Mean Absolute Deviation Calculator

Input a data set, and calculate the Mean Absolute Deviation instantly.

Calculate Mean Absolute Deviation

Result

What Is Mean Absolute Deviation?

The Mean Absolute Deviation (Average Absolute Deviation, AAD) represents the average absolute difference between each data point and the dataset's mean. It is an effective measure of dispersion, calculated using the following formula: \( \text{Mean Absolute Deviation} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}| \) Where:

\( n \) is the total number of data points,
\( x_i \) represents each data point,
\( \bar{x} \) is the mean of the dataset.

Steps to Calculate Mean Absolute Deviation

Find the Mean: Calculate the dataset's mean (\( \bar{x} \)).
Calculate Deviations: For each data point (\( x_i \)), find the absolute difference from the mean (\( |x_i - \bar{x}| \)).
Compute the Mean Absolute Deviation: Add all the absolute deviations and divide by the total number of data points (\( n \)).

Examples

Example 1: Compute the mean absolute deviation for the data set [4, 8, 6, 5, 3, 7]

Solution:

1. Calculate the mean (\( \bar{x} \)):

\( \bar{x} = \frac{4 + 8 + 6 + 5 + 3 + 7}{6} = 5.5 \)

2. Calculate absolute deviations:

|4 - 5.5| = 1.5

|8 - 5.5| = 2.5

|6 - 5.5| = 0.5

|5 - 5.5| = 0.5

|3 - 5.5| = 2.5

|7 - 5.5| = 1.5

3. Compute the AAD:

\( \text{AAD} = \frac{1.5 + 2.5 + 0.5 + 0.5 + 2.5 + 1.5}{6} = 1.5 \)

Result: The AAD for this dataset is 1.5.

Example 1: Compute the mean absolute deviation for the data set [10, 12, 23, 23, 16, 23, 21, 16]

Solution:

1. Calculate the mean (\( \bar{x} \)):

\( \bar{x} = \frac{10 + 12 + 23 + 23 + 16 + 23 + 21 + 16}{8} = 18 \)

2. Calculate absolute deviations:

|10 - 18| = 8

|12 - 18| = 6

|23 - 18| = 5

|16 - 18| = 2

|23 - 18| = 5

|21 - 18| = 3

|16 - 18| = 2

3. Compute the AAD:

\( \text{AAD} = \frac{8 + 6 + 5 + 5 + 2 + 5 + 3 + 2}{8} = 4.5 \)

Result: The AAD for this dataset is 4.5.