Introduction
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} |
Which colour is the mode?
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?
What is the mode of the data?
How many classmates exercised for less than three hours?
How many classmates exercised for at least three hours?
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 |
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."
"We know that 17 people played 7 or fewer video games."
"21 people played more than 4 but less than 15 video games."
"The modal class was 5-9 video games."
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.