Math 6 · N.Q.A.1

Misleading graphs

Every number on the graph is right. How can the picture still say something false, and how big is the false impression?

You need

Reading a graph: titles, labels and unitsGrade 3

Where you are 12 in Data displays

Statistics7

Data displays12

▴ 6 above Stem and leaf plots Choosing a display for a question

Misleading graphs

Box plots3 Two-way tables3 ▾ Show all 12 (1 more)

Why this exists

The eye reads a graph by comparing sizes before it reads any tick label. A bar twice as tall says “twice as much”; a line climbing evenly says “steady growth”. That is what makes a graph quick to read, and it is also what lets a graph mislead while every number on it is correct.

Reading a graph you did not draw starts with the title, the axes and the key. This lesson compares what the picture seems to say with what the numbers say, and measures the gap. “It looks 4 times bigger, but it is really 1.25 times bigger” is a claim anyone can check.

Definition

A graph is misleading when the comparison its picture makes (lengths of bars, areas of pictures, sizes of slices, steepness of a line) does not match the comparison the data make, even though each value shown is correct.

For two amounts, the two comparisons are two ratios:

the drawn ratio: the size of one drawn shape divided by the size of the other;
the true ratio: one value divided by the other.

On an honest bar graph or picture graph the two ratios are equal. When they differ, the drawn ratio is the size of the false impression.

Intuition

Figure 1 shows the same two sales figures twice. On the left the value axis starts at 1{,}100, so each bar shows only the part of its value above 1{,}100, and the eye compares those parts. On the right the axis starts at 0 and the bars are close in height, as the numbers are.

Figure 1The same two numbers drawn twice. On the left the axis starts at 1{,}100; on the right it starts at 0.

The other cases work the same way. A picture grown in both directions grows in area faster than in height. Unevenly spaced dates drawn at equal steps make a slowing climb look steady. A time window that starts at a low point turns a fall into a rise. A big group’s total beats a small group’s even when the small group does the thing more often for its size. A value axis with no numbers or units cannot be checked at all, and a tilted 3D pie draws its front slices bigger than a flat circle graph would.

Results

A cut value axis. If a bar graph’s value axis starts at s instead of 0, the bar for a value v has height v - s, so

\text{drawn ratio} = \frac{\text{bigger value} - s}{\text{smaller value} - s}

which is bigger than the true ratio whenever s is above 0 and below both values.

Pictures grown in both directions. A picture made k times as tall and k times as wide has k \times k times the area, so a true ratio of k looks like k \times k.

Groups of different sizes. To compare how often something happens in groups of different sizes, compare rates (per person, per 100, per 1{,}000), not totals.

Uneven scales. Equal steps on an axis must stand for equal amounts, or the shape of the graph is wrong.

Why it is true

The cut axis. A bar stands on the bottom of the axis, s, and covers the values from s up to v, a height of v - s. Suppose the bigger value is r times the smaller, with r above 1. Then

\begin{aligned} \text{bigger} - s &= r \times \text{smaller} - s \\ &> r \times \text{smaller} - r \times s \\ &= r \times (\text{smaller} - s) \end{aligned}

The middle step holds because r \times s is more than s when s is above 0, and taking away more leaves less. So the taller bar is more than r times the shorter: the cut always exaggerates, and only s = 0 makes the ratios equal.

The pictures. Area is width times height. Multiplying both by k multiplies the area by k \times k (Scale factor, area and volume).

The totals. A total is a rate times the size of the group, so a big group with a low rate can make a bigger total than a small group with a high rate (Rates and unit rates).

Worked examples

Example 1

A bar graph whose axis starts at 1,100

Gracjan sees an advert with a bar graph of last month’s sales of two phones (Figure 1, left). Phone A sold 1{,}500 and Phone B sold 1{,}200, and the value axis starts at 1{,}100. How many times as tall as Phone B’s bar is Phone A’s bar, and how many times as many phones did Phone A really sell?

Solution

Each bar starts at the bottom of the axis, 1{,}100, so its drawn height is the part of its value above 1{,}100:

1{,}500 - 1{,}100 = 400 \qquad 1{,}200 - 1{,}100 = 100

Phone A’s bar is 400 \div 100 = 4 times as tall. The true comparison divides the values: \displaystyle 1{,}500 \div 1{,}200 = \frac{5}{4} = 1.25. The picture says 4 times as many; the data say 1.25 times, which is 25 percent more. On the right of Figure 1 the bars stand in the true ratio.

