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# Histograms

## Visual representation of data on a histogram and the many graphs associated with this display

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Histograms

### Let's Think About It

License: CC BY-NC 3.0

Mrs. Alameda has given all her computer science classes an exam. She has created a frequency table for the scores, but needs a better way to display the data. How can she do this?

In this concept, you will learn how to create a histogram from a frequency table.

### Guidance

Data is a set of numerical or non-numerical information. Data can be analyzed in many different ways. In this concept we will analyze numerical data using frequency tables.

A frequency table shows the frequency, or amount of occurrences, of specific groups of data called bins. Said another way, a frequency table shows the number of times that a group of data occurred. A frequency table has two columns, one for the bins (or categories) and the other for the number of occurrences.

The range of a set of data is the difference between the largest and smallest values. The range identifies how far apart the values in the data set are.

A histogram is a vertical bar graph that illustrates the frequency of the data. A histogram is a graphical representation of a frequency table. In a histogram, the horizontal axis lists the bins, or categories, of the data. The vertical axis lists the frequency, or amount, of the occurrences, which is represented by the height of the columns. Unlike a bar graph, the columns within a histogram do not have space between them.

Let's look at an example.

The frequency table below shows the number of hours that twenty people said they slept each night. Create a histogram to display the data.

 Bins: Number of Hours Slept Frequency 5 1 6 2 7 4 8 3 9 3 10 3 11 2 12 2

First, draw the horizontal and vertical axis.

Next, label the horizontal axis. The horizontal axis lists the different categories of data. In this case, the category will be "Hours."

Next, label the vertical axis. The vertical axis lists the quantity or amount of the data. In this case, the category will be "Frequency."

Next, title the graph. The title of the graph should be short and clear. It should explain what data is presented in the graph. In this case, the title will be “Hours Slept Each Night.”

Then, determine the units on the vertical axis. To do this, start by reviewing the smallest and largest frequencies in the table. The smallest value is 1 and the largest is 4. Based on these values label the vertical axis from 0-4. Since the frequency values are always whole numbers, the units should be whole numbers. Since the range is small, 4, the vertical axis should use a unit of 1.

Next, draw the vertical columns. To do this, write each bin along the horizontal axis. Then draw each column vertically until it reaches the frequency for that time. For example, draw a vertical column to the number 1 for the bin 5. Do not leave spaces between each column.

The answer is the graph should look like the one below.

License: CC BY-NC 3.0

### Guided Practice

The frequency table below shows the amount of time (in minutes) that 20 middle school students spend on a computer each day. Create a histogram to display the data.

 Bins: Mins Spent on Computer Frequency 0-10 5 11-20 5 21-30 2 31-40 2 41-50 3 51-60 3

First, draw the horizontal and vertical axis.

Next, label the horizontal axis. The horizontal axis lists the different categories of data. In this case, the category will be "Minutes."

Next, label the vertical axis. The vertical axis lists the quantity or amount of the data. In this case, the category will be "Frequency."

Next, title the graph. The title of the graph should be short and clear. It should explain what data is presented in the graph. In this case, the title will be “Minutes Spent on Computer Each Day.”

Then, determine the units on the vertical axis. To do this, start by reviewing the smallest and largest frequencies in the table. The smallest value is 2 and the largest is 5. Based on these values label the vertical axis from 0-5. Since the frequency values are always whole numbers, the units should be whole numbers. Since the range is small, 5, the vertical axis should use a unit of 1.

Next, draw the vertical columns. To do this, write each bin along the horizontal axis. Then draw each column vertically until it reaches the frequency for that time. For example, draw a vertical column to the number 5 for the bin 0-10. Do not leave spaces between the columns.

The answer is the graph should look like the one below.

License: CC BY-NC 3.0

### Examples

#### Example 1

The frequency table below shows the height (in meters) a ball bounced after being dropped. Create a histogram to display the data. Then state two conclusions about the data.

 Bins: Height (meters) Frequency 4-5 3 6-7 4 8-9 3 10-11 2 12-13 2 14-15 1

First, draw the horizontal  and vertical  axis.

License: CC BY-NC 3.0

Next, label the horizontal axis. The horizontal axis lists the different categories of data. In this case, the category will be "Height (in meters)."

Next, label the vertical axis. The vertical axis lists the quantity or amount of the data. In this case, the category will be "Frequency."

Next, title the graph. The title of the graph should be short and clear. It should explain what data is presented in the graph. In this case, the title will be “Height of Ball.”

Then, determine the units on the vertical axis. To do this, start by reviewing the smallest and largest frequencies in the table. The smallest value is 1 and the largest is 4. Based on these values label the vertical axis from 0-4. Since the range is small the vertical axis should use a unit of 1.

Next, draw the vertical columns. To do this, write each bin along the horizontal axis. Then draw each column vertically until it reaches the frequency for that time. For example, draw a vertical column to the number 3 for the bin 4-5.

The first answer is the graph should look like the one below. The second answer is two conclusions that can be made are: the most frequent bounce heights were between six and seven meters and the least frequent bounce heights were between fourteen and fifteen meters.

License: CC BY-NC 3.0

#### Example 2

A triathlon club asked it's members how many hours they exercise in a typical week. The data collected is below. Create a histogram to represent the data and then state three conclusions.

8, 2, 4, 7.5, 10, 11, 5, 6, 8, 12, 11, 9, 6.5, 10.5, 13

Histograms are made from frequency tables. Before making a histogram, a frequency table must be created.

To create a frequency table, first draw a two column table.  The left hand column will be the hours exercised and the right will be the frequency, or the number of people who exercise the given amount.

