1.1: Analyzing Graphs
Introduction
A New Year of Big Changes
The students at Smith Middle School returned to classes and found out that there was a big surprise. There wasn’t any more wood shop. It had been cancelled from the curriculum.
“How could they have cancelled wood shop?” Kyle, a seventh grader asked looking at his schedule. “I have been waiting all year for this.”
In the past, the students at Smith Middle School had to be in seventh grade to take wood shop. Many students longed to be a part of wood shop for all of sixth grade.
“I know, I was waiting for it too,” Sarah said.
“There must be a mistake,” Tanisha commented.
However, there wasn’t a mistake. The administration was sure that there wasn’t enough student interest to continue having wood shop.
This set the students in the seventh and eighth grade on a mission. That very first day afterschool the students held their own meeting on the football field. They decided to gather their data and prove to the administration that woodshop was a necessary part of the curriculum.
Over the next few weeks, the students worked hard to gather data about wood shop. They learned that in 2008, there were 30 out of 100 seventh graders and 40 out of 100 eighth graders who had participated in wood shop. Then in 2009, the numbers had increased. There were 40 seventh graders and 58 eighth graders who had participated.
“This is great!” Kyle said. “Now we can prove that wood shop is wanted!”
“Yes, but I think we should draw a chart to show our results,” Tanisha said.
This is where you come in. In this lesson, you will learn how to draw different types of graphs to display data. Pay close attention because at the end of the lesson you will need to draw a bar graph to show the data that was collected.
What You Will Learn
By the end of this lesson you will be able to complete the following skills.
- Interpret given bar graphs.
- Make a frequency table given unorganized data.
- Make a histogram of data in a frequency table.
- Collect, organize, display and analyze real-world data using bar graphs, frequency tables and histograms.
Teaching Time
I. Interpret Given Bar Graphs
In the real world, we work with data all the time. What are we talking about when we use that word, “data”? We are referring to statistics or information that has been gathered about real events. Data is any information that is collected. When we conduct at survey or measure distances over time or record how trends change, these are examples of data. The big idea is that data is collected information and in this lesson you are going to learn how to organize, analyze and work with data.
Let’s start by thinking about bar graphs. You have probably been working with bar graphs since elementary school, but sometimes we forget what a bar graph actually is and what its purpose is in working with data. Let’s begin there.
What is a bar graph?
A bar graph is a graph that uses columns to show comparison of quantities or amounts. The key word here is comparison. If you can remember to think about comparing when working with a bar graph, it will help you to keep things straight.
Take a few minutes to write the definition of a bar graph down in your notebook. Be sure to underline the word “comparison” to help you in remembering a bar graph’s purpose.
Let’s look at an example and see how we can use a bar graph to answer questions.
Example
The bar graph below depicts the number of hours that high school students are permitted to spend working at a part-time job each week. Use the information from the bar graph to answer the questions below.
Take a good look at this graph. Now we are going to use the graph to answer the following questions.
How many more hours per week are seniors aloud to spend working than sophomores? Seniors are allowed to spend twenty hours per week working. Sophomores are allowed to work twelve hours per week. Therefore, seniors are allowed to work eight more hours each week than sophomores.
Are juniors allowed to work twice as many hours as freshman? Juniors are permitted to work fifteen hours per week. Freshmen are permitted to work eight hours per week. The amount of time freshmen are allowed to work doubled is sixteen. Therefore, juniors are permitted to work almost double the amount freshman are allowed to work, but not quite.
You can see how the visual display of the information is helpful in answering questions about the data. Let’s look at another example.
Example
The bar graph below compares the air travel costs to fly to numerous U.S. cities from Oakland, California on three airlines. Use the information on the bar graph to answer the questions below.
For which city did all three airlines have a similar price? All three airlines had a similar price for travel to Atlanta.
Which airline had the lowest price for three of the five cities? Coast to Coast Airlines had the lowest price ticket to Las Vegas, Houston, and Orlando.
For which city did Western Airlines and Coast to Coast Airlines have a difference in price of forty dollars? The price of ticket to Orlando on Western Airlines was forty dollars more than a ticket on Coast to Coast Airlines.
By now, you can begin to see the benefits of using a bar graph. Not only is it a visual display, but the information is organized in such a way that the comparisons between the data is very clear. When you want to show comparisons in a visual way, a bar graph is your best option.
However, there are other ways to examine data. Let’s look at using frequency tables.
II. Make a Frequency Table Given Unorganized Data
To understand what a frequency table is, let’s first look at the words themselves. The word frequency refers to how often something occurs. A table is a way of organizing information using columns. Therefore, a frequency table is way of summarizing data by depicting the number of times a data value occurs. To show this, a frequency table organizes the information into a table with three separate columns.
How do we create a frequency table?
First, you need to make a table with three separate columns.
