Mean Absolute Deviation

Mean Absolute Deviation is a statistical measure that quantifies the dispersion or variability of a set of data points. Its purpose is to provide a measure of how spread out the data is around the mean, regardless of the direction of the differences.

Mean Absolute deviation is calculated by finding the average absolute difference between each data point and the mean of the dataset. It helps to understand the average amount by which individual data points deviate from the mean, thus giving insight into the overall variability and stability of the data set.

Mean Absolute Deviation (MAD) determines the average deviation or dispersion within a data set. It is calculated by taking the absolute value of the differences between each data point and the mean, then finding the average of these absolute differences.

The usability of Mean Absolute Deviation lies in its ability to provide valuable insights into the variability and spread of data, regardless of the direction of the deviations. Unlike other measures like variance or standard deviation, MAD is not influenced by squared differences and is thus more robust to outliers.

By analyzing MAD, researchers and analysts can understand the typical deviation from the mean, which helps in assessing the data's consistency, stability, and predictability. MAD is widely used in various fields, such as finance, economics, and quality control, to assess data quality, compare different data sets, and make informed decisions based on the variability of the data.

The formula for calculating Mean Absolute Deviation is as follows:

MAD = Σ |xᵢ - μ| / n

Where:

• MAD represents the Mean Absolute Deviation.
• Σ denotes the summation symbol, indicating that one must sum the absolute differences for each data point.
• xᵢ represents each data point in the dataset.
• μ represents the mean of the dataset.
• n represents the total number of data points in the dataset.

Let us have a look at the examples to understand the concept better.

Example #1

Suppose one collects commute time data for a group of individuals over a week and want to understand the average deviation from the mean commute time.

Let's say the commute times for the group on Monday to Friday are as follows:

25 minutes, 30 minutes, 27 minutes, 40 minutes, and 35 minutes.

To calculate MAD, one would first find the mean by adding up all the commute times (25 + 30 + 27 + 40 + 35) and dividing by the total number of data points (5), which gives a mean of 31.4 minutes.

Next, one would find the absolute difference between each commute time and the mean:

|25 - 31.4| = 6.4, |30 - 31.4| = 1.4, |27 - 31.4| = 4.4, |40 - 31.4| = 8.6, |35 - 31.4| = 3.6.

The average of these absolute differences is:

(6.4 + 1.4 + 4.4 + 8.6 + 3.6) / 5 = 4.88 minutes

In this example, the MAD of the commute times in 4.88 minutes, indicating that, on average, the individual commute times deviate from the mean by approximately 4.88 minutes. This measure helps understand the dispersion or variability in commute times and can be used for comparisons or decision-making regarding scheduling or transportation planning.

Example #2

Let us look at an example of Mean Absolute Deviation (MAD) in a financial context:

In analyzing two investment portfolios, Portfolio A and Portfolio B, the MAD can be calculated to evaluate the dispersion of their returns. Let's consider the monthly returns for both portfolios:

Portfolio A: 5%, -2%, 3%, 1%, -4%

Portfolio B: 2%, 0%, 1%, -1%, -3%

By calculating the absolute difference between each return and its corresponding mean return and then finding the average of these absolute differences, the MAD can be determined for each portfolio.

For instance, if the mean return for Portfolio A is 0.6%, the absolute differences from the mean are 4.4%, 2.6%, 2.4%, 0.6%, and 4.6%. The average of these absolute differences represents the MAD for Portfolio A.

Let us look at advantages of MAD:

• Robust to outliers: MAD is less influenced by extreme values than other measures like variance or standard deviation.
• Intuitive interpretation: MAD represents the average absolute deviation from the mean, providing a straightforward understanding of variability.
• Applicable to non-normally distributed data: MAD can be used with datasets that do not follow a normal distribution.
• Useful for comparing datasets: MAD allows for easy comparisons of dispersion between different datasets, facilitating the identification of variations across groups or periods.

Let us look at disadvantages of MAD:

• Lack of mathematical properties: MAD does not possess certain mathematical properties like additivity, limiting its usefulness in some statistical analyses.
• Less efficient estimator: MAD is generally a less efficient estimator of dispersion compared to other measures, requiring a larger sample size for similar precision.
• Limited inferential power: MAD does not provide information about the shape of the distribution or relationships between variables, restricting certain types of inferences.
• Lack of theoretical basis: MAD does not have a strong theoretical foundation like other measures, making it less commonly used in certain statistical analyses.