Next, determine the size of the bins. The range of this data is from 2 to 13 minutes. Given this size of the range of data, create bins of size 3, starting at 0.

 Bins: Hours of Exercise Frequency 0-2 3-5 6-8 9-11 12-14

Next, calculate the frequency of each bin. To do this, count up how many data points are between 0-2, and place this value in the right column of the table. Then count up how many data points are between 3-5, and place this value in the table. Continue until the entire table is complete.

 Bins: Hours of Exercise Frequency 0-2 1 3-5 2 6-8 5 9-11 5 12-14 2

Next, create the histogram. To do this, first draw the horizontal and verticalaxes.

License: CC BY-NC 3.0

Next, label the horizontal axis. The horizontal axis lists the different categories of data. In this case, the category will be "Hours."

Next, label the vertical axis. The vertical axis lists the quantity or amount of the data. In this case, the category will be "Frequency."

Next, title the graph. The title of the graph should be short and clear. It should explain what data is presented in the graph. In this case, the title will be “Hours of Exercise in a Week.”

Then, determine the units on the vertical axis. To do this, start by reviewing the smallest and largest frequencies in the table. The smallest value is 1 and the largest is 5. Based on these values label the vertical axis from 0-5. Since the range is small the vertical axis should use a unit of 1.

Next, draw the vertical columns. To do this, write each bin along the horizontal axis. Then draw each column vertically until it reaches the frequency for that time. For example, draw a vertical column to the number 1 for the bin 0-2.

The first answer is the graph should look like the one below. The second answer is three conclusions that can be stated are: an equal number of people reported that they exercise between 6-8 and 9-11 hours in a typical week; two people reported they exercise between 3-5 hours in a typical week; and 0-2 hours a week has the smallest frequency.

License: CC BY-NC 3.0

#### Example 3

The histogram below shows the number of sodas that students in Ms. Jones class drink in one day. Analyze the graph and state three conclusions about the data.

License: CC BY-NC 3.0

First, analyze the graph by comparing the frequencies of the bins.

The answer is three conclusions that can be made from the graph are: there are 20 students in Ms. Jones class; the greatest number of students drink between 0 and 3 sodas per day; and one quarter of the class drink 8 or more sodas per day.

License: CC BY-NC 3.0

Remember Mrs. Alameda and her computer science classes?

Mrs. Alameda has created a frequency table representing the scores on the computer science exam for all her classes. The frequency table is difficult to read and interpret, so Mrs. Alameda wants to create a graphical display.

 Bins: Test Scores Frequency 0-50 17 51-60 9 61-70 35 71-80 81 81-90 123 91-100 81

Mrs. Alameda can make a histogram to represent the data. This is the best method for graphically illustrating frequency data.

To make a histogram, first draw the horizontal and vertical axes.

License: CC BY-NC 3.0

Next, label the horizontal axis. The horizontal axis lists the different categories of data. In this case, the category will be "Scores."

Next, label the vertical axis. The vertical axis lists the quantity or amount of the data. In this case, the category will be "Frequency."

Next, title the graph. The title of the graph should be short and clear. It should explain what data is presented in the graph. In this case, the title will be “Computer Science Exam Scores.”

Then, determine the units on the vertical axis. To do this, start by reviewing the smallest and largest frequencies in the table. The smallest value is 9 and the largest is 123. Based on these values label the vertical axis from 0-140. Since the range of the frequency values is large, the units should also be large. The vertical axis should use a unit of 20.

Next, draw the vertical columns. To do this, write each bin along the horizontal axis. Then draw each column vertically until it reaches the frequency for that score. For example, draw a vertical column to the number 17 for the bin 0-50. Do not leave spaces between the columns.

The answer is the graph should look like the one below.

License: CC BY-NC 3.0

### Explore More

Use what you have learned to complete each dilemma.

1. Create a histogram to display the data from the frequency table below.

Monthly Internet Purchases
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

2. The data collected depicts the number of letters in the last names of twenty people. Create a frequency table to display the data.

3. Create a histogram to display the data.

4. 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.

5. Create a histogram to display the data.

6. Write a few sentences to explain any conclusions that you can draw from the data.

7. Generate a question that you will use to survey twenty people.

8. Make a table to collect the answers.

9. Display the data on a frequency table

10. Create a histogram to display the data histogram.

Here is a list of the number of students who did not complete their homework in one month.

1, 1, 3, 3, 4, 3, 3, 5, 6, 1, 1, 1, 2, 2, 3

11. Create a frequency table of the data.

12. What is the most popular value?

13. What is the least popular value?

14. What is the range of values?

15. What is the average?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 11.12.

### Vocabulary Language: English

bar graph

bar graph

A bar graph is a plot made of bars whose heights (vertical bars) or lengths (horizontal bars) represent the frequencies of each category, with space between each bar.
frequency density

frequency density

The vertical axis of a histogram is labelled frequency density.
Frequency table

Frequency table

A frequency table is a table that summarizes a data set by stating the number of times each value occurs within the data set.
Histogram

Histogram

A histogram is a display that indicates the frequency of specified ranges of continuous data values on a graph in the form of immediately adjacent bars.
Interval

Interval

An interval is a range of data in a data set.
Range

Range

The range of a data set is the difference between the smallest value and the greatest value in the data set.
right-skewed distribution

right-skewed distribution

A right-skewed distribution has a peak to the left of the distribution and data values that taper off to the right.
unimodal

unimodal

If a data set has only 1 value that occurs most often, the set is called  unimodal.

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### Explore More

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