Mean Absolute Deviation

Definition

The mean absolute deviation (MAD) is a measure of statistical dispersion that calculates the average absolute difference between each data point and the mean of the dataset. It provides a way to quantify the typical magnitude of the deviations from the central tendency, giving an indication of the spread or variability within the data.

5 Must Know Facts For Your Next Test

1. The mean absolute deviation is calculated by taking the absolute value of the difference between each data point and the mean, and then finding the average of those absolute differences.
2. Unlike variance and standard deviation, the mean absolute deviation is less sensitive to outliers in the data, making it a more robust measure of spread.
3. The mean absolute deviation has the same units as the original data, making it easier to interpret the magnitude of the deviations compared to standard deviation.
4. Mean absolute deviation is commonly used in finance to measure the typical size of deviations from the average return of a financial asset or portfolio.
5. When the data follows a normal distribution, the mean absolute deviation is approximately 0.7979 times the standard deviation.

Review Questions

• Explain how the mean absolute deviation differs from variance and standard deviation as measures of statistical dispersion.
  • The key difference is that the mean absolute deviation calculates the average of the absolute differences from the mean, whereas variance and standard deviation use the squared differences. This makes the mean absolute deviation less sensitive to outliers in the data, as extreme deviations are not magnified to the same degree. Additionally, the mean absolute deviation retains the original units of the data, making it more intuitive to interpret the typical size of deviations compared to the unitless variance and the standard deviation, which has different units than the original data.
• Describe how the mean absolute deviation can be used to analyze the spread of returns in a financial portfolio.
  • In finance, the mean absolute deviation is commonly used to measure the typical size of deviations from the average return of a financial asset or portfolio. This provides a measure of the overall risk or volatility of the investment, as a higher mean absolute deviation indicates a greater degree of variability in the returns. Compared to standard deviation, the mean absolute deviation is less influenced by outlier returns, making it a more robust metric for understanding the typical magnitude of deviations an investor can expect from the average performance.
• Analyze how the relationship between the mean absolute deviation and standard deviation changes depending on the underlying distribution of the data.
  • When the data follows a normal distribution, the mean absolute deviation is approximately 0.7979 times the standard deviation. This relationship holds because the normal distribution is symmetric, and the average of the absolute deviations is proportional to the standard deviation. However, for non-normal distributions, this relationship may not hold. For skewed or heavy-tailed distributions, the mean absolute deviation and standard deviation can diverge, as the mean absolute deviation is less sensitive to extreme outliers. Understanding this context-dependent relationship is crucial when interpreting and comparing these measures of statistical dispersion.

Related terms

Variance:

A measure of statistical dispersion that calculates the average squared deviation of each data point from the mean.

Standard Deviation:

The square root of the variance, which provides a measure of the average absolute deviation from the mean in the original units of the data.

Outliers:

Data points that lie an abnormal distance from other values in a dataset, which can significantly impact measures of dispersion like the mean absolute deviation.