One column is designated for intervals. The amount of intervals is determined by the range in data values. Intervals are equal in size and do not overlap. If the range in data values is not that great, the intervals will be small. If the range in data values is great, the intervals will be larger. It is important that the intervals are of equal size and do not overlap.
Another column is created for tallied results. This is where you tally the number of times you see a data value from each interval. This is where you will see tally marks or lines that record the number of times a data value occurs.
In the last column, add the tally marks to determine the frequency results.
Let’s look at creating a frequency table for the following example.
Example
Create a frequency table to display the data below.
\begin{align*}43, 42, 45, 42, 39, 38, 50, 52, 36, 49, 38, 50, 40, 37, 35\end{align*}
Step 1: Make a table with three separate columns.
- Intervals
- Tallied results
- Frequency results
Since the range in data values is not that great, intervals will be in groups of five.
Step 2: Looking at the data, tally the number of times a data value occurs.
Step 3: Add the tally marks to record the frequency.
Example
The data values below depict student scores (out of 100%) on a recent math exam. Organize the data on a frequency table.
\begin{align*}92, 88, 75, 82, 95, 99, 84, 89, 90, 79, 68, 71, 88, 93, 87, 92, 77, 68, 71, 85\end{align*}
Step 1: Make a table with three separate columns.
- Intervals
- Tallied results
- Frequency results
Since the range in data is big (thirty-one), intervals will be in groups of ten.
Step 2: Looking at the data, tally the number of times a data value occurs.
Step 3: Add the tally marks to record the frequency.
Take a few minutes to write down the steps for creating a frequency table.
Thinking about the frequency that an event occurs can help you to understand and predict certain trends. Think about how useful the trend of grades could be if you were a teacher thinking about a student’s progress.
We can also create a histogram to display data. Histograms and bar graphs are often confused, but they are different. Let’s look at how.
III. Make a Histogram of Data in a Frequency Table
A histogram shows the frequency of data values on a graph. Like a frequency table, data is grouped in intervals of equal size that do not overlap. Like a bar graph, the height of each bar depicts the frequency of the data values. However, on a histogram the vertical columns have no space in between each other.
Example
Create a histogram to display the information on the frequency table.
Here are the steps for creating a histogram from data organized in a frequency table.
Step 1: Draw the horizontal \begin{align*}(x)\end{align*} and vertical \begin{align*}(y)\end{align*} axis.
Step 2: Give the graph the title “Frequency Table Data.”
Step 3: Label the horizontal axis “Hours.” List the intervals across the horizontal axis.
Step 4: Label the vertical axis “Frequency.” Since the range in frequencies is not that great, label the axis by ones.
Step 5: For each interval on the horizontal access, draw a vertical column to the appropriate frequency value. On a histogram, there is no space in between vertical columns.
Looking at the histogram, you can see that data values between thirty-six and forty were most frequent. Data values between forty-one and forty-five and forty-six and fifty occurred an equal number of times.
Example
The frequency table depicts student scores on a recent math exam. Use the information from the frequency table to create a histogram.
Use the steps listed above to create this histogram in your notebook. Then check your work.
Looking at the histogram, it is evident that the majority of scores fell between eighty-six and ninety-five percent. One fourth of the students earned between sixty-six percent and seventy-five percent. One fourth of the students earned between a seventy-six and eighty-five percent.
IV. Collect, Organize, Display and Analyze Real-World Data Using Bar Graphs, Frequency Tables and Histograms
We have already been working with some real-world data with bar graphs, frequency tables and histograms. Let’s look at how to work with these concepts and a few more examples.
Example
The data table below depicts the ages of twenty of our nation’s presidents at the time of Inauguration. Create a bar graph, frequency table, and histogram to display the data.
Recall that to create a bar graph, first draw a horizontal \begin{align*}(x)\end{align*} and vertical \begin{align*}(y)\end{align*} axis. Label the horizontal axis with the president’s last names. Label the vertical axis with intervals of two, beginning with the number thirty. Draw a vertical column to the appropriate value for each president.
Now, create a frequency table by drawing three columns. Designate the first column for intervals of four. The middle column is to tally the ages. The final column depicts the total frequency for each interval.
Finally, create the histogram. To create a histogram, first draw a horizontal \begin{align*}(x)\end{align*} and vertical \begin{align*}(y)\end{align*} axis. Label the horizontal axis with the intervals depicted on the frequency table. Label the vertical axis by ones. Draw a vertical column to the appropriate value for each interval on the horizontal axis. Recall that there is no space between the vertical columns on a bar graph.
Examining data using bar graphs, histograms and frequency tables can help us to understand information in a visual way. For people who learn in a visual way, using data displays is a great way to tackle a task!
Real-Life Example Completed
A New Year of Big Changes
Here is the original problem once again. Now it is time for you to apply what you have learned. Reread this problem and make notes on the information that the students collected for 2008 and 2009. Then create a bar graph to show your work.
The students at Smith Middle School returned to classes and found out that there was a big surprise. There wasn’t any more wood shop.
