Cumulative Frequency

Cumulative frequency gives the cut-off value or the phenomenon in which the application will work below a certain fixed limit. This tells the occurence of a phenomenon less than a reference value. This is also refereed by the name Frequency of  NON -Excedence. This is also some times referred as ogive.

Cumulative Frequency Definition


Cumulative frequency corresponding to a certain value is the sum of all frequencies that adds up diagonally to give the total value of the frequency.

Cumulative Frequency Graph

Using both the less than and greater than cumulative frequency distribution table, its possible to draw graph of which lets try to draw cumulative frequency polygon.

Less than cumulative frequency distribution

Lets consider this distribution

Class
Frequency
Cumulative frequency
-0.5 - 9.5 6 6
9.5 - 19.5 22 28
19.25 - 29.5 56 84
29.5 - 39.5 33 117
39.5 - 49.5 5 121
Total
121

When less than cumulative frequency distribution is used for graphing, always consider take the upper limit of the class intervals. So the table changes to

Upper limit
Cumulative frequency
9.5 6
19.5 28
29.5 84
39.5 117
49.5 121
Total

Hence the cumulative frequency graph will be as follows:

Cumulative Frequency Distribution Graph

Greater than cumulative frequency distribution

Lets consider this distribution
Class
Frequency
Cumulative frequency
-0.5-9.5 6 112
9.5-19.5 22 115
19.5-29.5 56 93
29.5-39.5 33 37
39.5-49.5 5 5
Total 121

When greater than cumulative frequency distribution is used for graphing, always consider take the lower limit of the class intervals. So the table changes to
Upper limit
Cumulative frequency
-0.5 112
9.5 115
19.5 93
29.5 37
39.5 5
Total

Hence the graph will be as follows:

Cumulative Frequency Graphs

Cumulative frequency is of more than or less than type. In order to construct a less than frequency we add up all the frequencies at or above the given class and for a more than type we add up all the frequencies from or below the class. It is always better to arrange the data classes in ascending order in order to determine the cumulative frequency.

Cumulative Frequency Graph

Cumulative Relative Frequency Distribution

The relative cumulative frequency can be defined as the quotient between the cumulative frequency of a that particular value and the total number of data taken. It can be expressed as a ratio or percentage.

Lets check an example with the following cumulative distribution

Class
Cumulative frequency
0 - 12 6
10 - 20 28
20 - 30 84
30 - 40 117
40 - 50 121

The relative frequency distribution is as follows

Class
Cumulative frequency
Relative cumulative frequency
0 - 10 6 $\frac{6}{121}$ = 0.049
10 - 20 28 $\frac{28}{121}$ = 0.231
20 - 30 84 $\frac{84}{121}$ = 0.694
30 - 40 117 $\frac{}{117121}$ = 0.967
40 - 50 121 $\frac{121}{121}$ = 1

Cumulative Frequency Distribution

Groups of data always have a less value when compared to data that are placed in some kind of order. The most common type of arrangement is ascending or descending order. Such group will be called as array or as distribution.

The total frequency of all classes that are less than its upper class boundary is called cumulative frequency distribution. It can also be defined as the total frequency of all classes that are greater than its lower class boundary is called cumulative frequency distribution.

Hence, cumulative frequency distribution can be used to determine the number of data that lie either above or below a particular value in a data set.

Given a frequency distribution table, its rather easy to calculate the cumulative frequency.

A tabular arrangement of data showing how many observations lie above, or below, certain values is called cumulative frequency distribution.

Determine the frequency distribution from the given cumulative distribution table.

Class IntervalCumulative frequency
15 - 2050
20 - 2545
25 - 3036
30 - 3523
35 - 4015
40 - 458

We can obtain the frequency of any class by taking the difference between its cumulative frequency and that of the next class.

Class IntervalFrequency
15 - 2050 - 45 = 5
20 - 2545 - 36 = 9
25 - 3036 - 23 = 13
30 - 3523 - 15 = 8
35 - 4015 - 8 = 7
40 - 458 - 0 = 8

As seen from the above table the frequency of class 30 - 35 is 8, class 35 - 40 is 7 and so on.

Cumulative Frequency Distribution Example

The following statistics problems explain the concept of cumulative frequency distribution.

Construct the cumulative frequency distribution for the data recorded from 35 families regarding number of hours spent watching television per day.

Class (Hours)Frequency (# of Families)Less than cumulative frequencyMore than cumulative frequency
05535
171230
251723
332018
452515
553010
62325
72343
81351

In the above table, number of families watching less than or equal to 3 hours of television is obtained by adding up all the frequencies above and at data value 3. So we get the cumulative frequency as 20 (5 + 7 + 5).

Similarly number of families watching more than or equal to 5 hours of television is obtained by adding all the frequencies below and at data value 5. So the more than cumulative frequency here is 10 (5 + 2 + 2 + 1).

If it is mentioned just to determine the cumulative frequency we assume it to be the less than type.

Cumulative Frequency Table

Construct cumulative frequency for the frequency distribution of wages of 50 workers in a factory.

Class (Wages)Frequency (Number of Workers)Cumulative frequency
22 - 2433
25 - 2736
28 - 30612
31 - 331224
34 - 36832
37 - 39638
40 - 42543
43 - 45447
46 - 48249
49 - 51150

From the above table we can very well say that the number of workers earning less than $30 is 12. Such information becomes very useful when a data set is huge.

We can also determine the individual frequency of a class if we are given just the cumulative frequency distribution table as shown in the next example.