Mean Absolute Deviation (MAD): Average of Absolute Deviations from the Mean

Mean Absolute Deviation (MAD) represents the average of absolute deviations from the mean, providing a measure of dispersion less sensitive to outliers compared to Standard Deviation.

Mean Absolute Deviation (MAD) is a statistical measure used to quantify the amount of variation or dispersion in a set of data points. It is the average of the absolute deviations from a central point, typically the mean of the data set. Unlike the standard deviation, which squares the deviations, MAD uses absolute values, making it less sensitive to extreme outliers.

Definition and Formula

Definition: Mean Absolute Deviation (MAD) represents the average of the absolute differences between each data point and the mean of the data set. It provides a simple, yet robust, measure of the variability of a dataset.

Formula:

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

Where:

• \( n \) = Number of data points
• \( x_i \) = Individual data points
• \( \mu \) = Mean of the data set

Calculation Steps

• Compute the Mean (\( \mu \)): Find the mean of the data set by summing all data points and dividing by the number of points.

  $$ \mu = \frac{\sum_{i=1}^{n} x_i}{n} $$

• Calculate Absolute Deviations: For each data point (\( x_i \)), calculate the absolute deviation from the mean (\( |x_i - \mu| \)).

• Determine the Mean Absolute Deviation (MAD): Sum all the absolute deviations and divide by the total number of data points (\( n \)) to get the MAD.

Examples

Example 1: Simple Data Set

Consider a small data set: {2, 4, 6, 8, 10}

• Mean Calculation:

  $$ \mu = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6 $$

• Absolute Deviations:

  $$ |2-6| = 4, \quad |4-6| = 2, \quad |6-6| = 0, \quad |8-6| = 2, \quad |10-6| = 4 $$

• MAD Calculation:

  $$ \text{MAD} = \frac{4 + 2 + 0 + 2 + 4}{5} = \frac{12}{5} = 2.4 $$

Applications

• Descriptive Statistics: MAD is used to describe data sets in terms of their variability.
• Robustness to Outliers: Unlike standard deviation, MAD is less affected by outliers, making it useful in datasets with extreme values.

Comparisons

MAD vs. Standard Deviation

• Sensitivity to Outliers:

  • MAD: Less sensitive to outliers as it uses absolute deviations.
  • Standard Deviation: More sensitive to outliers due to squaring deviations.

• Interpretation:

  • MAD: Provides a straightforward average of absolute deviations.
  • Standard Deviation: Provides a measure of dispersion assuming a normal distribution.

FAQs

Why use MAD instead of standard deviation?

MAD is preferred when dealing with datasets that contain outliers, as it provides a more robust measure of central tendency and dispersion.

Can MAD be used for skewed distributions?

Yes, because it is less sensitive to outliers and extreme values, MAD works well with skewed distributions.

Is MAD applicable in finance?

Yes, MAD can be used in finance to assess the risk and volatility of asset returns.

How does MAD relate to the mean?

MAD measures the average distance of each data point from the mean, providing insight into the dataset’s spread.

Summary

Mean Absolute Deviation (MAD) offers a robust and straightforward way to assess the variability in a dataset. By taking the average of absolute deviations, MAD provides a reliable measure less influenced by outliers than standard deviation. This makes it particularly useful in many applications where robustness and simplicity are desirable.

References:

1. Understanding Statistics: Concepts and Applications. John Wiley & Sons.
2. Statistical Methods for Data Analysis. Chapman & Hall.
3. Applied Multivariate Statistical Analysis. Pearson Education.