Mean absolute deviation

What Is Mean Absolute Deviation?

Mean Absolute Deviation (MAD) is a measure of statistical dispersion that describes the average distance between each data point in a data set and the arithmetic mean of that data set. As a fundamental concept within statistical analysis, the mean absolute deviation quantifies the typical variability or spread of values around the central point of a distribution. Unlike other measures of dispersion, the mean absolute deviation is particularly intuitive because it directly represents the average of the absolute differences from the mean, providing a clear indication of how scattered the data points are.

History and Origin

The concept of average deviation has roots in early statistical thought, with its principles appearing in the work of mathematicians and statisticians as they sought to better understand data distributions. One of the earliest known mentions of a related concept, the median absolute deviation, dates back to Carl Friedrich Gauss in 1816, in a paper concerning the accuracy of numerical observations. While the mean absolute deviation differs in its choice of central point (the mean versus the median), its underlying utility as a measure of typical deviation has been recognized for centuries. The development and adoption of various statistical measures often evolved based on computational ease and theoretical properties. Historically, the mean absolute deviation faced challenges in analytical calculations due to the involvement of absolute value functions, which made it less amenable to certain mathematical techniques compared to squared deviations.⁶

Key Takeaways

• Mean absolute deviation (MAD) measures the average distance between each data point and the mean of a data set.
• It provides a straightforward, easily interpretable measure of data dispersion.
• MAD is less sensitive to outliers than standard deviation because it uses absolute values rather than squared differences.
• A higher mean absolute deviation indicates greater variability within the data.
• Despite its simplicity, it is less commonly used in advanced statistical inference due to algebraic complexities.

Formula and Calculation

The formula for the mean absolute deviation for a given data set is:

Where:

• (\text{MAD}) = Mean Absolute Deviation
• (x_i) = Each individual data point in the data set
• (\bar{x}) = The arithmetic mean of the data set
• (|x_i - \bar{x}|) = The absolute deviation of each data point from the mean
• (n) = The total number of data points in the set

To calculate the mean absolute deviation, one first determines the mean of all data points. Next, the absolute difference between each data point and the mean is calculated. Finally, these absolute differences are summed, and the total is divided by the number of data points. This process provides the average absolute deviation from the mean.

Interpreting the Mean Absolute Deviation

Interpreting the mean absolute deviation is straightforward: it represents the average amount by which individual data points deviate from the data set's average. For instance, if a set of investment returns has a mean absolute deviation of 2%, it means that, on average, individual returns vary by 2 percentage points from the average return. A smaller mean absolute deviation indicates that the data points are clustered more closely around the mean, suggesting less variability and a more consistent data set. Conversely, a larger mean absolute deviation implies that the data points are more spread out from the mean, indicating higher dispersion. This measure's strength lies in its direct interpretability, as it uses the same units as the original data.

Hypothetical Example

Consider a small portfolio with the following monthly returns over five months: 2%, 5%, 3%, 8%, and 2%. Let's calculate the mean absolute deviation for these returns.

1. Calculate the Mean:
   (\bar{x} = \frac{2 + 5 + 3 + 8 + 2}{5} = \frac{20}{5} = 4%)

2. Calculate the Absolute Deviations from the Mean:

   • Month 1: (|2% - 4%| = |-2%| = 2%)
   • Month 2: (|5% - 4%| = |1%| = 1%)
   • Month 3: (|3% - 4%| = |-1%| = 1%)
   • Month 4: (|8% - 4%| = |4%| = 4%)
   • Month 5: (|2% - 4%| = |-2%| = 2%)

3. Sum the Absolute Deviations:
   (2% + 1% + 1% + 4% + 2% = 10%)

4. Divide by the Number of Data Points:
   (\text{MAD} = \frac{10%}{5} = 2%)

In this example, the mean absolute deviation of the portfolio performance is 2%. This means that, on average, the monthly returns for this portfolio deviate by 2 percentage points from the average monthly return of 4%. This provides a clear metric of the portfolio's recent volatility.

Practical Applications

While not as prevalent as standard deviation in certain advanced financial models, the mean absolute deviation finds practical applications in various contexts, particularly when a straightforward and easily understandable measure of spread is desired. In financial markets, for example, analysts might use mean absolute deviation to assess the consistency of a stock's daily price movements or the stability of bond yields. Investment firms, such as Morningstar, include "Absolute Deviation" as a data point in their custom calculation capabilities, defining it as the average deviation from a benchmark in absolute terms, highlighting its use in evaluating investment performance.⁵

Mean absolute deviation is also relevant in risk management to gauge the typical dispersion of potential outcomes, offering insights into the range of expected losses or gains. For instance, economists at institutions like the Federal Reserve might analyze the mean absolute deviation of economic indicators to understand the consistency of economic data or the typical deviation from trends.⁴ Its intuitive nature makes it useful for communicating variability to non-technical audiences, as it directly answers the question: "On average, how far do our data points spread from the center?"

Limitations and Criticisms

Despite its intuitive appeal, the mean absolute deviation has certain limitations that have led to its less frequent use compared to other measures like standard deviation, particularly in advanced quantitative analysis. The primary criticism stems from the use of the absolute value function in its calculation. While making it robust to outliers and easier to interpret, this function makes algebraic manipulation and further statistical analysis more complex. For instance, calculating the mean absolute deviation of a combination of variables or performing complex statistical inference becomes challenging.², ³

In many statistical methodologies, squaring deviations (as in variance and standard deviation) is preferred because squared terms have desirable mathematical properties that allow for easier differentiation and optimization in models like linear regression. The mean absolute deviation, while robust, lacks these convenient algebraic properties, which limits its integration into more complex statistical frameworks and probability distribution theories. Researchers have noted that while the mean absolute deviation is often more efficient in practice for non-normal distributions and less sensitive to errors, its analytical tractability remains a drawback.¹

Mean Absolute Deviation vs. Standard Deviation

Mean Absolute Deviation (MAD) and Standard Deviation are both measures of dispersion or variability within a data set, but they differ significantly in their calculation and properties.

The primary distinction lies in how they handle deviations from the mean. Mean absolute deviation uses the absolute value of these differences, treating all deviations equally regardless of their magnitude. In contrast, standard deviation squares the differences, which gives greater weight to larger deviations. This mathematical property makes standard deviation more suitable for certain statistical theories, especially those assuming a normal probability distribution. However, for a simple understanding of typical spread, mean absolute deviation often offers a clearer picture that aligns with common intuition.

FAQs

Why is the Mean Absolute Deviation sometimes preferred over Standard Deviation?

The mean absolute deviation can be preferred in situations where simplicity and direct interpretability are paramount, or when dealing with data sets that may contain outliers. Because it uses absolute differences rather than squared differences, it is less influenced by extreme values, providing a more robust measure of typical variability. Its value is also expressed in the original units of the data, making it easier to understand for a non-expert audience.

Can Mean Absolute Deviation be zero?

Yes, the mean absolute deviation can be zero. This occurs only when all data points in a data set are identical. If every value is the same, then each data point is equal to the arithmetic mean, resulting in zero deviation for every point, and thus a mean absolute deviation of zero.

Is Mean Absolute Deviation used in finance?

While the standard deviation is more commonly used for measures like volatility in complex financial models, mean absolute deviation is indeed used in finance for descriptive purposes. It can help assess the average dispersion of investment returns or price movements, especially when a more robust measure, less sensitive to extreme price swings, is desired for basic analysis or internal risk management reporting. Some financial data providers may also include it in their custom calculation offerings.