Relative frequency

Data and probability, grade 8 Lesson 3 of 14Learning

Relative frequency

Relative frequency

The relative frequency of an answer is its share of all the answers: its frequency divided by the total of all the frequencies.

\displaystyle \frac{f}{n}

$\displaystyle f$ – the answer’s frequency.
$\displaystyle n$ – the total of all the frequencies, that is, the number of answers.
There are 25 pupils in Noa’s class, and 10 of them chose football. The relative frequency of football: $\displaystyle \frac{10}{25}$ .

A fraction, a decimal and a percent

A relative frequency is written as a fraction, a decimal or a percent – three forms of the same number.
The favourite sport of the 25 pupils in Noa’s class:

Sport	Frequency	Relative frequency	As a percent
football	10	$\displaystyle 0.4$	$\displaystyle 40\%$
basketball	6	$\displaystyle 0.24$	$\displaystyle 24\%$
swimming	5	$\displaystyle 0.2$	$\displaystyle 20\%$
tennis	4	$\displaystyle 0.16$	$\displaystyle 16\%$

As a fraction: $\displaystyle {\frac{10}{25}}={\frac{2}{5}}$ , $\displaystyle \frac{6}{25}$ , $\displaystyle {\frac{5}{25}}={\frac{1}{5}}$ , $\displaystyle \frac{4}{25}$ .
As a decimal: expand the fraction to the denominator 100, $\displaystyle {\frac{6}{25}}={\frac{24}{100}}={0.24}$ .
As a percent: 24 hundredths are $\displaystyle 24\%$ .

The sum of the relative frequencies

All the relative frequencies always add up to 1, that is, to 100%: all the answers together are the whole group.

In Noa’s class: $\displaystyle {0.4+0.24+0.2+0.16}={1}$ , and in percent $\displaystyle {40\%+24\%+20\%+16\%}={100\%}$ .
This is also a check: if the sum is not 1, there is a mistake in the working.

Comparing groups of different sizes

To compare groups of different sizes, compare the relative frequencies, not the frequencies.
There are 25 pupils in Noa’s class, and 12 of them walk to school.
There are 40 pupils in Omer’s class, and 16 of them walk.

More pupils walk in Omer’s class: 16 against 12. But his class is bigger, too.
In Noa’s class: $\displaystyle {\frac{12}{25}}={\frac{48}{100}}={48\%}$ .
In Omer’s class: $\displaystyle {\frac{16}{40}}={\frac{4}{10}}={40\%}$ .
A larger part of Noa’s class walks to school: 48% against 40%.

From a relative frequency to a frequency

The relative frequency times the number of answers gives back the frequency.
There are 40 pupils in Omer’s class.
In a survey, 35% of them have a dog.

The number of dog owners: $\displaystyle {0.35\cdot40}={14}$ pupils.
A check: $\displaystyle {\frac{14}{40}}={\frac{35}{100}}={35\%}$ .

Exercise 1

20 pupils answered the question “How many books did you read in the summer holiday?”.

Books	0	1	2	3	4
Frequency	3	6	5	4	2

Write the relative frequency of each answer as a percent.

a

0 books

/ – 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

1 book

/ – 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

2 books

/ – 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

d

3 books

/ – 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

e

4 books

/ – 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

f

Add up the five relative frequencies you found.

/ – 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

56 pupils went on the annual trip.
14 of them forgot to bring a hat.
What is the relative frequency of the pupils who forgot a hat?

a

As a fraction in lowest terms.

/ – 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

As a decimal.

/ – 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

As a percent.

/ – 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 3

There are 30 pupils in class 8E, and 18 of them have a younger brother or sister.
There are 40 pupils in class 8F, and 21 of them have a younger brother or sister.

a

What is the relative frequency of the pupils with a younger brother or sister in 8E? Write it as a percent.

/ – 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

And in 8F?

/ – 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

In which class does a larger part of the pupils have a younger brother or sister? Explain.

Exercise 4

Find the frequency.

a

There are 360 pupils in the school, and 15% of them come to school by bike.
How many pupils come by bike?

/ – 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

There are 32 pupils in a class.
The relative frequency of the pupils who have ever flown abroad is $\displaystyle 0.375$ .
How many pupils have flown abroad?

/ – 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 5

80 pupils answered the question “What do you do in the long break?”.
One relative frequency is missing from the table:

Answer	football	talking with friends	reading	phone
Relative frequency	$\displaystyle 0.3$	$\displaystyle 0.25$	?	$\displaystyle 0.1$

a

What is the relative frequency of reading?

/ – 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 many pupils read in the long break?

/ – 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 6

Yossi ran a survey and worked out the relative frequencies of its four answers:
$\displaystyle 0.3$ , $\displaystyle 0.4$ , $\displaystyle 0.2$ , $\displaystyle 0.2$ .
How can you tell, without seeing the survey, that Yossi made a mistake?

Formulas: data and probability, grade 8

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
$\displaystyle n\cdot P(A)$