Assumed Mean Method: Formula, Steps & Examples

Content Writer

Assumed Mean Method is used to calculate the arithmetic mean of grouped data. It helps to reduce the calculations and results in small numerical values, which makes it easy to calculate the mean easily with no long calculations.

Assumed mean method is recommended if the given data is large.
The direct method for calculating the mean is not suitable in the case.
The method depends on assuming the mean and rounding to an easy value to calculate.
The value obtained is subtracted from all the sample values.
A central class is chosen when the samples are converted into equal-size ranges or class intervals and the calculations are performed.
Assumed Mean can be calculated using the formula

a + (Σf_id_i /Σf_i).

where,

a denotes the assumed mean
fi is the frequency of the ith class
d_i is the deviation of the ith class
Σf_i refers to the total number of observations
x_i is the class mark.

Related Links

Table of Content

Assumed Mean Method
Assumed Mean Method Formula
How to Calculate Mean using Assumed Mean Method?
Solved Examples on Assumed Mean Method
Things to Remember
Sample Questions

Key Terms: Assumed Mean Method, Mean, Assumed Mean, Arithmetic Mean, Step-deviation Method, Grouped Data, Class Mark, Class Interval

Assumed Mean Method

[Click Here for Sample Questions]

Assumed mean method is used for calculating the arithmetic mean in Statistics. The mean of grouped data refers to the average of a set of data that are grouped together in different categories.

There are three main methods of calculating the mean of grouped data, namely the direct method, the assumed mean method, and the step deviation method.
Assumed Mean Method is the most feasible option to calculate the mean.
It is also used to find out the standard deviation of a data set.
Assumed mean method simplifies the calculation of the mean of the grouped data in comparison to the direct method.
The method is generally used for data sets with large values or samples.

Methods to find arithmetic mean

Methods to Find Arithmetic Mean

The video below explains this:

Assumed Mean Method Detailed Video Explanation:

Read More:

Relevant Topics
Difference between Mean and Median	Mode	Median
Median of Grouped Data	Mean Absolute Deviation	Statistics Formulas

Assumed Mean Method Formula

[Click Here for Sample Questions]

Assumed Mean Method Formula helps to calculate the mean for large data values. Let x₁, x₂, x₃,…, and x_n be mid-points or class marks of n class intervals. f₁, f₂, f₃, …, and f_n are the respective frequencies. The assumed mean method formula is:

\(\bar{x} = a + \frac{\sum f_id_i}{\sum f_i}\)

Where

a: Assumed Mean
f_i: Frequency of ith Class
d_i = x_i – a: Deviation of ith Class
Σf_i= n: Total Number of Observations
x_i = Class Mark: (Upper-Class Limit + Lower-Class Limit) / 2

Read More: Weighted Mean Formula

How to Calculate Mean using Assumed Mean Method?

[Click Here for Sample Questions]

In order to calculate the mean of grouped data using the assumed mean method, one needs to follow the given steps:

First, calculate the midpoint or x_i for the class interval.
Take the central value from the class marks as the assumed mean and mark it as a.
Now, calculate the deviation di = x_i - a for each i.
Calculate the product of d_if_i for each i.
Now, find the total of f_i.
Lastly, calculate the mean by using the assumed mean method = a + ∑d_if_i / ∑f_i.

Read More: Frequency Distribution Table

Solved Examples on Assumed Mean Method

[Click Here for Sample Questions]

Given below are some examples of calculating the mean of grouped data by the assumed mean method:

Example 1: The table given below shows the marks obtained by 110 students in class. What will be the mean marks of the students? Use the assumed mean method.

Class	0-10	10-20	20-30	30-40	40-50
Frequency	12	28	32	25	13

Solution: The given data is:

Class (C_i)	Frequency (f_i)	Class mark (x_i)	d_i = x_i – a	f_id_i
0-10	12	5	5 – 25 = – 20	-240
10-20	28	15	15 – 25 = – 10	-280
20-30	32	25 = a	25-25 = 0	0
30-40	25	35	35-25 = 10	250
40-50	13	45	45-25 = 20	260
Total	Σf_i =110			Σf_id_i = -10

Let us take Assumed mean = a = 25
Using the Assumed Mean Method
Mean = a + (Σf_id_i / Σf_i)
25 + (-10 / 110)
25 – ( 1 / 11)
(275 – 1) / 11
274 / 11 = 24.9

Thus, the mean marks of the students in the class are 24.9.

Example 2: A group of students surveyed 20 homes in a locality on the number of plants they have in their homes.

Number of Plants	0 - 2	2 - 4	4 - 6	6 - 8	8 - 10	10 - 12	12 - 14
Number of Houses	1	2	1	5	6	2	3

Find the mean number of plants per household using the assumed mean method of calculating mean.

