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Graphs that mislead

Statistics and data · level 4, lesson 6 of 6

A question to start

Two juice companies show you a graph of how many people prefer their juice. Company Kiwi's bar is twice as tall as Company Mango's. Kiwi must be twice as popular, right?

Look at the numbers on the side: Kiwi 52 people, Mango 50 people. That is barely different. The graph is not lying about the numbers; every number on it is true. It lies with its shape. The vertical axis starts at 48 instead of 0, so only the tops of the bars are shown, and the tops are what your eye compares.

A graph is supposed to let your eyes see the truth faster than reading a table. That same power can be turned around. In this lesson you learn four classic tricks, why each one fools the eye, and how to redraw the graph honestly.

The idea

The rule behind honest pictures

Almost every honest graph follows one rule:

The size you see should be in proportion to the number it stands for.

If one number is twice another, its bar should be twice as long, its slice twice as big, its picture twice the area. Every trick in this lesson breaks this rule in some way.

Trick 1: a bar graph axis that does not start at 0

A bar's length stands for its number, measured from 0. If the axis starts at 48, a bar for 50 shows only the piece from 48 to 50 (2 units), and a bar for 52 shows the piece from 48 to 52 (4 units). The second looks twice as tall, though 52 is only \(\frac{2}{50} = 4\%\) more than 50.

4850525456 MangoKiwi axis starts at 48 0204060 MangoKiwi axis starts at 0

The fix: bar graphs start at 0. If small differences really matter (and they sometimes do), show the numbers clearly, or graph the differences themselves.

A note on line graphs: a line graph shows change, and its height is read against the scale, not as a length from zero. So a line graph of body temperature from 36 °C to 39 °C, with the axis starting at 35, can be perfectly honest, as long as the axis is clearly labelled. A broken axis is often marked with a small zigzag near the bottom to warn the reader. The rule "start at 0" is strict for bars, because bars are read as lengths.

Trick 2: unequal intervals

The bottom axis of a line graph or histogram is a number line. Equal distances along it must stand for equal amounts of time or value.

Suppose a town's population is given for the years 1990, 2000, 2010, 2020, 2021, 2022, and the graph puts those six years equally spaced. The last three points are one year apart, but drawn as far apart as the ten-year gaps. Any growth in the decade-long gaps is squashed, and a change from 2020 to 2022 looks as slow (or as fast) as a change over a whole decade. The shape of the line is wrong.

In a histogram, unequal intervals cause a different problem. If one interval is 0-10 and the next is 10-50, the wide interval collects more values just because it is wider, so its bar is taller even if values are no more common there.

The fix: space the axis by the actual numbers (2020 to 2021 is one tenth of the gap from 2000 to 2010), and keep histogram intervals equal.

Trick 3: 3D pie charts

A circle graph drawn as a tilted 3D disc looks impressive, but tilting changes what you see. The slices at the front show their thick edge as well as their top, and with perspective the near side is drawn larger than the far side. Slices at the back are squashed.

25%25%25%25% tilted 3D flat: four equal quarters

All four slices are exactly 25%, but in the 3D version the front (orange) slice looks the biggest, because you also see its side, and the back (grey) one looks the thinnest. A company could put its own product in the front slice and let the drawing do the boasting.

The fix: draw circle graphs flat, and write the percent on every slice. (Or use a bar graph: lesson 3.)

Trick 4: pictures scaled in two directions

A pictograph might use a drawing of a bag of rice to show sales: 100 bags in year 1, 200 bags in year 2. The sneaky version draws the year 2 bag twice as tall and twice as wide.

year 1: 100 year 2: 200 looks 4 times bigger year 1 year 2: two bags misleading honest

Why is it misleading? The eye judges a picture by its area (how much ink), not its height. Doubling the height and the width multiplies the area by \(2 \times 2 = 4\): four small bags fit inside the big one, as the dashed lines show. So a doubling looks like a fourfold jump. If the picture is drawn as a 3D object (a box, a barrel) and grows in all three directions, the volume it suggests is \(2 \times 2 \times 2 = 8\) times bigger.

The fix: keep every picture the same size and use more of them (a pictograph with a key, as in level 2), or use bars, which grow in only one direction.

