Assumed Mean Method

Assumed Mean Method is a statistical technique that is used to calculate the arithmetic mean of a group of data. It is particularly helpful when dealing with large numbers in grouped data. This method involves selecting a central value, known as the assumed mean, and then adjusting the calculations around this value to make the arithmetic more manageable. This technique is used in data analysis to estimate the central tendency of a dataset when the exact mean is not known.

For instance, if you have a data set with class intervals and their respective frequencies, the assumed mean method allows you to break down the problem into simpler steps, making it easier to find the mean without any hard calculations.

What is Mean?

Mean, often referred to as the average, is a measure of central tendency that provides a single value representing the center of a data set.

It is calculated by summing all the values in the data set and then dividing by the number of values. There are three different methods to find the mean of a group of data. These 3 methods are as follows:

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

In this article, we will discuss the Assumed Mean Method in detail.

What is Assumed Mean Method?

Assumed Mean Method, also known as the "Shortcut Method". Assumed Mean Method works by choosing an assumed mean (A) close to the actual mean, and then calculating deviations from this assumed mean to calculate the actual mean of the given dataset.

Assumed Mean Method Formula

Assumed Mean Method simplifies the calculation of the mean by using an assumed mean (A). The formula for the Assumed Mean Method is:

x̄ = a + ∑ƒ_id_i /∑ƒ_i

Where,

• a is assumed mean,
• ƒ_i is frequency of i^th class,
• d_i = x_i - a is derivation of i^th class,
• ∑ƒ_i = n is total number of observations
• x_i is class mark and is equal to (upper class limit + lower class limit)/2

Steps to Find Mean using Assumed Mean Method

For process for calculating mean by using assumed mean method, are discussed below:

Step 1: For each class interval, we have to calculate the class mark by using the formula,

x_i = (upper limit + lower limit)/2

Step 2: Choose an approximate and suitable value of mean, and denote it by "a" .

Step 3: Calculate the deviations using the following formula,

d_i = (x_i - a) for each i

Step 4: Calculate the product by,

(ƒ_i × d_i ), for each i

Step 5: Find total frequency

n = ∑ƒ_i

Step 6: Finally, calculate the mean by using the following formula,

x̄ = a + ∑ ƒ_id_i/∑ƒ_i

Let's consider an example for better understanding.

Example: Find mean using assumed mean method for following data:

Solution:

Step 1: Calculate mid-point for each class-interval:

Class Interval	Midpoint (xi​)	Frequency (f)
10 - 20	15	3
20 - 30	25	7
30 - 40	35	12
40 - 50	45	15
50 - 60	55	8
60 - 70	65	5

Step 2: Select a midpoint close to the center of the data. Let's assume a = 45.

Step 3: For each class interval, calculate the deviation from the assumed mean using d_i = (x_i - a) for each i.

Step 4: Multiply each deviation by the corresponding frequency.

Class Interval	Midpoint (xi​)	Frequency (fi)	Deviation (di​)	fi​ × di​
10 - 20	15	3	15 - 45 = -30	3 × (-30) = -90
20 - 30	25	7	25 - 45 = -20	7 × (20) = -140
30 - 40	35	12	35 - 45 = -10	12 × (-10) = -120
40 - 50	45	15	45 - 45 = 0	15 × 0 =0
50 - 60	55	8	55 - 45 = 10	8 × 10 =80
60 - 70	65	5	65 - 45 = 20	5 × 20 = 100
		∑ƒi = 50		∑ƒidi = -170

Step 5: Calculate the mean using formula: x̄ = a + ∑ ƒ_id_i/∑ƒ_i

x̄= 45 + (-170)/50

⇒ x̄= 45 - 3.4 = 41.6

Thus, mean of given dataset is 41.6

Difference between Direct, Assumed Mean and Step Deviation Method

Direct Method, Assumed Mean Method, and Step Deviation Method are three different techniques for calculating the mean (average) of a dataset. Each method has its own way of simplifying the calculations. Some of the common differences among these methods are:

Also Read,

• Relation between Mean, Median and Mode
• Mode of Grouped Data
• Statistics Formula
• Calculation of Median in Discrete Series

Solved Examples on Assumed Mean Method

Example 1: The following table gives information about the marks obtained by 120 students in an examination.

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

Solution:

Let assume mean of the given data be 25 i.e., a = 25.

The formula,

x̄ = a + ∑ƒ_id_i/∑ƒ_i

⇒ x̄ = 25 + (-10/120)

⇒ x̄ = 25 - 1/12

⇒ x̄ = (300-1)/12

⇒ x̄ = 299/12

⇒ x̄ = 24.91

Therefore, the mean marks of the students are 24.91 .

Example 2: A group of students surveyed as a part of their environmental awareness

The program in which they collected the following data of plants in 20 homes in a area. Find the mean number of plants per household using the assumed mean method.

Solution:

x̄ = a+ (Σƒ_id_i /Σƒ_i)

⇒ x̄ = 7+(32/20)

⇒ x̄ = 7+(8/5)

⇒ x̄ = 8.6

∴ Mean number of plants per household = 8.6

Practice Questions on Assumed Mean Method: Unsolved

Problem 1: The following table contains information on the grades received by 160 students in an examination. Find the mean marks of the students using the assumed mean method.

Problem 2: Find the mean of the following data using the assumed mean method formula.

Problem 3: Find the arithmetic mean using the assumed-mean method:

Problem 4: The following table contains, distribution of daily wages of 60 worker of a factory.

Find the mean daily wages of the workers of the factory by using an appropriate method.

Problem 5: Find the mean of the following data using assumed mean method.

Conclusion

Assumed Mean Method is a valuable tool in statistics for estimating the mean of a dataset when exact values are unavailable. By leveraging an assumed mean, analysts can perform essential calculations and make informed decisions despite incomplete data. This method is particularly useful in fields such as quality control and financial analysis, where precise data may be hard to obtain. Whether you're dealing with preliminary data or refining your analysis, the assumed mean provides a practical approach to navigating uncertainty and improving decision-making.