Arithmetic Mean Formula of Grouped Data Using Direct Method | Solved Examples, Important Questions, FAQs

Arithmetic Mean for Grouped Data in Statistics | Class 11 Notes

Written by Neeraj Anand | Published by ANAND TECHNICAL PUBLISHERS

Introduction:

In statistics, the Arithmetic Mean for grouped data is an essential concept used to determine the central tendency of a dataset. For students of Class 11 Mathematics, especially those preparing for Board Exams, JEE Mains, and Advanced, mastering this concept is crucial for solving complex problems involving grouped frequency distributions.

This article explains the formulas, methods, and examples necessary to calculate the arithmetic mean for grouped data using direct method.

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Arithmetic Mean (Average) in Statistics

Arithmetic Mean or the average of the given observations defined as the sum of the product of observations (x_i) and their corresponding frequencies (f_i) divided by the sum of all the frequencies (f_i).

The mean of the data is generally represented by the notation x̄. If x₁, x₂, x₃, …x_n are the number of observations with respective frequencies f₁, f₂, f₃, … f_n, then

The sum of observations = f₁x₁+ f₂x₂ + f₃x₃ + ….+ f_nx_n.

The total number of observations = f₁+f₂+… + f_n.

Therefore, the mean of the data, x̄ = (f₁x₁+ f₂x₂ + f₃x₃ + ….+ f_nx_n)/ ( f₁+f₂+… + f_n).

In short, the above form can be represented using the summation (Σ).

\(\begin{array}{l}\bar{x}=\frac{\sum_{i=1}^{n}f_{i}x_{i}}{\sum_{i=1}^{n}f_{i}}\end{array} \)

Where, “i” varies from 1 to n.

Example.1 : Find the mean for the following distribution.

xi	11	14	17	20
fi	3	6	8	7

Solution:

For the given data, we can find the mean using the direct method.

xi	fi	fixi
11	3	33
14	6	84
17	8	136
20	7	140
	∑fi = 24	∑fi xi = 393

Mean = ∑f_ix_i/∑f_i = 393/24 = 16.4

Example.2 : The marks scored by 30 students of class 10 of a certain school in the Maths paper consisting of 100 marks is given below in the tabular form. Find the mean of the marks obtained by the class 10 students.

Here

x_i = Marks obtained

f_i = Number of students

Solution:

To find the mean of the marks obtained by the students in the Mathematics paper, we need to find the product of each x_i and their corresponding frequency f_i.

Marks Obtained (xi)	Number of students (fi)	fixi
10	1	10
20	1	20
36	3	108
40	4	160
50	3	150
56	2	112
60	4	240
70	4	280
72	1	72
80	1	80
88	2	176
92	3	276
95	1	95
Total	Σfi = 30	Σfixi = 1779

Table 1

Thus, by using the formula,

\(\begin{array}{l}\bar{x}=\frac{\sum_{i=1}^{n}f_{i}x_{i}}{\sum_{i=1}^{n}f_{i}}\end{array} \)

, we get

x̄ = 1779/30

x̄ = 59.3

Hence, the mean of the marks obtained is 59.3.

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Grouped Data in Stastics

Grouping of data plays a significant role when we have to deal with large data. Data formed by arranging individual observations of a variable into groups, so that a frequency distribution table of these groups provides a convenient way of summarizing or analyzing the data is termed as grouped data.

Group data as is expressed in the form of class intervals and not as an individual unit is termed as grouped data. The observations are grouped together to form intervals, which then are assigned frequencies pertaining to the number of times all the units belonging to that particular interval appear in the given data set. Such intervals make it very easy to analyze the data set on hand and help interpret and communicate effectively and quickly.

How to determine the class size?

In order to avoid confusion on the size of the class intervals that we need to take while grouping the data, one must follow the below steps.

Step 1: Identify the highest and the lowest (least) data values in the given observations.

Step 2: Find the difference between the highest and least value.

Step 3: Now, assume the number of class intervals we need (usually 5 to 20 classes are suggested to take based the number of observations).

Step 4: Divide the difference of highest and least value by the number of classes, this result in the size of the class interval.

Step 5: In case of any decimal number obtained as a class size take the nearest whole number greater than the obtained decimal as the class size.

Example:.1

A teacher assigned with the task of marking 60 students’ papers (out of 100 marks) can divide the data set in 10 groups (100-0)/10 = 10, 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 (100-0)/20=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

This method of grouping data makes it so much easier to calculate the measures of central tendency, especially when the data set is large, like in the above case.

Three Methods to Find the Mean of Grouped Data

In many cases, the data is large and to make a meaningful study, the data has to be condensed as grouped data. So, in those cases, we have to convert the ungrouped data into a grouped data and then find the mean. The three methods to find the mean of the grouped data is:

1. Direct Method
2. Assumed Mean Method
3. Step-deviation Method.

Direct Method

Consider the Example.2 as given above. Now, convert the ungrouped data into grouped data by forming a class interval of width 15.

