How to find Mean of grouped data by direct method?

Grouped data is data that is organized into groups or class intervals instead of listing every individual value. This makes large sets of data easier to understand and analyze.

When there are many observations, writing each value separately can be difficult. So the data is combined into intervals (ranges), and the number of observations in each interval is counted. This count is called the frequency.

Example:

A teacher assigned with the task of marking 60 students' papers (out of 100 marks) can divide the data set in 10 groups, like students who have scored between 0 and 10 would be put under 0- 10 class interval, those who got between 10 and 20 would be put in 10- 20 interval, and so on until the last group (interval) becomes 90- 100. Such division is shown as follows:

Marks Scored	Number of Students
0 - 10	5
10 - 20	10
20 -30	3
30 - 40	10
40 - 50	4
50 - 60	7
60 - 70	9
70 - 80	6
80 - 90	4
90 - 100	2

Alternatively, the teacher could have made 5 class intervals by choosing aa class size of 20, which is shown as follows:

Marks Scored	Number of Students
0 - 20	15
20 - 40	13
40 - 60	11
60 - 80	15
80 - 100	6

Grouping data like this simplifies large datasets and makes it easier to calculate measures such as the mean, median, and mode.

Arithmetic Mean for Grouped Data

The following steps are required in order to calculate the arithmetic mean for grouped data:

• Calculate the midpoints of the class intervals in the given data set. The midpoints, or class marks, denoted by 'm,' are computed by adding up the lower and upper class limits and dividing the said sum by 2. In other words, one needs to find the average of the upper and lower class limits of a particular class to get the midpoints.

Example:

Class Intervals	Class Marks/ Mid- points
0 - 10	= 05
10 - 20	= 15
20 - 30	= 25
30 - 40	= 35
40 - 50	= 45

• Multiply the frequencies of the given class intervals, denoted by 'f,' with their respective class marks, denoted by 'm.'. After multiplying all the f with the respective m, add up all these results to depict it as Σfm.
• Add up all the frequencies together, and denote it with Σf.
• Divide the sum of frequencies (Σf) with the sum of the product of midpoints and frequencies (Σfm).
• The number so obtained is the arithmetic mean for the given data set.

Hence, arithmetic mean for a given data set where class marks are m, and frequencies are f, through direct method is calculated using the following formula:

Sample Questions

Question 1. Calculate the arithmetic mean for the following data set using direct method:

For the computation of mean, we need to calculate the class intervals of the given class intervals. This is done as follows:

Marks	Number of Students(f)	Mid- Points(m)	fm
0 - 10	5	5	25
10 - 20	12	15	180
20 - 30	14	25	350
30 - 40	10	35	350
40 - 50	9	45	405
Σf = 50	Σfm = 1310

Mean = X̄ =  =  = 26.2

Hence, the mean of the given data set is 26.2

Question 2. Calculate the arithmetic mean for the following data set using the direct method:

For the computation of mean, we need to calculate the class intervals of the given class intervals. This is done as follows:

Class Intervals	Frequency(f)	Mid- Points(m)	fm
0 - 2	2	1	2
2 - 4	4	3	12
4 - 6	6	5	30
6 - 8	8	7	56
8 - 10	10	9	90
Σf = 30	Σfm = 190

Mean = X̄ =  =  = 6.33

Hence, the mean of the given data set is 6.33

Question 3. Calculate the arithmetic mean for the following data set using the direct method:

For the computation of mean, we need to calculate the class intervals of the given class intervals. This is done as follows:

Class Intervals	Frequency(f)	Mid- Points(m)	fm
10 - 20	5	15	75
20 - 30	3	25	75
30 - 40	4	35	140
40 - 50	7	45	315
50 - 60	2	55	110
60 - 70	6	65	390
70 - 80	13	75	975
Σf = 40	Σfm = 2080

Mean = X̄ =  =  = 52

Hence, the mean of the given data set is 52.

Question 4. Calculate the arithmetic mean for the following data set using the direct method:

For the computation of mean, we need to calculate the class intervals of the given class intervals. This is done as follows:

Class Intervals	Frequency(f)	Mid- Points(m)	fm
100 - 120	4	110	440
120 - 140	6	130	780
140 - 160	10	150	1500
160 - 180	8	170	1360
180 - 200	5	190	950
Σf = 33	Σfm = 5030

Mean = X̄ =  =  = 152.42

Hence, the mean of the given data set is 152.42

Practice Problems

Problem 1: Calculate the arithmetic mean for the following data set using the direct method

Problem 2: Calculate the mean for the following data set

Problem 3: Determine the mean for the following data set

Problem 4: Calculate the mean for the following data set

Problem 5: Find the mean between the following two variables