11.5: Histograms
Introduction
The Finish Line
Jasper is curious about how many days it takes a musher to finish the Iditarod. Looking online, he has discovered that the average is from 10 – 15 days, but that isn’t specific enough for him.
“I want to know more details about it,” he tells Mr. Hawkins first thing on Monday morning.
“Well, you have to narrow down your findings. I would suggest you look at the final standings from 2010. Then you can create a frequency table and a histogram.”
“Alright, that’s a good idea,” Jasper says.
Jasper begins his research on the Iditarod website. He makes notes on the number of days that it took the mushers in the 2010 Iditarod to finish. Here is the frequency table that he created with his findings.
Days | Tally | Frequency |
---|---|---|
8 | 1 | 1 |
9 | 11111 | 18 |
11111 | ||
11111 | ||
111 | ||
10 | 11111 | 16 |
11111 | ||
11111 | ||
1 | ||
11 | 11111 | 6 |
1 | ||
12 | 11111 | 9 |
1111 | ||
13 | 1111 | 4 |
Next, Jasper began making his histogram. But as soon as he started to draw it, something did not look right.
Jasper could use some help. In this lesson, you will learn how to take a frequency table and make a histogram out of it. Pay close attention and at the end of this lesson you will be able to help Jasper create his visual display.
What You Will Learn
By the end of this lesson, you will be able to:
- Make a frequency table to organize and display given data.
- Make a histogram given a frequency table.
- Make a frequency table and a histogram given unorganized data.
- Collect, organize, display and analyze real-world data using frequency tables and histograms.
Teaching Time
I. Make a Frequency Table to Organize and Display given Data
You have been learning all about the different ways to display data. In this lesson, you will learn about frequency tables and histograms. Let’s start by looking at frequency tables.
What is a frequency table?
A frequency table is another way of summarizing data. A frequency table depicts the number of times a data value occurs.
A frequency table is created by making a table with three separate columns. One column is designated for intervals. The range of each interval is determined by the range in data values. 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.
In the last column, add the tally marks to determine the frequency results.
Let’s look at how we can apply this information with an example.
Example
Twenty people were asked to state the number of hours they sleep each night. The results of the survey are listed below. Create a frequency table to display the data.
7, 8, 6, 9, 10, 12, 5, 7, 8, 9, 10, 11, 12, 7, 6, 7, 8, 10, 11, 9
Step 1: Make a table with three separate columns.
- Intervals
- Tallied results
- Frequency results
In this case, there is not a wide range in data values, therefore the intervals will be displayed by ones.
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.
Number of Hours Slept | Tally | Frequency |
---|---|---|
5 | I | 1 |
6 | I I | 2 |
7 | I I I I | 4 |
8 | I I I | 3 |
9 | I I I | 3 |
10 | I I I | 3 |
11 | I I | 2 |
12 | I I | 2 |
Now you can see how arranging the data in this way makes it much easier to follow.
Example
The data below depicts the amount of time (in minutes) 20 middle school students spent on the computer each day. Arrange the data on a frequency table.
10, 32, 8, 55, 5, 0, 30, 20, 25, 45, 40, 60, 45, 15, 5, 56, 47, 12, 15, 20
Step 1: Make a table with three separate columns.
- Intervals
- Tallied results
- Frequency results
In this case, there is a moderate range in data values, therefore the intervals will be displayed by fives.
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.
Number of Minutes on the Computer | Tally | Frequency |
---|---|---|
0 – 5 | I I I | 3 |
6 – 10 | I I | 2 |
11 – 15 | I I I | 3 |
16 – 20 | I I | 2 |
21 – 25 | I | 1 |
26 – 30 | I | 1 |
31 – 35 | I | 1 |
36 – 40 | I | 1 |
41 – 45 | I I | 2 |
46 – 50 | I | 1 |
51 – 55 | I | 1 |
56 – 60 | I I | 2 |
Once again, the tally marks in the frequency table can give you a clear picture of the data.
11G. Lesson Exercises
Look at the frequency table above and answer the following questions.
- How many students spent 51 – 55 minutes on the computer?
- How many students spent 0 – 5 minutes on the computer?
- How many students spent 41 – 45 minutes on the computer?
Take a few minutes to check your work with a partner.
II. Make a Histogram given a Frequency Table
Frequency tables are a great way to record and organize data. Once you have created a frequency table, you can make a histogram to present a visual display of the information in the frequency table.
What is a histogram?
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. A histogram differs from a bar graph in that the vertical columns are drawn with no space in between them.
Now let’s look at creating a histogram from a frequency table.
Example
Create a histogram using the results on the frequency table below.
