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The term “mean” refers to the arithmetic average of the data set and is derived by dividing the total number of data points by the number of data points in the set of data. It is a point that represents the average of all the data points in the set. The arithmetic mean of the grouped data is called the assumed mean. The arithmetic mean is calculated using the assumed mean approach in statistics. Mean may be calculated quickly from a data set using this approach. This approach depends on guessing the mean and rounding to a convenient result. Once more, this value is calculated from each sample value. Calculations are carried out after selecting a central class and dividing the data into equal-sized ranges or class intervals. Read the article to know more!
Contents
Different Methods of Finding Mean
Depending on the amount of the data, there are three ways to find the mean for grouped data. As follows:
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Direct Method
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Assumed Mean Method
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Step-deviation Method
The Direct Method of Mean
The arithmetic mean may be determined using the direct technique by adding all observations in the data set and dividing by the total number of observations. This is shown as
\[\bar{X}=\dfrac{\sum{f_ix_i}}{\sum{f_i}}\]
Where
\[\sum{f_ix_i}=\] Sum of all observations
\[\sum{f_i}=\] Number of frequencies
What is Assumed Mean?
The arithmetic mean of the grouped data is called the assumed mean. The arithmetic mean is calculated using the assumed mean approach in statistics. Let \[{{f}_{1}},{{f}_{2}},{{f}_{3}},…,{{f}_{n}}\] be the corresponding frequencies and \[{{x}_{1}},{{x}_{2}},{{x}_{3}},…,{{x}_{n}}\] be the midpoints or class marks of \[n\] class intervals. The assumed mean method formula is as follows:
\[\bar{x}=a+\dfrac{\sum{{{f}_{i}}{{d}_{i}}}}{\sum{{{f}_{i}}}}\]
Where
\[a=\] assumed frequency
\[{{f}_{i}}=\] frequency of ith class
\[{{d}_{i}}={{x}_{i}}-a=\] derivation of ith class
\[\sum{{{f}_{i}}}=n=\] total number of observations
\[{{x}_{i}}=\] class marks = \[\dfrac{\text{(upper class limit + lower class limit)}}{2}\]
The Step Deviation Method
Step Deviation Method \[=A +\dfrac{h(\sum{{{u}_{i}}{{f}_{i}})}}{\sum{{{f}_{i}}}}\]
Where,
A is the assumed mean
h is the class size
\[{{u}_{i}}={{d}_{i}}/h\]
\[{{f}_{i}}\] is a frequency
\[{{d}_{i}}={{x}_{i}}-A\]
\[{{x}_{i}}\] is the midpoint of the class interval
Assumed mean is taken in step deviation method. This process is known as the “change of origin” or “scale” method. Three methods are used to determine the mean: the direct method, the assumed or short-cut method and the step deviation method. The step deviation method is considered an extension of the assumed mean since we employ the deviation formula.
The step deviation method may be used to get the mean of large numbers that can be divided by a single common factor. By reducing all of the data by a common factor, the variances will decrease to a smaller amount. In other words, the step deviation method is used when the deviations of the class marks from the predicted mean are significant and they all have a similar component.
Interesting Fact
This approach may be used to quicken the computation of the mean by hand. When the assumed mean is chosen correctly, the variances are negligible and practically cancel out, making the addition faster.
Important Questions
1. Calculate the mean of the following data, using classes of width 15.
48, 9, 14, 24, 39, 11, 25, 5, 18, 3, 22, 37, 19, 27, 18, 42, 4, 10, 32, 50
Solution:
Let us classify the data into the following classes.
\[0-10,\text{ }10-20,\text{ }20-30,\text{ }30-40,\text{ }40-50\]. The required values may then be entered into a table, as illustrated below. Assume a mean of 25.
\[\bar{x}=a+\dfrac{\sum{{{f}_{i}}{{d}_{i}}}}{\sum{{{f}_{i}}}}\]
\[=25+(80/125)\]
\[=50+(16/25)\]
\[=50+(16/25)\]
\[=50+(16/25)\]
Thus, the mean of the given data is 50.64
2. How do you calculate the assumed mean?
Solution: The assumed mean method’s formula is as follows:
\[\bar{x}=a+\dfrac{\sum{{{f}_{i}}{{d}_{i}}}}{\sum{{{f}_{i}}}}\]
Where,
\[a=\] assumed frequency
\[{{f}_{i}}=\] frequency of ith class
\[{{d}_{i}}={{x}_{i}}-a=\]derivation of ith class
\[\sum{{{f}_{i}}}=n=\] total number of observations
\[{{x}_{i}}=\] class marks = (upper-class limit \[+\] lower class limit)\[/2\]
3. The table below contains information on the grades received by 150 students in an examination.
Find the mean marks of the students using the assumed mean method.
Solution:
\[\bar{x}=a+\dfrac{\sum{{{f}_{i}}{{d}_{i}}}}{\sum{{{f}_{i}}}}\]
\[=50+(580/150)\]
\[=50+(58/15)\]
\[=50+(3.8667)\]
\[=53.87\]
Thus, the mean marks of the students are 53.87
Conclusion
In this article, we learned how to find assumed mean. Depending on the amount of data, there are three ways to find the mean for grouped data. As follows: The direct method, the assumed or shortcut method of mean, and the step deviation method. The arithmetic mean of the grouped data is called the assumed mean. The arithmetic mean is calculated using the assumed mean approach in statistics.
Practice Questions:
1. Assumed mean is taken in which method?
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Short cut method of mean
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Stepping deviation method of mean
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Direct method of mean
2. In the below formula, ‘a’ stands for?
\[\bar{x}=a+\dfrac{\sum{{{f}_{i}}{{d}_{i}}}}{\sum{{{f}_{i}}}}\]
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Frequency of ith class
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Deviation of ith class
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Class Marks
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Assumed frequency
Answers
1) B
2) D
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