Drawing conclusions from information

Science and society, grade 10 (3 units) Lesson 1 of 15Learning

Drawing conclusions from information

What a chart shows

Before drawing a conclusion from a chart, read exactly what it shows:

what was measured – and in what units;
who was measured – and how many;
when it was measured;
who published the data.

A secondary school asked 600 pupils, in one week in November, how many hours they sleep on the night before a school day.
The chart shows the mean in each year group:

What was measured: the mean hours of sleep a night, in hours.
Who: 600 pupils of one school, in Years 7–12.
When: one week in November.
Who published: the school itself.

A conclusion the data support

A sound conclusion says only what the data show: about those who were measured, and about what was measured.

“In this school, in the week measured, the older pupils slept less on average” – the data support it.
“Every Year 12 pupil sleeps 7 hours” – no: 7 is a mean. Some Year 12 pupils sleep 5 hours, and some sleep 9.
“Teenagers in Israel sleep less as they grow older” – the data are not enough: one school was measured, not all the country’s teenagers.
“Year 12 pupils sleep less because of homework” – the data say nothing about the reason: nobody was asked about it.

A rate per 1,000 people

To compare groups of different sizes, compare a rate – how many cases there are for every 1,000 people (or 100, or 100,000) – not the number of cases itself.

\displaystyle \frac{k}{N}\cdot1000

$\displaystyle k$ – the number of cases, $\displaystyle N$ – the number of people.

In one year:
Town A has 60,000 residents, and had 120 road accidents with injuries.
Town B has 250,000 residents, and had 300 such accidents.

Town A: $\displaystyle {\frac{120}{60000}\cdot1000}={2}$ accidents per 1,000 residents.
Town B: $\displaystyle {\frac{300}{250000}\cdot1000}={1.2}$ accidents per 1,000 residents.

Town B had more accidents, but far more people live there. For the number of residents, town A has more accidents.

A third factor

When two things rise and fall together, one of them does not necessarily cause the other: often a third thing causes both.

A seaside town counted, each month, how many hundred ice creams the beach kiosk sold, and how many times the lifeguards pulled swimmers out of the water:

In the months when more ice creams were sold, there were also more rescues.
But ice cream does not cause drowning: the heat causes both. In the hot months more people buy ice cream, and more people go into the sea.
To know that one thing causes another takes more than a chart: an experiment, for example, where only one thing is changed and the result is watched.

Exercise 1

The mean household water use per person per day in one town, from the town’s water company:

In which year was the use highest?

/ – 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

By how many litres did the use fall from 2020 to 2024?

/ – 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

Is the conclusion right: “In 2024 every resident of the town used 136 litres of water a day”? Explain.

Exercise 2

Clinic A has 1,500 registered patients, and last winter 45 of them had flu.
Clinic B has 4,000 registered patients, and 80 of them had flu.

How many patients had flu at clinic A for every 100 patients?

/ – 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

How many patients had flu at clinic B for every 100 patients?

/ – 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

At which clinic was flu more common? Explain.

Exercise 3

A newspaper wrote: “In towns where more doctors work, more people die each year.”

Do the doctors cause the deaths? What is the third factor? Explain.

Exercise 4

In a survey of 400 users of a fitness app, 70% said they work out at least three times a week.

Do the data support the conclusion? Explain each answer.

280 of the people in the survey said they work out at least three times a week.
70% of Israelis work out at least three times a week.
The app makes people work out more.

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

Open as a page

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
$\displaystyle n\cdot P(A)$