Average the absolute deviations from a chosen center (mean or median).
Key Concepts
- Center choice: mean or median
- Deviation: absolute difference |x_i − center|
- MAD variants: mean-based or median-based MAD
Steps
- Action: Choose center (mean or median)
- Action: Compute d_i = |x_i − center|
- Action: Compute MAD by chosen method
Formula
MAD from mean:
$$MAD_{text{mean}} = frac{1}{n} sum_{i=1}^{n} |x_i – overline{x}|$$
MAD from median:
$$MAD_{text{median}} = frac{1}{n} sum_{i=1}^{n} |x_i – mathrm{med}(x)|$$
Example
Data: x = [2, 4, 7, 9, 12]
Mean: (bar{x} = frac{1}{5}(2+4+7+9+12) = 6.8)
Deviations: |2−6.8| = 4.8, |4−6.8| = 2.8, |7−6.8| = 0.2, |9−6.8| = 2.2, |12−6.8| = 5.2
MAD from mean: (frac{1}{5}(4.8+2.8+0.2+2.2+5.2) = 3.04)
Median: med(x) = 7
Deviations from median: 5, 3, 0, 2, 5
MAD from median (average): (frac{1}{5}(5+3+0+2+5) = 3)
MAD from median (robust): median(|x_i − med|) = median(5,3,0,2,5) = 3
Notes: Use MAD_mean to measure variability around the mean; MAD_median or median(|…|) is a robust alternative less affected by outliers.