Chapter 10 · Measurement and data
Frequency Tables, Stem-and-Leaf Plots, and Data Problems
Organize data in a frequency table or a stem-and-leaf plot, read values back, and combine them to solve problems with more than one step.
Suppose 12 students each write down how many books they read last month. You get a list of 12 numbers in no special order. A list like this is data. Data means information that someone has collected. Each number in the list is one value.
A jumbled list is hard to read. You cannot quickly see which value comes up most or which value is the largest. In this lesson you will learn two ways to organize data so that it is easy to read. Then you will use organized data to solve problems that take more than one step.
The frequency of a value is the number of times that value appears in the data. A frequency table shows each different value next to its frequency.
To make one, write the different values in order, one in each row. Then go through the list one number at a time and count how many times each value appears. Cross off each number as you count it, so that you do not miss one or count one twice.
Each row pairs a value with a count. A row with the value 3 and the frequency 5 tells you that the value 3 appears 5 times in the data.
Worked example
Make a frequency table and read it
Twelve students wrote down how many books they read last month: 3, 2, 1, 3, 2, 4, 3, 2, 3, 1, 2, 3. Make a frequency table. Then find the most common number of books and how many students read 3 or more books.
Find the different values. The list only has the numbers 1, 2, 3 and 4. Write them in order, one in each row, so the table has four rows.
Count how many times each value appears. Cross off each number in the list as you count it, so that you count every number exactly once. The value 1 appears 2 times, the value 2 appears 4 times, the value 3 appears 5 times and the value 4 appears 1 time. These counts are the frequencies.
Check the counts. You counted every student in exactly one row, so the frequencies must add up to the 12 students in the list.
Find the most common number of books. The most common value is the one with the greatest frequency. The greatest frequency is 5, and it is in the row for 3 books.
Find how many students read 3 or more books. In this table that means 3 books or 4 books, so add the frequencies in those two rows.
Answer
The frequencies are 2, 4, 5 and 1 for 1, 2, 3 and 4 books. The most common number of books is 3, and 6 students read 3 or more books.
| Number of books | Frequency (number of students) |
|---|---|
1 | 2 |
2 | 4 |
3 | 5 |
4 | 1 |
Some data has many different values, such as the number of minutes each student read last night. A frequency table would need a row for each different value, so it could get very long. A stem-and-leaf plot is another way to organize data like this. It groups the values into rows and still shows every value.
You split each value into two parts. In this lesson every value is a two-digit whole number, so the stem is the tens digit and the leaf is the ones digit. The value 32 has stem 3 and leaf 2, because 32 is 3 tens and 2 ones.
Values with the same stem share a row. You write the stem once on the left and the leaves to its right. A key is a short note beside the plot that shows how to join a stem and a leaf. You will often see a key written with a bar, like this: means 32.
Worked example
Make a stem-and-leaf plot and read it
Twelve students wrote down how many minutes they read last night: 32, 18, 25, 41, 25, 30, 17, 36, 29, 44, 21, 38. Make a stem-and-leaf plot. Then find the longest time and how many students read for 20 minutes or more.
Choose the stems. The smallest value is 17 and the largest is 44, so the tens digits go from 1 to 4. Write the stems 1, 2, 3 and 4 in a column, one in each row.
Place each value. Go through the list in order and write the ones digit of each value in the row for its tens digit. The first value, 32, is 3 tens and 2 ones, so its leaf 2 goes in the row for stem 3. After you place every value, the rows are stem 1: 8, 7; stem 2: 5, 5, 9, 1; stem 3: 2, 0, 6, 8; stem 4: 1, 4.
Put the leaves in each row in order from least to greatest. Keep both 5s in the row for stem 2, because two students read for 25 minutes and each leaf stands for one student. The rows become stem 1: 7, 8; stem 2: 1, 5, 5, 9; stem 3: 0, 2, 6, 8; stem 4: 1, 4. Count the leaves to check. There must be 12, one for each value in the list.
Read the longest time. The stems and the leaves are now in order, so the largest value is the last leaf in the last row. That is stem 4 with leaf 4, which means 4 tens and 4 ones.
Count the students who read for 20 minutes or more. In this plot those values have stem 2, stem 3 or stem 4, so count the leaves in those three rows.
Answer
The plot has four rows: stem 1 with leaves 7, 8; stem 2 with leaves 1, 5, 5, 9; stem 3 with leaves 0, 2, 6, 8; and stem 4 with leaves 1, 4. The longest time is 44 minutes, and 10 students read for 20 minutes or more.
| Stem (tens) | Leaves (ones) |
|---|---|
1 | 7 8 |
2 | 1 5 5 9 |
3 | 0 2 6 8 |
4 | 1 4 |
Key concept
Know what each number stands for
A frequency table and a stem-and-leaf plot both organize a list of data without changing it. To read them, know what each number stands for.
- In a frequency table, each row pairs a value with its frequency, the number of times that value appears.
- In a stem-and-leaf plot, a stem and one leaf join to make one value.
When you make either one correctly, you use every value in the list exactly once. So the frequencies add up to the number of values in the list, and the number of leaves equals the number of values in the list. Use these two facts to check your work.
Organized data helps you answer bigger questions. A multi-step problem is a problem that takes more than one calculation to solve. This plan keeps you on track:
- Decide what the question asks for.
- Read each value you need from the table or graph, and be sure what each number counts.
- Combine the values, one calculation at a time.
You already know how to read a value from a bar on a bar graph or from a point on a line graph. After you read the values, you combine them in the same way.
Worked example
Solve a multi-step problem from a table
Use the frequency table of books read last month. Its rows are: 1 book, frequency 2; 2 books, frequency 4; 3 books, frequency 5; 4 books, frequency 1. The 12 students had a goal of reading 40 books in all. How many more books did they need to read to reach the goal?
Decide what you need. The question asks for the gap between the goal of 40 books and the number of books the students read. The table does not show that total, so find it first.
Start with one row. The row for 3 books has frequency 5. That means 5 students read 3 books each, which is 5 groups of 3 books.
Do the same for the other rows. In the row for 1 book, 2 students read 1 book each. In the row for 2 books, 4 students read 2 books each. In the row for 4 books, 1 student read 4 books.
Add the books from the four rows. This is the total number of books that all 12 students read. You can check it by adding the 12 numbers in the original list, which gives the same total.
Subtract the total from the goal. The difference is the number of books the students still needed.
Answer
The students read 29 books in all, so they needed 11 more books to reach their goal of 40 books.
A frequency table and a stem-and-leaf plot organize data without changing it. In a frequency table, each row gives a value and how many times it appears. In a stem-and-leaf plot, each leaf joins its stem to make one value. To solve a problem with more than one step, first be sure what each number in the table or graph counts. Then read the values you need and combine them one calculation at a time.