Example 2

Picture graphs: height against area

Gudrun sees an advert comparing two phones with pictures (Figure 2). Phone A sold 1{,}200 and Phone B sold 800. Phone B’s picture is 2 cm wide and 4 cm tall; Phone A’s is 3 cm wide and 6 cm tall. How many times as big does Phone A look, and how many times as many did it sell?

Figure 2Phone A is 1.5 times as tall and 1.5 times as wide, so it covers 2.25 times the area.

Solution

The true ratio is 1{,}200 \div 800 = 1.5, and the heights match it: 6 \div 4 = 1.5. But the widths grew too, 3 \div 2 = 1.5, and the eye takes in the whole picture, its area:

2 \times 4 = 8 \text{ cm}^2 \qquad 3 \times 6 = 18 \text{ cm}^2

Phone A’s picture covers 18 \div 8 = 2.25 times the area, because 1.5 \times 1.5 = 2.25. An honest version keeps the width at 2 cm and makes the height 6 cm, an area of 12 \text{ cm}^2, and 12 \div 8 = 1.5. Another uses equal small pictures, each for 400 phones: three for Phone A, two for Phone B, as in a pictograph.

Example 3

Choosing the window

An advert for a phone shows its weekly sales for weeks 7 to 10 under the words “sales climbing every week”. Guido finds all ten weeks:

Week	1	2	3	4	5
Phones sold	300	280	260	240	210

Week	6	7	8	9	10
Phones sold	170	150	165	180	195

Compare the change the advert shows with the change over all ten weeks.

Solution

Sales did rise each week from week 7 to week 10: from 150 to 195, up 45, and 45 \div 150 = 0.3, a rise of 30 percent.

Over all ten weeks sales went from 300 to 195, down 105, and 105 \div 300 = 0.35, a fall of 35 percent. The window starts at week 7, the lowest week of all, so it shows the recovery and hides the fall. Even the window’s best week, 195, is below each of the first five weeks.

Example 4

Dates spaced evenly when they are not

A streaming service’s poster shows its subscribers, in thousands, at five dates placed at equal steps (Figure 3, left): January 20, February 26, March 32, June 38, December 44. The line is straight. Hakon wants to know whether the growth was steady.

Figure 3Left, the poster spaces five dates evenly. Right, the same points spaced by month.

Solution

Each step rises by 6 thousand, which is why the line is straight. But the steps cover different lengths of time, so find the growth per month, in thousands:

Stretch	Months	Rise	Per month
Jan to Feb	1	6	6
Feb to Mar	1	6	6
Mar to Jun	3	6	2
Jun to Dec	6	6	1

Growth fell from 6 thousand a month to 1 thousand a month, so the last stretch is 6 times slower than the first. Spaced by month (Figure 3, right), the line climbs steeply and then flattens.

Example 5

Totals from towns of different sizes

A poster about two towns’ bicycle hire says “Town A loves cycling most” and shows a bar graph of one month’s rides. Halina finds the town sizes:

Town	Rides	Residents
A	6,000	30,000
B	4,000	10,000

Compare the towns by total and by rate.

Solution

By total, Town A had 6{,}000 \div 4{,}000 = 1.5 times as many rides. But Town A has three times as many people, so compare rides per 100 residents (a rate per hundred). Town A has 30{,}000 \div 100 = 300 hundreds of residents and Town B has 100:

\text{A: } 6{,}000 \div 300 = 20 \qquad \text{B: } 4{,}000 \div 100 = 40

For its size, Town B rides 40 \div 20 = 2 times as much. The totals graph does not just exaggerate: it points the wrong way.

Example 6

Working back to the cut

An advert shows two phones’ sales as bars. Phone A sold 1{,}400 and Phone B sold 1{,}000, and Phone A’s bar is drawn exactly 3 times as tall as Phone B’s. The axis labels are too small to read. Where must the value axis start?

Solution

Call Phone B’s drawn height one part. Phone A’s bar is three parts, so the bars differ by two parts. They also differ by 1{,}400 - 1{,}000 = 400, because the cut takes the same amount off both bars. Two parts are 400, so one part is 200.

Phone B’s bar shows 200 of its 1{,}000, so the axis starts at 1{,}000 - 200 = 800. Check: Phone A’s bar shows 1{,}400 - 800 = 600, and 600 = 3 \times 200. The bars say 3 times as many; the data say 1{,}400 \div 1{,}000 = 1.4 times.

