Cumulative Frequency

Cumulative frequency is obtained by adding the frequency of a class interval and the frequencies of the preceding intervals up to that class interval.

The whole frequency of all classes less than the upper class boundary of a specified class is called the cumulative frequency of that class.

If cumulative frequencies are represented in a table then it is called as cumulative frequency distribution. Now let’s see the types of cumulative frequency distribution.

If instead of taking the simple cumulative frequency, as in the case of ogive, we take the ordinate as the percentage cumulative frequency, we shall get a percentage cumulative frequency curve. To draw such a curve, first of all the simple frequency must be expressed as percentage of the total frequency. Then such percentages are cumulated and plotted as in the case of an ogive. Such a curve is useful for comparing different frequency distributions as they are adjusted to a uniform standard. It is to be noted that the ogive for a discrete series is drawn on the assumption that the data is continuous. When the class frequencies run up to a maximum at one end of the range, they form a J-shaped curve.

To form the cumulative percentage frequency distribution table ensure the given steps

• Frequency distribution table is constructed.
• Calculate the cumulative frequency for the given individual data according to their frequencies.
• Substitute the cumulative frequency in the given formula and calculate the cumulative percentage values.

How to calculate the cumulative percentage frequency?

Cumulative frequency% = $\left(\frac{(Cumulative\ Frequency)}{Total\ Number (N)} \times 100\right )$

where N is total number of frequencies.

Example- Cumulative percentage frequency distribution

Form a cumulative percentage frequency distribution table with the given following data

4, 3, 1, 5, 6, 3, 4, 3, 4, 1, 5, 6, 4, 2, 1, 3, 5.

Solution:

Step 1- Cumulative percentage frequency distribution

Form the frequency distribution table as follows,

Classes	Class boundaries	Frequency
1	0.5- 1.5	3
2	1.5- 2.5	1
3	2.5-3 .5	4
4	3.5- 4.5	4
5	4.5- 5.5	3
6	5.5-6.5	2

Step 2 - Cumulative percentage frequency distribution

Form the cumulative frequency for the given table,

Classes	Class boundaries	Frequency	Cumulative Frequency
1	0.5-1.5	3	3
2	1.5-2.5	1	3 + 1 = 4
3	2.5-3.5	4	4+ 4 = 8
4	3.5-4.5	4	8 + 4 =12
5	4.5-5.5	3	12 + 3 = 15
6	5.5-6.5	2	15 + 2 = 17

N = 17

Step 3 -Cumulative percentage frequency distribution

Calculate the cumulative percentage using the formula,

cumulative frequency %= $\left(\frac{(Cumulative\ Frequency)}{Total\ Number (N)} \times 100\right )$

where N is total number of frequencies.

Classes	Class boundaries	Frequency	Cumulative Frequency	C. F % = c.f / N * 100
1	0.5-1.5	3	3	3/ 17 * 100 = 17
2	1.5-2.5	1	3 + 1 = 4	4/ 17 * 100 = 23
3	2.5-3.5	4	4+ 4 = 8	8/ 17 * 100 = 47
4	3.5-4.5	4	8 + 4 =12	12/ 17 * 100 = 70
5	4.5-5.5	3	12 + 3 = 15	15/ 17 * 100 = 88
6	5.5-6.5	2	15 + 2 = 17	17/ 17 * 100 = 100

N = 17

Thus a cumulative percentage frequency distribution table is created.

Step 4 -Cumulative percentage frequency distribution

we can plot the graph for the cumulative percentage frequency as follows,

An Ogive (cumulative frequency graph) is a graph that represents the cumulative frequencies of the classes in a frequency distribution. It shows the data below or above a particular value. The two types of Ogives are less than Ogive and greater than or more than Ogive.

Steps for constructing a less than Ogive chart (less than Cumulative frequency graph): -

1. Draw and label the horizontal and vertical axes.

2. Take the cumulative frequencies along the y axis (vertical axis) and the upper class limits on the x axis (horizontal axis)

3. Plot the cumulative frequencies against each upper class limit.

4. Join the points with a smooth curve.

Steps for constructing a greater than or more than Ogive chart (more than Cumulative frequency graph): -

1. Draw and label the horizontal and vertical axes.

2. Take the cumulative frequencies along the y axis (vertical axis) and the lower class limits on the x axis (horizontal axis)

3. Plot the cumulative frequencies against each lower class limit.

4. Join the points with a smooth curve.

Example

Draw the less than cumulative frequency curve for the following data

Class	20-25	25-30	30-35	35-40	40-45	45-50	50-55
F	10	12	8	20	11	4	5

Solution

First lets find the less than cumulative frequency corresponding to each class. For this the frequencies of all preceding classes are added to the frequency of a class. The less than cumulative frequency table is given below.

Upper Limit	Frequency	Less than Cumulative frequency
25	10	10
30	12	10 + 12 = 22
36	8	22 + 8 = 30
40	20	30 + 20 = 50
45	11	50 + 11 = 61
50	4	61 + 4 = 65
55	5	65 + 5 = 70

Now we draw the horizontal and vertical axes and label it. Plot the cumulative frequencies corresponding to the upper limit of each class and join the points using a smooth curve.

The less than cumulative frequency curve is shown below.


When the cumulative frequencies plotted against the class limits are joined by straight lines, we get a cumulative frequency polygon.

The cumulative frequency polygon of the above problem is given by


Relative cumulative frequency of a class is the frequency obtained by dividing cumulative frequency by the total frequency.

The relative frequency calculation is illustrated below

Upper Limit	Frequency	Less than Cumulative frequency	Relative Cumulative Frequency
25	10	10	10 / 70 = 0.143
30	12	10 + 12 = 22	22 / 70 = 0.314
35	8	22 + 8 = 30	30 / 70 = 0.429
40	20	30 + 20 = 50	50 / 70 = 0.714
45	11	50 + 11 = 61	61 / 70 =0. 871
50	4	61 + 4 = 65	65 / 70 = 0.929
55	5	65 + 5 = 70	70 / 70 =1
Total	70

Example:

The following frequency distribution table gives the marks obtained by 40 students:

Table (a)

Note: The frequencies can be added, as indicated by the arrows, to obtain the cumulative frequency.

In the table(a), it is observed that 4 students got marks 'less than 10', 9 students got marks 'less than 20' and so on.

Therefore, the above distribution is called 'less than' cumulative frequency distribution.

Table (a) can be re-written as table (b).

Class	Cumulative Frequency
Less than 10	4
Less than 20	9
Less than 30	21
Less than 40	32
Less than 50	40

Table (b)

In the same way 'more than' cumulative frequency distribution can be obtained by adding to the other frequencies in the reverse order.

Table (c)

Note: The frequencies can be added, as indicated by the arrows, to obtain the cumulative frequency.

Table (c) can be re-written as table (d)

Class	Cumulative Frequency (c.f.)
More than 0	40
More than 10	36
More than 20	31
More than 30	19
More than 40	8

Table (d)