Average Deviation Formula: Step-wise Calculation and Examples

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The average deviation of a set of data is the average value of all the deviations from the central point of that set. The average deviation value is also referred to as Mean absolute deviation or Average absolute deviation. It is an effective way to analyse the variability in a given set of data. In statistics, average deviation is an important concept and is used in various sectors which involve the handling of data and numbers.

Table of Content

What is Average Deviation?
Average Deviation Formula
Solved Examples
Previous Years’ Questions
Things to Remember
Sample Questions

Key Terms: Deviation, Average Deviation, Mean, Median, Mode, Statistics, Data, Set

What is Average Deviation?

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The average deviation is the process through which the difference between mean and median can be measured. It is the deviation of all the data in the set from its central point. Here,

Mean is defined as the average value of all the values in the data set.
Median is defined as the middle value when the data set is arranged in ascending order, i.e., from lowest to highest.

Average Deviation Calculation

Average Deviation Calculation

Also Read:

Related Topics
Median of Grouped Data	Mean, Median and Mode	Mode Formula
Measures of Central Tendency	Variance and Standard Deviation	Relation between Mean, Median and Mode

Average Deviation Formula

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The average of all the deviations from mean, median and mode is called the average deviation or mean deviation. The average deviation formula is given as:

Average Deviation Formula = \( \frac{1}{N}\displaystyle\sum_{i=1}^{10} |x_i - m|\)

where,

N is the total number of terms

\( \frac{1}{N}\displaystyle\sum_{i=1}^{10} |x_i - m|\) is the sum of modulus ofdeviation from the mean

The steps to calculate the average deviation of a data set are illustrated below.

Steps to Calculate Average Deviation

Step 1: Calculate the median or mean of the data set.

Step 2: Calculate the deviation of each data value from the obtained mean/median.

Step 3: Calculate the total sum of obtained deviations.

Step 4: Divide the calculated sum of all deviations by the total number of values and find the average.

The obtained value is the average deviation or the mean deviation.

Solved Examples

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To understand the concept of the average deviation method, let us see a few examples.

Example 1: Calculate the average deviation for the given data: 4, 6, 8, 10, 12, 14.

Solution: Following the steps, the mean of the data set will be

Mean = (4 + 6 + 8 + 10 + 12 + 14)/6 = 9

Now let’s calculate the absolute deviations as: (9 - 4), (9 - 6), (9 - 8), (9 - 10), (9 - 12) and (9 - 14)

The absolute deviations are (taking the positive values) 5, 3, 1, 1, 3, 5

Now the average deviation = ( 5 + 3 + 1 + 1 + 3 + 5)/6

The average deviation for given data set is 3.

Example 2: Calculate the average deviation for the given data: 5, 7, 9, 11, 12, 16.

Solution: Following the steps, the mean of the data set will be

Mean = (5 + 7 + 9 + 11 + 12 + 16)/6 = 10

Calculating the absolute deviations as: (10 - 5), (10 - 7), (10 - 9), (10 - 11), (10 - 12) and (10 - 16)

The absolute deviations are (taking the positive values) 5, 3, 1, 1, 2, 6

Now the average deviation = ( 5 + 3 + 1 + 1 + 2 + 6)/6

The average deviation of the data set is 3.

Previous Year Questions

If the value of mode and mean is 60 and 66… [JEE Advanced 2013]
The mean of 5 observations is 5 and their variance is… [JEE Mains 2016]
The mean of five observations is 5 and their variance is 9.20… [Jee Mains 2019]
The mean and variance of a random variable X having… [JKCET 2014]
The mean of 100 items was 60. Later it was found that… [JKCET 2016]
If M. D. is 12, the value of S.D. will be… [BITSAT 2014]
The average speed of a car running at the rate of 15 km/hour during…
The average weight of 25 boys was calculated to be 78.4kg…
The mean and standard deviation of 100 observations were calculated…
The mean and standard deviation of marks obtained by 50 students of...

Things to Remember

For a given data set the average deviation is the average value of all the deviations from the central point of that set.
The process to find the difference between the mean and median of a data set is called the average deviation.
Mean deviation is the deviation of all the values in the set from the central point.
In a data set median is the middle value when the data set is arranged in ascending order.
For a given data set mean is the average value of all the values in the data set.

