D. Statistical Analysis · One-Variable Data AnalysisHow Statistical Summaries Can Mislead
D1.5About 70 minutes
Learning goals- I can explain how a choice of summary statistic (mean, median, a single number without spread) can misrepresent data.
- I can identify misleading graphs: truncated axes, distorted pictographs, 3-D effects, unequal classes and selective time periods.
- I can tell percentage points from percent change, and absolute change from relative change.
- I can redraw or restate a misleading summary honestly and justify my changes.
Words to know
truncated axis
A vertical axis that starts above zero, which exaggerates differences between bar heights.
percentage point
The unit for the difference between two percents: from 4% to 6% is a change of 2 percentage points.
relative (percent) change
(new − old) ÷ old × 100%. From 4% to 6% is a 50% relative increase.
cherry-picking
Choosing only the data, time period or summary that supports a conclusion, and leaving out the rest.
lie factor
The size of the effect shown in a graph divided by the size of the effect in the data. An honest graph has a lie factor close to 1.
Numbers don't lie, but...
Every statistic in a news story, an advertisement or a social media post was chosen by someone. A summary can be completely accurate and still leave a false impression: the mean instead of the median, a graph whose axis starts at 46 instead of 0, a percent change without the starting value, an average that hides a group whose experience is very different.
The curriculum asks you to "describe how statistical summaries ... can be used to misrepresent one-variable data". This lesson is a field guide to the most common tricks, so you can spot them and avoid them in your own work. The unit Two-Variable Data Analysis and Validity goes further, with how the media, advertisers and pollsters use and misuse statistics, including misleading claims about relationships between variables.
Trick 1: choosing the convenient centre
In skewed data, the mean and median can be far apart (lesson 4). An employer advertising jobs may quote the mean salary (pulled up by a few executives); employees asking for a raise may quote the median. Both numbers are accurate; each is chosen to support a point of view. Honest practice: report the summary that answers the question, and when the data is skewed, report both and say why they differ.
Trick 2: one number with no spread, no size and no margin
"The average student in our program gets a job paying $68 000." How many students? How spread out are the salaries? Was it a random sample, or only the graduates who answered a survey? A single statistic without its spread, its sample size and (for a sample) its margin of error cannot be judged. Honest practice: always give n, a measure of spread, and how the data was collected.
Trick 3: the truncated axis
Bar heights are compared by eye. If the vertical axis starts at 46% instead of 0%, a bar for 52% is drawn 6 units tall and a bar for 48% is drawn 2 units tall, so the larger bar looks 3 times as big even though 52 ÷ 48 is only about 1.08. Line graphs of change over time sometimes need a non-zero start to show the detail, but then the axis must be clearly labelled (often with a break symbol), and bar graphs should always start at zero.
Worked exampleExample 1: Measuring the distortion
An ad shows two bars: "Our brand 52%" and "Their brand 48%", with the axis starting at 46%. (a) How many times taller does the 52% bar look? (b) How many times larger is 52% than 48%? (c) Find the lie factor, using (apparent ratio − 1) ÷ (actual ratio − 1).
- (a) Visible heights: 52 − 46 = 6 and 48 − 46 = 2, so the bar looks 6 ÷ 2 = 3 times as tall.
- (b) 52 ÷ 48 ≈ 1.083: about 8.3% larger.
- (c) Shown effect: a 200% increase (3 − 1 = 2). Real effect: an 8.3% increase (0.083). Lie factor ≈ 2 ÷ 0.083 ≈ 24.
The graph exaggerates the difference by a factor of about 24. An honest graph has a lie factor near 1.
Trick 4: pictographs that grow in two (or three) directions
If sales double and the artist doubles both the width and the height of a picture, the picture's area is 4 times as large, and a 3-D drawing (a box, a barrel) has 8 times the volume. Readers judge size by area, so the growth looks far bigger than it is. Honest practice: use bars, or repeat identical symbols (one symbol = 1 000 boxes).
Trick 5: other graph tricks
- 3-D and tilted circle graphs: slices at the front look bigger than equal slices at the back.
- Unequal class widths in histograms (lesson 3): a wide class collects more values just because it is wide.
- Selective time periods: choosing the start or end date to make a trend look better or worse ("crime is up 20% since last March!", when last March was unusually low).
- Missing labels, units or sources: a graph without them cannot be checked.
- Inconsistent scales: tick marks that are not evenly spaced, or two graphs side by side on different scales.
Trick 6: percentage points vs percent change
If a mortgage rate goes from 4% to 6%, it rose by 2 percentage points, which is a 50% relative increase (2 ÷ 4). If a party's support goes from 30% to 33%, that is 3 points but a 10% increase. Both descriptions are true; choosing the more dramatic one is a common way to spin a result.
