Grade by Grade
D. Statistical Analysis · Two-Variable Data Analysis and Validity

Statistics in the Media and Advertising

D3.1About 70 minutes
Learning goals
  • I can interpret statistics and graphs presented in the media and in advertising.
  • I can explain common ways graphs mislead, including truncated axes and distorted pictures.
  • I can explain the difference between percent and percentage points, and between relative and absolute change.
  • I can explain how averages, weak correlations and missing groups can be used to mislead.

Words to know

truncated axis
A vertical axis that starts at a value other than zero, which can make small differences look large.
lie factor
A term from statistician Edward Tufte: the size of an effect shown in a graphic divided by the size of the effect in the data. Honest graphs have a lie factor close to 1.
percentage point
The difference between two percentages. A rise from 4% to 5% is 1 percentage point, but a 25% increase.
relative change
The change compared with the starting value: (new − old) ÷ old.
absolute change
The actual size of the change: new − old.
cherry-picking
Choosing only the data that support a claim and leaving out the rest.

"Satisfaction is FOUR TIMES higher at our store!"

An ad shows two bars: one tiny, one tall. The tall one belongs to the store that paid for the ad. Look closely at the numbers, though: 82% and 88%. How did a 6-point difference become "four times"?

Statistics appear in news stories, ads, social media posts and political speeches every day. The people presenting them may want to inform you, persuade you or sell you something. The curriculum asks you to interpret these statistics and explain how they are used and misused. This lesson is a toolkit.

Axis starts at 80%80%82%84%86%88%90%82%Store A88%Store BB looks 4 times as bigAxis starts at 0%0%20%40%60%80%100%82%Store A88%Store BB is 1.07 times as big
The same two numbers. Starting the vertical axis at 80% (left) makes B's bar 4 times as tall as A's. With the axis starting at 0 (right), the true difference is clear.

Misleading graphs

  1. Truncated vertical axis. Starting the axis above zero exaggerates differences. This is one of the examples of misuse named in the Ontario curriculum. (For line graphs that show change over time, a non-zero start can be acceptable if it is clearly marked; for bar graphs, where the length of the bar is the message, bars should start at zero.)
  2. Distorted pictures. If a picture is scaled in two directions to show a doubling, its area grows four times.
  3. Uneven scales. Unequal gaps between years or values on an axis bend the trend.
  4. 3-D effects. Tilted pie charts make front slices look bigger.
  5. Cherry-picked time windows. Starting a graph at an unusually low point makes any recovery look like a boom.
  6. Two vertical axes. Two series with different scales can be made to look like they move together (or not) just by choosing the scales.
Worked example

Example 1: How big is the exaggeration?

In the truncated graph, the axis runs from 80% to 90%. Store A is 82% and Store B is 88%. Calculate the apparent and actual ratios and the lie factor.

  1. Bar heights above the axis start: A = 82 − 80 = 2 units, B = 88 − 80 = 8 units. Apparent ratio: 8 ÷ 2 = 4.
  2. Actual ratio: 88 ÷ 82 ≈ 1.07.
  3. Effect shown: B's bar is (8 − 2) ÷ 2 = 300% taller. Effect in the data: (88 − 82) ÷ 82 ≈ 7.3% higher.
  4. Lie factor ≈ 300 ÷ 7.3 ≈ 41.

The graph exaggerates the difference about 41 times. A fair statement: "88% of customers were satisfied at Store B compared with 82% at Store A, a difference of 6 percentage points" (and we would still want to know how the survey was done).

Syrup production doubled: two ways to draw it20252026Misleading: height AND width doubled,so the picture is 4 times the area20252026Honest: same-size symbols,twice as many
Doubling both the height and width of a picture makes it look four times as big. Honest pictographs repeat same-size symbols.

Misleading numbers

  • Percent vs percentage points. If a party's support goes from 20% to 30%, that is a rise of 10 percentage points, but a 50% increase. Both are true; a writer may pick whichever sounds more dramatic.
  • Relative vs absolute risk. "Doubles your risk!" may mean a risk rising from 1 in 100 000 to 2 in 100 000.
  • Which average? The mean income of a group can be pulled up by a few very high earners; the median is often more typical. Choosing the average that suits your message is a classic trick.
  • Weak correlations, general claims. The curriculum's example: drawing a broad conclusion from a weak correlation ("People who drink more water get better marks!" when r = 0.15).
  • Missing groups. Another curriculum example: using general population statistics when no data were collected for a minority group. A statistic about "Ontarians" may say nothing reliable about a particular community.
  • Vague wording. "Up to 50% off", "helps reduce", "9 out of 10 recommend" (out of how many asked? who were they?).
  • Small or self-selected samples. Online polls where anyone can vote measure only who chose to answer. (You studied sampling and bias in the unit Organization of Data for Analysis.)
Worked example

Example 2: Percent and percentage points

A bank's interest rate on a savings account rises from 2% to 3%. One headline says "rates up 1 point", another says "rates jump 50%". Which is correct?

  1. Absolute change: 3% − 2% = 1 percentage point.
  2. Relative change: (3 − 2) ÷ 2 = 0.5 = 50%.
  3. Both describe the same change. "Jump 50%" sounds bigger. On $1000 for a year, the extra interest is only about $10.

