Suppose that your physical education teacher asked each of the members of your gym class to do as many pullups as possible. If you recorded the number of pullups completed by each person, would you be able to represent this data with a stemandleaf plot? How about with a histogram? In this Concept, you'll learn how to create these two types of graphs so that you can analyze and interpret data.
Guidance
Understanding data is a very important mathematical ability. You must know how to use data and interpret the results to make informed decisions about politics, food, and income. This Concept will show two ways to graph data:
 as a stemandleaf plot and
 as a histogram
A stemandleaf plot is an organization of numerical data into categories based on place value. The stemandleaf plot is a graph that is similar to a histogram but it displays more information. For a stemandleaf plot, each number will be divided into two parts using place value.
The stem is the lefthand column and will contain the digits in the largest place. The righthand column will be the leaf and it will contain the digits in the smallest place.
Example A
In a recent study of male students at a local high school, students were asked how much money they spend socially on prom night. The following numbers represent the amount of dollars of a random selection of 40 male students.
Represent this data in a stemandleaf plot.
Solution:
The stems will be arranged vertically in ascending order (smallest to largest) and each leaf will be written to the right of its stem horizontally in order from least to greatest.
Stem  Leaf 

2  5, 8 
3  4, 5, 5, 5 
4  0, 0, 2, 7, 9 
5  0, 0, 0, 0, 5, 5, 8 
6  0, 0, 4, 5 
7  0, 0, 0, 5, 5 
8  0, 0, 0, 4, 5 
9  0, 0, 3, 5, 5 
10  0 
11  0 
12  0 
This stemandleaf plot can be interpreted very easily. By looking at stem 6, you see that 4 males spent 60 ‘some dollars’ on prom night. By counting the number of leaves, you know that 40 males responded to the question concerning how much money they spent on prom night. The smallest and largest data values are known by looking and the first and last stemandleaf. The stemandleaf is a ‘quick look’ chart that can quickly provide information from the data. This also serves as an easy method for sorting numbers manually.
Interpreting and Creating Histograms
Suppose you took a survey of 20 algebra students, asking their number of siblings. You would probably get a variety of answers. Some students would have no siblings while others would have several. The results may look like this.
1, 4, 2, 1, 0, 2, 1, 0, 1, 2, 1, 0, 0, 2, 2, 3, 1, 1, 3, 6
We could organize this many ways. The first way might just be to create an ordered list, relisting all numbers in order, starting with the smallest.
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 6
Another way to list the results is in a table.
Number of Siblings  Number of Matching Students 

0  4 
1  7 
2  5 
3  2 
4  1 
5  0 
6  1 
A table showing the number of times a particular category appears in a data set is a frequency distribution .
You could also make a visual representation of the data by making categories for the number of siblings on the axis and stacking representations of each student above the category marker. We could use crosses, stickmen or even photographs of the students to show how many students are in each category.
This graph is called a histogram .
A histogram is a bar chart that describes a frequency distribution.
The horizontal axis of the histogram is separated into equal intervals. The vertical bars represent how many items are in each interval.
Example B
Jim collected data at a local fair. The above histogram relates the number of children of particular ages who visited the facepainting booth. What can you conclude using this histogram?
Solution:
 You can find the sum of the heights of each bar to determine how many children visited the face painting booth.
 The tallest bar is at age seven. There were more sevenyear olds than any other age group.
 There is no bar at one or ten. This means zero tenyearolds and zero oneyearolds had their faces painted.
Creating Histograms Using a Graphing Calculator
Drawing a histogram is quite similar to drawing a scatter plot. Instead of graphing two lists, and , you will graph only one list, .
Example C
The following unordered data represents the ages of passengers on a train carriage. Make a histogram of the data on a graphing calculator.
35, 42, 38, 57, 2, 24, 27, 36, 45, 60, 38, 40, 40, 44, 1, 44, 48, 84, 38, 20, 4,
2, 48, 58, 3, 20, 6, 40, 22, 26, 17, 18, 40, 51, 62, 31, 27, 48, 35, 27, 37, 58, 21.
Solution:
Begin by entering the data into [LIST 1] of the [STAT] menu.
Choose and to enter into the [STATPLOT] menu.
Press [WINDOW] and ensure that Xmin and Xmax allow for all data points to be shown. The Xscl value determines the bin width.
Press [GRAPH] to display the histogram.
Video Review
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Guided Practice
Example: Studies (and logic) show that the more homework you do the better your grade in a course will be. In a study conducted at a local school, students in grade 10 were asked to check off what box represented the average amount of time they spent on homework each night. The following results were recorded:
Time Spent on Homework (Hours)  Tally  Frequency (# of students) 

12  
23  
34  
26  
5  
0 
Convert this data into a histogram.
Solution:
Explore More
Sample explanations for some of the practice exercises below are available by viewing the following videos. Note that there is not always a match between the number of the practice exercise in the videos and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: StemandLeaf Plots (6:45)
CK12 Basic Algebra: Histograms (6:08)
 What is the stem in a stemandleaf plot? What is a leaf ? What is an advantage to using a stemandleaf plot?
 Describe a histogram . What is an advantage of using a histogram?