Boundaries

The drawn-ratio test is for displays where a length or an area stands for an amount: bars, pictures, slices. A line graph or a dot plot is read by position, so its value axis may honestly start above 0, as long as it is labeled. The steepness of a line graph depends on the scale chosen, and no one steepness is correct; what must hold is that equal steps stand for equal amounts.

A break symbol on an axis admits the cut, but the eye still compares the cut lengths. And a rate is the fair comparison for “how often”, but a total answers a question about totals: how many bicycles a town needs depends on its total rides.

Where this goes next

A histogram with bins of different widths raises the area question again, which making a histogram avoids with equal bins. Choosing a display picks the picture whose sizes answer the question. Later, data can mislead before any graph is drawn, when a sample does not stand for its population (Populations, samples and census).

Exercises

Bars on a cut axis

An advert’s bar graph shows that Phone C sold 900 and Phone D sold 600, on a value axis starting at 500. How many times as tall as Phone D’s bar is Phone C’s? How many times as many did Phone C really sell?
The same two values, 900 and 600, are drawn on an axis starting at 300. How many times as tall is the taller bar now?
A streaming service’s poster shows 52 thousand subscribers this year and 48 thousand last year, on a value axis starting at 46 thousand. How many times as tall is this year’s bar, and what is the true ratio?
In Figure 4, how many times as tall as Monday’s bar is Saturday’s bar? How many times as many rides were there really?
In Figure 4, compare Sunday’s bar with Monday’s in the same two ways.
In Figure 4, compare Saturday’s bar with Sunday’s in the same two ways.

Figure 4A poster’s graph of bicycle hire rides, for Exercises 4 to 6.

Pictures

An advert says Phone E sold 3 times as many as Phone F, and draws Phone E’s picture 3 times as tall and 3 times as wide. How many times the area does Phone E’s picture cover?
A streaming service had 40 thousand subscribers last year and 50 thousand this year. A poster draws last year as a screen 4 cm wide and 3 cm tall, and this year as a screen 5 cm wide and 3.75 cm tall. How many times as big does this year look, and what is the true ratio?
An advert draws sales as boxes in 3D, the second box 2 times as long, 2 times as wide and 2 times as tall as the first, for 2 times the sales. How many of the small boxes would fit in the big one?
Phone G sold 1{,}000 and Phone H sold 2{,}500. An advert draws both phones 2 cm wide, Phone G 2 cm tall and Phone H 5 cm tall. Compare the areas with the sales. Is this picture honest?

Windows and scales

A streaming service’s subscribers, in thousands, for months 1 to 12 were 50, 49, 47, 45, 42, 40, 38, 36, 35, 36, 37, 38. A poster shows months 9 to 12 under “growing again”. What change does the poster show, and what was the change over the whole year?
An advert graphs a shop’s running total of phones sold at Week 1, Week 2, Week 4 and Week 8, equally spaced: 100, 130, 160, 190. Find the phones sold per week in each of the three steps.
A bar graph’s value axis has equally spaced ticks labeled 0, 100, 200, 400, 800. (a) Bars for 100 and 400: how many times as tall does the 400 bar look, and what is the true ratio? (b) The same for bars of 400 and 800.

Totals and rates

Town C had 9{,}000 bicycle hire rides and has 45{,}000 residents. Town D had 5{,}000 rides and has 20{,}000 residents. Find each town’s rides per 100 residents. Which town rides more for its size?
A poster says streaming Plan Gold had 3 times as many complaints as Plan Silver: 300 against 100. Plan Gold has 30{,}000 subscribers and Plan Silver 5{,}000. Find complaints per 1{,}000 subscribers for each.
Phone model E needed 120 repairs out of 4{,}000 sold; model F needed 90 out of 1{,}500 sold. Find repairs per 100 phones sold. Which model needs repair more often?

Working backwards

Bars for 750 and 500 are drawn with the taller exactly 2 times as tall. Where does the value axis start?
Bars for 1{,}800 and 1{,}500 are drawn with the taller exactly 4 times as tall. Where does the value axis start?
A poster’s bars are 5 cm and 2 cm tall, the axis starts at 30, and the short bar’s value is 40. What is the tall bar’s value? Give the drawn ratio and the true ratio.

Explaining

A poster’s line graph of subscribers climbs from the bottom corner to the top corner, but the value axis has no numbers and no units. Explain why the picture cannot tell you how much subscribers grew, giving two sets of numbers it could stand for.
Explain why a 3D pie chart drawn tilted, as if lying on a table, can make one slice look bigger than another slice with the same percent.
A line graph of a person’s body temperature over a day has a value axis from 36 to 38 degrees Celsius. Explain why that is not misleading, though a bar graph with a cut axis is.
Explain why a value axis starting at s, above 0 and below both values, always makes the bigger bar look more than its true number of times the smaller one.

