Computation of mean deviation: The mean deviation of a data set is used to calculate how far the values are from the centre point. Mean, median, and mode are the central tendencies of the given data. In other words, the mean deviation helps calculate the average of the absolute deviations from the central point of the data. For both grouped and ungrouped data, the mean deviation can be calculated. In comparison to standard deviation, the mean deviation is a more straightforward measure of variability. The mean deviation is used to find the average deviation from a centre point. 

Definition of Mean Deviation

The mean deviation is a measure used in statistics and mathematics to find the difference between the observed and predicted value of a variable. The mean deviation is the distance from the centre point in simple terms. Similarly, the mean deviation is used to determine how far values deviate from the middle data of a set. The mean deviation is abbreviated as MAD.

The term ‘mean deviation’ refers to the average absolute deviation. The average absolute deviation is defined as the average of the absolute deviations from the central point of the data. Mean, mode, and median are the central points.

Computation of Mean Deviation for Ungrouped Data

The data which has not been sorted or classified and is still in its raw form is called ungrouped data.

The formula for calculating the mean deviation for ungrouped data is

\(\operatorname{MAD}=\frac{\sum_{1}^{n}\left|x_{i}-\bar{x}\right|}{n}\)

Here,

• \(x_{i}\)-represents the \(i^{\text {th }}\) observation
• \(\bar{x}\)-represents the central point (mean, median, or mode)
• \(n\) – the number of observations in the data set.

Computation of Mean Deviation for Grouped Data

Grouped data is data that has been organised and classified. Discrete and continuous frequency distributions are used to group data. The following are the mean deviation formulas for grouped data.

Continuous Frequency Distribution

Class intervals make up this type of grouped data. The continuous frequency distribution gives the frequency of repetition of observation within each interval. The formula for mean deviation is as follows.

\(\operatorname{MAD}=\frac{\sum_{1}^{n} f_{i}\left|x_{i}-\bar{x}\right|}{\sum_{1}^{n} f_{i}}\)

Here, \(f_{i}\) is the frequency of repetition, and \(x_{i}\) is the mid-value of the class interval.

Discrete Frequency Distribution

Individual data points are specified in this type of data, as is the frequency they occur. The formula for calculating the mean deviation of a discrete frequency distribution is as follows:

\(\mathrm{MAD}=\frac{\sum_{1}^{n} f_{i}\left|x_{i}-\bar{x}\right|}{\sum_{1}^{n} f_{i}}\)

Here, \(f_{i}\) is the frequency of occurrence, and \(x_{i}\) is the specified individual value.

Mean Deviation About the Mode

The value of the data that occurs the most frequently in a given data set is defined as the mode. The following are the formulas for calculating the mean deviation about mode.

• Ungrouped data

\(\mathrm{MAD}=\frac{\sum_{1}^{n}\left|x_{i}-\operatorname{mode}\right|}{n}\)

• Grouped data

\(\mathrm{MAD}=\frac{\sum_{1}^{n} f_{i}\left|x_{i}-\operatorname{mode}\right|}{\sum_{1}^{n} f_{i}}\)

Here,

Mode of the ungrouped data is the most frequent value

For grouped data, it is given by: Mode \(=l+\frac{f_{1}-f_{0}}{2 f_{1}-f_{2}-f_{0}} \times h\)

\(f_{1}\) is the frequency of modal class

\(f_{0}\) and \(f_{2}\) are the frequency of the preceding and succeeding the modal classes

\(h\) is the size of the class

\(l\) is the lower limit of modal class.

Mean Deviation About the Mean

A data set’s mean is also known as its expected value. The mean is defined as the ratio of sum of all observations and the total number of observations. The following are the formulas for mean deviation from the mean:

• Ungrouped data

The mean and computation of mean deviation about mean of ungrouped data is given below:

\(\mu=\frac{x_{1}+x_{2}+x_{3}+\cdots \ldots \ldots \ldots+x_{n}}{n}\)

\(\operatorname{MAD}=\frac{\sum_{1}^{n}\left|x_{i}-\mu\right|}{n}\)

• Grouped data

The mean and computation of mean deviation about mean of grouped data is given below:

\(\mu=\frac{\sum_{1}^{n} x_{i} f_{i}}{\sum_{1}^{n} f_{i}}\)

\(\mathrm{MAD}=\frac{\sum_{1}^{n} f_{i}\left|x_{i}-\mu\right|}{\sum_{1}^{n} f_{i}}\)

Mean Deviation About the Median

The median is the value of the data that separates the lower and upper halves of the data. The following are the various formulas for mean deviation from the median.

• Ungrouped data

\(\mathrm{MAD}=\frac{\sum_{1}^{n}\left|x_{i}-M\right|}{n}\)

• Grouped data

\(\operatorname{MAD}=\frac{\sum_{1}^{n} f_{i}\left|x_{i}-M\right|}{\sum_{1}^{n} f_{i}}\)

Where, \(M\) is the median of the given data.

