Stem-and-Leaf Plots

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

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

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.

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 5 9 4 2 2 7 7 1 6 8 9 5 4 8 2 4 9 9 8 7 0 5 9 6 9 8 3 5 6 2 3 0 9 Note: a stem may have one or more digits. A leaf always has just one digit. 8|2 represents a score of 82.

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 2 2 4 5 9 7 1 4 5 6 7 8 9 8 0 2 4 5 6 7 8 9 9 9 9 0 2 3 3 5 6 8 9

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 2 2 4 5 9 7 1 4 5 6 7 8 9 8 0 2 4 5 6 7 8 9 9 9 9 0 2 3 3 5 6 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

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 360. Thus you would plot 356 using stem 3 and leaf 6. What would be the stem-and-leaf of 499? 5 and 0

More than 2 digits Truncated—To truncate means to cut off, so truncate 356 as 350. Thus you would plot 356 using stem 3 and leaf 5. What would be the stem and leaf of 499? 4 and 9

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.

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

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 5 5 5 5 1 1 21 0 1 4 4 4 5 9 5 0 22 5 6 23 2 1 1 24 1 1 1 7 25 3 2 26 2 3 27 28

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 5 5 5 5 1 1 21 0 1 4 4 4 5 9 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.

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 5 5 5 5 1 1 21 0 1 4 4 4 5 9 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.

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 5 5 5 5 1 1 21 0 1 4 4 4 5 9 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

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 5 5 5 5 1 1 21 0 1 4 4 4 5 9 5 0 22 5 6 23 2 1 1 24 1 1 1 7 25 3 2 26 2 3 27 28 Answer: NO

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

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 7 3 1 0 1 0 0 3 6 9 4 0 2 3 8 2 2 7 Answer: 2350-2449

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 7 3 1 0 1 0 0 3 6 9 4 0 2 3 8 2 2 7 Answer: 1900-1999 Since each stem represents 1,000; for example 3|8 represents 3,000-3,999