Number of Hours Slept | Tally | Frequency |
---|---|---|
5 | I | 1 |
6 | I I | 2 |
7 | I I I I | 4 |
8 | I I I | 3 |
9 | I I I | 3 |
10 | I I I | 3 |
11 | I I | 2 |
12 | I I | 2 |
To create a histogram:
1. Draw the horizontal \begin{align*}(x)\end{align*} and vertical \begin{align*}(y)\end{align*} axis.
2. Give the graph the title “Hours Slept Each Night.”
3. Label the horizontal axis “Hours.” List the intervals across the horizontal axis.
4. Label the vertical axis “Frequency.” Since the range in frequencies is not that great, label the axis by halves.
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.
Take a few minutes to copy down the steps for creating a histogram in your notebook.
Example
Create a histogram to display the data on the frequency table below.
Number of Minutes on the Computer | Tally | Frequency |
---|---|---|
0 – 5 | I I I | 3 |
6 – 10 | I I | 2 |
11 – 15 | I I I | 3 |
16 – 20 | I I | 2 |
21 – 25 | I | 1 |
26 – 30 | I | 1 |
31 – 35 | I | 1 |
36 – 40 | I | 1 |
41 – 45 | I I | 2 |
46 – 50 | I | 1 |
51 – 55 | I | 1 |
56 – 60 | I I | 2 |
To create a histogram:
1. Draw the horizontal \begin{align*}(x)\end{align*} and vertical \begin{align*}(y)\end{align*} axis.
2. Give the graph the title “Minutes Spent on the Computer.”
3. Label the horizontal axis “Minutes.” List the intervals across the horizontal axis.
4. Title the vertical axis “Frequency.” Label the axis by halves (0.5).
5. For each interval on the horizontal access, draw a vertical column to the appropriate frequency value. Recall that on a histogram, there are no spaces in between vertical columns.
III. Make a Frequency Table and a Histogram given Unorganized Data
Sometimes, you will be given a set of data that you will need to organize. This data will be unorganized. To work with it, you will have to organize it by creating a frequency table. Then you can use that frequency table to create a histogram.
Let’s look at an example.
Example
Fifteen people were asked to state the number of hours they exercise in a seven day period. The results of the survey are listed below. Make a frequency table and histogram to display the data.
8, 2, 4, 7.5, 10, 11, 5, 6, 8, 12, 11, 9, 6.5, 10.5, 13
First arrange the data on a frequency table. Recall that a table with three columns needs to be drawn: one for intervals, one for tallied results, and another for frequency results. The range in values for this set of data is eleven. Therefore, data will be tallied in intervals of three.
Hours of Exercise | Tally | Frequency |
---|---|---|
0 – 2 | I | 1 |
3 – 5 | I I | 2 |
6 – 8 | I I I I I | 5 |
9 – 11 | I I I I I | 5 |
12 – 14 | I I | 2 |
Next, the data needs to be displayed on a histogram. Recall that a horizontal \begin{align*}(x)\end{align*} and vertical \begin{align*}(y)\end{align*} axis needs to be drawn. List the intervals across the horizontal axis. Name this axis “Hours of Exercise.” Label the vertical axis by ones. Title the vertical axis “Frequency.” For each set of intervals, draw vertical columns the appropriate frequency. Color in the vertical columns and ensure that no space is between them. Title the graph “Hours of Exercise.”
Now let’s make some conclusions based on the information displayed in the histogram.
Looking at the histogram above, you can that equal numbers of people reported that they exercise between six and eight and nine and eleven hours each week. Two people stated that they exercise between three and five hours per week. Two people reported that they exercise between twelve and fourteen hours per week. Zero to two is the interval with the least frequency.
11H. Lesson Exercises
Look at this frequency table and use it to complete the following.
Number of Sodas | Tally | Frequency |
---|---|---|
0 – 3 | I I I I I I I I | 8 |
4 – 7 | I I I I I I I | 7 |
8 – 11 | I I I | 3 |
12 – 15 | I I | 2 |
- Which category is the most popular?
- Which category is the least popular?
- Create a histogram that shows the data.
'Take a few minutes to check your work with a partner. Compare histograms. Do they both accurately display the data from the frequency table?
IV. Collect, Organize, Display and Analyze Real-World Data using Frequency Tables and Histograms
In the past few sections, we have been working with frequency tables and histograms. Much of the work that we have been doing has been with real-world data. Statistics makes the most sense when it involves real-world information.
Here is a different type of example using a ball.
Example
The data on the table below depicts the height (in meters) a ball bounces after being dropped from different heights. Create a frequency table and histogram to display the data.
\begin{align*}6 \quad 9 \quad 4 \quad 12 \quad 11 \quad 5 \quad 7 \quad 9 \quad 13 \quad 5 \quad 6 \quad 10 \quad 14 \quad 7 \quad 8\end{align*}
First arrange the data on a frequency table. Recall that a table with three columns needs to be drawn: one for intervals, one for tallied results, and another for frequency results. The range in values for this set of data is ten. Therefore, data will be tallied in intervals of two.
