Cumulative Frequency Distribution

By cumulative frequency corresponding to a particular value of the variable, we either mean the number of observations less than or equal to ( $\leq$ ) that value or the number of observations greater than or equal to ( $\geq$ ) that value. The former is known as the cumulative frequency of less than type and the latter is known as the cumulative frequency of greater than type.

Thus, cumulative frequency of less than type for a particular value of the variable is obtained by cumulating or adding the frequencies of all values less than that value upto the frequency that particular value, i.e., by adding its frequency to the frequencies of all the values smaller than that value.

Similarly, cumulative frequency of greater than type for a particular value of the variable is obtained by cumulating or adding the frequencies of all values greater than that value, starting from the frequency that particular value, i.e., by adding its frequency to the frequencies of all the values greater than that value.

Specification of the different values of the variable together with the corresponding cumulative frequencies is called a cumulative frequency distribution which is generally represented in the form of a table where the values of the variable are written in order.

Cumulative Relative Frequency is the same as Cumulative Frequency; just that in case of Cumulative Relative Frequency, as the name suggests; we add or cumulate the relative frequencies instead of the simple frequencies.

The purposed of a cumulative frequency distribution are as follows:

The first purpose is to find the number of observations or proportion or percentage of observations below/more than a certain value and in between two given values.
Secondly, a cumulative frequency distribution helps us to find values of different measures like medians, modes etc.

Cumulative Frequency Distribution of Discrete Variable

Let us refer to the following frequency distribution:

Number of car accidents	Frequency
3	5
4	9
5	11
6	4
7	1
Total	30

The cumulative frequency distribution will be:

Number of car accidents	Frequency	Cumulative Frequency		Relative Frequency	Cumulative Relative Frequency
Number of car accidents	Frequency	(< type)	(> type)	Relative Frequency	(< type)	(> type)
3	5	5	30	0.17	0.17	1.00
4	9	14	25	0.30	0.47	0.83
5	11	25	16	0.37	0.84	0.53
6	4	29	5	0.13	0.97	0.16
7	1	30	1	0.03	1.00	0.03
Total	30	–	–	1.00	–	–

The Cumulative Frequency (< type) corresponding to the value 3 is 5 which means that the number of values less than or equal to 3 is 5. Similarly the Cumulative Frequency (< type) corresponding to the value 6 is 29 which means the number of values $\leq 6$ is 5+9+11+4=29, i.e., there are 29 values less than or equal to 6. Similarly, the Cumulative Frequency (> type) corresponding to the value 6 is 5 which means the number of values $\geq 6$ is 4+1=5, i.e., there are 5 values greater than or equal to 6.

If we want to find out the proportion of values less or greater than a particular value we refer to the Cumulative Relative Frequency columns. The Cumulative Relative Frequency (< type) corresponding to 4 is 0.47 which means that the number of values less than or equal to 4 is 0.47 part of the total number of values. Again, Cumulative Relative Frequency (> type) corresponding to 4 is 0.83 which means that the number of values greater than or equal to 4 is 0.83 part of the total number of values.

If the distribution of the discrete variable be a grouped frequency distribution the cumulative frequencies, instead of corresponding to individual values will correspond to the class boundaries. The Cumulative Frequency (< type) corresponds to the upper class boundaries of each class and the Cumulative Frequency (> type) corresponds to the lower class boundaries of each class.

It should be noted that the Cumulative Frequency (< type) corresponding to the highest value and the Cumulative Frequency (> type) corresponding to the lowest value are both equal to the total frequency which is 30.

Cumulative Frequency Distribution of Continuous Variable

Let us refer to the following frequency distribution:

Class Intervals(Temperatures in $^{\circ}C$ )	Frequency
17-20	17
21-24	7
25-28	10
29-32	9
33-36	7
Total	50

The cumulative frequency distribution will be:

Class Intervals(Temperatures in $^{\circ}C$ )	Class Boundaries	Frequency	Cumulative Frequency		Relative Frequency	Cumulative Relative Frequency
Class Intervals(Temperatures in $^{\circ}C$ )	Class Boundaries	Frequency	(< type)	(> type)	Relative Frequency	(< type)	(> type)
17-20	16.5-20.5	17	17	50	0.34	0.34	1.00
21-24	20.5-24.5	7	24	33	0.14	0.48	0.66
25-28	24.5-28.5	10	34	26	0.20	0.68	0.52
29-32	28.5-32.5	9	43	16	0.18	0.86	0.32
33-36	32.5-36.5	7	50	7	0.14	1.00	0.14
Total	–	50	–	–	1.00	–	–

In case of a continuous variable, the distribution will always be grouped. Hence, for a continuous variable the cumulative frequencies, instead of corresponding to individual values correspond to class boundaries. The same rule is followed as in the case of grouped frequency distribution of a discrete variable. Cumulative Frequency(< type) corresponds to upper class boundaries and Cumulative Frequency(> type) corresponds to lower class boundaries.

Here, he Cumulative Frequency(< type) corresponding to the class boundary 20.5 is 17 which means that the number of values less than or equal to 20.5 is 17. Similarly, the Cumulative Frequency(< type) corresponding to the class boundary 28.5 is 34 which means the number of values $\leq 29.5$ is 17+7+10=34, i.e., there are 34 values less than or equal to 28.5. Similarly, the Cumulative Frequency(> type) corresponding to the value 28.5 is 16 which means the number of values $\geq 28.5$ is 9+7=16, i.e., there are 16 values greater than or equal to 28.5.

If we want to find out the proportion of values less or greater than a particular value we refer to the Cumulative Relative Frequency columns. The Cumulative Relative Frequency (< type) corresponding to the class boundary 28.5 is 0.68 which means that the number of values less than or equal to 28.5 is 0.68 part of the total number of values. Again, Cumulative Relative Frequency (> type) corresponding to the class boundary 28.5 is 0.32 which means that the number of values greater than or equal to 28.5 is 0.32 part of the total number of values.

It should be noted that the Cumulative Frequency (< type) corresponding to the upper class boundary of the highest class and the Cumulative Frequency (> type) corresponding to the lower class boundary of the lowest class are both equal to the total frequency which is 50.

Exercise

1. Convert the following frequency distribution into a more than type cumulative frequency distribution:

Weekly Wages (Rs.)	30	60	90	120	150	180
Number of Workers	22	74	132	171	189	200

2. Calculate the greater than type and less than type cumulative frequencies for the following frequency distributions:

Weight of persons (in pound)	Number of persons
125-134	10
135-144	24
145-154	36
155-164	50
165-174	36
175-184	20
185-194	14
195-204	10

3. Obtain the frequency distribution from the data on the weights (in Kg.) of 30 college students:

Weights (in Kg.)	Number of students
Below 40	1
Below 50	6
Below 60	16
Below 70	24
Below 80	30

Cumulative Frequency Distribution of Discrete Variable

Cumulative Frequency Distribution of Continuous Variable

Exercise

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