Mathwords logoReference LibraryMathwords

Mean Absolute Deviation (MAD)

Mean Absolute Deviation (MAD) is a measure of how spread out a set of numbers is. You find it by calculating how far each data point is from the mean, ignoring negative signs, and then averaging those distances.

The Mean Absolute Deviation (MAD) of a data set is the arithmetic mean of the absolute deviations from the data's mean. It provides a single number that describes the typical distance between each data value and the center of the data set. Unlike variance, MAD keeps values in the same units as the original data, making it straightforward to interpret.

Key Formula

MAD=1n∑i=1n∣xi−xˉ∣\text{MAD} = \frac{1}{n}\sum_{i=1}^{n} |x_i - \bar{x}|
Where:
  • nn = the number of data points
  • xix_i = each individual data value
  • xˉ\bar{x} = the mean of the data set
  • ∣  ∣|\;| = absolute value (distance from zero, always positive)

Worked Example

Problem: Find the Mean Absolute Deviation of this data set: 2, 6, 7, 9, 11.
Step 1: Find the mean: Add all the values and divide by how many there are.
xˉ=2+6+7+9+115=355=7\bar{x} = \frac{2 + 6 + 7 + 9 + 11}{5} = \frac{35}{5} = 7
Step 2: Find each absolute deviation: Subtract the mean from each data point and take the absolute value.
∣2−7∣=5,∣6−7∣=1,∣7−7∣=0,∣9−7∣=2,∣11−7∣=4|2-7|=5,\quad |6-7|=1,\quad |7-7|=0,\quad |9-7|=2,\quad |11-7|=4
Step 3: Find the mean of those absolute deviations: Add up all the absolute deviations and divide by the number of data points.
MAD=5+1+0+2+45=125=2.4\text{MAD} = \frac{5 + 1 + 0 + 2 + 4}{5} = \frac{12}{5} = 2.4
Answer: The Mean Absolute Deviation is 2.4. On average, each data point is 2.4 units away from the mean of 7.

Visualization

Why It Matters

MAD gives you a simple way to describe how consistent or varied a set of data is. For example, if two basketball players both average 15 points per game but one has a MAD of 1 and the other has a MAD of 6, you know the second player's scoring is much less predictable. Scientists and analysts use MAD as a quick, intuitive measure of spread that's easier to understand than standard deviation.

Common Mistakes

Mistake: Forgetting to take the absolute value, so negative and positive deviations cancel out.
Correction: Without absolute values, the deviations from the mean always sum to zero. The whole point of MAD is to use absolute value so every distance counts as positive.
Mistake: Dividing by the wrong number when finding the mean of the deviations.
Correction: Divide by the total number of data points in the original set — the same number you used when calculating the mean.

Related Terms

  • Mean — MAD is calculated around the mean
  • Absolute Value — Used to make all deviations positive