Bounce Height | Tally | Frequency |
---|---|---|
3 – 4 | I | 1 |
5 – 6 | I I I I | 4 |
7 – 8 | I I I | 3 |
9 – 10 | I I I | 3 |
11 – 12 | I I | 2 |
13 – 14 | I I | 2 |
Next, the data needs to be displayed on a histogram. Recall that a horizontal \begin{align*}(x)\end{align*} and vertical \begin{align*}(y)\end{align*} axis needs to be drawn. List the intervals across the horizontal axis. Name this axis “Bounce Height.” Label the vertical axis by ones. Title the vertical axis “Frequency.” For each set of intervals, draw vertical columns the appropriate frequency. Color in the vertical columns and ensure that no space is between them. Title the graph “Bounce Heights.”
Now what conclusions can we draw from the frequency table and histogram?
You can see that the most frequent bounce heights were between five and six meters. The least frequent bounce heights were between three and four meters. Three balls bounced between seven and eight meters and nine and ten meters. Two balls bounced between eleven and twelve meters and thirteen and fourteen meters.
Real Life Example Completed
The Finish Line
Here is the original problem once again. Reread it and then look at the histogram created from the frequency table.
Jasper is curious about how many days it takes a musher to finish the Iditarod. Looking online, he has discovered that the average is from 10 – 15 days, but that isn’t specific enough for him.
“I want to know more details about it,” he tells Mr. Hawkins first thing on Monday morning.
“Well, you have to narrow down your findings. I would suggest you look at the final standings from 2010. Then you can create a frequency table and a histogram.”
“Alright, that’s a good idea,” Jasper says.
Jasper begins his research on the Iditarod website. He makes notes on the number of days that it took the mushers in the 2010 Iditarod to finish. Here is the frequency table that he created with his findings.
Days | Tally | Frequency |
---|---|---|
8 | 1 | 1 |
9 | 11111 | 18 |
11111 | ||
11111 | ||
111 | ||
10 | 11111 | 16 |
11111 | ||
11111 | ||
1 | ||
11 | 11111 | 6 |
1 | ||
12 | 11111 | 9 |
1111 | ||
13 | 1111 | 4 |
Next, Jasper began making his histogram. But as soon as he started to draw it, something did not look right.
Then Jasper realized that he needed to put the number of mushers on the \begin{align*}y\end{align*} axis and the number of days on the \begin{align*}x\end{align*} axis. He included each day instead of a range of days since there were only six possible options for days to finish.
Here is Jasper’s final histogram.
Vocabulary
Here are the vocabulary words found in this lesson.
- Frequency Table
- a table that keeps track of the number of times a data value occurs
- Histogram
- a type of bar graph that shows frequency and distribution of data. The bars in a histogram are not spaced apart, but they are found right next to each other.
Technology Integration
Time to Practice
Directions: Use each set of data to answer the following questions.
Here is a list of the number of days that 7th grade students at Marrimack Middle School bought lunch.
0 Days = 15 students
1 Day = 13 students
2 Days = 30 students
3 Days = 21 students
4 Days = 35 students
5 Days = 60 students
1. How many students are in the \begin{align*}7^{th}\end{align*} grade?
2. What is the most popular number of days?
3. What is the least popular number of days?
4. Create a frequency table to show the data.
5. Were any students left out of the count?
6. How do you know?
7. Create a frequency table to display the data below.
2, 5, 3, 1, 6, 5, 7, 8, 3, 1
8. Were there any numbers not represented?
9. What is the most popular number or numbers?
10. Create a histogram to display the data from the frequency table below.
Data Values | Tally | Frequency |
---|---|---|
0 – 3 | I I I | 3 |
4 – 7 | I I I I | 4 |
8 – 11 | I | 1 |
12 – 15 | I I | 2 |
11. The data collected depicts the number of letters in the last names of twenty people. Create a frequency table to display the data.
\begin{align*}12 \quad 3 \quad 5 \quad 9 \quad 11 \quad 2 \quad 7 \quad 5 \quad 6 \quad 8 \quad 14 \quad 4 \quad 8 \quad 7 \quad 5 \quad 10 \quad 5 \quad 9 \quad 7 \quad 15\end{align*}
12. Create a histogram to display the data.
13. The data collected depicts the number of hours twelve families traveled this summer to their vacation destination. Create a frequency table to display the data.
\begin{align*}7 \quad 3 \quad 10 \quad 5 \quad 12 \quad 9 \quad 8 \quad 4 \quad 3 \quad 11 \quad 3 \quad 9\end{align*}
14. Create a histogram to display the data.
15. Write a few sentences to explain any conclusions that you can draw from the data.
16. Generate a question that you will use to survey twenty people.
17. Make a table to collect the answers.
18. Display the data on a frequency table.
19. Create a histogram to display the data.
20. Write a brief summary to describe the data.