Misleading graphs

Science and society, grade 10 (3 units) Lesson 2 of 15Learning

Misleading graphs

An axis that does not start at 0

When the axis of a bar chart does not start at 0, the ratio of the bars’ heights is not the ratio of the values: a small difference looks huge.

An advert for phone A showed this chart: how many hours the battery of three phones lasts while playing videos.

In the chart phone A’s bar is 7 times as tall as phone C’s. But the axis starts at 9.5, so only the part above 9.5 hours shows:

Phone A: $\displaystyle {10.2-9.5}={0.7}$
Phone C: $\displaystyle {9.6-9.5}={0.1}$

The same data, with the axis starting at 0:

The real difference: $\displaystyle {10.2-9.6}={0.6}$ hours, and $\displaystyle {\frac{0.6}{9.6}}={0.0625}$ – phone A’s battery lasts only about 6% longer.

An axis stretched upwards

In a line graph too, where the axis starts and where it ends decide how steep the line looks. The real change is measured in numbers: by what percent the value changed.

The mean rent of a three-room flat in one town, in shekels, the same data twice – once with the axis starting at 5,000, once at 0:

In the first graph the rent seems to have “soared”; in the second, to have risen a little.
The rise over the four years: $\displaystyle {5600-5000}={600}$ ₪.
In percent: $\displaystyle {\frac{600}{5000}}={0.12}$ , a rise of 12% in four years.

A period chosen on purpose

Whoever chooses the year to compare from can reach almost any conclusion they like. So check a longer period too.

The number of burglaries of homes in one town each year:

The mayor said in a speech: “Since 2021 burglaries have fallen by 20%.”

The calculation is right: $\displaystyle {\frac{380-304}{380}}={\frac{76}{380}}={0.2}$ .
But 2021 was an unusual year, with more burglaries than any other.
Compared with 2018, 2024 had slightly more burglaries: 304 against 300.

A picture that grows in both directions

When a quantity is shown by a picture, and the picture is made larger both in height and in width, its area grows far more than the quantity.

A council leaflet showed the number of people using shared bicycles: 2,000 last year and 4,000 this year. It drew last year as a small bicycle, and this year as a bicycle twice as tall and twice as wide.

The number of users doubled.
The picture’s area grew $\displaystyle {2\cdot2}={4}$ times – and the eye compares areas.
A fair picture: two bicycles of the same size beside one – each bicycle 2,000 users.

Exercise 1

An internet provider published the mean download speed of three providers, in megabits per second:

a

How many times as tall as provider C’s bar is provider A’s, in the chart?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

By what percent is provider A’s speed really higher than provider C’s?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A newspaper article headed “Monthly bus pass price soars” showed this graph of the price of a monthly bus pass, in shekels:

a

By what percent did the price rise from 2021 to 2024?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Does the word “soars” fit? What in the graph makes the rise look large?

Exercise 3

The number of tourists who visited a small town, in thousands:

a

By what percent did the number of tourists grow from 2021 to 2024?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

By what percent did the number of tourists grow from 2019 to 2024?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

Two headlines:
“Tourism in the town up 120% since 2021”.
“Tourism in the town back to its level before the pandemic”.

Which headline describes the data more fairly? Explain.

Exercise 4

Last year 3,000 trees were planted in a town, and this year 6,000.
A poster drew two trees: this year’s tree is twice as tall and twice as wide as last year’s.

a

How many times as large as the small tree’s area is the large tree’s area on the poster?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How should the poster be drawn so that it does not mislead?

Exercise 5

A survey asked 500 teenagers which apps they use every day. More than one answer could be chosen.

App	A	B	C	D
Percent of those who answered	55%	40%	25%	15%

A news site showed the results in a pie chart. What is wrong with a pie chart for these data? Explain with a calculation.

Formulas: science and society, grade 10 (3 units)

Information and representations

$\displaystyle p$ percent of $\displaystyle n$
$\displaystyle p\%\cdot n$ $\displaystyle =$ $\displaystyle \frac{p}{100}\cdot n$
A rate per 1,000 people – $\displaystyle k$ the number of cases, $\displaystyle N$ the number of people
$\displaystyle \frac{k}{N}\cdot1000$
The average rate of change between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle \frac{y_2-y_1}{x_2-x_1}$
A linear relation – $\displaystyle m$ the change in $\displaystyle y$ when $\displaystyle x$ grows by 1, $\displaystyle b$ the value of $\displaystyle y$ when $\displaystyle {x}={0}$
$\displaystyle y=mx+b$

Measures of centre

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
A weighted mean – $\displaystyle w_1, \dots, w_k$ the weights of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{w_1x_1+w_2x_2+\dots+w_kx_k}{w_1+w_2+\dots+w_k}$
The mean of grouped data – $\displaystyle m_1, \dots, m_k$ the class middles, $\displaystyle f_1, \dots, f_k$ their frequencies
$\displaystyle \bar{x}\approx\frac{m_1f_1+m_2f_2+\dots+m_kf_k}{f_1+f_2+\dots+f_k}$
The place of the median in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$

Probability

Equally likely outcomes – $\displaystyle k$ the outcomes in the event $\displaystyle A$ , $\displaystyle n$ all the outcomes
$\displaystyle P(A)=\frac{k}{n}$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
A path of a tree diagram – the product of the probabilities $\displaystyle p_1$ , $\displaystyle p_2$ on its branches
$\displaystyle P=p_1\cdot p_2$
An event of several paths – the sum of their probabilities
$\displaystyle P(A)=P_1+P_2+\ldots$
Estimating a probability from relative frequency – $\displaystyle A$ happened $\displaystyle f$ times in $\displaystyle n$ experiments
$\displaystyle P(A)\approx\frac{f}{n}$
How many times an event $\displaystyle A$ is expected to happen in $\displaystyle n$ experiments
$\displaystyle n\cdot P(A)$