The median of the ungrouped and grouped data is given by

• Ungrouped data

If \(n\) is odd, \(M=\left(\frac{\mathrm{n}+1}{2}\right)^{t h}\) term

If \(n\) is even \(M=\frac{\left(\frac{n}{2}+\frac{n+1}{2}\right)^{t h}}{2}\) term

• Grouped data

\(M=l+\frac{\left(\frac{n}{2}-c f\right)}{f} \times h\)

Here,

\(cf\) is the cumulative frequency of the preceding the median class

\(l\) is the lower limit of the median class

\(h\) is the size of the class

\(f\) is the frequency of the median class

Computation of Standard Deviation and Variance

Standard deviation is an essential tool for measuring dispersion in a distribution. The square root of the arithmetic mean of the squares of deviations of observations from their mean value is the standard deviation. It is commonly denoted by sigma, i.e. \(σ\). It should be noted that, unlike mean deviation, which can be measured using the mean, median, and mode, Standard deviation can only be measured using the mean.

Standard deviation can be found by using the formula:

\(\sigma=\sqrt{\frac{\sum f D^{2}}{N}}\)

Where,

\(D\) is the deviation of an item related to the calculated mean
\(f\) is the frequency corresponding to the particular observation
\(N\) is the sum of frequencies

Although the standard deviation is the most important tool for determining dispersion, it is essential to understand that it is derived from the variance. It is better compared to mean deviation because it employs the square of deviations. 

Keeping this in mind, statisticians employ the square root of the variance, also known as standard deviation. The standard deviation is effectively the square root of the variance. We can easily calculate variance as the square of standard deviation if we know how to calculate that.

\(\text {Variance} =(\text { Standard deviation })^{2}=\sigma^{2}\)

Mean Deviation vs Standard Deviation

The mean deviation and standard deviation help determine the variability of data. The table below shows the differences between standard deviation and mean deviation.

Mean Deviation	Standard Deviation
To find the mean deviation, we will use the central point such as mean, mode or median.	To calculate the standard deviation, we use the mean only.
To find the mean deviation, we will consider the absolute value of the deviations.	We will use the square of the deviations to get the standard deviation
It is less commonly used.	It is the commonly used measure of variability.
Mean deviation is used when there are a large number of outliers in the data.	Standard deviation is used when there are fewer outliers in the data.

Solved Examples of Computation of Mean Deviation

Below are a few solved examples that can help in getting a better idea.

Q.1. Find the mean deviation of the given data about the median value.

Class intervals	Frequency
\(15-25\)	\(12\)
\(25-35\)	\(6\)
\(35-45\)	\(9\)
\(45-55\)	\(4\)
\(55-65\)	\(2\)

Ans: Median class of the given data \(=25-35\), whose value is greater than \(\frac{N}{2}=\frac{33}{2}=16.5\).

So, \(l=25\)

\(h=10\)

\(cf=12\)

\(f=6\)

\(\frac{N}{2}=16.5\)

Median \(=M=l+\frac{\left(\frac{n}{2}-c f\right)}{f} \times h=25+\frac{(16.5-12)}{6} \times 10=32.5\)

Class	\(f\)	\(cf\)	\({x_i}\)	\(|x-M|\)	\(f.|x-M|\)
\(15-25\)	\(12\)	\(12\)	\(20\)	\(12.5\)	\(150\)
\(25-35\)	\(6\)	\(18\)	\(30\)	\(2.5\)	\(15\)
\(35-45\)	\(9\)	\(27\)	\(40\)	\(7.5\)	\(67.5\)
\(45-55\)	\(4\)	\(31\)	\(50\)	\(17.5\)	\(70\)
\(55-65\)	\(2\)	\(33\)	\(60\)	\(27.5\)	\(55\)
Total	\(N=33\)				\(357.5\)

Mean deviation about median \(=\frac{\sum_{1}^{n} f_{i}\left|x_{i}-M\right|}{\sum_{1}^{n} f_{i}}\)

\(=\frac{357.5}{33}\)

\(=10.83\)

Hence, the mean deviation about the median is \(10.83\).

Q.2. Find the mean deviation for the data set: \(302,140,352,563,455,215,213\)

Ans: Given: \(302,140,352,563,455,215,213\)

Mean, \(\mu=\frac{302+140+352+563+455+215+213}{7}\)

\(\therefore \mu=320\)

Mean deviation of the given data is \(\frac{\sum_{1}^{n}\left|x_{i}-\mu\right|}{n}\)

\(=\frac{(|302-320|+|140-320|+|352-320|+|563-320|+|455-320|+|215-320|+|213-320|)}{7}\)

\(=117.14\)

Hence, the mean deviation of the given data is \(117.14\).