Solution: The data is given as:

No. of Plants	No.of Houses (f_i)	X_i	d_i= x_i - a	f_id_i
0-2	1	1	1-7=-6	-6
2-4	2	3	3-7=-4	-8
4-6	1	5	5-7=-2	-2
6-8	5	7=a	7-7=0	0
8-10	6	9	9-7=2	12
10-12	2	11	11-7=9	18
12-14	3	13	13-7=6	18
Total	Σf_i =20			Σf_id_i = 32

We have taken 7 as the assumed mean here.
Using the assumed mean method,
Mean = a + (Σf_id_i / Σf_i)
7 + (32 / 20)
7+ (8 / 5)
8.6

The required answer is 8.6.

Read More:

Related Topics
Measures of Central Tendency	Graphical Representation of Data and Central Tendency	Mode of Grouped Data
Relation Between Mean Median and Mode	Mode Formula	Measures of Dispersion

Things to Remember

Mean is the average or central value of a set of numbers that is used to measure the central tendency of the data.
There are three methods to calculate the mean of grouped data namely direct method, assumed mean method, and step-deviation method.
Assumed Mean Method is used when data is large, and we can not use the direct method.
Assumed Mean is calculated using the formula a + (Σf_id_i /Σf_i).
The method results in small numerical values that make the calculations easy for data sets with large values.

Sample Questions

Ques. The table depicts information about the percentage distribution of female employees in a company of various branches and a number of departments. Find the mean percentage of female employees using the assumed mean method. (3 Marks)

Solution: We will calculate the class mark and deviation as follows:

Percentage of Female Employees	Number of Departments (f_i)	Class Mark (x_i)	d_i = x_i – a	f_id_i
5-15	1	10	-30	-30
15-25	2	20	-20	-40
25-35	4	30	-10	-40
35-45	4	40 = a	0	0
45-55	7	50	10	70
55-65	11	60	20	220
65-75	6	70	30	180
Total	Σf_i =35			Σf_id_i = 360

Let the assumed mean = a = 40
Using the Assumed Mean Method Formula,
Mean = a + (Σf_id_i/Σf_i)
40+ (360/35)
40+(72/7)
40 + 10.28
50.28

Therefore, the mean percentage of female employees is calculated as 50.28.

Ques. How do you define the assumed mean method? (2 Marks)

Ans. In statistics, the assumed mean is defined as a method that is used for calculating the arithmetic mean and standard deviation of a data set. It is used for the calculation of the mean for data with larger values or samples.

Ques. Calculate the mean of the following data using the assumed mean method. (3 Marks)

Ans. Using the data given above, we get

Class Interval	(f_i)	(x_i)	d_i= x_i - A	f_id_i
0 - 10	12	5	- 20	- 240
10 - 20	15	15	- 10	- 150
20 - 30	10	25 (a)	0	0
30 - 40	25	35	10	250
40 - 50	20	45	20	400
Total	82			260

Here, the assumed mean is taken as 25.
Using the Assumed Mean Method Formula,
Mean = a+ (Σf_id_i /Σf_i)
25 + 260/82
25 + 3.170
28.17.

Thus, the mean for the data is 28.17.

Ques. What are the different methods to calculate the mean of a given frequency? (3 Marks)

Ans. There are different methods to calculate the mean, however, the most prominent methods to calculate the mean are the direct method, assumed mean method, and step deviation method.

The direct method is used for classes with small frequencies and class marks.
The assumed mean method and step deviation method can be used for large data structures.

Ques. Find the arithmetic mean for the following data using the assumed mean method. (3 Marks)

Ans. Take the assumed mean A = 15

x_i	f_i	d_i = x - A	f_id_i
5	4	-10	-40
10	5	-5	-25
15	7	0	0
20	4	5	20
25	3	10	30
30	2	15	30
Total	∑f_i = 25		∑f_id_i = 15

Using the assumed mean method formula,
Arithmetic mean = a + (Σf_id_i/Σf_i)
15 + (15/25)
15 + (3/5)
(75 + 3)/5
78/5
15.6

Thus, the arithmetic mean of the given data is 15.6.

Ques. What will be the mean for the following data? (3 Marks)

Ans. We will use the assumed mean method to solve the question. Let us take the assumed mean a = 80

xi	f_i	d_i = x - A	f_id_i
65	6	-15	-90
70	11	-10	-110
75	3	-5	-15
80	5	0	0
85	4	5	20
90	7	10	70
95	10	15	150
100	4	20	80
Total	∑fi = 50		∑f_id_i = 115

Arithmetic Mean = a+ (Σf_id_i /Σf_i)
80 + (115/50)
80 + (23/10)
80 + 2.3
82.3

Hence, the mean for the given data is 82.3.

