Mean absolute deviation

 

Mean absolute deviation (MAD) is a measure of how spread out a set of values is. It is calculated by finding the absolute differences between each value in the set and the mean of the set, then taking the average of those absolute differences.

The formula for calculating MAD is:

$���=\frac{1}{�}{\sum }_{�=1}^{�}\mathrm{\mid }{�}_{�}-\overline{�}\mathrm{\mid }$

Where:

• $���$ is the mean absolute deviation.
• $�$ is the number of values in the set.
• ${�}_{�}$ represents each individual value in the set.
• $\overline{�}$ is the mean (average) of the set.

Here's a step-by-step example of how to calculate MAD:

1. Let's say we have a set of numbers: {10, 12, 15, 18, 20}.
2. First, find the mean of the set: $\overline{�}=\frac{10+12+15+18+20}{5}=\frac{75}{5}=15$
3. Next, calculate the absolute differences between each value and the mean:
   • Absolute difference for 10: $\mathrm{\mid }10-15\mathrm{\mid }=5$
   • Absolute difference for 12: $\mathrm{\mid }12-15\mathrm{\mid }=3$
   • Absolute difference for 15: $\mathrm{\mid }15-15\mathrm{\mid }=0$
   • Absolute difference for 18: $\mathrm{\mid }18-15\mathrm{\mid }=3$
   • Absolute difference for 20: $\mathrm{\mid }20-15\mathrm{\mid }=5$
4. Add up these absolute differences: $5+3+0+3+5=16$
5. Finally, divide this sum by the number of values in the set to get the mean absolute deviation: $���=\frac{16}{5}=3.2$

So, the mean absolute deviation for the set {10, 12, 15, 18, 20} is 3.2. This value gives us an idea of how much the individual values in the set deviate from the mean on average.