Note, that while taking the frequencies to each class interval, students falling in the upper-class limit will be considered in the next class interval.

Therefore, the grouped frequency distribution table for the above-given example is as follows:

Class Interval	10-25	25-40	40-55	55-70	70-85	85-100
Number of Students	2	3	7	6	6	6

Now, for each class interval, we need to find the midpoint (classmark) that serves as the representative of the whole class.

For example, for the first class interval, 10-25, the class mark is:

Class Mark = (Upper class limit + lower class limit)/2

Class Mark = (25+10)/2 = 17.5

Similarly, find the classmark for all the intervals.

Therefore, the mean of the marks obtained by the students is given as:

Class Interval	Number of students (fi)	Class Mark (xi)	fixi
10-25	2	17.5	35
25-40	3	32.5	97.5
40-55	7	47.5	332.5
55-70	6	62.5	375
70-85	6	77.5	465
85-100	6	92.5	555
Total	Σfi = 30		Σfixi = 1860

Table 2

Therefore, Mean, x̄ = 1860/30 = 62

The mean value obtained using the direct method is 62.

If you compare the mean obtained from Table 1 and Table 2, 59. 3 being the exact mean, whereas 62 is the approximate mean, because of the midpoint assumption in Table 2.

Solved Examples of Arithmetic Mean of Grouped Data (Direct Method)

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

Marks	Number of Students
0 – 10	5
10 – 20	12
20 – 30	14
30 – 40	10
40 – 50	9

Solution:

To Calculate the arithmetic 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 = Σfm/Σf​  = 1310/50​  = 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:

Class Intervals	Frequency
0 – 2	2
2 – 4	4
4 – 6	6
6 – 8	8
8 – 10	10

Solution:

To Calculate the arithmetic 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 = Σfm/Σf​  = 190/30​  = 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:

Class Intervals	Frequency
10 – 20	5
20 – 30	3
30 – 40	4
40 – 50	7
50 – 60	2
60 – 70	6
70 – 80	13

Solution:

To Calculate the arithmetic 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 = Σfm/Σf  = 2080/40 = 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:

Class Intervals	Frequency
100 – 120	4
120 – 140	6
140 – 160	10
160 – 180	8
180 – 200	5

Solution:

To Calculate the arithmetic 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 = Σfm/Σf = 5030/33 = 152.42

Hence, the mean of the given data set is 152.42

How to find Mean of grouped data by direct method – Practice Problems

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

Class Intervals	Frequency
10-20	6
20-30	8
30-40	10
40-50	5
50-60	7

Problem 2: Calculate the median for the following data set

Class Intervals	Frequency
0-10	5
10-20	15
20-30	20
30-40	10
40-50	8

Problem 3: Determine the mode for the following data set

Class Intervals	Frequency
5-15	12
15-25	18
25-35	20
35-45	10
45-55	5

Problem 4: Calculate the standard deviation for the following data set

Class Intervals	Frequency
0-5	4
5-10	6
10-15	8
15-20	5
20-25	7

Problem 5: Find the correlation between the following two variables

Variable X (Class Intervals)	Variable Y (Frequency)
1-10	10
11-20	14
21-30	8
31-40	6
41-50	12

How to Find Mean of Grouped Data by Direct Method – Frequently Asked Questions(FAQs)

Q1

What is grouped data and ungrouped data?

Grouped data means the data (or information) given in the form of class intervals such as 0-20, 20-40 and so on. Ungrouped data is defined as the data given as individual points (i.e. values or numbers) such as 15, 63, 34, 20, 25, and so on.

Q2

What is grouped data example?

Suppose we have a data ranges from 0 to 50 like 2, 17, 0, 1, 8, 19, 43, 2, 1, 32, and so on. In this case, we can group the data into classes such as 0-10, 10-20,…,40-50. This is a simple example of grouped data.

Q3

What are the advantages of grouping data?

The main advantages of grouping data are:
Assist us in concentrating on essential subgroups mainly and overlooks trivial ones
Helps in increasing the efficiency and correctness of the required estimation

Q4

How do you group data into a class?

An important technique used for grouping the given data is tally marks. With the help of tally marks table it is possible to convert the data into classes without any confusion. Then find the height (or size) of the class interval by dividing the difference of the highest and the least data value by the number of classes we want (in case of the decimal value, the nearest whole number defines the class size).

Q5

How many classes does a grouped data can have?

For an ideal grouped data it is suggested to have the number of class intervals as minimum 5 and maximum 20. But we can also observe grouped data with less than 5 class intervals in many situations.

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