# Cumulative Frequency – Explanation & Examples

The definition of cumulative frequency is:

“The cumulative frequency is the frequency of data points that lie up to a certain value in your data.”

In this topic, we will discuss the cumulative frequency from the following aspects:

• What is the cumulative frequency in statistics?
• How to find cumulative frequency?
• Cumulative frequency formula.
• Practical questions.

## What is the cumulative frequency in statistics?

The cumulative frequency is the frequency of data points that lie up to a certain value in your data. Cumulative frequency is used to determine the number of data points that lie above (or below) a certain value in a data set.

The cumulative frequency of a certain data point is the sum of all previous frequencies up to that data point in a frequency table.
The last cumulative frequency value will always be equal to the total number of data points. The data points can be categorical or numeric data.

### – Example 1 of categorical data

The following are the smoking habits of 10 participants from a certain survey. Each individual chooses his smoking habit as “Never smoker”, “Current or former < 1y”, for current or former smokers who quit smoking for less than 1 year, or “Former >= 1y” for former smokers who quit smoking for more than or equal to 1 year.

 participant Smoking habit 1 Never smoker 2 Never smoker 3 Current or former < 1y 4 Never smoker 5 Current or former < 1y 6 Never smoker 7 Never smoker 8 Former >= 1y 9 Former >= 1y 10 Former >= 1y

We can list the occurrences of different smoking habits in the following frequency table.

 Smoking habit frequency Never smoker 5 Current or former < 1y 2 Former >= 1y 3

We see that the most frequent smoking habit is “Never smoker” with 5 occurrences and the least frequent smoking habit is “Current or former < 1y” smoking habit with only 2 occurrences.

We can add a third column for the cumulative frequency.

 Smoking habit frequency cumulative frequency Never smoker 5 5 Current or former < 1y 2 7 Former >= 1y 3 10
• The cumulative frequency for the first smoking habit “Never smoker” is the same as its frequency = 5.
• The cumulative frequency for the second smoking habit “Current or former < 1y” = frequency of previous smoking habit “Never smoker + frequency of second smoking habit ”Current or former < 1y” = 5+2 = 7.
• The cumulative frequency for the third smoking habit “Former >= 1y” = frequency of “Never smoker” + frequency of “Current or former < 1y” + frequency of “Former >= 1y” = 5+2+3 = 10.
• The last number of cumulative frequencies is the same as the total data points which are 10.

The following line graph can be used to plot the cumulative frequency where we plot the categories on the x-axis and the cumulative frequency on the y-axis.

We see that:

• The largest cumulative frequency is 10 so our data points are 10 or 10 participants.
• The cumulative frequency of the first category, never smoker, is 5. This means that its frequency is 5.
• The cumulative frequency of the second category, Current or former < 1y, is 7. This means that the total frequency of never smokers and Current or former < 1y smokers is 7. The individual frequency of the current or former < 1y smokers = current cumulative frequency – previous cumulative frequency = 7-5 = 2.
• The cumulative frequency of the last category, Former >= 1y, is 10. This means that the total frequency of never smokers, Current or former < 1y smokers, and Former >= 1y is 10. The individual frequency of the Former >= 1y smokers is 10-7 = 3.

### – Example 2 of categorical data

The following is the frequency table for the marital status of 100 participants from a certain survey.

 marital status frequency No answer 0 Never married 29 Separated 1 Divorced 14 Widowed 20 Married 36

We see that the most frequent marital status is “Married” with 36 occurrences.

We can add a third column for the cumulative frequency.

 marital status frequency cumulative frequency No answer 0 0 Never married 29 29 Separated 1 30 Divorced 14 44 Widowed 20 64 Married 36 100
• The cumulative frequency for the first marital status “No answer” is the same as its frequency = 0.
• The cumulative frequency for the second marital status “Never married” = frequency of first marital status + frequency of second marital status = 0+29 = 29.
• The cumulative frequency for the third marital status “Separated” = frequency of first marital status + frequency of second marital status + frequency of third marital status = 0+29+1 = 30.
• The cumulative frequency for the fourth marital status “Divorced” = frequency of first marital status + frequency of second marital status + frequency of third marital status+ frequency of fourth marital status = 0+29+1+14 = 44, and so on.
• The last number of cumulative frequency is the same as the total data points which are 100.

The following line graph can be used to plot the cumulative frequency.

We see the same information that we concluded from the table.

### – Example 3 of numerical data

The following is the frequency table for the number of cylinders of 32 different car models in 1973-1974.

 Number of cylinders frequency 4 11 6 7 8 14

We see that the most frequent number of cylinders is 8 with 14 occurrences or 14 different cars have this number of cylinders. The least frequent number is 6 with only 6 cars having this number.

We can add a third column for the cumulative frequency.

 Number of cylinders frequency cumulative frequency 4 11 11 6 7 18 8 14 32
• The cumulative frequency for the first number of cylinders “4” is the same as its frequency = 11.
• The cumulative frequency for the second number “6” = frequency of 4 + frequency of 6 = 11+7 = 18.
• The cumulative frequency for the third number “8” = frequency of 4 + frequency of 6 + frequency of 8 = 11+7+14 = 32.
• The last number of cumulative frequency is the same as the total data points which are 100.

The following line graph can be used to plot the cumulative frequency.

We see the same information that we concluded from the table.

### – Example 4 of numerical data

The following is the frequency table for the weights of 100 participants (in Kg) from a certain survey.

 Weight frequency 43.5 1 45.8 1 49 1 50.4 1 51 1 53 3 53.6 1 54 1 55 2 55.5 1 55.8 1 56.4 1 56.6 1 56.8 1 57 1 58 1 59 1 60 2 60.3 1 61 2 62 1 63 1 63.4 1 64 3 65 2 65.5 1 66 4 67 4 67.5 1 68 3 69 4 70 5 71 1 71.5 1 72 2 72.4 1 73 2 74 1 75 4<