Understanding Mean Absolute Deviation (MAD)

In statistics, Mean Absolute Deviation (MAD) is a way to measure how spread out or scattered a set of data is. It tells us the average distance between each data point and the mean of the data set.

A larger MAD means the data points are far away from the mean (more spread out). A smaller MAD means the data points are clustered closely around the mean (less spread out).

How to Calculate MAD

Finding the MAD takes three simple steps:

1. Find the mean (average) of the data set.
2. Find the distance of each data point from the mean. This is the absolute value of the difference: $| \text{data point} - \text{mean} |$.
3. Find the mean of those distances by adding them all up and dividing by the number of data points.

Example Problems

Example 1: Find the MAD of 2, 4, 6, 8, 10

Step 1: Find the mean. $\text{Mean} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6$

Step 2: Find the distance of each point from the mean (6).

• $|2 - 6| = 4$
• $|4 - 6| = 2$
• $|6 - 6| = 0$
• $|8 - 6| = 2$
• $|10 - 6| = 4$

Step 3: Find the mean of these distances. $\text{MAD} = \frac{4 + 2 + 0 + 2 + 4}{5} = \frac{12}{5} = 2.4$

The Mean Absolute Deviation is $2.4$.

Example 2: Calculate the MAD for 10, 12, 8, 15, 5

Step 1: Find the mean. $\text{Mean} = \frac{10 + 12 + 8 + 15 + 5}{5} = \frac{50}{5} = 10$

Step 2: Find the distances from the mean (10).

• $|10 - 10| = 0$
• $|12 - 10| = 2$
• $|8 - 10| = 2$
• $|15 - 10| = 5$
• $|5 - 10| = 5$

Step 3: Average the distances. $\text{MAD} = \frac{0 + 2 + 2 + 5 + 5}{5} = \frac{14}{5} = 2.8$

The Mean Absolute Deviation is $2.8$.

Interpreting MAD

Question: Dataset A has a MAD of $1.5$ and Dataset B has a MAD of $4.2$. Which is more spread out?

Answer: Dataset B is more spread out. Because the MAD represents the average distance from the mean, a higher number ($4.2 > 1.5$) indicates that the values in Dataset B are scattered further away from their average compared to Dataset A.