Both are correct. Careful writers say "percentage points" for the difference between two percentages, to avoid confusion.

Worked example

Example 3: Relative and absolute risk

A news story says a food additive "increases the risk of a rare illness by 50%". The underlying study found 6 cases per 100 000 people who ate the additive and 4 cases per 100 000 people who did not. Find the relative and absolute increases.

  1. Relative increase: (6 − 4) ÷ 4 = 0.5 = 50%.
  2. Absolute increase: 6 − 4 = 2 more cases per 100 000 people, which is 0.002 percentage points.
  3. So for each person, the risk went from 0.004% to 0.006%.

The 50% figure is accurate but sounds alarming; the absolute change is tiny. Good reporting gives both, and also says whether the study was an experiment or an observational study (and so whether the additive actually causes the illness).

Questions to ask about any media statistic

  1. Who produced it, and what do they want?
  2. What exactly was measured? Percent of what?
  3. How many people or items, and how were they chosen?
  4. Is a change given as relative or absolute? As percent or percentage points?
  5. Does the graph's axis start at zero? Are the scales even? Are pictures scaled fairly?
  6. Which average is used, and why?
  7. Is a correlation being presented as a cause?
  8. Who is missing from the data?
Just for fun

Many cereal ads have called a cereal "part of a balanced breakfast." Technically, so is the bowl. Ads choose words that sound like evidence without promising anything you could test.

Hands-on activity

Statistics scavenger hunt

You need: flyers, cereal or snack boxes, a newspaper or news website, and social media or ads you see this week.

  1. Find five statistics or graphs used to inform or persuade (for example "4 out of 5", "30% less sugar", a chart in a news story, a sports statistic in a broadcast).
  2. For each, record where you found it and who produced it.
  3. Use the questions in the tip box to rate each one: fair, missing information, or misleading. Explain why in one or two sentences.
  4. Choose the most misleading one and rewrite it as a fair statement (or redraw the graph fairly).

What to look for: common finds are comparisons with no base ("30% less sugar" than what?), unknown sample sizes ("9 out of 10"), truncated axes in news charts, and relative changes with no absolute numbers.

Your turn

1

A bar graph's vertical axis starts at 50. Bars show 52 and 58. How many times as tall does the second bar look? How many times as large is the value?

2

Unemployment falls from 8% to 6%. Express the change in percentage points and as a percent.

3

An ad says "Doubles your chance of winning!" The chance went from 1 in 1 000 000 to 2 in 1 000 000. Is the ad honest? Is it useful?

4

A company says its average employee salary is $95 000. The median is $52 000. Which better describes a typical employee? Why might the company quote the mean?

5

A pictograph shows the number of students in a program doubling by drawing a student icon twice as tall and twice as wide. What is wrong?

6

A headline says "Study: kids who eat fish are smarter" based on r = 0.12. What two problems can you see?

7

Why is an online poll on a news site not a reliable measure of public opinion?

More practice: warm-up

1

Support rises from 30% to 36%. How many percentage points is that?

2

In the previous question, what is the relative increase?

3

Should a bar graph's axis start at 0?

4

A picture is doubled in height and width. By what factor does its area change?

5

What is cherry-picking?

More practice: on your own

1

A line graph of a company's stock price shows only the last 3 weeks, during which it rose 12%. Over the past year it fell 40%. Explain the issue and give a fairer presentation.

2

A province reports that 85% of residents have a family doctor. Explain why this might hide a problem for some communities.

3

A toothpaste ad says "recommended by 4 out of 5 dentists surveyed". List three things you would want to know.

4

A graph shows rainfall for 2018, 2019, 2020, 2023 and 2026 at equal spacing on the time axis. Why is that misleading?

5

A drug reduces the risk of a heart attack over 5 years from 2% to 1.5%. Give the absolute and relative risk reductions.

Challenge

1

A graph shows sales of 120 and 150 units with the axis starting at 100. Find the lie factor.

2

A company says, "Our product is 99% fat-free!" Explain why this could still be a high-fat food relative to similar products, using percentages.

3

A government announces that average household income rose 4% last year. Describe a situation in which most households' incomes fell while the average still rose.

Check your understanding

Quiz

1.

A bar graph's axis starts at 90%. Bars show 92% and 96%. Visually, the second bar looks:

2.

A rate rises from 5% to 6%. Which statement is correct?

3.

A risk rises from 2 in 10 000 to 3 in 10 000. The relative increase is:

4.

Which is an example of misuse named in the Ontario curriculum?

5.

A pictograph shows a quantity tripling by enlarging an icon 3 times in height and width. The icon's area grows by a factor of:

6.

A company reports its mean salary but not its median. Why might that mislead?

For parents and teachers

Big idea (D3.1): media and advertising statistics are often true but misleading. Your teen should check axes, pictures, percent vs percentage points, relative vs absolute changes, which average is used, the strength of correlations, the sample, and who is missing.

How to help: Do the scavenger hunt together; flyers and cereal boxes are full of examples. When a news story gives a statistic at dinner, ask, "Percent of what? Compared with what?"

Common difficulties:

  • Mixing up percent change and percentage points.
  • Not noticing where a vertical axis starts.
  • Assuming a technically true statement cannot mislead.
  • Accepting "4 out of 5" without asking how many were asked and how they were chosen.