“How could they have cancelled wood shop?” Kyle, a seventh grader asked looking at his schedule. “I have been waiting all year for this.”
In the past, the students at Smith Middle School had to be in seventh grade to take wood shop. Many students longed to be a part of wood shop for all of sixth grade.
“I know, I was waiting for it too,” Sarah said.
“There must be a mistake,” Tanisha commented.
However, there wasn’t a mistake. The administration was sure that there wasn’t enough student interest to continue having wood shop.
This set the students in the seventh and eighth grade on a mission. That very first day afterschool the students held their own meeting on the football field. They decided to gather their data and prove to the administration that woodshop was a necessary part of the curriculum.
Over the next few weeks, the students worked hard to gather data about wood shop. They learned that in 2008, there were 30 out of 100 seventh graders and 40 out of 100 eighth graders who had participated in wood shop. Then in 2009, the numbers had increased. There were 40 seventh graders and 58 eighth graders who had participated.
“This is great!” Kyle said. “Now we can prove that wood shop is wanted!”
“Yes, but I think we should draw a chart to show our results,” Tanisha said.
Now work on drawing a bar graph to represent the data. Remember that the bar graph that you draw needs to show the data for 2008 and 2009.
Solution to Real – Life Example
To create a bar graph, remember you will need a vertical axis and a horizontal axis. Now because there are 100 students in both the seventh and the eighth grade, you will be able to figure out the intervals for the vertical axis. The horizontal axis should show the numbers of seventh and eighth graders for both 2008 and 2009.
Here is a bar graph to show the data.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Data
- information that has been collected regarding an occurrence or an event.
- Bar Graph
- is a graph that uses columns to compare quantities or amounts.
- Frequency Table
- Summarizing data by depicting the number of times that a data value occurs.
- Histogram
- Showing the frequency of data values on a graph.
Time to Practice
Directions: Use the data table below to complete the following tasks.
State | Year |
---|---|
West Virginia | 1863 |
Nevada | 1867 |
Colorado | 1876 |
North Dakota | 1889 |
South Dakota | 1889 |
Montana | 1889 |
Washington | 1889 |
Idaho | 1890 |
Wyoming | 1890 |
Utah | 1896 |
Oklahoma | 1907 |
New Mexico | 1912 |
Arizona | 1912 |
Alaska | 1959 |
Hawaii | 1959 |
- Create a bar graph of the data.
- Create a frequency table of the data.
- Create a histogram of the data.
- Looking at the bar graph, identify the last two states to join the union.
- In what year did they join the union?
- Identify the states that joined in 1889.
- During which years did twenty percent of the states join the union?
- Show how you converted the fraction to a percent to define your answer for question seven.
- How many states joined the union after 1904?
- What percent is that?
Directions: Use the table to complete each problem listed below.
Type of Dog | Weight (in pounds) |
---|---|
Pug | 18 |
Jack Russell Terrier | 20 |
Dachshund | 20 |
Corgi | 38 |
Border Collie | 40 |
English Springer Spaniel | 45 |
Golden Retriever | 75 |
Labrador | 75 |
Akita | 130 |
Bull Mastiff | 145 |
- Create a bar graph of the data.
- Create a frequency table of the data.
- Create a histogram of the data.
- Identify the types of dogs that weigh less than 50 pounds.
- What fraction of the dogs weigh less than 50 pounds?
- What percent of the dogs weigh less than 50 pounds?
- Identify the types of dogs that weigh more than 50 pounds.
- What fraction of the dogs weigh more than 50 pounds?
- What percent of the dogs weigh more than 50 pounds?
- What is the heaviest of the dogs? What is its weight?
Directions: The data table below depicts the magnitude of the last ten earthquakes that occurred in Lima, Peru. Use the data to answer the following questions.
Date | Magnitude of Quake |
---|---|
July \begin{align*}12^{th}\end{align*} | 5.1 |
July \begin{align*}18^{th}\end{align*} | 3.2 |
August \begin{align*}4^{th}\end{align*} | 4.1 |
August \begin{align*}10^{th}\end{align*} | 3.9 |
August \begin{align*}19^{th}\end{align*} | 2.5 |
September \begin{align*}2^{nd}\end{align*} | 3.9 |
September \begin{align*}8^{th}\end{align*} | 4.1 |
September \begin{align*}16^{th}\end{align*} | 4.8 |
October \begin{align*}1^{st}\end{align*} | 5.0 |
October \begin{align*}10^{th}\end{align*} | 3.5 |
- Create a bar graph of the data.
- Create a frequency table of the data.
- Create a histogram of the data.
- On what dates did the earthquakes with the largest and smallest magnitude occur?
- What percent of the earthquakes were between 4.1 and 5.1 in magnitude?
- What fraction did you use in question 25 to find the accurate percentage?
- On which date was there an earthquake measuring 5.0?
- On which dates were there earthquakes measuring 3.9?