For each of the following examples, describe why you would likely use a histogram.
 Frequency of the favorite drinks for the first 100 people to enter the school dance
 Frequency of the average time it takes the people in your class to finish a math assignment
 Frequency of the average distance people park their cars away from the mall in order to walk a little more
 Prepare a histogram using the following scores from a recent science test. Offer four conclusions you can make about this histogram? Include mean, median, mode, range, etc.
Score (%)  Tally  Frequency 

5060  4  
6070  6  
7080  11  
8090  8  
90100  4 

A research firm has just developed a streakfree glass cleaner. The product is sold at a number of local chain stores and its sales are being closely monitored. At the end of one year, the sales of the product are released. The company is planning on starting up an advertisement campaign to promote the product. The data is found in the chart below.
 Display the sales of the product before the ad campaign in a stemandleaf plot.
 How many chain stores were involved in selling the streakfree glass cleaner?
 In stem 1, what does the number 11 represent? What does the number 8 represent?
 What percentage of stores sold less than 175 bottles of streakfree glass cleaner?

Using the following data, answer the questions that follow. Data: 607.4, 886.0, 822.2, 755.7, 900.6, 770.9, 780.8, 760.1, 936.9, 962.9, 859.9, 848.3, 898.7, 670.9, 946.7, 817.8, 868.1, 887.1, 881.3, 744.6, 984.9, 941.5, 851.8, 905.4, 810.6, 765.3, 881.9, 851.6, 815.7, 989.7, 723.4, 869.3, 951.0, 794.7, 807.6, 841.3, 741.5, 822.2, 966.2, 950.1 A. Create a stemandleaf plot. Round each data point to the nearest tens place. Use the hundreds digit as the stem and the tens place as the leaf.
 What is the mean of the data?
 What is the mode of the data?
 What is the median of the data?
B. Make a frequency table for the data. Use a bin width of 50. C. Plot the data as a histogram with a bin width of:
 50
 100
 The women from the senior citizen’s complex bowl every day of the month. Lizzie had never bowled before and was enjoying this newfound pastime. She decided to keep track of her best score of the day for the month of September. Here are the scores that she recorded: In order for Lizzie to see how well she is doing, create a stemandleaf plot of her scores.
 It is your job to entertain your younger sibling every Saturday morning. You decide to take the youngster to the community pool to swim. Since swimming is a new thing to do, your little buddy isn’t too sure about the water and is a bit scared of the new adventure. You decide to keep a record of the length of time he stays in the water each morning. You recorded the following times (in minutes): 12, 13, 21, 27, 33, 34, 35, 37, 40, 40, 41. Create a stemandleaf plot to represent this data. List two conclusions you can make from this graph.

The following stemandleaf plot shows data collected for the speed of 40 cars in a 35 mph limit zone in Culver City, California.
 Find the mean, median, and mode speed.
 Complete a frequency table, starting at 25 mph with a bin width of 5 mph.
 Use the table to construct a histogram with the intervals from your frequency table.

The following histogram displays the results of a largerscale survey of the number of siblings. Use it to find:
 The median of the data
 The mean of the data
 The mode of the data
 The number of people who have an odd number of siblings
 The percentage of the people surveyed who have four or more siblings
 Use the stemandleaf plot to devise a situation that follows the data.
 An oil company claims that its premiumgrade gasoline contains an additive that significantly increases gas mileage. To prove their claim, the company selected 15 drivers and first filled each of their cars with 45 L of regular gasoline and asked them to record their mileage. The company then filled each of the cars with 45 L of premium gasoline and again asked them to record their mileage. The results below show the number of kilometers each car traveled.
640  570  660  580  610 

540  555  588  615  570 
550  590  585  587  591 
659  619  639  629  664 

635  709  637  633  618 
694  638  689  589  500 
Display each set of data to explain whether or not the claim made by the oil company is true or false.
Mixed Review
 How many ways can a nineperson soccer team line up for a picture if the goalie is to be in the center?
 Graph using its intercepts.
 Simplify .
 Rewrite in standard form: .
 Graph .
 A ball is dropped from a height of 10 meters. When will it reach the ground?
 Solve the following system: .