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Lesson Explainer: Statistical Diagrams Mathematics

In this explainer, we will learn how to display and analyze bar graphs to answer questions.

Suppose we did a survey to find the favorite fruit of each student in a class. We might summarize the data in a table like the one below.

BananaAppleOrangeStrawberryKiwiGrapes
6541031

From the table, we can see that four students chose oranges as their favorite fruit. Sometimes, using a table to represent our data is really useful. However, often it does not really give us a clear picture of what is going on. A better way is to represent it visually; this can be done using a bar graph, also sometimes called a bar chart. Let us draw a bar graph using the fruit preferences data.

The first thing we should do is draw our axes. We can label the horizontal axis with the different kinds of fruit. And on the vertical axis we put the number of students. To choose what number to start our vertical axis from and what number to finish with, we look for the highest and lowest numbers in our table. In this case, the highest is 10 and the lowest is 1. So, if we start our vertical axis at the minimum possible value, which is zero, and choose, say, 12 for the maximum, we will cover the whole range of our data.

Note that, in this case, our categories are types of fruit and we can put these on the horizontal axis in any order we like.

Now let us begin to fill in the bar graph. To help make sure our bars will be the correct heights, we draw in some horizontal grid lines, or ruled lines across from the vertical axis. If we begin at the left, with the Banana category, we know that 6 students chose bananas as their favorite fruit. So counting up to 6 on the vertical axis (marked in pink), this is the height of our first bar. The bar can be drawn above the Banana label on the horizontal up to a height of 6.

Continuing in this way for the rest of the fruit categories, we can complete our bar chart.

Notice that for the Apple category, the height of the bar is midway between 4 and 6, since 5 students chose apples as their preferred fruit, and similarly for Kiwi (3, between 2 and 4) and Grapes (1, between 0 and 2). We can see quite clearly from the bar graph that the most popular fruit (the one with the highest bar) was strawberries and the least popular (the one with the shortest bar) was grapes.

The bars in a bar graph do not all have to be the same color. Sometimes we use a different color to highlight important features as long as they are the correct height and the same width for each category. So, for example, we could have colored our Fruit bar chart as follows.

Let us now look at some examples of how we can use bar graphs to answer questions.

Example 1: Journey Time

The graph shows a student’s journey time to school on each day of one week. On which day did it take 20 minutes?

Answer

To find out on which day the student’s journey took 20 minutes, we look up 20 on the vertical axis and map across to see which bar or bars have a height of exactly 20.

We can see that the bar above Tuesday is the only one with a height of exactly 20 on our scale. So, the student’s journey took 20 minutes on Tuesday.

Note also that because the bar above Tuesday is the tallest, with a height of 20 (minutes), we can also say that the student’s longest journey time was on Tuesday.

In this next example, we will use a bar graph to read off the frequency for each category.

Example 2: Using a Bar Graph to Complete a Frequency Table

The graph shows the number of students who joined various activity clubs in a school. Complete the given table.

ActivityMusicSportsArtTheatreScouting
Number

Answer

To fill in the table, we will need to find the heights of the bars, since for each particular activity, the height of the bar represents the number of students who joined that activity club. So, for example, if we look at the bar above the Music category and map across to the vertical axis, we can see that 25 students joined the Music Club (see diagram (a) below). If we do the same for the Sports category (diagram (b)), we see that 30 students joined the Sports Club.

In this way, we can find the number of students who joined each club and complete the table.

ActivityMusicSportsArtTheatreScouting
Number2530452540

Note

We can work out the total number of students who joined activity clubs in the school by adding up all the numbers for the activity clubs in the table; that is, 25+30+45+25+40=165 students in total.

Let us look at another example of how to read information from a bar graph.

Example 3: Reading Bar Graphs

The graph shows the time it takes students to travel to school. How many students take more than 15 minutes?

Answer

On the horizontal axis, there are four different journey times with bars above them: 5 minutes, 10 minutes, 15 minutes, and 20 minutes. Only one of these corresponds to a journey of more than 15 minutes: the 20 minutes category. The students in this category are the only students who took longer than 15 minutes to travel to school. And we can work out how many students are in this category by reading off the height of the bar for the 20 minutes category, from our bar graph.

The height of the 20 minutes bar is 40. This means that 40 students take more than 15 minutes to travel to school.

The next example of reading information from a bar graph has more information within the graph.

Example 4: Reading Bar Graphs

The bar graph shows the favorite superheroes for a group of children. How many more children prefer Wonder Woman than the superhero that is the second most popular?

Answer

We know that Wonder Woman is the most popular superhero since she has the tallest bar (with a height of 33, shown above the bar). If we work out which is the second most popular superhero, then we can find out how many more children prefer Wonder Woman to that superhero.

The second most popular superhero is the one with the next tallest bar. If we look at both the heights and the numbers above the bars, we can see that the second most popular superhero is Wolverine, with a height of 29.

Now we know that Wonder Woman is the favorite superhero of 33 children and Wolverine is the favorite superhero of 29 children. So, 3329=4 more children who prefer Wonder Woman than the second most popular superhero, Wolverine.

Our final example also involves using information we read from a bar graph to calculate something.

Example 5: Population Increase from a Bar Graph

The graph shows the approximated population of a village in Egypt. Giving your answer in thousands, by how much has the population increased from 1950 to 2000?

Answer

To find out how much the population increased between 1950 and 2000, we need first to work out what the population was for both of those years.

Reading from the graph, by the height of the 1950 bar, we can see that the 1950 population is halfway between 0 and 10 thousand. So, the population in 1950 was 5‎ ‎000. Similarly, for the year 2000, we can see from the graph that the population is halfway between 80 and 90 thousand. So, the population in the year 2000 was 85‎ ‎000.

To calculate the population increase, we subtract the population in 1950 from the population in 2000: populationincrease=850005000=80000. So, the village population increased by 80 thousand between the years 1950 and 2000.

We have seen that bar graphs can be very useful in giving us a picture of data that has been counted for different categories (this is often called frequency data). Let us summarize some of the important points.

Key Points

  • Normally, the frequencies or counts will be displayed on the vertical axis and cover the whole range of possible values of the numbers counted for each category.
  • The categories are normally displayed on the horizontal axis, and unless they are numbers, for example, years, it usually does not matter what order we put them in.
  • A bar is placed above each category, and the height of each bar represents frequency (or count) for that category.
  • The bars should have equal widths.

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