Mean Deviation

We define the mean deviation of the data set as the value which tells us how far each data is from the centre point of the data set. The centre point of the data set can be the Mean, Median or Mode. Thus, the mean of the deviation of all the data in a set from the centre point of the data set is called the mean deviation of the data set. We can calculate the mean deviation for both Grouped data and ungrouped data. Mean deviation measures the arbitrary change in the values of the data set from the centre point of the data set.

What is Mean Deviation?

The mean deviation of a given standard distribution is the average of the deviation from the central tendency. Central Tendency can be computed using the Arithmetic Mean, Median, or Mode of the data. It is used to show how far the observations are situated from the central point of the data (the central point can be either mean, median or mode).

Mean Deviation Definition

We simply define the mean deviation of the given data distribution as the mean of the absolute deviations of the observations from a suitable central value. This suitable central value can be the mean, median, and mode of any one of the central tendencies of the data. 

Mean Deviation Example

If we have to find the mean deviation of the data set, {4, 5, 6, 7, 8} about the mean of the data set. Then we first find the mean of the data set,

Mean = (4 + 5 + 6 + 7 + 8)/5 = 6

Now we subtract the mean from each data set to obtain the deviation from the mean.

Values	Absolute Deviation from Mean
4	2
5	1
6	0
7	1
8	2

Now we take the mean of the deviation value so obtained. As,

Mean Deviation = (2+1+0+1+2)/5

⇒ Mean Deviation = 1.2

This gives the mean deviation of the given data set.

Mean Deviation Formula

There are various types of mean deviation formulas used depending upon the types of data given and the central point chosen for the given data set. We have different formulas for grouped data, and ungrouped data, also the mean deviation formula is different for deviation about mean and the deviation about median or mode. The image added below shows the mean deviation formula in various cases.

Mean Deviation Formula for Ungrouped Data

For ungrouped data or data that is not properly arranged that is the given data is in raw form, the mean deviation is calculated using the formula,

Mean Deviation = ∑_iⁿ (x_i – x̄) / n

where,
x_i represents the ith observation
x̄ represents any central point (mean, median, or  mode)
n represents the number of observations present in the data set

Mean Deviation for Discrete Frequency Distribution

In the discrete series, the data of each individual value is represented and their individual frequency is represented in the next column of the table. The table added below shows the data in a discrete frequency distribution table.

Wages	Number of Workers (Frequency)
2500	7
3000	9
4000	5
4500	6
5000	3

The formula  used to calculate the mean deviation in this type of data set is,

Mean Deviation = ∑_iⁿ f_i(x_i – x̄) / ∑_iⁿf_i

Where,

• x_i represents the specific ith value, 
• x̄ represents any central point (mean, median, or  mode), and 
• f_i represents the frequency of each class interval.

Mean Deviation for Continuous Frequency Distribution

In a continuous series, data is arranged in certain definite classes. The data items contained in classes lose their individual identity, and the individual items are merged into one or the other class group. The classes have continuity, that is the end of the first class is marked by the beginning of the next class. Thus, the name continuous series. For instance, the continuous series is depicted using the following data,

Age Group	Frequency
10-20	15
20-30	10
30-40	13
40-50	12

The formula used to calculate the mean deviation in this type of data set is,

Mean Deviation = ∑_iⁿ (x_i – x̄) / ∑_iⁿ f_i

Where,

• x_i represents the specific ith value, 
• x̄ represents any central point (mean, median, or  mode), and 
• f_i represents the frequency of each class interval.

Mean Deviation about Mean

We can easily calculate the mean deviation from the mean of the given data set. The mean of the data set is simply the arith mean also called the average of the data set. It is calculated by taking the sum of all observations divided by the number of observations.

