Mean Absolute Deviation (original) (raw)

Last Updated : 23 Jul, 2025

Mean Absolute Deviation is one of the metrics of statistics that helps us find out the average spread of the data i.e., Mean Absolute Deviation shows the average distance of the observation of the dataset from the mean of the dataset. It is helpful in the analysis of data and understanding of the data with a much better understanding.

Mean Absolute Deviation is one of the measures of the spread which include other measures i.e., range, quartiles, interquartile range, standard deviation, and variance. In this article, we will learn about the measure of spread which is Mean Absolute Deviation, and other than this we will also learn about the formula to find it.

Table of Content

• What is Mean Absolute Deviation?
• What is Measure of Spread?
• Mean Absolute Deviation Formula
• How to Calculate Mean Absolute Deviation?
• Mean Absolute Deviation vs. Standard Deviation

What is Mean Absolute Deviation?

Mean Absolute Deviation (MAD) of a data set is the average distance between each data point of the data set and the mean of data. i.e. it represents the amount of variation that occurs around the mean value in the data set. It is also a measure of spread. It is calculated as the average of the sum of the absolute difference between each value of the data set and the mean.

What is Measure of Spread?

The measure of spread represents the amount of dispersion in a data set. i.e., how spread out are the values of the dataset around the central value (example- **mean/mode/median). It tells how far away the data points tend to fall from the central value.

• The lower value of the measure of spread reflects that the data points are close to the central value. In this case, the values in a data set are more consistent.
• Further, the distance of the data points from the central value, the greater the spread. whereas here, the values are not much consistent.

Using the above diagram, we can infer that the narrow distribution represents a lower spread, and the broad distribution represents a higher spread.

Mean Absolute Deviation Formula

As Mean Absolute Deviation is the average of the absolute value of deviation about the mean of the data, its formula for grouped as well as ungrouped data is given as follows:

For Ungrouped Data

The Mean Absolute Deviation Formula for ungrouped data is given as follows:

\bold{\text{MAD} = \frac{\sum |x_i - \mu|}{n}}

where,

• **x irepresents the each observation of the dataset,
• **μ is the mean of the data set, and
• **n is the number of observations in the data set.

For Grouped Data

The Mean Absolute Deviation Formula for grouped data is given as follows:

\bold{\text{MAD} = \frac{\sum{f_i\left|x_i - \bar{x}\right|}}{\sum {f_i}}}

Where,

• **x irepresents the each observation of the dataset,
• \bar{x} is mean of dataset
• **f irepresents frequency of corresponding observation **x i,
• **1 < i < n and **n is the number of data points in the data set.

How to Calculate Mean Absolute Deviation?

To calculate the mean absolute deviation for a set of values, we can use the following steps:

**Step 1: Identify whether the data set is either grouped or ungrouped and calculate the Mean.

**Step 2: Calculate the absolute difference between each data point and the mean.

**Step 3: Add the Absolute Difference calculated for each data point in the step 2.

**Step 4: Dividing the sum of absolute difference by the number of data points given to calculate the mean abosolute deviation.

Using these steps, we can calculate the Mean Absolute Deviation of any dataset either grouped or ungrouped.

Mean Absolute Deviation vs. Standard Deviation

There are some differences between Mean Absolute Deviation and Standard Deviation, which are as follows:

**Read More,

• Mean Deviation
• Standard Deviation Formula
• Variance and Standard Deviation

Solved Examples on Mean Absolute Deviation

**Example 1: The data set is 11, 15, 18, 17, 12, and 17. Calculate the mean absolute deviation of the given data set.

**Solution:

**Step 1: Calculating the mean

x̅ = (x1 + x2 + x3 + …… + xn) / n

x̅ = (11 + 15 + 18 + 17 + 12 + 17 ) / 6

x̅ = 15

**The mean of the given data = 15

**Step 2: Calculating the absolute difference between each data-point and mean.

Data-Point	Absolute Difference from the mean
11	|11 - 15
12	|12 - 15
15	|15 - 15
17	|17 - 15
17	|17 - 15
18	|18 - 15

**Step 3: Adding the Absolute Difference together

(∑ |xi - mean| ) = 4 + 3 + 0 + 2 + 2 + 3

****(∑ |x** i **- mean| ) = 14

**Step 4: Dividing the sum of absolute difference and the number of data-points.

MAD = (∑ |xi - mean|) ÷ n

MAD = 14/6

**MAD = 2.33

**Hence, we can conclude that, on average, each data-point is 2 distance away from the mean.

**Example 2: The following table shows the number of oranges that grew on Nancy's orange tree each season

**Find the mean absolute deviation (MAD) of the data set.

**Solution:

**Step 1: Calculating the mean

x̅ = (x1 + x2 + x3 + …… + xn) / n

x̅ = (5 + 17 + 24 + 10) / 4

x̅ = 56/4

**The mean of the given data = 14

**Step 2: Calculating the absolute difference between each data-point and mean

**Step 3:Adding the Absolute Difference together

(∑ |xi - mean| ) = 9 + 3 + 10 + 4

****(∑ |xi - mean| ) = 26**

**Step 4: Dividing the sum of absolute difference and the number of data-points

MAD = (∑ |xi - mean| ) ÷ n

MAD = 26 / 4

**MAD = 6.5

**Example 3: Consider the following data.

**Calculate the mean absolute deviation of the given data.

**Solution:

**Step 1: Calculating the mean

x̅ = (x1 + x2 + x3 + …… + xn) / n

x̅ = (90 + 74 + 80 + 92) / 4

x̅ = 336/4

**The mean of the given data = 84

**Step 2: Calculating the absolute difference between each data-point and mean

**Step 3: Adding the Absolute Difference together

(∑ |xi - mean| ) = 6 + 10 + 4 + 8

****(∑ |xi - mean| ) = 28**

**Step 4: Dividing the sum of absolute difference and the number of data-points

MAD = (∑ |xi - mean|) ÷ n

MAD = 28 / 4

**MAD = 7