Relative frequency: a fraction, a decimal and a percentage

Data and probability, grade 7 Lesson 3 of 9Learning

Relative frequency: a fraction, a decimal and a percentage

Relative frequency

The relative frequency of a value is its part of all the data: its frequency, divided by the number of all the data.

\displaystyle \frac{f}{n}

$\displaystyle f$ – the value’s frequency.
$\displaystyle n$ – the number of all the data.

40 pupils answered the question “How many brothers and sisters do you have?”:

Brothers and sisters	0	1	2	3	4 or more
Frequency	6	14	11	7	2

11 of the 40 pupils have 2 brothers and sisters: the relative frequency of 2 is $\displaystyle \frac{11}{40}$ .

A fraction, a decimal and a percentage

The same relative frequency is written in three ways:

\frac{11}{40}=0.275=27.5\%

From a fraction to a decimal: divide the numerator by the denominator, $\displaystyle {11:40}={0.275}$ .
From a decimal to a percentage: multiply by 100, $\displaystyle {0.275\times100}={27.5}$ , that is 27.5%.
From a percentage to a fraction: $\displaystyle {27.5\%}={\frac{27.5}{100}}={\frac{275}{1000}}={\frac{11}{40}}$ .

The sum of the relative frequencies

This is the whole table, with the relative frequencies in percent:

Brothers and sisters	0	1	2	3	4 or more
Frequency	6	14	11	7	2
Relative frequency	15%	35%	27.5%	17.5%	5%

The relative frequencies of all the values add up to 1, that is 100%:

\displaystyle \frac{f_1}{n}+\frac{f_2}{n}+\dots+\frac{f_k}{n}=1

$\displaystyle f_1, f_2, \dots, f_k$ – the frequencies of all the values. They add up to $\displaystyle n$ , so the fractions add up to $\displaystyle {\frac{n}{n}}={1}$ .
Here: $\displaystyle {15\%+35\%+27.5\%+17.5\%+5\%}={100\%}$ . Any other sum means a mistake in the working.

A rounded relative frequency

Sometimes the division does not end. A class has 30 pupils, and 7 of them are in the drama club:
$\displaystyle {\frac{7}{30}}={0.2333\ldots}$ , and you round, say to a tenth of a percent: $\displaystyle \frac{7}{30}\approx23.3\%$ .

The sign $\displaystyle \approx$ means “is about”.
When several relative frequencies are rounded, their sum can come out a little under 100% or a little over, say 99.9%. That is no mistake: it is the rounding.

Several values together

What part of the pupils in the table have at least 3 brothers and sisters? Add the frequencies of 3 and of “4 or more”: $\displaystyle {7+2}={9}$ , and the relative frequency is $\displaystyle {\frac{9}{40}}={22.5\%}$ .

You can also add their relative frequencies: $\displaystyle {17.5\%+5\%}={22.5\%}$ .

The pie chart

A pie chart shows the relative frequencies as parts of a circle. The whole circle is all the data, 100%, and each value gets a part as large as its relative frequency.
The 160 pupils of grade 7 said how they get to school:

Half the circle is 50%, and a quarter is 25%. You see at once that fewer than half of the pupils walk, and that more than a quarter come by bus.
Walk: $\displaystyle {\frac{64}{160}}={40\%}$ . From the percentage back to the frequency by multiplying: $\displaystyle {40\%\times160}={64}$ pupils.
A pie chart shows each value’s part of the whole; a bar chart makes it easier to compare two values that are close.

Exercise 1

80 pupils answered a survey, and 14 of them said they play a musical instrument.
What is the relative frequency of the pupils who play?

Write it as a fraction in its 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