The formula used to calculate the mean deviation about the mean of the data set is,

For Ungrouped Data,

Mean Deviation = ∑_iⁿ (x_i – x̄) / n

Where x̄ = (x₁ + x₂ + x₃ + … + x_n)/n

For Continuous and Discrete Frequency Distribution,

Mean Deviation = ∑_iⁿ (x_i – x̄) / ∑_iⁿ f_i

Where x̄ = ∑_iⁿ x_if_i / ∑_iⁿf_i

Mean Deviation about Median

The middle point of the data set when arranged in ascending or descending order is called the median of the data set. it is the middle value of the data set which divides the data set into two equal halves. The formula to calculate the mean deviation of the data set about the mean is,

For Ungrouped Data,

Mean Deviation = ∑_iⁿ (x_i – M) / n

Where M represents the middle point or median of the data set and is calculated as,

• For n = odd terms, 
  • M = [(n + 1)/2]^th observation
• For n = even terms, 
  • M = [(n/2)^th + (n/2 + 1)^th] / 2

For Discrete Frequency Distribution,

Mean Deviation = ∑_iⁿ f_i(x_i – M) / ∑_iⁿ f_i

Where M represents the middle point or median of the data set and is calculated in the same way as above.

For Continuous Frequency Distribution,

Mean Deviation = ∑_iⁿ f_i(x_i – M) / ∑_iⁿ f_i

Where M represents the middle point or median of the data set and is calculated as,

M = l + {[∑_iⁿ f_i/2 – cf] / f}×h

Where,

• cf is the cumulative frequency of the class preceding the median class,
• l is the lower value of the median class,
• h is the length of the median class, and
• f is the frequency of the median class.

Mean Deviation about Mode

The term with the highest frequency in the data set is the mode of the data set. The formula to calculate the mean deviation of the data set about the mode is,

For Ungrouped Data,

Mean Deviation = ∑_iⁿ (x_i – M) / n

Where M represents the mode of the data set.

For Discrete Frequency Distribution,

Mean Deviation = ∑_iⁿ f_i(x_i – M) / ∑_iⁿ f_i

Where M represents the mode of the data set.

For Continuous Frequency Distribution,

Mean Deviation = ∑_iⁿ f_i(x_i – M) / ∑_iⁿ f_i

where,
M represents the mode of the data set and is calculated as,

Mode = l + [(f – f₁) / (2f – f₁ – f₂)] × h

Where,

• l is the lower value of the modal class,
• h is the size of the modal class, 
• f is the frequency of the modal class,
• f₁ is the frequency of the class preceding the modal class, and 
• f₂ is the frequency of the class succeeding the modal class.

How to Calculate Mean Deviation?

We can calculate the mean deviation of the given data set by following the steps given below,

Step 1: Calculate the value of the mean, median, or mode of the series.

Step 2: Calculate the absolute deviations about the mean, median, or mode.

Step 3: Multiply the deviations of the given data set with their respective frequency. It is required if the frequency of the data is given.

Step 4: Find the sum of all the deviations and divide it by the number of observations to get the mean deviation.

This can be understood using the example,

Example: Find the mean deviation of the given data about their mean.

{4, 6, 7, 3, 5, 5}

Solution:

Step 1: Find the mean of the given data.

Mean = (4 + 6 + 7 + 3 + 5 + 5) / 6 = 30 / 6 = 5

Step 2: Find the absolute deviation

Given Data Set	Absolute Deviation of Mean
4	4-5 = |-1| = 1
6	6-5 = |1| = 1
7	7-5 = |2| = 2
3	3-5 = |-2| = 2
5	5-5 = |0| = 0
5	5-5 = |0| = 0

Step 3: Find the mean deviation of the absolute value so obtained.

Mean Deviation = (1+1+2+2+0+0)/6 = 6/6 = 1

Thus, the mean deviation from the mean of the given data set is 1.

Mean Deviation and Standard Deviation

Mean deviation and standard deviation are measures of central tendency that are highly used for finding the various measures of the data set. The basic difference between the mean deviation and the standard deviation is discussed in the table below,

Mean Deviation	Standard Deviation
All the central points (mean, median and mode) are used to find the mean deviation.	Mean is only used to find the standard deviation.
Absolute value of the deviation is used to find the mean deviation.	Square of the deviation is used to find the standard deviation.
Mean deviation is a less frequently used data measure.	Standard Deviation is a highly used data measure that is used to find various central measures.

Merits and Demerits of Mean Deviation

Mean deviation is a highly used data measure and the various merits and demerits of the mean deviation are,

Merits of Mean Deviation

Various merits of the mean deviation are,

• Mean deviation is very easily calculated and easy to understand.
• Change in extreme values does not affect the mean deviation drastically.
• Its gets fluctuated the least as compared to other statistical measures. 
• We use this measure in various business and economic activities.

