Assumed mean method finds the actual mean of the data by first assuming a mean value. The term “mean” refers to the average value of a set of data, which is derived by dividing the total number of counts by all the data. The mean, or total average, is easily determined by adding all the numbers together, then dividing by the entire number of numbers. The mean value in a set of data is a determined average that lies halfway between the highest and lowest values.

What is Assumed Mean Method?

Assumed mean method gives us smaller numbers to work with making calculations easier and is thus suitable if your data set has large values. When calculating the mean using the direct mean method, you obtain significantly bigger numbers. The likelihood of making calculating errors is decreased when utilizing the assumed mean approach, also known as a shift of origin because it gives you smaller numbers to work with (as well as negative numbers that lower the sum). You can obtain even lower values by utilizing an assumed mean, a shifted origin and a scale change.

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Assumed Mean Method Formula

In the assumed mean method, we assume the value of a mean then we calculate the deviation and adjust the mean accordingly. It is suitable for calculating the mean or average for tables involving largely spaced limits.

We can calculate mean using the assumed mean method by following the below steps:

• Step 1: Sort your data set from smallest to largest. For example, assume your data set is 73, 75, 76, 78 and 79.
• Step 2: Assume a mean. This should be a number that you feel is a close representation of your data set. In a simple example, take the number in the center of your data set; in this case 76.
• Step 3: Subtract your assumed mean from each data entry. In our example, 73 − 76 = −3, 75 − 76 = −1, 76 − 76 = 0, 78 − 76 = 2 and 79 − 76 = 3.
• Step 4: Add together these differences from the mean. Example: (−3) + (−1) + 0 + 2 + 3 = 1.
• Step 5: Divide the sum of the differences from assumed mean by the number of data points. Example: 1 ÷ 5 = 0.2.
• Step 6: Add the result of the division to your assumed mean. Assumed mean = 76 + 0.2 = 76.2.

Types of Assumed Mean Method

Quantitative Data can be categorized into 2 categories:

• Grouped data are those created by grouping individual observations of a variable so that a frequency distribution table of these groups offers a practical way to summarise or analyze the data.
• Ungrouped data is the data you initially collect from an experiment or study. The information is unorganized—that is, it hasn’t been classified, categorized or in any other way grouped. An ungrouped set of data is basically a list of numbers.

Assumed Mean Method for Grouped Data

Grouped data are data formed by aggregating individual observations of a variable into groups so that a frequency distribution of these groups serves as a convenient means of summarizing or analyzing the data. Grouped data is of the form 1−10,11−20,21,−30 and so on. Hence, we find its class mark. The midpoint of each class interval is the definition of a class mark. Calculating a class 

mark is as follows:

Where fi is class frequency. The frequency of a class interval is the number of observations that occur in a particular predefined interval. So, for example, if 30 people of weight 55 to 60 appear in our study’s data, the frequency for the 55 – 60 interval is 30.

Step Deviation Method

The Step Deviation Method is a simplified way to calculate the mean (average) of grouped data in statistics. It is especially helpful when the class intervals and frequencies are large and difficult to handle manually.

Why use it?

When direct or assumed mean methods become lengthy due to large numbers, the Step Deviation Method reduces the calculations by scaling down the data, making it quicker and easier to find the mean.

Steps to Apply the Step Deviation Method

Difference Between Assumed Mean Method and Step Deviation Method

Step deviation method is the extended version of the shortcut or assumed method for calculating the mean of large values. These deviation values can be divided by a common factor that has been scaled down to a smaller amount. Change of origin or scale method is another name for the step deviation method.

Learn about Rolle’s Theorem

Assumed Mean Method Examples

Now let’s see some solved examples on the assumed mean method.

Solved Examples on Assumed Mean Method

Now let’s see some solved examples on the assumed mean method.

Example 1: Find the mean of the following data using direct method, assumed mean method and step deviation method. \(40, 50, 55, 78, 58\).

Solution:

Direct Method:

\(\bar{x}=\frac{\sum{x}}{N}=\frac{281}{5}\)

\(\bar{x}=56.2\)

Assumed Mean Method:

\(\bar{x}=A+\frac{\sum{d}}{N}=40+\frac{81}{5}=40+16.2\)

\(\bar{x}=56.2\)

Step Deviation Method:

\(\bar{x}=A+\frac{\sum{d^\prime}}{N}\times{c}=40+\frac{8.1}{5}\times{10}\)

\(\bar{x}=40+1.62\times{10}\)

\(\bar{x}=40+16.2\)

\(\bar{x}=56.2\)

Example 2: Calculate the arithmetic mean for the following data using Assumed Mean Method.

Solution: Now we have to use the formula given above to find the arithmetic mean using Assumed Mean Method.

Take the assumed mean \(A = 80\)

Arithmetic Mean \(= A + [\frac{{\sum}fd}{N]\)

\(= 80 + (\frac{115}{50})\)

\(= 80 + (\frac{23}{10})\)

\(= 80 + 2.3\)

\(= 82.3\)

Example 3: Consider the following data set and calculate the mean using Assumed Mean Method.

Solution: Let the assumed mean for this data be \(a = 150

Mean for this data, [latex]\bar{x} = a + \frac{{\sum {f_i}{d_i}}}{{\sum {f_i}}}

\(\Rightarrow\bar{x} = 150 + \frac{{60}}{{20}}\)

\(\Rightarrow\bar{x} = 150 + 3\)

\(\therefore\bar{x} = 153\)

Example 4: The following table shows the weight of 15 students. Calculate using Assumed Mean Method:

Solution: Let the assumed mean be \(A = 49\)

We have, \({\sum}f_i = 15, {\sum}f_id_i = −12\) and A = 49\)

We know,

\(Mean = A + \frac{\sum_{i=1}^nf_id_i}{\sum_{i=1}^nf_i}\)

\(= 49 + \frac{-12}{15}\)

\(= 49 – 0.8\)

\(= 48.02\)

Hence, mean weight = \(48.2\) kg.

Hope this article on the Assumed Mean Method examples was informative. Get some practice of the same on our free Testbook App. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam: