13.02 Displaying data

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 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:

1. Gathering the data
2. Organising the data
3. Interpreting the data

We 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 those data.

Melanie has answered her initial question, but she realises she can use the same data to answer other questions about the colours of cars in her neighbourhood.

a) What percentage of the cars were black?

When comparing quantities of a large population, it is easier to compare them by using percentages, rather than their exact quantities. 

We can read from the table that $3$3 cars were black. Since Melanie recorded the colour of $20$20 cars, this means that $3$3 out of $20$20 of the cars were black. We can express this as the fraction $\frac{3}{20}$320​. To convert this to a percentage we can multiply the numerator and denominator by $5$5:

b) What was the most common colour of car?

Looking at the table, we can see that the result with the highest frequency is the colour white, so this was the most common colour. This means that the mode of the data is "white".

Frequency tables

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

As seen in the exploration, 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.

Practice question

Question 1

In the practice question above, Thomas found that there were no classmates who exercised for $4$4 hours. Instead of leaving the frequency blank, Thomas put $0$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.

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.

As we can see, grouped frequency tables are useful when the data are more spread out. While the teacher could have obtained the same information from a normal frequency table, the grouping of the results condensed the data into an easier to interpret form.

However, the drawback of a grouped frequency table is that the data becomes less precise, since we have grouped multiple data points together rather than looking at them individually.

Practice questions

Question 2

Question 3

Representing data can be tricky because we want to make it easy to interpret without losing any information. Most of the time we are forced to compromise, either making the data simpler to express in a simpler manner or instead having more complex ways to express our data.

The line plot

The line plot is a useful way to express discrete data in a visually simple manner. The main advantages of the line plot are that we can find the most common score and range very easily, as well as quickly see how the data is distributed. 

The line plot is particularly suited to discrete data where the frequency of results are often greater than one.

Let's have a look at an example of a line plot.

Have a go at this in the practice question below.

The stem-and-leaf plot

The stem-and-leaf plot is an example of a way to express data in a more complicated way so that we can express more information visually. In particular, the stem-and-leaf plot is used when we have lots of numerical data points.

A stem-and-leaf plot is made up of two components, the stem and the leaf. The stem is usually used to represent the tens part of a score while the leaf is used to represent the ones part of the score.

For example, the score $52$52 would be expressed on a stem-and-leaf plot like so:

As we can see, we expressed the score by writing the ones digit in the row corresponding to its tens digit. In other words, we attached the leaf, $2$2, to its stem, $5$5, to make the score $52$52.

What is useful about the stem-and-leaf plot is that we can record as many scores as we like by writing the leaves in the appropriate rows. As such, we could express the data set:

$52,46,31,57,49,51,52,30$52,46,31,57,49,51,52,30

with this stem-and-leaf plot:

As we can see, each score has been expressed on the plot as a ones digit written in its tens row.

Notice that the leaves have been arranged in ascending order from left to right. We need to do this so that we can find the median without jumping back and forth across our rows.

It is also worth noting that if there is more than one of the same score, in this case $52$52 appears twice, each score should have its own leaf.

Using the key for stem-and-leaf plots

While stem-and-leaf plots are used primarily to store data of two digit numbers, there are some cases where the stem and leaf might mean something different. It is for this reason that we should always check the key before translating the leaves into scores.

For example, in the stem-and-leaf plot below the stem represents the whole number of kilometres while the leaf represents tenths of a kilometre.

There are also cases of the stem representing the number of tens as usual, except it uses two digit numbers in the stem to express three digit numbers. In this case, the score $128$128 is represented by the leaf "$8$8" attached to the "$12$12" stem.

In both cases, we need the key to tell us how to interpret the stem-and-leaf plot since the data is different from our usual two digit scores.