Demerits of Mean Deviation

Various demerits of the mean deviation are,

• This measure of central tendency is not rigid and we can use the mean, median or mode of any central value to find the mean deviation.
• It uses absolute values and thus inaccuracies in the data are highly incurred.
• Various studies do not this measure to find the deviation in the data.

Read More,

• Geometric Mean
• Harmonic Mean

Examples on Mean Deviation

Example 1: Find the mean deviation of the following data.

7, 5, 1, 3, 6, 4, 10

Solution:

For Finding the mean deviation,

First, calculate the mean value of the given data

Mean = 

Mean = 5

Now finding the absolute deviation.

Given Value	Absolute Deviation of Mean
7	7 – 5 = |2| = 2
5	5 – 5 = |0| = 0
1	1 – 5 = |-4| = 4
3	3 – 5 = |-2| = 2
6	6 – 5 = |1| = 1
4	4 – 5 = |-1| = 1
10	10 – 5 = |5| = 5

Further the mean of Absolute Values

= 2, 0, 4, 2, 1, 1, 5

Mean = 

Mean = 2.14

Therefore,

Mean deviation for 7, 5, 1, 3, 6, 4, 10 is 2.14

Example 3: Find the mean deviation of the following data.

11, 9, 7, 3, 2, 8, 10, 12, 15, 13

Solution:

For Finding the mean deviation,

First, calculate the mean value of the given data

Mean = 

Mean = 8

Now finding the absolute deviation.

Given Value	Absolute Deviation of Mean
11	11 – 8 = |3| = 3
9	9 – 8 = |1| = 1
7	7 – 8 = |-1| = 1
3	3 – 8 = |-5| = 5
2	2 – 8 = |-6| = 6
8	8 – 8 = |0| = 0
10	10 – 8 = |2| = 2
12	12 – 8 = |4| = 4
15	15 – 8 = |7| = 7
13	13 – 8 = |5| = 5

Further find the mean of these values obtained are 3, 1, 1, 5, 6, 0, 2, 4, 7, 5.  

⇒

⇒ Mean = 3.4

Therefore, Mean deviation for 11, 9, 7, 3, 2, 8, 10, 12, 15, 13 is 3.4.

Example 3: Find the mean deviation of the following data table,

Class Interval	Frequency
05-15	8
15-25	12
25-35	6
35-45	4

Solution:

The mean of the following data is,

Class Interval	Frequency(fi)	Mid Point(xi)	fixi
05-15	8	10	80
15-25	12	20	240
25-35	6	30	180
35-45	4	40	160
	∑fi = 30		∑fixi = 660

x̄ = ∑f_ix_i / ∑f_i = 660/30

   = 22

Class Interval	Frequency(fi)	Mid Point(xi)	|xi – x̄|
05-15	8	10	12
15-25	12	20	2
25-35	6	30	8
35-45	4	40	18
	∑fi = 30		∑|xi – x̄| = 40

Mean Deviation = 40/30 = 1.33

FAQs on Mean Deviation

Q1: What is Mean Deviation in Statistics?

Answer:

Mean deviation is the measure of the central tendency that gives the average of absolute deviation with respect to any central tendencies, i.e. Mean, Median, or Mode.

Q2: What are the Advantages of Mean Deviation?

Answer:

The advantages of using mean deviation are:

• It is based on all the data values given, and hence it provides a better measure of dispersion.
• It is easy to understand and calculate.

Q3: What is the Formula of Mean Deviation?

Answer:

The formula to calculate the mean deviation of the data set along the mean is,

Mean Deviation = ∑_iⁿ (x_i – x̄) / n

where,
x̄ is the mean of the data set and is calculated as
x̄ = (x1 + x2 + x3 + … + xn)/n

Q4: How to Calculate Mean Deviation?

Answer:

We can easily calculate the mean deviation using the steps discussed below,

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

Step 2: Subtract the mean from each data value and find the absolute deviation.

Step 3: Find the mean of the absolute deviation to get the mean deviation.