Similarly, relative change without the starting value can mislead: "Risk doubles!" means something very different if the risk went from 1 in 10 to 2 in 10 than if it went from 1 in a million to 2 in a million.
Worked exampleExample 2: Points or percent?
A headline says: "Youth unemployment soars 25%!" The rate went from 8% to 10% in a region. Rewrite the claim honestly in two ways.
- Absolute change: 10% − 8% = 2 percentage points.
- Relative change: (10 − 8) ÷ 8 = 0.25 = 25%. The headline is technically true.
- An honest statement gives both and the starting value.
"The youth unemployment rate rose from 8% to 10%, an increase of 2 percentage points (a 25% relative increase)."
Trick 7: an average that hides a group
A single summary for a whole population can hide very different experiences in subgroups. A province's average wait time for a medical procedure may look acceptable while wait times in northern or rural communities are much longer. National statistics can leave out groups for which data was never collected, or groups too small to report separately, such as some First Nations, Inuit and Métis communities, which is one reason First Nations, Inuit and Métis organizations in Canada call for data about their communities to be collected and governed by and for those communities (for example, the First Nations principles of OCAP®: ownership, control, access and possession). Honest practice: where it matters, break the data down by relevant groups (and say when data for a group is missing).
Worked exampleExample 3: An average that hides a subgroup
A school reports that its students' mean commute is 18 minutes. The data: 300 students in town average 12 minutes; 60 students on rural bus routes average 48 minutes. Check the overall mean and explain what it hides.
- Overall mean = (300 × 12 + 60 × 48) ÷ 360 = (3 600 + 2 880) ÷ 360 = 6 480 ÷ 360 = 18 minutes ✓.
- The 18-minute figure describes neither group well: town students take about 12 minutes and rural students about 48, four times as long.
- Any decision based only on the overall mean (for example, about start times or bus routes) would overlook the rural students.
The mean is correct but hides a subgroup with a very different experience. Report the groups separately.
A checklist for any statistic you read
- Who collected the data, and why? 2. How many individuals, and how were they chosen? 3. Which summary was used: mean or median? Is the spread given? 4. Does the graph start at zero? Are the scales and labels honest? 5. Percent or percentage points? What was the starting value? 6. Is a group being hidden by an overall average? 7. Is there a margin of error, and is the reported difference bigger than it?
Just for funDid you hear about the graph with the axis starting at 99%? It made a tiny rise in sales look like a mountain. The marketing department called it "growth". The statistics teacher called it "Exhibit A".
Hands-on activityMisleading-graph hunt
You need: flyers, newspapers, magazines or websites (with a parent's help for online sources), paper and a pencil.
- Find three graphs or statistics used to persuade (advertising, opinion pieces, social media posts, political flyers).
- For each, use the checklist above: write the claim, the summary or graph used, and any trick you can spot (truncated axis, mean vs median, missing n, percent vs points, distorted pictograph...).
- Choose the most misleading one and redraw or restate it honestly. If numbers are missing, say what you would need to know.
- Explain in two sentences why your version is fairer.
What to look for: the redrawn graph should start bar axes at zero, use equal scales and clear labels, and the restated claim should include the starting value, the units (points vs percent) and the sample size if available. If a source contains no tricks at all, that is a good finding too: say what it did well.
Your turn
1A bar graph shows 61 and 64 with the axis starting at 60. How many times taller does the 64 bar look? What is the actual ratio?
Visible heights 1 and 4: it looks 4 times as tall. Actual ratio: 64 ÷ 61 ≈ 1.05.
2An interest rate rises from 3% to 4.5%. Express the change in percentage points and as a percent change.
1.5 percentage points; (4.5 − 3) ÷ 3 = 0.5 = 50% increase.
3A company says, "Our average customer saves $500 a year!" List three questions you should ask.
Any three of: Is it the mean or median? How many customers (n), and how were they chosen (all customers, or only those who responded)? What is the spread (do many save nothing)? Compared with what? Over what time period?
4A pictograph shows a small house for 2015 prices and a house twice as wide and twice as tall for 2025 prices. What impression does it give, and what increase does it really show?
The 2025 house has 4 times the area, suggesting prices quadrupled. The real increase shown is a doubling.
5Why should a bar graph's axis start at 0, but a line graph of daily temperatures may not?
Bar graphs are read by length, so lengths must be proportional to values. A line graph is read by position and change; a non-zero start can be fine if it is clearly labelled, and temperatures in °C have no natural zero anyway.
