Collecting data

Data and probability, grade 8 Lesson 1 of 14Learning

Collecting data

A question and its data

Every investigation starts with a question, such as: “How do the pupils in our class get to school?”

To answer it we collect data: facts that were counted, measured or asked. Here the data are the pupils’ answers – one answer from each pupil.

Survey

A survey asks a group of people the same question and writes down each one’s answer.

A good survey question is clear to everyone and has a short answer: “How many brothers and sisters do you have?”
To a question like “Tell us about your family” everyone answers differently, and the answers cannot be counted.

Population and sample

The population is everyone the question is about. The sample is the part of the population that is actually asked.

The question: how many hours a night do the school’s 8th graders sleep? The population – all 150 pupils of the grade.
We ask only 30 of them: they are the sample. That saves time, and still tells us a lot about the whole grade.
A good sample is like the population: pupils from every class, boys and girls.
If we ask only the running team, who train at six in the morning, the sample is not like the grade: the runners get up earlier than everyone else.

Kinds of data

There are two kinds of data:

Categorical data – an answer that is a word, a kind or a name: your favourite sport, the colour of your phone, the way you get to school.
Numerical data – a number that was measured or counted: height, the number of brothers and sisters, minutes in front of a screen a day.
A number that is only a name – a bus line, the number on a player’s shirt – is categorical: there is no sense in adding it to other numbers.

Raw data

The 25 pupils of class 8D answered the question “How do you get to school?”.
These are the answers, in the order they were written down:

walk, bike, car, bus, walk, walk, bike, bus, bus, bus, car, bus, walk, walk, car, car, walk, walk, bus, bike, bus, walk, bike, bike, walk.

A list like this is raw data: all the answers, in no order at all. It is hard to see from it how many pupils chose each way.

Frequency table

The fifth mark crosses the four before it, so counting in fives is easy.

A frequency table puts the data in order: a row for each answer, holding the number of times the answer came up – its frequency.

Go through the list in order, and for each answer draw a mark in its row.
At the end count the marks in each row: that is the frequency.

Way to school	Tally	Frequency
walk	$\displaystyle \cancel{\vert\vert\vert\vert}\;\vert\vert\vert\vert$	9
bus	$\displaystyle \cancel{\vert\vert\vert\vert}\;\vert\vert$	7
bike	$\displaystyle \cancel{\vert\vert\vert\vert}$	5
car	$\displaystyle \vert\vert\vert\vert$	4
Total		25

The frequencies add up to the number of answers: $\displaystyle {9+7+5+4}={25}$ , the number of pupils who answered. That is how you check that no answer was left out.

Exercise 1

For each piece of data, write whether it is categorical or numerical:

Your favourite subject.
Your shoe size.
How many messages you sent yesterday.
The town you were born in.
The number of the bus line you take.

Exercise 2

The pupils’ council wants to know where the middle-school pupils would like to go on the annual trip.
There are 450 pupils in the middle school.
The council asked 60 of them.

What is the population, and what is the sample?
Dana suggested asking only the 60 pupils of the nature club. Is that a good sample? Explain.

Exercise 3

24 pupils answered the question “How many brothers and sisters do you have?”.
The answers:
3, 2, 2, 2, 1, 4, 3, 1, 1, 1, 4, 1, 1, 2, 2, 3, 1, 0, 3, 1, 0, 0, 2, 2

Make a frequency table with a tally column.
Add up the frequencies, and check that the total is the number of pupils.

/ – 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 4

30 pupils chose their favourite subject.
One frequency is missing from the table:

Subject	Frequency
maths	8
English	6
PE	?
art	5
science	4

How many pupils chose PE?

/ – 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

Formulas: data and probability, grade 8

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Relative frequency

The relative frequency of an answer – $\displaystyle f$ its frequency, $\displaystyle n$ the total of all the frequencies
$\displaystyle \frac{f}{n}$

Measures of centre and the range

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The mean from a frequency table – $\displaystyle f_1, \dots, f_k$ the frequencies of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}$
The place of the median in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$
The range – the largest value minus the smallest
$\displaystyle R=x_{\max}-x_{\min}$

Probability

Equally likely outcomes – $\displaystyle k$ the outcomes in the event $\displaystyle A$ , $\displaystyle n$ all the outcomes
$\displaystyle P(A)=\frac{k}{n}$
An impossible event – 0, a certain event – 1
$\displaystyle 0\le P(A)\le1$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
How many times an event $\displaystyle A$ is expected to happen in $\displaystyle n$ experiments
$\displaystyle n\cdot P(A)$