# Cumulative Frequency

Cumulative frequency is defined as a running total of frequencies i.e., the sum of all previous frequencies up to the current point.

**Prerequisite/Revise this:**

↪ The cumulative frequency at a certain point is found by adding the frequency at the present point to the cumulative frequency of the previous points.

↪ Cumulative frequency can be used to determine the number of observations that lie below (or above) a particular value in a data set. This is very helpful in finding the median for large data.

↪ It is usually observed by constructing a cumulative frequency table.

**Example** – Data of marks, out of 50, obtained by 100 students in a test is arranged in a frequency table as below:

We can calculate the number of observations or cumulative frequency up to the current point(value) as following :

∴ Cumulative frequency table for given data can be written as:

↪ The cumulative frequency for the first data point is the same as its frequency since there is no cumulative frequency before it.

↪ The last value will always be equal to the total number of observations (Σ*f*), since all frequencies will already have been added to the previous total.

## Cumulative Frequency Distribution For Grouped Data

Cumulative frequency for a class is the number of observations that lie below or above of that particular class in a data set.

**↪ Cumulative frequency distribution of the less than type**

The cumulative frequency for a class is the sum of frequencies less than the upper class limit and found by adding the frequency of the present class to the cumulative frequency of the previous class.

It increases as we move down the classes.

**↪ Cumulative frequency distribution of the more than type**

The cumulative frequency for a class is the sum of frequency more than or equal to the lower class limit and found by adding the frequency of the present class to the cumulative frequency of the next class. Or it can be found by subtracting the respective frequency from cumulative frequency of the previous class.

It decreases as we move down the classes.

**Ex**– A grouped frequency distribution of marks obtained, out of 100, by 53 students, in a certain examination, is as follows:

↪ We can calculate the cumulative frequency of less than type for each class as:

∴ The cumulative frequency distribution table of less than type for the given data can be finally written as:

↪ We can calculate the cumulative frequency of more than type for each class as:

∴ The cumulative frequency distribution table of more than type for the given data can be finally written as:

# Cumulative Frequency Graph (Ogive)

A cumulative frequency graph, or an Ogive, is a curve showing the cumulative frequency for a given set of data.

*The term ‘ogive’ is pronounced as ‘ojeev’ and is derived from the word ogee. It has a shape of an elongated ‘S’ and is sometimes called a double curve with one portion being concave and the other being convex. In architecture, the ogee shape is one of the characteristics of the 14th and 15th century Gothic styles.*

## Ogive For Ungrouped data

Variables and Cumulative frequencies are plotted on x-axis and y-axis on a suitable scales respectively. Data points(x, y) are plotted and joined to obtain the graph.

## Ogive For Grouped Data

### Less than Ogive

Upper Class limits and Cumulative frequencies are plotted on x-axis and y-axis on a suitable scales respectively. Data points(x, y) are plotted and joined to obtain the graph.

### More than Ogive

Lower Class limits and Cumulative frequencies are plotted on x-axis and y-axis on a suitable scales respectively. Data points(x, y) are plotted and joined to obtain the graph.