Mean Absolute Deviation Calculator

Created:March 24, 2025

The Mean Absolute Deviation (MAD) Calculator helps you measure the average distance between each data point and the mean. Unlike variance or standard deviation, MAD uses absolute differences which makes it less sensitive to outliers and provides a more intuitive measure of dispersion in the original units of your data. It's particularly useful for financial analysis, quality control, and analyzing datasets where you want to understand typical deviations without overemphasizing extreme values.

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Mean Absolute Deviation (MAD)

Definition

Mean Absolute Deviation measures the average absolute difference between each data point and the mean of all data points. It's a measure of dispersion in the original units of measurement, making it more intuitive to interpret than variance.

Formula

Mean Absolute Deviation Formula:

\text{MAD} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|

Where:

$n$ = number of observations
$x_i$ = the value of the $i$ -th observation
$\bar{x}$ = the mean of all observations
$|x_i - \bar{x}|$ = the absolute difference between each value and the mean

Interpretation Guidelines

Lower MAD values indicate data points are closer to the mean (less spread)

Higher MAD values indicate data points are further from the mean (more spread)

MAD is measured in the same units as the original data, making it directly interpretable

Key Advantages

Less sensitive to outliers than standard deviation
Uses the same units as the original data
Easier to interpret for non-statisticians

Step by Step Example

Let's calculate the mean absolute deviation for a dataset of daily temperatures (in °C) for a week:

Day	Temperature (°C)
Monday	22
Tuesday	20
Wednesday	25
Thursday	21
Friday	23
Saturday	24
Sunday	19

Mean Absolute Deviation Calculation

Step 1: Calculate the mean temperature:

\bar{x} = \frac{22 + 20 + 25 + 21 + 23 + 24 + 19}{7} = \frac{154}{7} = 22°C

Step 2: Calculate the absolute deviations from the mean:

Day	Temperature (°C)	Deviation from Mean	Absolute Deviation
Monday	22	22 - 22 = 0	\|0\| = 0
Tuesday	20	20 - 22 = -2	\|-2\| = 2
Wednesday	25	25 - 22 = 3	\|3\| = 3
Thursday	21	21 - 22 = -1	\|-1\| = 1
Friday	23	23 - 22 = 1	\|1\| = 1
Saturday	24	24 - 22 = 2	\|2\| = 2
Sunday	19	19 - 22 = -3	\|-3\| = 3

Step 3: Calculate the mean of the absolute deviations:

\text{MAD} = \frac{0 + 2 + 3 + 1 + 1 + 2 + 3}{7} = \frac{12}{7} = 1.71°C

Interpretation: A MAD of 1.71°C indicates the typical deviation from the average temperature during the week. This is particularly useful for understanding temperature variations in concrete, interpretable units.

MAD vs. Standard Deviation

While both MAD and standard deviation measure dispersion, they have key differences that make each suitable for different situations:

Feature	Mean Absolute Deviation	Standard Deviation
Formula	Average of absolute deviations from the mean	Square root of the average squared deviations from the mean
Units	Same as original data	Same as original data
Outlier Sensitivity	Less sensitive to outliers	More sensitive to outliers (squares the deviations)
Mathematical Properties	Less suitable for further statistical inference	Better mathematical properties for inference
Interpretation	More intuitive for non-technical audiences	More widely used in advanced statistics

When to Use MAD Instead of Standard Deviation

When communicating with non-technical audiences who find averages easier to understand
When dealing with data that may contain outliers that shouldn't have outsized influence
In financial applications where absolute deviations (like dollars) are more relevant than squared terms