Mean Deviation

In statistics and mathematics, the deviation is a measure that is used to find the difference between the observed value and the expected value of a variable. In simple words, the deviation is the distance from the center point. Similarly, the mean deviation is used to calculate how far the values fall from the middle of the data set. In this article, let us discuss the definition, formula, and examples in detail.

Mean Deviation Definition

The mean deviation is defined as a statistical measure that is used to calculate the average deviation from the mean value of the given data set. The mean deviation of the data values can be easily calculated using the below procedure. 

Step 1: Find the mean value for the given data values

Step 2: Now, subtract the mean value from each of the data values given (Note: Ignore the minus symbol)

Step 3: Now, find the mean of those values obtained in step 2.

Mean Deviation Formula

The formula to calculate the mean deviation for the given data set is given below.

Mean Deviation = [Σ |X – µ|]/N

Here, 

Σ represents the addition of values

X represents each value in the data set

µ represents the mean of the data set

N represents the number of data values

| | represents the absolute value, which ignores the “-” symbol

Mean Deviation for Frequency Distribution

To present the data in the more compressed form we group it and mention the frequency distribution of each such group. These groups are known as class intervals.

Grouping of data is possible in two ways:

1. Discrete Frequency Distribution
2. Continuous Frequency Distribution

In the upcoming discussion, we will be discussing mean absolute deviation in a discrete frequency distribution.

Let us first know what is actually meant by the discrete distribution of frequency.

Mean Deviation for Discrete Distribution Frequency

As the name itself suggests, by discrete we mean distinct or non-continuous. In such a distribution the frequency (number of observations) given in the set of data is discrete in nature.

If the data set consists of values x1,x2, x3………xn each occurring with a frequency of f1, f2… fn respectively then such a representation of data is known as the discrete distribution of frequency.

To calculate the mean deviation for grouped data and particularly for discrete distribution data the following steps are followed:

Step I: The measure of central tendency about which mean deviation is to be found out is calculated. Let this measure be a.

If this measure is mean then it is calculated as,

where 

$\begin{array}{l}�=\sum _{�=1}^{�}{�}_{�}\end{array}$

If the measure is median then the given set of data is arranged in ascending order and then the cumulative frequency is calculated then the observations whose cumulative frequency is equal to or just greater than N/2 is taken as the median for the given discrete distribution of frequency and it is seen that this value lies in the middle of the frequency distribution.

Step II: Calculate the absolute deviation of each observation from the measure of central tendency calculated in step (I)

StepIII: The mean absolute deviation around the measure of central tendency  is then calculated by using the formula

If the central tendency is mean then,

In case of median

Let us look into the following examples for a better understanding.

Mean Deviation Examples

Example 1: 

Determine the mean deviation for the data values 5, 3,7, 8, 4, 9.

Solution:

Given data values are 5, 3, 7, 8, 4, 9.

We know that the procedure to calculate the mean deviation.

First, find the mean for the given data:

Mean, µ = ( 5+3+7+8+4+9)/6

µ = 36/6

µ = 6

Therefore, the mean value is 6.

Now, subtract each mean from the data value, and ignore the minus symbol if any

(Ignore”-”)

5 – 6 = 1

3 – 6 = 3

7 – 6 = 1

8 – 6 = 2

4 – 6 = 2

9 – 6 = 3

Now, the obtained data set is 1, 3, 1, 2, 2, 3.

Finally, find the mean value for the obtained data set

Therefore, the mean deviation is 

= (1+3 + 1+ 2+ 2+3) /6

= 12/6

= 2

Hence, the mean deviation for 5, 3,7, 8, 4, 9 is 2.

Example 2:

In a foreign language class, there are 4 languages, and the frequencies of students learning the language and the frequency of lectures per week are given as:

Language	Sanskrit	Spanish	French	English
No. of students(xi)	6	5	9	12
Frequency of lectures(fi)	5	7	4	9

Calculate the mean deviation about the mean for the given data.

Solution: The following table gives us a tabular representation of data and the calculations

&

Win