6A report says, "Average household income in the region is $112 000." The region includes a small neighbourhood of very wealthy households. What is misleading, and what would you report instead?
The mean is pulled up by the few very high incomes (right skew), so it overstates a typical household's income. Report the median (and perhaps the mean as well, explaining the difference).
More practice: warm-up
1From 20% to 25% is how many percentage points?
2From 20% to 25% is what percent increase?
3Should a bar graph's frequency axis start at 0?
4If a square's side is tripled, by what factor does its area grow?
5What two pieces of information should come with a sample mean?
Any two of: the spread (s), the sample size (n), the margin of error, how the sample was chosen.
More practice: on your own
1A graph shows a town's population: 2016: 9 800, 2026: 10 300, with the axis from 9 700 to 10 400. The headline says "Booming town!" Find the visual and actual growth.
Visible heights: 100 and 600, so it looks 6 times as large. Actual growth: 500 ÷ 9 800 ≈ 5.1% over ten years.
2A medication "cuts the risk of a rare side effect by 50%": from 2 in 10 000 to 1 in 10 000. Restate this with the absolute change.
The risk falls from 0.02% to 0.01%, a drop of 0.01 percentage points (1 fewer case per 10 000 people), which is a 50% relative reduction.
3A news report says the average class size in a board is 24. Explain how it is still possible for many students to be in classes of more than 30.
A mean of 24 is consistent with many small classes (for example, special programs with 10 to 15 students) balancing many large classes of more than 30. The mean hides the distribution; the board should report the spread or the percent of classes over 30.
4A school compares this year's pass rate (91%) with "the year with the lowest pass rate in a decade" (82%) and says results "improved by 9 points". What trick is this, and what is a fairer comparison?
Cherry-picking the comparison year. A fairer comparison uses last year, or the average of the past several years, and shows the whole time series.
5Explain why a 3-D circle graph tilted away from the reader can mislead.
The front slices are drawn larger and closer, and their sides show, so they look bigger than slices of equal size at the back. Area no longer matches the percent.
Challenge
1An honest graph-maker draws the 48% vs 52% comparison with the axis starting at 0. What is its lie factor? Then find the axis starting value a that would make the lie factor exactly 2.
From 0, the bars look 52 ÷ 48 times as tall, so the lie factor is 1. With start a, apparent ratio = (52 − a) ÷ (48 − a). We need (52 − a) ÷ (48 − a) − 1 = 2 × (52 ÷ 48 − 1) = 2 × (1/12) = 1/6. So 4 ÷ (48 − a) = 1/6 → 48 − a = 24 → a = 24.
2A company's average (mean) salary is $82 000 and its median is $61 000. A union says, "Most workers earn less than $61 000." Is that claim justified? What true claim about "most workers" could the union make instead?
No. By the definition of the median, at most about half of the workers earn less than $61 000, so "most" is not justified. A true claim: "Half of the workers earn $61 000 or less." Also, because the mean is far above the median (right skew), it is very likely that most workers earn less than the $82 000 average, which the union could check from the data and then state.
3Write a misleading but technically true one-sentence headline from this data, then an honest one: a city's snow-clearing budget rose from $14.0 million to $14.7 million, while the number of kilometres of road to clear rose from 1 000 to 1 100.
Misleading: "Snow-clearing budget hits record $14.7 million!" (true, but ignores the extra roads). Honest: "The snow-clearing budget rose 5% (from $14.0 million to $14.7 million), but roads to clear rose 10%, so spending per kilometre fell from $14 000 to about $13 364."
Quiz
1. A bar graph of 70 and 75 starts its axis at 65. How many times taller does the 75 bar look than the 70 bar?
2. A rate rises from 5% to 7%. Which statement is correct?
3. A pictograph doubles the height and width of a picture to show that a quantity doubled. The picture's area is
4. Which is the most honest way to report incomes in a strongly skewed data set?
5. An overall average hides a subgroup when
6. Which claim needs a sample size and margin of error before you can judge it?
Big idea (D1.5): accurate numbers can still mislead through the choice of summary, the design of a graph, or the way a change is described. Your teen should be able to spot the common tricks and fix them.
How to help: Make the misleading-graph hunt a family game: whoever finds the most misleading graph in a week's flyers or news wins. Ask your teen to explain the trick and redraw it honestly.
Common difficulties:
- Confusing percentage points with percent change.
- Thinking a non-zero axis is always dishonest (it is a problem for bar graphs, and must be clearly marked on line graphs).
- Accepting a single average without asking about spread, sample size and subgroups.
- Calling every persuasive statistic "false"; often it is accurate but selectively presented.