Average Deviation is an effective way to analyze the variability in the given data. It is the average of all deviations from a central point. Average Deviation measures the distance from mean or median. It is also called Average Absolute Deviation (or) Mean Absolute Deviation.
Mean– It is the average value of all the values in the data.
Median– It is the middle value in the data when the data is sorted from low to high.
What Is Average Deviation Formula?
The formula for average deviation is utilized to determine how much individual observations differ from the mean of a data set. Presented below is the formula for computing the average deviation across n observations:
Average Deviation Formula
Average Deviation = 1/n x Σ|xi – x̅|
In this formula:
xi represents individual data points in the set.
x̅ denotes the mean of the data.
n is the count of data points in the set.
To enhance understanding, let’s explore a few examples demonstrating how to apply the average deviation formula effectively.
Steps to calculate Average Deviation
Step 1. Calculate the mean or median value for given data.
Mean can be calculated by adding all the numbers in the given data and this total sum is divided by the count of all digits.
Median value can be calculated by sorting the data from low to high or high to low and picking the middle value. If the total count of numbers in given data is odd then the middle value will be the median. If the total count is even then there will be two middle numbers for these 2 numbers calculate the average of those two and the resultant value is the median of that dataset.
Step 2. Calculate the deviation from mean/median.
The deviation value can be calculated by calculating the absolute difference between each value in the data and the result of Step-1 i.e., mean or median value.
Step 3. Find the sum of all deviations of the result from the above step.
Step 4. Find the average of all deviations using the resultant sum of deviations and count of all values/deviations in the dataset. The final resultant average is the Average Deviation of the given data.
Let’s look at the few examples to understand better.
Examples on Average Deviation Formula
Question 1: Find the Average Deviation for the data 1,2,3,4,9,8,7,6.
Solution:
Step 1: Find the mean for the given data.
Mean=(Sum of all values)/(count of all values)
Mean = (1+2+3+4+9+8+7+6)/8
= 40/8 => 5
Step 2: Find absolute deviations from data using mean.
Data
Mean
Deviation
1
5
abs(1-5) = 4
2
5
abs(2-5) = 3
3
5
abs(3-5) = 2
4
5
abs(4-5) = 1
9
5
abs(9-5) = 4
8
5
abs(8-5) = 3
7
5
abs(7-5) = 2
6
5
abs(6-5) = 1
Step 3: Sum of all deviations=4+3+2+1+4+3+2+1=20
Step 4: Find Average Deviation = sum of all deviations/count of values in data
= 20/8 => 5/4 => 1.25
So Average Deviation within the given data is 1.25
Question 2: Find the Average Deviation for the data 1,2,3,4,9,8,7,6. (Use median to find central point)
Solution:
Step 1: Find median for the given data.
To find the median first we need to sort the given data either in ascending order or descending 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
Step 2: Find absolute deviations from data using median.
Data
Median
Deviation
1
5
abs(1-5) = 4
2
5
abs(2-5) = 3
3
5
abs(3-5) = 2
5
5
abs(4-5) = 1
9
5
abs(9-5) = 4
8
5
abs(8-5) = 3
7
5
abs(7-5) = 2
6
5
abs(6-5) = 1
Step 3: Sum of all deviations = 4+3+2+1+4+3+2+1=20
Step 4: Find Average Deviation = sum of all deviations/count of values in data
= 20/8=> 5/4 => 1.25
So Average Deviation within the given data is 1.25
Question 3: Find the Average Deviation for the data 10,25,30,14,39,18,17. (Use median to find central point)
Solution:
Step 1: Find median for the given data.
To find median first we need to sort the given data either in ascending order or descending order.
Sorted data- 10,14,17,18,25,30,39
Here the size of data set is odd i.e., count=7.
So we have only one middle value18 which is median.
Step 2: Find absolute deviations from data using 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
Step 3: Sum of all deviations = 8+4+1+0+7+12+21=53
Step 4: Find Average Deviation=sum of all deviations/count of values in data
=53/7=> 7.57
So Average Deviation within the given data is 7.57
Question 4: Find the Average Deviation for the data 10,20,30,40,39,28,17,10,20,26.
Solution:
Step-1 Find mean for the given data.
Mean=(Sum of all values)/(count of all values)
Mean=(10+20+30+40+39+28+17+10+20+26)/10
=240/10=>24
Mean=24
Step-2 Find 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
Step 3: Sum of all deviations=14+4+6+16+15+4+7+14+4+2=86
Step 4: Find Average Deviation=sum of all deviations/count of values in data
=86/10=> 8.6
So Average Deviation within the given data is 8.6
Question 5: Find the Average Deviation for the data 10,20,30,40,50 (Use mean/median to find central point)
Solution:
Step 1: Find the center point for the given data.
As data is already in sorted order it is preferred to use the median to find the central point.
Here the size of the data set is odd i.e., count=5.
So we have only one middle value 30 which is the median.
Step 2: Find 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
Step 3: Sum of all deviations=20+10+0+10+20 =60
Step 4: Find Average Deviation=sum of all deviations/count of values in data
=60/5=>12
So Average Deviation within the given data is 12.
Average Deviation Formula – FAQs
What is the average deviation formula?
The average deviation formula is used to measure the average difference between each data point in a set and the set’s mean. It is calculated as: Average Deviation = 1/n * Σ|xi – x̅|, where xi are the data points, x̅ is the mean, and n is the number of data points.
How do you calculate average deviation by hand?
To calculate average deviation manually, first find the mean of the data set. Then subtract the mean from each data point, take the absolute values of these differences, sum them up, and finally divide by the number of data points.
What is the purpose of calculating average deviation?
Calculating average deviation helps in understanding the variability or dispersion of a data set. It shows how spread out the data points are from the mean, providing insights into the consistency of the data.
Can average deviation be negative?
No, average deviation cannot be negative because it is calculated using the absolute values of differences between each data point and the mean, which are always non-negative.
What is the difference between average deviation and standard deviation?
Average deviation uses the absolute differences from the mean for its calculation, whereas standard deviation squares these differences. This makes standard deviation more sensitive to outliers and typically larger than the average deviation.
When should you use average deviation?
Average deviation is useful when you need a straightforward measure of dispersion that is less affected by extreme values. It’s particularly good for non-normal distributions or for educational purposes where simplicity is key.
How does average deviation affect data analysis?
Knowing the average deviation of a data set provides clarity on the reliability of the mean. A small average deviation indicates that the data points are closely clustered around the mean, suggesting higher consistency within the data set.
Is average deviation suitable for all types of data?
Average deviation can be applied to any numerical data set but is most informative in distributions where extreme values are less common, as it is not as robust to outliers as other measures of dispersion like the standard deviation.