Question

mean absolute deviation

Answer 1

Mean Absolute Deviation (MAD) is a measure of the dispersion or variability in a dataset. It indicates how much individual data points deviate from the mean of the dataset, on average. MAD is particularly useful because it provides a clear and intuitive measure of variability without being affected by outliers, as it considers absolute values.

### How to Calculate Mean Absolute Deviation:

1. **Find the Mean**: Calculate the mean (average) of the data set.

\[
\text{Mean} (\mu) = \frac{\sum_{i=1}^{n} x_i}{n}
\]

where \(x_i\) represents each individual data point and \(n\) is the total number of data points.

2. **Calculate Deviations**: For each data point, calculate the absolute deviation from the mean:

\[
|x_i - \mu|
\]

3. **Mean of Absolute Deviations**: Finally, calculate the mean of these absolute deviations:

\[
\text{MAD} = \frac{\sum_{i=1}^{n} |x_i - \mu|}{n}
\]

### Example:

Consider the following data set: \( [2, 4, 4, 4, 5, 5, 7, 9] \)

1. Calculate the mean:
\[
\mu = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = \frac{40}{8} = 5
\]

2. Calculate the absolute deviations:
- |2 - 5| = 3
- |4 - 5| = 1
- |4 - 5| = 1
- |4 - 5| = 1
- |5 - 5| = 0
- |5 - 5| = 0
- |7 - 5| = 2
- |9 - 5| = 4

3. Calculate the MAD:
\[
\text{MAD} = \frac{3 + 1 + 1 + 1 + 0 + 0 + 2 + 4}{8} = \frac{12}{8} = 1.5
\]

### Interpretation:

In this example, the mean absolute deviation is 1.5, indicating that, on average, the individual data points deviate from the mean by 1.5 units.

MAD is widely used in statistics and data analysis, particularly in fields such as finance, engineering, and environmental science, where understanding variability is crucial.