1. Stem-and-Leaf Plots

2. Objectives
Display and interpet data on a stem-and-leaf plot

3. Application
Mr Servello wants to study the distribution of the scores for a 100-point unit exam given in his first period chemistry class. The scores of the 35 students in the class are listed below. 82 77 49 84 44 98 93 71 76 65 89 95 78 69 64 88 54 87 91 80 85 55 62 79 90 86 75 74 99 96

4. Organizing Data
He can organize and display the scores in a compact way by using a stem-and-leaf plot. In a stem-and-leaf plot, the greatest common place value of the data is used to form the stems. The numbers in the next greatest common place-value position are then used to form the leaves. In the list previously given, the greatest place value is tens. Thus, the number 82 would have stem 8 and leaf 2.

5. To make a stem-and-leaf plot
First, make a vertical list of the stems. Since the test scores range from 44 to 99, the stems range from 4 to 9. Then, plot each number by placing the units digit (leaf) to the right of its correct stem. Thus, the scores 82 is plotted by placing leaf 2 to the right of the stem 8. The complete stem-and-leaf plot is shown at the right.
Stem
Leaf 4
9 4 4 5
4 5 6 7 8 9
Note: a stem may have one or more digits. A leaf always has just one digit.
8|2 represents a score of 82.

6. To make a stem-and-leaf plot
A second stem-and-leaf plot can be made to arrange the leaves in numerical order from least to greatest as shown at the right. This will make it easier for Mr. Juarez to analyze the data.
Stem
Leaf 4
4 4 9 5
4 5 6 7 8 9

7. Ex. 1: Use the information in the stem-and-leaf plots above to answer each question.
What were the highest and the lowest scores?
Stem
Leaf 4
4 4 9 5
4 5 6 7 8 9
99 and 44
Which test score occurred most frequently?
89 (3 times)
In which 10-point interval did the most students score?
80-89(10 students)
How many students received a score of 70 or better?
25 students

8. More than 2 digits
Sometimes the data for a stem-and-leaf plot are numbers that have more than two digits. Before plotting the numbers, they may need to be rounded or truncated to determine each stem and leaf. Suppose you want to plot 356 using the hundreds digit for the stem.
Rounded: Round 356 to Thus you would plot 356 using stem 3 and leaf 6. What would be the stem-and-leaf of 499?
5 and 0

9. More than 2 digits
Truncated—To truncate means to cut off, so truncate 356 as Thus you would plot 356 using stem 3 and leaf 5. What would be the stem and leaf of 499?
4 and 9

10. Back-to-back stem-and-leaf
A back-to-back stem-and-leaf plot is sometimes used to compare two data or rounded and truncated values of the same data set. In a back-to-back plot, the same stem is used for the leaves of both plots.

11. Ex. 2: The average annual pay for workers in selected states are listed below. Make a back-to-back stem-and-leaf plot of the average annual pay comparing rounded values and truncated values. Then answer each question.
State
Avg. Annual Pay
Alaska
$28,033
Michigan
$24,193
California
$24,126
Minnesota
$21,481
Colorado
$21,472
New Jersey
$25,748
Connecticut
$26,234
New York
$26,347
Delaware
$21,977
Ohio
$21,501
Illinois
$23,608
Pennsylvania
$21,485
Maryland
$22,515
Texas
$21,130
Massachusetts
$24,143
Virginia
$21,053

12. Ex. 2: Since the data range from $21,053 to $28,033, the stems range from 21 to 28 for both plots.
State
Avg. Annual Pay
Alaska
$28,033
Michigan
$24,193
California
$24,126
Minnesota
$21,481
Colorado
$21,472
New Jersey
$25,748
Connecticut
$26,234
New York
$26,347
Delaware
$21,977
Ohio
$21,501
Illinois
$23,608
Pennsylvania
$21,485
Maryland
$22,515
Texas
$21,130
Massachusetts
$24,143
Virginia
$21,053
Rounded
Stem
Truncated 21
5 0 22 5 6 23
2 1 1 24
1 1 1 7 25
3 2 26
2 3 27 28

13. What does 21|5 represent in each plot?
Ex. 2: Since the data range from $21,053 to $28,033, the stems range from 21 to 28 for both plots.
What does 21|5 represent in each plot?
Rounded
Stem
Truncated 21
5 0 22 5 6 23
2 1 1 24
1 1 1 7 25
3 2 26
2 3 27 28
Answer: It represents $21,450 to $21,549 for rounded data and $21,500 to $21,599 for truncated data.

14. What is the difference between the highest lowest average annual pay?
Ex. 2: Since the data range from $21,053 to $28,033, the stems range from 21 to 28 for both plots.
What is the difference between the highest lowest average annual pay?
Rounded
Stem
Truncated 21
5 0 22 5 6 23
2 1 1 24
1 1 1 7 25
3 2 26
2 3 27 28
Answer: About $6,900 for rounded data and $7,000 for truncated data.

15. Did more of the states have average annual pay above or below $25,000?
Ex. 2: Since the data range from $21,053 to $28,033, the stems range from 21 to 28 for both plots.
Did more of the states have average annual pay above or below $25,000?
Rounded
Stem
Truncated 21
5 0 22 5 6 23
2 1 1 24
1 1 1 7 25
3 2 26
2 3 27 28
Answer: Below $25,000

16. Ex. 2: Since the data range from $21,053 to $28,033, the stems range from 21 to 28 for both plots.
Does there appear to be any significant difference between the two stem-and-leaf plots?
Rounded
Stem
Truncated 21
5 0 22 5 6 23
2 1 1 24
1 1 1 7 25
3 2 26
2 3 27 28
Answer: NO

17. Ex. 3: Set up the problem
The enrollment of several small colleges in the Center City area are listed below. Make a back-to-back stem-and-leaf plot of enrollments comparing rounded values and truncated values. Then answer each question.
College
Enrollment Figure
Miller Business School
1342
Capital College
1685
Para-Professional Institute
1013
Parke College
2350
State Community
3781
Fashion Institute
1096
College of Art and Design
1960
Franklin Community College
3243

18. Ex. 3: Since the data range from 1013 to 3781, we an use stems that represent 1,000 each.
What range of student enrollment is represented on the rounded side by 2|4?
Rounded
Stem
Truncated 1
4 0 2 3
8 2
2 7
Answer:

19. Ex. 3: Since the data range from 1013 to 3781, we an use stems that represent 1,000 each.
What range of student enrollment is represented on the truncated side by 1|9?
Rounded
Stem
Truncated 1
4 0 2 3
8 2
2 7
Answer:
Since each stem represents 1,000; for example 3|8 represents 3,000-3,999