One common way of collecting data is through a survey. Conducting a survey involves choosing a question to ask and then recording the answer. This is great for collecting information, but at the end we are left with a long list of answers that can be difficult to interpret.

This is where tables come in. We can use various tables to organise our data so that we can interpret it at a glance.

Tables, frequencies, and modes

When conducting a survey, the three main steps are:

Gathering the data
Organising the data
Interpreting the data

We have looked at what questions we should ask when gathering different types of data . Now we are going to look at how tables can be used to help us organise and interpret data.

The frequency of a result is the number of times that it appears in the list of data.

The mode of a data set is the result with the highest frequency. If there are multiple results that share the highest frequency then there will be more than one mode.

Examples

Example 1

Yvonne asks 15 of her friends what their favourite colour is. She writes down their answer. Here is what she wrote down:

Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow,

Green, Blue, Yellow, Pink, Yellow, Pink, Pink

Count the number of each colour and complete the table.

\text{Colour}	\text{Number of} \\\ \text{Friends}
\text{Pink}
\text{Green}
\text{Blue}
\text{Yellow}

Worked Solution

Create a strategy

Count the number of times Yvonne wrote each colour.

Apply the idea

\text{Colour}	\text{Number of} \\\ \text{Friends}
\text{Pink}	6
\text{Green}	2
\text{Blue}	3
\text{Yellow}	4

Which colour is the mode?

Worked Solution

Create a strategy

Choose the colour that has the largest number.

Apply the idea

The mode of the data is pink because it occurred the most often.

Idea summary

The frequency of a result is the number of times that it appears in the list of data.

The mode of a data set is the result with the highest frequency. If there are multiple results that share the highest frequency then there will be more than one mode.

Frequency tables

When representing the frequency of different results in our data, we often choose to use a frequency table.

A frequency table communicates the frequency of each result from a set of data. This is often represented as a column table with the far-left column describing the result and any columns to the right recording frequencies of different result types.

Frequency tables can help us find the least or most common results among categorical data. They can also allow us to calculate what fraction of the data a certain result represents.

When working with numerical data, frequency tables can also help us to answer other questions that we might have about how the data are distributed.

To calculate the total number of data points we can add up all the frequencies. Then to calculate the total for "less than" some number, we add up the frequencies for the results that are less than that number. Similarly, when calculating the total for "at least" some number, we add up the frequencies that are greater than or equal to that number.

Examples

Example 2

Thomas conducted a survey on the average number of hours his classmates exercised per day and displayed his data in the table below.

\text{No. exercise} \\ \text{hours}	\text{ Frequency}
0	2
1	12
2	7
3	5
4	0
5	3

How many classmates did Thomas survey?

Worked Solution

Create a strategy

To find the number of classmates surveyed, add all the frequencies of the results.

Apply the idea

\displaystyle \text{Classmates surveyed}	\displaystyle =	\displaystyle 2+12+7+5+0+3	Add all the frequencies
	\displaystyle =	\displaystyle 29	Evaluate

Reflect and check

Thomas found that there were no classmates who exercised for 4 hours. Instead of leaving the frequency blank, Thomas put 0 as the frequency. If he had left this information out of the table then we would not know how many classmates fit this category.

What is the mode of the data?

Worked Solution

Create a strategy

Choose the result with the highest frequency in the table.

Apply the idea

1 hour of exercise is the mode because it has the highest frequency.

How many classmates exercised for less than three hours?

Worked Solution

Create a strategy

Add the frequencies of the results that represent less than three hours of exercise.

Apply the idea

\displaystyle \text{No. classmates}	\displaystyle =	\displaystyle 2+12+7	Add the frequencies
	\displaystyle =	\displaystyle 21	Evaluate

How many classmates exercised for at least three hours?

Worked Solution

Create a strategy

Add the frequencies of the results representing three or more hours of exercise.

Apply the idea

\displaystyle \text{No. classmates}	\displaystyle =	\displaystyle 5+0+3	Add the frequencies
	\displaystyle =	\displaystyle 8	Evaluate

Idea summary

A frequency table communicates the frequency of each result from a set of data. Typically the far left column describes the result or data value and any columns to the right represent frequencies or how many times a result occurred.

Grouped frequency tables

When the data are more spread out, sometimes it doesn't make sense to record the frequency for each separate result and instead we group results together to get a grouped frequency table.

A grouped frequency table combines multiple results into a single group. We can find the frequency of a group by adding all the frequencies of the results contained in that group.

The modal class in a grouped frequency table is the group that has the highest frequency. If there are multiple groups that share the highest frequency then there will be more than one modal class.

Consider the following heights in centimetres: 189,\,154,\,146,\,162,\,165,\,156,\,192,\,175,\,167,\,174, \\ 161,\,153,\,184,\,177,\,155,\,192,\,169,\,166,\,148,\,170, \\ 168,\,151,\,186,\,152,\,195,\,169,\,143,\,164,\,170,\,177

\text{Height (cm)}	\text{ Frequency}
140-149	3
150-159	6
160-169	9
170-179	6
180-189	3
190-199	3

We could display this data in a grouped frequency table with class sizes of 10 cm as shown.

The modal class would be 160 - 169 because it has a frequency of 9 which is the highest frequency in the table.

Examples

Example 3

A survey of 30 people asked them how many video games they had played in the past month. The results are shown in the table below:

\text{Number of video } \\\ \text{games played}	\text{Frequency}
0-4	5
5-9	12
10-14	9
15-19	4

Determine whether each of the following statements are true or false:

"We know that 25 people played 10 or more video games."

Worked Solution

Create a strategy

Choose classes that match the description and see if the sum of frequency also matches the description.

Apply the idea

The two classes where people played 10 or more video games are 10 -14 and 15-9.

\displaystyle \text{Total frequency}	\displaystyle =	\displaystyle 9+4	Add the frequency of the two classes
	\displaystyle =	\displaystyle 13	Evaluate

The statement is false, because the total frequency of the two classes is 13 which is less than 25.

"We know that 17 people played 7 or fewer video games."

Worked Solution

Create a strategy

Choose the class that a person belongs to if the person has played 6 video games and tell the difference of the class.

Apply the idea

The statement is false, because 7 lies in the middle of a class, we can't tell how many people played 7 or fewer video games.

"21 people played more than 4 but less than 15 video games."

Worked Solution

Create a strategy

Choose classes that match the description and see if the sum of frequency also matches the description.

Apply the idea

The two classes where people played more than 4 but less than 15 video games are 5-9 and 10-4.

\displaystyle \text{Total frequency}	\displaystyle =	\displaystyle 12+9	Add the frequency of the two classes
	\displaystyle =	\displaystyle 21	Evaluate

The statement is true, because the total frequency of the two classes is 21 which is the same as the given in the statement.

"The modal class was 5-9 video games."

Worked Solution

Create a strategy

The class with the highest frequency is the modal class.

Apply the idea

The statement is true, because in the table, 12 is the highest frequency which is in the class of 5-9.

Idea summary

A grouped frequency table combines multiple results into a single group. We can find the frequency of a group by adding all the frequencies of the results contained in that group.

Outcomes

MA4-19SP

collects, represents and interprets single sets of data, using appropriate statistical displays

18.02 Displaying data

Introduction

Ideas

Tables, frequencies, and modes

Examples

Example 1

Frequency tables

Examples

Example 2

Grouped frequency tables

Examples

Example 3

Outcomes

MA4-19SP

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