts slice proportions (visual distortion) [1]; A — no axis labels (pie charts don't have axes but this signals low transparency) [1]. Award 2 marks for any 2 well-explained problems.

Q8 (4 marks): The graph uses a truncated axis starting at 90, making a 4-unit increase (92 to 96) look like a huge jump [1]. The "400%" claim is false: real increase = (96−92)÷92×100% ≈ 4.3%, not 400% [1]. The company likely calculated the percentage of the visible bar height increase, not the actual value increase — this is deliberately deceptive [1]. An honest graph would: start at 0, show both bars with their actual values (e.g., 92 and 96 on a 0–100 scale), have a neutral title like "Customer Energy Levels Before and After", and clearly label axes with units and a source [1].

Stretch Challenge · +25 XP, +10 coins

The Graph Detective

Find a real graph online or in a newspaper (any topic: sports, politics, health, finance). Apply the full STALC checklist and write a detailed critique. Identify every misleading technique used (if any), calculate the actual differences where possible, and describe what an honest version of the graph would look like. Share your findings.

Reveal guide

Common finds: news channel graphs with truncated axes, company annual reports showing only best quarters, political party leaflets using cherry-picked periods, health product ads using 3D pie charts. A full STALC critique should address all 5 letters with specific evidence from the graph. An honest graph would: zero-based axis, equal intervals, neutral title, labelled axes with units, full time period, flat 2D design.

Quick Review

Truncated axis

Starts above zero; makes differences look huge

3D effects

Front slices look bigger; flat charts are more honest

Inconsistent scale

Unequal intervals distort the shape of trends

Cherry-picking

Only showing the best period hides the full picture

STALC checklist

Scale, Title, Axes, Legend, Context

Data correct ≠ honest

Real numbers can still create a false impression

Interactive: Graph Fixer

Spot the misleading elements in a graph and use the controls to fix them. Apply the STALC checklist and see how the graph changes when made honest.

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