Also Read:

Related Topics
Dispersion	Variance	Measures of Dispersion
Absolute Values	Harmonic Mean	Degree of Polynomial

Sample Questions

Ques: Find the Average Deviation for the data 1,2,3,4,9,8,7,6. (Use median to find central point). [2 marks]

Ans: Sorting the data in ascending order

Sorted data- 1,2,3,4,6,7,8,9

Here the size of the data set is even i.e., count=8.

So we got two middle values 4 & 6.

Find the average of these two numbers to find the median value.

Median = (4+6)/2 = 5

Now the absolute deviations, (5 - 1), (5 - 2), (5 - 3), (5 - 4), (5 - 6), (5 - 7), (5 - 8), (5 - 9)

The absolute deviations are 4, 3, 2, 1, 1, 2, 3, 4

The average deviation = (4 + 3 + 2 + 1 + 1 + 2 + 3 + 4 )/8

Average deviation = 1.25

Ques: Find the Average Deviation for the data 1,2,3,4,9,8,7,6. [2 marks]

Ans: (Sum of all values)/(count of all values)

Mean = (1+2+3+4+9+8+7+6)/8

= 40/8 => 5

Now, absolute deviations = 4, 3, 2, 1, 4, 3, 2, 1

Average deviation = (4 + 3 + 2 + 1 + 4 + 3 + 2 + 1)/8

Average deviation = 1.25

Ques: Find the Average Deviation for the data 10,25,30,14,39,18,17. (Use median to find central point). [2 marks]

Ans: To find the median first we need to sort the given data

Sorted data- 10,14,17,18,25,30,39

Here the size of the data set is odd i.e., count = 7.

So the middle value18 is the median.

Absolute deviation from the median

abs(10-18) = 8

abs(14-18) = 4

abs(17-18) = 1

abs(18-18) = 0

abs(25-18) = 7

abs(30-18) = 12

abs(39-18) = 21

Adding all deviations = 8+4+1+0+7+12+21=53

The average deviation=sum of all deviations/count of values in data = 53/7=> 7.57

The average Deviation within the given data is 7.57

Ques: Find the Average Deviation for the data 10,20,30,40,39,28,17,10,20,26. [2 marks]

Ans: Mean for the given data.

Mean = (10+20+30+40+39+28+17+10+20+26)/10

Mean = 24

The absolute deviations from data using mean.

abs(10-24) = 14

abs(20-24) = 4

abs(30-24) = 6

abs(40-24) = 16

abs(39-24) = 15

abs(28-24) = 4

abs(17-24) = 7

abs(10-24) = 14

abs(20-24) = 4

abs(26-24) = 2

Addition of all deviations=14 + 4 + 6 + 16 + 15 + 4 + 7 + 14 + 4 + 2 = 86

Hence, average deviation=sum of all deviations/count of values in data = 86/10 => 8.6

The average deviation within the given data is 8.6

Ques: Find the Average Deviation for the data 10,20,30,40,50 (Use mean/median to find central point). [2 marks]

Ans: As data is already in sorted order it is preferred to use the median to find the central point.

The size of the data set is odd i.e., count = 5.

Thus the middle value 30 is the median.

The absolute deviations from data using the median.

abs(10-30)=20

abs(20-30)=10

abs(30-30)=0

abs(40-30)=10

abs(50-30)=20

Adding all deviations=20+10+0+10+20 =60

The average deviation=sum of all deviations/count of values in data = 60/5 =>12

So Average Deviation within the given data is 12.

Ques: What is the difference between standard deviation and average deviation? [2 marks]

Ans: While standard deviation is to calculate the spread of data over the mean, the average deviation is the distance of the data from the mean value. Roughly, the standard deviation is equal to th average absolute deviation.

Ques: What are the uses of average deviation? [2 marks]

Ans: Average deviation is often used by statistics to calculate the dispersion among the measures in a given population. Many data scientists and analysts use average deviation to calculate the variability of large data sets.

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Average Deviation Formula: Step-wise Calculation and Examples

What is Average Deviation?

Average Deviation Formula

Steps to Calculate Average Deviation

Solved Examples

Previous Year Questions

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

3.
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

4.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

5.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

6.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

Similar Mathematics Concepts

Comments

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Average Deviation Formula: Step-wise Calculation and Examples

What is Average Deviation?

Average Deviation Formula

Steps to Calculate Average Deviation

Solved Examples

Previous Year Questions

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

3.A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

4.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

5. Find: The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

6.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

3.
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

4.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

5.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

6.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).