Mean Absolute Deviation

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21 Aug, 2024

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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.

What Is Mean Absolute Deviation?

Mean Absolute Deviation (MAD) measures the dispersion or variability within a dataset by calculating the average absolute difference between each data point and the mean.
It is robust to outliers, making it a reliable measure when dealing with datasets that may contain extreme values.
MAD has an intuitive interpretation as the average absolute deviation from the mean, allowing for easier understanding and communication of data spread.
MAD is useful for comparing datasets, identifying variations across groups or time periods, and supporting informed decision-making based on data variability.

Mean Absolute Deviation Explained

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.

Formula

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.

Calculation Examples

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.

Advantages And Disadvantages

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.

Mean Absolute Deviation vs Standard Deviation

Let us take a look at differences between mean absolute deviation and standard deviation

Mean Absolute Deviation (MAD)	Standard Deviation
Robust to outliers	Sensitive to outliers
Intuitive interpretation	Widely used and understood
Applicable to non-normally distributed data	Assumes normal distribution
Less efficient estimator	More efficient estimator
Less commonly used in statistical analyses	Commonly used in statistical analyses
Provides an absolute measure of dispersion	Dependent on the units of measurement

Mean Absolute Deviation vs Variance

Let us have a look at the differences between mean absolute deviation and variance

Mean Absolute Deviation (MAD)	Variance
Robust to outliers	Sensitive to outliers
Provides the average absolute deviation from the mean	Measures the average squared deviation from the mean
Intuitive interpretation	Requires squaring of deviations, which may be less intuitive
The absolute measure of dispersion	Measures dispersion relative to the mean
Less commonly used in some statistical analyses	Widely used in statistical analyses
Not influenced by the units of measurement	Dependent on the units of measurement

Frequently Asked Questions (FAQs)

1. How to calculate mean absolute deviation in Excel ?

One can use the AVERAGE and ABS functions to calculate the mean absolute deviation in Excel. Subtract the mean from each data point, take the absolute value of each difference, and then find the average of those absolute differences.

2. Is mean absolute deviation, a measure of center ?

No, mean absolute deviation is not a measure of center. It measures dispersion or variability in a dataset, quantifying the average absolute difference between each data point and the mean.

3. Why is mean absolute deviation important?

Mean absolute deviation is important because it provides insights into the variability or spread of data, helping to assess data quality, compare datasets, and make informed decisions. It is robust to outliers and provides a straightforward interpretation of average deviation.

4. How are mean absolute deviation and interquartile range similar?

Mean absolute deviation and interquartile range are similar in measuring the variability or spread in a dataset. However, they differ in the way they calculate and represent that spread. Mean absolute deviation considers the average absolute difference from the mean, while interquartile range focuses on the range between the 25th and 75th percentiles.