Answers

4 times as tall; really 1.5 times. The bars show 900 - 500 = 400 and 600 - 500 = 100, so 400 \div 100 = 4, while 900 \div 600 = 1.5.
2 times: 900 - 300 = 600 and 600 - 300 = 300, and 600 \div 300 = 2, still more than 1.5.
3 times as tall, since the bars show 52 - 46 = 6 and 48 - 46 = 2 thousand; the true ratio is \displaystyle 52 \div 48 = \frac{13}{12}, about 1.08.
4 times as tall; really 1.3 times. The bars show 260 - 180 = 80 and 200 - 180 = 20, so 80 \div 20 = 4, while 260 \div 200 = 1.3.
3 times as tall; really 1.2 times: 60 \div 20 = 3, while 240 \div 200 = 1.2.
About 1.33 times as tall; really about 1.08 times: \displaystyle 80 \div 60 = \frac{4}{3} and \displaystyle 260 \div 240 = \frac{13}{12}.
9 times the area for 3 times the sales, because 3 \times 3 = 9.
It looks about 1.56 times as big; the true ratio is 50 \div 40 = 1.25. The areas are 4 \times 3 = 12 and 5 \times 3.75 = 18.75 \text{ cm}^2, and 18.75 \div 12 = 1.5625 = 1.25 \times 1.25.
8, because 2 \times 2 \times 2 = 8. A solid-looking box can look 8 times as big for 2 times the sales.
Honest. The areas are 2 \times 2 = 4 and 2 \times 5 = 10 \text{ cm}^2, and 10 \div 4 = 2.5, the same as 2{,}500 \div 1{,}000 = 2.5.
The poster shows a rise of 3 thousand, from 35 to 38. Over the year subscribers fell by 12 thousand, from 50 to 38.
30, 15 and 7.5 phones per week. Each step adds 30, over 1, 2 and 4 weeks. The straight line hides that sales slowed.
(a) It looks 3 times as tall (400 sits 3 steps up, 100 sits 1), but 400 \div 100 = 4. (b) It looks 4 \div 3, about 1.33 times as tall, but 800 \div 400 = 2. This uneven scale shrinks differences.
Town C: 9{,}000 \div 450 = 20; Town D: 5{,}000 \div 200 = 25 rides per 100 residents. Town D rides more for its size, though Town C had more rides in total.
Gold: 300 \div 30 = 10; Silver: 100 \div 5 = 20 complaints per 1{,}000 subscribers. Silver’s rate is 2 times Gold’s.
E: 120 \div 40 = 3; F: 90 \div 15 = 6 repairs per 100 sold. Model F, twice as often, though E had more repairs in total.
At 250. The bars differ by one short bar and by 750 - 500 = 250, so the axis starts at 500 - 250 = 250. Check: 750 - 250 = 500 = 2 \times 250.
At 1{,}400. The bars differ by three short bars and by 300, so one short bar is 100 and the axis starts at 1{,}500 - 100 = 1{,}400. Check: 1{,}800 - 1{,}400 = 400 = 4 \times 100.
55. The short bar shows 40 - 30 = 10 in 2 cm, so each cm is 5; the tall bar shows 5 \times 5 = 25, and 30 + 25 = 55. Drawn ratio 5 \div 2 = 2.5; true ratio \displaystyle 55 \div 40 = \frac{11}{8} = 1.375.
With no numbers, the bottom and top of the axis could be anything. With an axis from 10 to 20 thousand, subscribers doubled; with an axis from 1{,}000 to 1{,}010 they grew by 1 percent. With no unit the line might not even count people.
On a flat pie each slice’s angle and area are its share. Tilting squashes the circle into an oval, so front and back slices open wider than side slices. A thick 3D pie shows a side wall under its front slices only, extra ink, and drawing in perspective makes near slices bigger. So equal percents look unequal.
A line graph is read by where its points sit and how they move, not by a length from 0, and the labels show the range. An axis from 0 would press every reading into a thin band near the top. A bar’s length stands for its amount, so a cut changes what the length says.
Say the bigger value is r times the smaller, r above 1. The bigger bar shows r \times \text{smaller} - s. Because r \times s is more than s, this is more than r \times \text{smaller} - r \times s = r \times (\text{smaller} - s), which is r times the smaller bar.