A checklist for any graph

  1. Read the title, the axis labels and the units.
  2. Do the bars start at 0?
  3. Are the steps on each axis equal?
  4. Is the graph flat, with sizes in proportion to the numbers?
  5. Read the actual numbers and compare them yourself: how much bigger is one really, as a fraction or percent?
  6. Who made the graph, and is anything left out (missing years, missing groups)?

Worked examples

1. How big does it look, how big is it? A bar graph's axis starts at 90. Bar P is 95 and bar Q is 100.

  1. Visible lengths: P shows \(95 - 90 = 5\) units, Q shows \(100 - 90 = 10\) units. Q looks twice as big.
  2. Real comparison: Q is \(100 - 95 = 5\) more than P, and \(\frac{5}{95}\) is about 5% more.
  3. Honest redraw: axis from 0 to 100; the bars are almost the same height.

2. A shrinking picture. A poster shows coins: last year's coin has diameter 4 cm and this year's has diameter 2 cm, for savings of 400 and 200 dollars. The real ratio is \(200 : 400\), a half. The picture's area ratio is \((2 \times 2) : (4 \times 4) = 4 : 16\), a quarter. The savings look as if they fell to a quarter when they halved.

3. Squeezed time. A graph of a club's members shows 2015: 20, 2020: 40, 2021: 44, 2022: 48, with the four years equally spaced. Real rates: from 2015 to 2020, \(40 - 20 = 20\) new members in 5 years, that is 4 a year. From 2020 to 2022, \(48 - 40 = 8\) in 2 years, also 4 a year. With honest spacing the line is a straight line: steady growth. With equal spacing, the first piece looks very steep (20 in one step) and the rest look slow, suggesting the club stalled. It did not.

4. Reading a 3D pie. A 3D pie shows "Brand A" in front looking about half the pie, with labels 35%, 30%, 20%, 15% and A labelled 35%. Trust the label, not the look: A is just over a third, and the pie overstates it because of its front position.

Try it

  1. A bar graph has an axis starting at 1000. Bars show 1010 and 1040. How many times taller does the second look? How much bigger is it really, in percent?
  2. A pictograph doubles a cow's height and width to show milk production going from 50 to 100 litres. How many times bigger does the second cow look?
  3. If a 3D box picture is tripled in height, width and depth, how many times bigger does it look?
  4. Why is it acceptable for a line graph of a fever (37 to 39 °C) to start its axis at 36 °C, but not for a bar graph?
  5. A graph shows test scores for the years 2019, 2020, 2023, 2024, equally spaced. What is misleading?
  6. Redraw in words: a bar chart of 3 classes' book totals (120, 125, 130) with an axis from 115 to 130. What would an honest version look like?
  7. Find a graph in a newspaper, a website or an advert, and run the six-point checklist on it. Write one sentence on whether it is fair.
Answers
  1. Visible lengths \(1010 - 1000 = 10\) and \(1040 - 1000 = 40\): it looks 4 times taller. Really it is \(1040 - 1010 = 30\) more, and \(\frac{30}{1010}\) is about 3% more.
  2. Area grows by \(2 \times 2 = 4\): it looks 4 times bigger, though production only doubled.
  3. \(3 \times 3 \times 3 = 27\) times.
  4. A line graph is read against its scale to see change, so a labelled axis starting at 36 is fine. Bars are read as lengths from 0, so cutting the axis distorts the comparison of lengths.
  5. The gap 2020 to 2023 (three years) is drawn the same as one-year gaps, so change over those three years looks as if it happened in one year.
  6. Axis from 0 to about 140, in steps of 20. The three bars look nearly equal (120, 125, 130), which is true: the largest is only \(\frac{10}{120}\), about 8%, more than the smallest.
  7. Answers will vary. A good answer names the trick (if any) and states the true comparison in numbers.

Common mistake

Comparing what the bars look like instead of what the numbers say. Always do the arithmetic: "52 against 50 is 2 more, about 4%", and only then look at the picture.

Calling every cut axis a lie. Bars must start at 0; a clearly labelled line graph need not. Judge the graph by whether the shape you see matches the numbers.

Where it leads