**Cumulative Frequency**

Cumulative frequency refers to the running total of frequencies in a data set. It provides valuable information about the distribution of the data and allows for easy interpretation and analysis. Cumulative frequency can be presented in two different types: less-than type and more-than type.

**Less-Than Type Cumulative Frequency**

In the less-than type cumulative frequency, each value is associated with the frequency of values less than or equal to it. It is calculated by adding up the frequencies of all the values that are less than or equal to a given value.

For example, let's consider a data set of exam scores: 60, 70, 80, 90, 95. The frequencies of these scores are 3, 5, 7, 9, and 10 respectively. The less-than type cumulative frequency for each score would be:

- For the score of 60, the cumulative frequency is 3, as there are 3 scores less than or equal to 60.
- For the score of 70, the cumulative frequency is 5, as there are 5 scores less than or equal to 70.
- For the score of 80, the cumulative frequency is 7, as there are 7 scores less than or equal to 80.
- For the score of 90, the cumulative frequency is 9, as there are 9 scores less than or equal to 90.
- For the score of 95, the cumulative frequency is 10, as there are all the scores in the data set.

**More-Than Type Cumulative Frequency**

In the more-than type cumulative frequency, each value is associated with the frequency of values greater than or equal to it. It is calculated by subtracting the less-than type cumulative frequency from the total frequency.

Using the same example of exam scores, the more-than type cumulative frequency for each score would be:

- For the score of 60, the more-than type cumulative frequency is 10 - 3 = 7.
- For the score of 70, the more-than type cumulative frequency is 10 - 5 = 5.
- For the score of 80, the more-than type cumulative frequency is 10 - 7 = 3.
- For the score of 90, the more-than type cumulative frequency is 10 - 9 = 1.
- For the score of 95, the more-than type cumulative frequency is 10 - 10 = 0.

**Both Types of Cumulative Frequency**

In conclusion, cumulative frequency refers to both the less-than type and more-than type. It provides a comprehensive overview of the distribution of data, allowing for comparison and analysis of different values within the data set. Both types of cumulative frequency are useful in various statistical analyses and graphical representations.

Both