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:
- Direct Method
- Assumed Mean Method
- Step-Deviation Method
In this article, we will discuss the Assumed Mean Method in detail.
Table of Content
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 + ∑ƒidi /∑ƒi
Where,
- a is assumed mean,
- ƒi is frequency of ith class,
- di = xi - a is derivation of ith class,
- ∑ƒi = n is total number of observations
- xi 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,
xi = (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,
di = (xi - a) for each i
Step 4: Calculate the product by,
(ƒi × di ), for each i
Step 5: Find total frequency
n = ∑ƒi
Step 6: Finally, calculate the mean by using the following formula,
x̄ = a + ∑ ƒidi/∑ƒi
Let's consider an example for better understanding.
Example: Find mean using assumed mean method for following data:
| Class Interval | 10 - 20 | 20 - 30 | 30 - 40 | 40 - 50 | 50 - 60 | 60 - 70 |
|---|---|---|---|---|---|---|
| Frequency (f) | 3 | 7 | 12 | 15 | 8 | 5 |
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 di = (xi - 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 + ∑ ƒidi/∑ƒi
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:
Direct Method | Assumed Mean Method | Step Deviation Method |
|---|---|---|
Best method for simple problems and small data set. | Best method for large data sets with potentially large values. | Best method for large data sets with uniform class intervals. |
This method is straightforward and easy to understand. | It simplifies large arithmetic calculations | Further simplifies calculations and arithmetic complexity by working with smaller numbers. |
In this method we calculate midpoints(xi) of each class intervals and multiply them with corresponding frequency(ƒ), i.e., ƒixi . | In this method we calculate deviation(di) of each data point and multiply them with corresponding frequency(ƒ), i.e., ƒidi . | In this method we calculate step deviation(ui) of each class intervals and multiply them with corresponding frequency(ƒ), i.e., ƒiui . |
The formula of this method is Mean = ∑ƒixi / ∑ƒi | Formula for assumed mean method is Mean = a + ∑ ƒidi / ∑ ƒi | Formula for step deviation method is Mean = a + h × ∑ ƒiui / ∑ ƒi |
Also Read,
Solved Examples on Assumed Mean Method
Example 1: The following table gives information about the marks obtained by 120 students in an examination.
Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
Frequency | 14 | 30 | 34 | 27 | 15 |
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.
Class | Frequency | Class mark (xi) | di = (xi - a) | ƒidi |
|---|---|---|---|---|
0-10 | 14 | 5 | 5 - 25 = -20 | -280 |
10-20 | 30 | 15 | 15 - 25 = -10 | -300 |
20-30 | 34 | 25 = a | 25 - 25 = 0 | 0 |
30-40 | 27 | 35 | 35 - 25 = 10 | 270 |
40-50 | 15 | 45 | 45 - 25 = 20 | 300 |
∑ƒi = 120 | ∑ƒidi = -10 |
The formula,
x̄ = a + ∑ƒidi/∑ƒi
⇒ 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
Number of plants | 0 - 2 | 2 - 4 | 4 - 6 | 6 - 8 | 8 - 10 | 10 - 12 | 12 - 14 |
Number of houses | 1 | 2 | 2 | 4 | 3 | 2 | 6 |
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:
No. of Plants | No. of houses/ Frequency (ƒi) | Class mark (xi) | di = (xi - a) | ƒidi |
|---|---|---|---|---|
0-2 | 1 | 1 | -6 | -6 |
2-4 | 2 | 3 | -4 | -8 |
4-6 | 2 | 5 | -2 | -4 |
6-8 | 4 | 7 = a | 0 | 0 |
8-10 | 3 | 9 | 2 | 6 |
10-12 | 2 | 11 | 4 | 8 |
12-14 | 6 | 13 | 6 | 36 |
∑ ƒi = 20 | ∑ƒidi = 32 |
x̄ = a+ (Σƒidi /Σƒi)
⇒ 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.
Class | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |
|---|---|---|---|---|---|
Frequency | 20 | 30 | 45 | 35 | 30 |
Problem 2: Find the mean of the following data using the assumed mean method formula.
Marks | Number of students |
|---|---|
0-10 | 5 |
10-20 | 3 |
20-30 | 4 |
30-40 | 3 |
40-50 | 3 |
50-60 | 4 |
60-70 | 7 |
70-80 | 9 |
80-90 | 7 |
Problem 3: Find the arithmetic mean using the assumed-mean method:
Class interval | 100-120 | 120-140 | 140-160 | 160-180 | 180-200 |
|---|---|---|---|---|---|
Frequency | 20 | 30 | 15 | 10 | 5 |
Problem 4: The following table contains, distribution of daily wages of 60 worker of a factory.
Daily Wages | 100-120 | 120-140 | 140-160 | 160-180 | 180-200 |
|---|---|---|---|---|---|
Number of Workers | 15 | 12 | 10 | 13 | 10 |
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.
Class Interval | Frequency |
|---|---|
0-10 | 8 |
10-20 | 12 |
20-30 | 10 |
30-40 | 12 |
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.