Skill: Evaluate misleading data displays and statistical claims

A data display can use correct numbers but still create a misleading impression. To evaluate it, check:

1. The axis: Does it start at $0$? Are the intervals equal and clearly labeled?
2. The visual size: Do bars, pictures, or 3-D effects make a difference look larger than it really is?
3. The actual numbers: Find both the absolute change and the relative change.
4. The claim: Consider the sample size, how people were chosen, and whether the data show causation or only an association.

Worked example

A school surveys $40$ students about how often they recycle.

• Before a recycling campaign: $12$ students recycled.
• After the campaign: $18$ students recycled.

A bar graph shows the two numbers, but its vertical axis starts at $10$ instead of $0$.

Step 1: Check the numbers

The number of students increased by

$$18-12=6.$$

So, $6$ more students recycled.

The relative increase is

$$\frac{6}{12}=0.5=50\%.$$

The percentage increase is $50\%$, but the number increased from $12$ to only $18$ students.

Step 2: Check the axis

Because the graph starts at $10$, the first bar rises only from $10$ to $12$, while the second rises from $10$ to $18$. The second bar may look about four times as tall as the first:

$$\frac{18-10}{12-10}=\frac{8}{2}=4.$$

However, the actual number of recyclers is only

$$\frac{18}{12}=1.5$$

times as large. The truncated axis exaggerates the visual difference.

Step 3: Evaluate the claim

The school says, “The campaign caused recycling to increase dramatically for all students.”

A better evaluation is:

• The data show that recycling increased by $6$ students, or $50\%$.
• The graph is misleading because its axis does not start at $0$.
• Only $40$ students were surveyed, so the results may not represent the entire school.
• The data show an increase after the campaign, but they do not prove that the campaign caused it. Other changes could have happened at the same time.

A fair conclusion would be: “Among the $40$ surveyed students, the number who recycled increased from $12$ to $18. The graph exaggerates the difference because its axis starts at$10.”