# Lesson Explainer: Double Bar Graphs Mathematics • 6th Grade

In this explainer, we will learn how to use a double bar graph to display two sets of related data using bars of different colors and heights.

A helpful way to display data for a nonnumerical (categorical) variable is in a bar graph. For example, suppose we have collected data on the preferred platform used by a group of core gamers. The variable is “platform,” which has 3 categories: “smart phone,” “console,” and “PC.” And we have counted the number of gamers who chose each category.

The height of a bar, which we can read off the vertical axis, is the number of gamers who chose that option. So, for example, we can see that as the height of the bar above the “PC” option is 80, we know that 80 gamers preferred a PC for gaming.

When we have two categorical variables, we can use a double (or multiple) bar graph to gain information about the two variables. Let us look at an example.

### Example 1: Constructing and Reading Information from a Double Bar Graph

The data in the table below represents the preferred gaming platform for a group of gamers broken down by gender.

1. Illustrate the data in a double bar graph.
2. From your bar graph, determine which is the most popular platform among female gamers.

### Answer

Part 1

To illustrate the data in a double bar graph, as with the single bar graph, we put the “platform” categories on the horizontal axis and the frequency (number of gamers) on the vertical axis. We note that the highest frequency in the table is 50, corresponding to male gamers who prefer smartphones as their gaming platform. We use a slightly larger number than this, say, 60, as our maximum vertical marker. Above each “platform” category, we place two bars: one for male gamers and one for female gamers.

The height of each bar represents the number of gamers who chose that category of platform. The bars for female and male gamers are different colors so that we can easily distinguish between them, and we have a key or legend on the graph telling us which color corresponds to which gender.

Part 2

To determine from the bar graph which is the most popular platform among female gamers, we simply look for the tallest green bar, since the green bars represent the female gamers.

The highest green bar is above the “smart phone” category. Hence, the most popular platform for gaming among female gamers was “smart phone.”

In our next example, we will use a double bar graph to gain information from data.

### Example 2: Reading Information from a Double Bar Graph

The bar graph shows the number of runs that two teams achieved in each game of a baseball playoff. How many runs did the Hawks score in game 3?

### Answer

To find how many runs the Hawks scored in game 3, we note first from the key that the Hawks’data is represented by the blue bars. The blue bar above game 3 on the horizontal axis has a height of 7.

Hence, in game 3, the Hawks scored 7 runs.

The next examples demonstrate how we can use information from double bar graphs in calculations.

### Example 3: Using Information from a Double Bar Graph in Calculations

The double bar graph shown represents the numbers of tickets sold at a movie theater. Suppose tickets for adults cost \$6, and those for children cost \$4. On which day did the movie theater make more than \$207 in ticket sales?

### Answer

To find on which day the movie theater made more than \$207, we will need to work out how much was made on each day. To do this, for each day we calculate how much was made from adult tickets and how much was made from child tickets and then add these amounts together. But, first, we will need to read off the number of tickets sold from the bar graph.

Starting with Thursday, the number of adult tickets sold was 8 (represented by the green bar above Thursday), and the number of child tickets sold (the red bar above Thursday) was 13.

Multiplying each of these by their ticket price (adult = \$6 and child = \$4) gives the total for each. And their sum gives us the total amount made for ticket sales on Thursday of \$100. If we do the same calculation for each of the days, we get the following:

We can see from the totals column in the table that the only day on which the movie theater made more than \$207 dollars was Friday, where a total of \$228 was made in ticket sales.

### Example 4: Comparing Categories Using a Double Bar Graph

The number of students in each grade of a middle school is shown in the graph. Does the total number of students in a grade decrease by a constant amount as the grades increase?

### Answer

To find out whether or not the total number of students in a grade decreases by a constant amount as the grades increase, we will first need to read the relevant figures off from our graph and calculate the total number of students for each grade. We can then determine whether or not the decrease in student numbers was constant or not by comparing grade 6 with grade 7 and grade 7 with grade 8.

In total, there were 334 students in grade 6, 300 in grade 7, and 268 in grade 8, and we can see from both the figures and the bar graph that the numbers decrease as the grade number increases. If the decrease was by a constant amount, we would have the same difference between grades 6 and 7 as that between grades 7 and 8. Let us see if this is the case:

We can see that student numbers decreased by 34 between grades 6 and 7 (the negative sign indicates a decrease as opposed to an increase) and decreased by 32 between grades 7 and 8. These are not the same; hence, the decrease in student numbers is not constant as the grades increase.

### Note

We could equally well have calculated the differences by taking the earlier grade first; for example, The number is the same, 34; however, the sign is now positive, but since we know that grade 7 had the lower number of students, we now need to make sure we state explicitly that there was a decrease of 34 between grades 6 and 7.

### Example 5: Working Out the Mean from a Double Bar Graph

The bar graph shows the highest and lowest temperatures in some of the world’s capital cities in a month. Calculate the mean of the highest temperatures in Cairo and London.

### Answer

To find the mean of the highest temperatures in Cairo and London, we first read off the highest temperatures in Cairo and in London from our double bar graph. The key tells us that high temperatures are represented by the green bars, so we must find the heights of the green bars for Cairo and for London.

The highest temperature in Cairo was 25 degrees and the highest in London was 10 degrees. Recalling that the mean is the sum of the numbers divided by the number of numbers, we can now calculate the mean of the highest temperatures in Cairo and London:

The mean high temperature in Cairo and London is therefore 17.5 degrees.

Finally, let us recap on the key points for displaying data in and interpreting double bar graphs.

### Key Points

• A double bar graph can be a useful way to display related data for two categorical variables.
• The categories of one of the variables are represented on one of the axes (usually the horizontal axis, but not always).
• The different colors or patterns of the bars represent the categories of the second variable.
• A key or legend indicates which categories are represented by which color or pattern.
• The frequency (or count or number of observations) is represented on the other axis (usually the vertical axis but not always).
• The height of each bar represents the frequency for that specific combination of categories of the two variables.
• The two double bar graphs below represent the same data set: breakfast choices for the guests of a particular hotel. There are two variables, “breakfast drink” and “breakfast food.” Breakfast drink has 2 categories: “coffee” and “tea.” Breakfast food has 3 categories: “toast,” “cereal,” and “pancakes.” In each graph, we can see, for example, that the number (or frequency) of hotel guests who had “coffee” and “pancakes” for breakfast is 50.
We could equally well change the graph around so that the three breakfast food options are represented by bars and the drinks are on the axes. The graphs below show the same data as the graphs above, but now the two drinks categories are on the axes and the three food categories are represented by the bars.

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