Ques. Given below is the data about the number of boys of a particular age in a class of 40 students. Calculate the mean age of the students in the class. (3 Marks)

Ans. Taking the assumed mean a = 16

x_i	f_i	d_i = x - A	f_id_i
13	3	-3	-9
14	8	-2	-16
15	9	-1	-9
16	11	0	0
17	6	1	6
18	3	2	6
Total	∑f_i= 40		∑f_id_i = -22

Mean = a+ (Σf_id_i /Σf_i)
16 + (-22/40)
16 - 0.55
15.45

Thus, the mean age of the students is 15.45 years.

Ques. What will be the arithmetic mean of the following data? Use the assumed mean method to calculate the same. (3 Marks)

Ans. We will take the assumed mean as 45.

xi	f_i	d_i= x - A	f_id_i
15	12	-30	-360
25	20	-20	-400
35	15	-10	-150
45	14	0	0
55	16	10	160
65	11	20	220
75	7	30	210
85	8	40	320
Total	∑f_i = 103		∑f_id_i = 0

Using the assumed mean method,
Mean = a + (Σf_id_i /Σf_i)
Mean = 45 + (0/103)
Mean = 45

Ques. What is the formula to calculate the mean through the assumed mean method? (3 Marks)

Ans. Mean through the assumed mean method is calculated by

Mean = a + (Σf_id_i /Σf_i)

Here,

a refers to the assumed mean.
d refers to deviation (x – A).
f refers to frequency.

In the assumed mean method, the data is given in the frequency distribution table. A value from the data is assumed as the mean value for calculation. On the basis of this assumed mean, the deviation is calculated by using d = x – A. Then we put all the values in the formula and compute it.

Ques. What is mean? (2 Marks)

Ans. Mean is one of the measures of central tendency along with the mode and median. Mean is the average of the given set of values. Mean denotes the equal distribution of values for a given data set or values. To calculate the mean, we need to add the total values and divide the sum by the total number of values.

Ques. The following table gives information about the marks obtained by 150 students in an examination. (3 marks)

Class	0-10	10-20	20-30	30-40	40-50
Frequency	10	20	30	40	50

Find the mean marks of the students using the assumed mean method.

Ans.

Class (CI)	Frequency (f_i)	Class mark (x_i)	d_i = x_i – a	f_id_i
0-10	10	5	5 – 25 = – 20	-200
10-20	20	15	15 – 25 = – 10	-200
20-30	30	25 = a	25-25 = 0	0
30-40	40	35	35-25 = 10	400
40-50	50	45	45-25 = 20	1000
Total	Σf_i =150			Σf_id_i = 1000

Assumed mean = a = 25
Mean of the data: a + ( Σf_id_i / Σf_i)
25 + (1000 / 150)
25 + (6.67 )
31.66

Hence, the mean marks of the students are 31.66.

Ques. The following table shows the weight of 15 students. Calculate using Assumed Mean Method. (3 marks)

Weight ( in kg )	Number of students
47	4
48	6
49	1
50	4
51	4

Ans. Let the assumed mean be A = 49

Weight ( in kg ) x_i	Number of students f_i	d_i=x_i−A =x_i− 49	f_id_i
47	4	-2	-8
48	6	-1	-6
49	1	0	0
50	4	1	4
51	4	2	8
–	∑f_i=19	–	Σf_id_i = – 2

Assumed mean = a = 49
Mean of the data: a + ( Σf_id_i / Σf_i)
49 + (- 2 / 19)
49 – 0.105
48.89

Hence, the mean weight of the students are 48.89.

Check-Out:

Class 9 & 10 Study Guides
NCERT Solutions Class 6 to 12	NCERT Solutions for Class 10 Maths	NCERT Solutions for Class 9 Maths
Biology Study Notes	Biology MCQs	NCERT Solutions for Class 9 Science
Physics Study Notes	Choice-based questions in physics	Topics with relation in physics
Topics for Comparison in Physics	NCERT Solutions for Class 10 Science	Important Derivations in Physics
Physics MCQs	Formulas in Physics	Important Physics Constants and Units
Chemistry Study Notes	Comparison topics in Chemistry	Chemistry MCQs
Maths MCQs	Important Chemistry Formulas	Important Named Reactions
Comparison Topics in Maths	Important Maths Formulas	Comparison Topics in Biology

Assumed Mean Method: Formula, Steps & Examples

Assumed Mean Method

Assumed Mean Method Detailed Video Explanation:

Assumed Mean Method Formula

How to Calculate Mean using Assumed Mean Method?

Solved Examples on Assumed Mean Method

Things to Remember

Sample Questions

CBSE X Related Questions

1.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

4.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

5.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

6.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

Assumed Mean Method: Formula, Steps & Examples

Assumed Mean Method

Assumed Mean Method Detailed Video Explanation:

Assumed Mean Method Formula

How to Calculate Mean using Assumed Mean Method?

Solved Examples on Assumed Mean Method

Things to Remember

Sample Questions

CBSE X Related Questions

1.Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

2.In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

4.Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

5.Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

6.An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

4.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

5.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

6.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is