Q.3. Calculate the absolute mean deviation for the following data

\(x\)	\(f\)
\(12\)	\(7\)
\(9\)	\(3\)
\(6\)	\(8\)
\(18\)	\(1\)
\(10\)	\(2\)

Ans:

We know that Mean \(=\frac{\sum f_{i} x_{i}}{\sum f_{i}}\)

\(=\frac{197}{21}\)

\(=9.381\)

\(x\)	\(f\)	\(x.f\)	\(|x-\bar{x}|\)	\(f \cdot|x-\bar{x}|\)
\(12\)	\(7\)	\(84\)	\(2.619\)	\(18.33\)
\(9\)	\(3\)	\(27\)	\(0.381\)	\(1.143\)
\(6\)	\(8\)	\(48\)	\(3.381\)	\(27.048\)
\(18\)	\(1\)	\(18\)	\(8.619\)	\(8.619\)
\(10\)	\(2\)	\(20\)	\(0.619\)	\(1.238\)
Total	\(N=21\)	\(197\)		\(56.378\)

Mean deviation \(=\frac{\sum_{1}^{n} f_{i}\left|x_{i}-\mu\right|}{\sum_{1}^{n} f_{i}}\)

\(=\frac{56.378}{21}\)

\(=2.684\)

Hence, the mean deviation of the given data is \(2.684\)

Q.4. If the mean of the data is \(31\). Determine the mean absolute deviation of the given data: \(30,14,35,55,45,21, x\).

Ans: Given: \(30,14,35,55,45,21, x\) and mean \(=31\).

We know that, mean \(=\frac{\sum f_{i} x_{i}}{\sum f_{i}}\)

\(31=\frac{30+14+35+55+45+21+x}{7}\)

\(31 \times 7+x=217\)

\(x=217-200\)

\(\therefore x=17\)

\(\mathrm{MAD}=\frac{\sum_{1}^{n}\left|x_{i}-\mu\right|}{n}\)

\(=\frac{|30-31|+|14-31|+|35-31|+|55-31|+|45-31|+|21-31|+|17-31|}{7}\)

\(=\frac{84}{7}\)

\(\therefore \mathrm{MAD}=12\)

Hence, the value of mean deviation of the data is \(12\).

Q.5. Calculate the standard deviation and variance of the given data:

\(x\)	\(2\)	\(4\)	\(6\)	\(8\)	\(10\)
\(f\)	\(3\)	\(5\)	\(9\)	\(5\)	\(3\)

Ans:

\(x\)	\(f\)	\(fx\)	\(D\)	\({D^2}\)	\(f{D^2}\)
\(2\)	\(3\)	\(6\)	\(-4\)	\(16\)	\(48\)
\(4\)	\(5\)	\(20\)	\(-2\)	\(4\)	\(20\)
\(6\)	\(9\)	\(54\)	\(0\)	\(0\)	\(0\)
\(8\)	\(5\)	\(40\)	\(2\)	\(4\)	\(20\)
\(10\)	\(3\)	\(30\)	\(4\)	\(16\)	\(48\)
	\(\sum f=25\)	\(\sum f x=150\)			\(\sum f D^{2}=136\)

Mean \(=\frac{\sum f x}{\sum f}\)

\(=\frac{150}{25}\)

\(=6\)

Variance, \(\sigma^{2}=\frac{\sum f D^{2}}{N}\)

\(\sigma^{2}=\frac{136}{25}\)

\(\therefore \sigma^{2}=5.44\)

Standard deviation \(=\sqrt{\text { variance }}\)

\(\sigma=\sqrt{5.44}\)

\(\therefore \sigma=2.33\)

Summary of Computation of Mean Deviation

A standard deviation is an essential tool for measuring dispersion in a distribution. The square root of the arithmetic mean of the squares of deviations of observations from their mean value is the standard deviation. The mean deviation of a data set is used to calculate how far the values are from the central tendency values, such as the mode and median.

The mean average deviation (MAD) is a measure used in Statistics and Mathematics to find the difference between the observed and predicted value of a variable. There are different formulas to find the mean deviation of ungrouped data, grouped data, mode and median.

FAQs on Computation of Mean Deviation

Students might be having many questions with respect to the Computation of Mean Deviation. Here are a few commonly asked questions and answers.

Q.1. What is a mean deviation in statistics?
Ans: The mean deviation is used in statistics to describe the spread of data around a central point (mean, median or mode). It is an example of an average absolute deviation.

Q.2. What is the formula for mean deviation?
Ans: The formulas for mean deviation are listed below:
Ungrouped data:
\(\mathrm{MAD}=\frac{\sum_{1}^{n}\left|x_{i}-\bar{x}\right|}{n}\)
Grouped data:
\(\mathrm{MAD}=\frac{\sum_{1}^{n} f_{i}\left|x_{i}-\bar{x}\right|}{\sum_{1}^{n} f_{i}}\)
Here, \(\bar{x}\) is the mean, mode or median value.

Q.3. What is the difference between standard deviation and mean deviation?
Ans: Mean deviation is calculated with any central point such as mean, mode or median, whereas standard deviation is calculated with only the mean.

Q.4. What is the coefficient of mean deviation?
Ans: The coefficient of variation (CV) is the ratio of the standard deviation to the mean. 

Q.5. Is mean deviation a dispersion measure?
Ans: Dispersion is measured by mean deviation. It aids in determining the variability of data in relation to the central measures of the given data set.

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