# 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

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

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

Step 3

**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

**Frequency**column.

Step 5

**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:

# 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.