Median Absolute Deviation Calculator

Input a data set, and compute the Median Absolute Deviation (MAD) of them.

Compute the Median Absolute Deviation

Result

What Is Median Absolute Deviation?

The Median Absolute Deviation (MAD) is a robust statistic used to evaluate data variability. Unlike standard deviation, MAD is based on the median, making it resistant to extreme values or outliers. It is ideal for analyzing datasets with irregularities.

For a dataset \( x_1, x_2, \dots, x_n \), the MAD is calculated as follows:

Find the Median: Determine the median (\( M \)) of the dataset.
Compute Absolute Deviations: Calculate the absolute difference between each data point and the median (\( |x_i - M| \)).
Find the MAD: Compute the median of these absolute deviations.

The formula for MAD is: \( MAD = \text{median}(|x_i - M|) \) Where:

\( M \) is the median of the original dataset.
\( |x_i - M| \) represents the absolute deviation of each data point from \( M \).

Examples

Example 1: Compute the median absolute deviation for the data set [1, 2, 3, 4, 5]

Solution:

1. Find the median (\( M \)):

\( M = 3 \)

2. Compute absolute deviations:

|1 - 3| = 2

|2 - 3| = 1

|3 - 3| = 0

|4 - 3| = 1

|5 - 3| = 2

3. Find the median of the absolute deviations:

\( \text{Sorted deviations: } 0, 1, 1, 2, 2 \)

Median = 1

Result: The MAD for this dataset is 1.

Example 2: Compute the median absolute deviation for the data set [10, 15, 14, 20, 18]

Solution:

1. Find the median (\( M \)):

\( M = 15 \)

2.Compute absolute deviations:

|10 - 15| = 5

|15 - 15| = 0

|14 - 15| = 1

|20 - 15| = 5

|18 - 15| = 3

3. Find the median of the absolute deviations:

\( \text{Sorted deviations: } 0, 1, 3, 5, 5 \)

Median = 3

Result: The MAD for this dataset is 3.