The truncated axis.
Same three numbers, two impressions. Cutting the axis is not always dishonest — sometimes small changes genuinely matter — but the reader has to be told, and a headline claiming a tripling would not be.
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- Grade 7-12 Math Review
- Recognizing Misleading Statistics and Graphs
Recognizing Misleading Statistics and Graphs
Almost every misleading graph contains only true numbers.
Spot the problem
A bar chart shows sales of 4,900, 5,000 and 5,100 units, with the vertical axis starting at 4,800. The last bar looks three times the first. What is wrong?
Starting the axis at 4,800 cuts off almost every bar, so only the tiny remaining tips get compared. Sales rose about 4%, but the picture suggests tripling. B is the thing to resist — the numbers are perfectly honest. The distortion is entirely in the scale, which is what makes it hard to spot and easy to defend.
Uneven intervals on the time axis.
Years shown
1990, 2000, 2010, 2020, 2022, 2023
evenly spaced on the page
The effect
recent years get equal width
a recent change looks steep
Honest version
spacing proportional to time
the last gaps become slivers
A line graph's slope means nothing if the horizontal spacing does not match the actual intervals. Always read the axis labels rather than trusting the shape.
Choosing the flattering average.
salaries: 18k × 7, plus 300k
eight people
mean ≈ ₱53,250
what the advert says
median = ₱18,000
what seven of the eight earn
"Average" is not one number
Both figures are correct. With skewed data the mean can describe nobody in the room, which is why an unqualified "average" is worth questioning — ask which one.
Percentages with no base number.
"Risk doubles"
from 1 in 100,000 to 2
technically true, barely news
"80% preferred ours"
of how many?
eight people, perhaps
"Up 300%"
from 1 to 4
the base decides everything
A relative change and an absolute change are different claims. Headlines almost always quote the one that sounds larger.
The five questions.
| Ask | Because |
|---|---|
| Where does the axis start? | a cut axis exaggerates |
| Are the intervals equal? | uneven spacing fakes a trend |
| Which average is it? | mean and median can differ wildly |
| How many were asked, and how? | a percentage hides its base |
| Is a cause being claimed? | a link is not a cause |
Ice-cream sales and drowning deaths rise together every summer. Neither causes the other — the weather drives both. A third factor lurking behind a correlation is the usual explanation.
Pictures that grow two ways, and data collected badly.
A picture is only honest if the AREA you see is in proportion to the number. The collection is only honest if the sample is large, fair and asked neutral questions.
A school canteen's rice sales doubled from 200 to 400 kg. The poster draws the second sack twice as tall and twice as wide. How much bigger does it look?
height × 2, width × 2
both dimensions scaled
area × (2 × 2) = × 4
what the eye compares
looks 4 times bigger; really 2 times
400 ÷ 200 = 2
A tilted 3-D pie chart
front slices show more surface
near slices look larger
A tiny sample
a handful of people for a whole region
cannot represent it
A one-sided sample
asking only one barangay, one school, one fan page
biased
A leading question
"Don't you agree that …?"
steers the answer
Who paid for it
the seller funds the study
a possible conflict of interest
Reading the picture, not the number
A student who compares the two sacks answers "sales quadrupled". The labels say 200 and 400 — a doubling. When icons change size, read the numbers, never the shapes.
Variations: one huge value, points versus percent, a cut axis measured.
Three checks you can do with arithmetic, not just suspicion.
Five students' weekly allowances: ₱50, ₱60, ₱60, ₱70 and ₱760. Find the mean and the median.
mean = 1,000 ÷ 5 = ₱200
the total is 1,000
median = ₱60
the middle of five sorted values
With five values, one outlier lifts the mean to ₱200, while four of the five get ₱70 or less. The median describes them.
A barangay's unemployment rate rose from 4% to 6%. Two reports describe it.
6% − 4% = 2 percentage points
the absolute change
2 ÷ 4 = 0.5 = a 50% increase
the relative change
Both are true. "Up 50%" sounds far worse than "up 2 points" — ask which one is being quoted.
Bars show 96 and 100 on an axis that starts at 92. How big is the change really?
drawn heights 4 and 8
96 − 92 and 100 − 92
looks like × 2
8 ÷ 4
real change 4 ÷ 96 ≈ 4.2%
from the labels
Subtract the axis start from each bar to see what the eye compares; divide the real values to see the truth.
On the exam
Expect a graph or headline with true numbers and a misleading picture (a cut axis, uneven intervals, resized icons), a choice between mean and median for skewed data, and a flaw in how data were collected or a cause claimed from a correlation. The fastest route is the five questions: where the axis starts, whether the intervals are equal, which average, how many were asked and how, and whether a cause is claimed. The 10-second check: compute the real change from the labels and compare it with what the picture suggests.
Now you try
Practice problem
Judge the claim
A report notes that towns with more fire engines have more fire damage, and concludes that fire engines cause damage. What is the flaw?
Bigger towns have more fires and more fire engines, so town size drives both — a lurking third variable. The correlation is real; the causal claim is not. D goes too far: the two really do move together, which is exactly why the claim is tempting. Reversing it would be just as unfounded.
Hint: when two things rise together, look for a third thing that could be raising both.