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How to Make a Cumulative Frequency Table (Mathematics Lesson)

Making a Cumulative Frequency Table

A cumulative frequency table is a great way to present a lot of data.

Imagine a teacher wanted an easy way to present the test scores of their mathematics class:

How to Make a Cumulative Frequency Table

Making a cumulative frequency table is easy.

An Example Question

Construct a cumulative frequency table for the numbers below.

Step 1
Construct a table with four columns:



Before we make a cumulative frequency table, we need to make a frequency table (the first three columns).

Read about how to make a frequency table

Step 2
In the Score column, write each number that appears in the set of numbers down the column, from the smallest to largest.




Step 3
Go through the set of numbers in the Question. For each number, cross it out and put a tally mark in the Tally column, in the row for that number.
  • The first number is 8. Cross it out and put a tally mark in the Tally column in the 8 row.


  • The second number is 10. Cross it out and put a tally mark in the Tally column in the 10 row.


  • Continue until all numbers have been crossed out and had tally marks placed in the Tally column.


Step 4
For each row, count the tally marks. Write the number of tally marks in the Frequency column.




Step 5
Complete the Cumulative frequency column. The cumulative frequency in each row is found by adding all the entries in the Frequency column from the top row to that row.
  • The cumulative frequency in the first row is equal to the frequency:


  • The cumulative frequency in the second row is found by adding the frequency in this row to all the frequencies above:



    Note: It is also found by adding the frequency in this row to the cumulative frequency above.

  • The cumulative frequency in the third row is found by adding the frequency in this row to all the frequencies above:



    Note: It is also found by adding the frequency in this row to the cumulative frequency above.

  • The cumulative frequency table is complete when all the cumulative frequencies have been entered:

This is the cumulative frequency table for the set of numbers given in the question.

A Real Example of How to Make a Cumulative Frequency Table

The slider below gives another example of making a cumulative frequency table.
Note

What Is a Cumulative Frequency Table?

A cumulative frequency table is a frequency table with a running total of the frequencies.

A frequency table shows how often - how frequently - each number appears in a list of numbers.

What Are Tally Marks?

Tally marks are a way of helping you to count, especially when you have to count a large set of numbers.

A vertical line is made for the first four numbers:






To make five, a diagonal bar is drawn across the four vertical lines:



After five, single bars are added again:




Once ten is reached, two groups of five tally marks are used:



Finally, it is easy to count the total number of tally marks. For example, consider the tally marks below:



Count up the groups of five tally marks in fives:



Finally add, the number of single marks:



There are 22 tally marks.

Top Tip

Quick Check 1

When there are a large number of numbers to put into a frequency table, it is easy to go wrong counting them.

Crossing the numbers off as you enter them into the Tally column is one way of ensuring you check each number once only.

Another useful check is to add all the numbers in the Frequency column. They should sum to the number of numbers there were to begin with.

For example, if there are six numbers:



You obtain the following frequency table:



Summing up the frequency column gives
2 + 3 + 2 = 7
This is not equal to the six numbers. This tells you that a mistake has been made. (In the example above, there is only one 3, not two).

Try to find the mistake or repeat the exercise.

Quick Check 2

The final entry in the Cumulative frequency column must equal the total of the Frequency column.

Add up the Frequency column to see if it is the same as the last number you write in the Cumulative frequency column:

The Cumulative Frequency Is Increasing

The numbers in the Cumulative frequency column must increase as you go down the rows.



This is because each time you go down a row, you add another frequency to the running total, so it must get larger.

Sometimes a the cumulative frequency will stay the same. What must the frequency be in that row?

Note

Counting the Numbers

In the test scores example, the numbers have been counted one at a time going from left to right.

This is not the only way to do it. You could:
  • Go through and count all the 5s. Then all the 6s. Then all the 7s etc.

  • Order the numbers and group like numbers together.
Find the way you find best to count the numbers without making mistakes. Feel free to try different ways until you find the one that suits you best.