Chapter 13 · Statistics and data displays
Creating and Interpreting Frequency Tables
Organize a list of data into a frequency table by tallying and counting, then use the table to answer questions about the data.
Twelve students each answered one question: How do you get to school? Here are their answers, in the order the students gave them:
bus, walk, car, bus, bike, walk, car, bus, walk, bike, car, bus
Which way is the most common? That is hard to see at a glance. You have to count.
Data are pieces of information that you collect, like these answers. Each single answer is one data value. A data value can be a word, like bus, or a number. The frequency of a value is the number of times it appears in the data. In this list, bike appears times, so the frequency of bike is .
A frequency table organizes data so you can see the counts right away. In this lesson, each row of the table stands for one of the different values in the data, and the table shows that value's frequency beside it.
To build one, read the data one value at a time. For each value, draw one tally mark, a short stroke, in the matching row. When you finish, count the marks in each row. That count is the frequency. Here is the frequency table for the answers.
| Way to school | Tally | Number of students (frequency) |
|---|---|---|
Bus | ||
Walk | ||
Car | ||
Bike |
Read the headers first. They are the labels at the top of the columns, and they tell you what the table counts. Each row stands for one way to get to school. The tally column has one mark for each student who gave that answer. The frequency column gives the same count as a number of students.
So the row for Bus tells you that students ride the bus. Bus is the most common answer, because its frequency, , is greater than every other frequency in the table.
You can also check the table. Each student gave exactly one answer, so each student has a mark in exactly one row. That means the frequencies must add up to the number of students, which is . They do:
Key concept
What a frequency table counts
A frequency table shows how often each value appears in a set of data. In the tables in this lesson, each row stands for one value, and the frequency in that row counts the data values that match it. The headers tell you what the table counts, such as students or games.
When you count every data value exactly once, in exactly one row, the frequencies add up to the total number of data values. Use this fact to find how many data values a table holds and to check a table you make. A sum that does not match means something went wrong.
Worked example
Create a frequency table
A soccer team played games. Here are the goals it scored in each game, in order: , , , , , , , , , , , , , . Make a frequency table for these data.
Find the different values. The least value in the list is 0 and the greatest is 4, and every whole number from 0 to 4 appears. Give each of these five values its own row, in order from least to greatest, so any value is easy to find.
Write headers so a reader knows what the table counts. The first column shows a number of goals in one game. The middle column is for tally marks. The last column is the frequency. It counts the games in which the team scored exactly that many goals.
Make the tally marks. Read the list in order. For each value, draw one tally mark in the matching row, then cross that value off the list. Crossing off helps you give each game exactly one mark.
Count the tally marks in each row and write the count as that row's frequency. The row for 0 has 3 marks, the row for 1 has 4, the row for 2 has 3, the row for 3 has 2, and the row for 4 has 2.
Check the table. Each of the 14 games got one mark, in exactly one row, so the frequencies, which count games, must add up to 14. Adding from left to right gives 7, then 10, then 12, then 14. A different sum would show a mistake, such as a game skipped or counted twice. A matching sum cannot catch a mark put in the wrong row, so tally with care.
Answer
The frequency table has one row for each number of goals from to . The frequencies are , , , and . They add up to , the number of games. You can see the finished table next.
| Goals in a game | Tally | Number of games (frequency) |
|---|---|---|
In the goals table, the first and last columns both hold whole numbers, so the headers matter. The row that starts with ends with . It tells you the team scored exactly goals in games. It does not tell you the team scored goals in games.
A finished frequency table often leaves out the tally marks and keeps only the frequencies. The next table shows the results of a reading survey. Each row stands for a number of books. The frequency in that row counts the students who read exactly that many books last month.
| Books read | Number of students (frequency) |
|---|---|
Worked example
Interpret a frequency table
Use the table of books read last month. (a) How many students answered the survey? (b) Which number of books is the most common answer? (c) How many students read at least books? (d) How many times as many students read exactly books as read exactly books?
Read the headers before the numbers. Each row stands for one answer, a number of books. The frequency in that row counts the students who gave that answer. For example, the row for 1 book has frequency 9, so 9 students read exactly 1 book.
For (a), each student gave exactly one answer, so the table counts each student in exactly one row. Add all five frequencies to count every student. Adding from left to right gives 13, then 25, then 31, then 34.
For (b), the most common answer is the one that the most students gave, so look for the greatest frequency. The greatest frequency is 12, and only one row has it. That row is for 2 books. The answer is the value 2, not the frequency 12.
For (c), at least 3 books means 3 books or more. In this table, the rows that fit are the rows for 3 books and 4 books. Each of those frequencies counts students, so add the two frequencies.
For (d), compare two frequencies that both count students: 12 students read exactly 2 books and 3 students read exactly 4 books. Divide 12 by 3 to find how many groups of 3 students fit in 12 students. Multiplication undoes division, and 4 groups of 3 students make 12 students, so the answer 4 is correct.
Answer
(a) students answered the survey. (b) The most common answer is books. (c) students read at least books. (d) times as many students read exactly books as read exactly books.
A frequency table turns a long list of data into a short set of counts. In the tables in this lesson, each row stands for one value, and its frequency counts the data values that match it. Read the headers first, so you know what each column counts. When the table counts every data value exactly once, in exactly one row, the frequencies add up to the number of data values. That sum tells you how many data values the table holds, and it helps you check a table you make. Next, you will read and create bar graphs, which show counts like these as bars.