The 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
The 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, the assumed mean method allows you to break down a data set with class intervals and their respective frequencies into simpler steps, making it easier to determine the mean without any hard calculations.
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
Steps to Find Mean Using Assumed Mean Method
The process for calculating the mean by using the assumed mean method is 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 the mean using the assumed mean method for the 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
Direct vs. Assumed Mean vs. Step Deviation Method
Direct Method, Assumed Mean Method, and Step Deviation There are three different methods 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 interval and multiply them with the 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 the step deviation (ui) of each class interval and multiply it with the 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 |
Solved Examples
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 an 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
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 the distribution of daily wages of 60 workers 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 the assumed mean method.
Class Interval | Frequency |
|---|---|
0-10 | 8 |
10-20 | 12 |
20-30 | 10 |
30-40 | 12 |