Which of the following statements about the mean absolute deviation ($\mathrm{MAD}$) is the most accurate?

• A. The mean absolute deviation is the average of the absolute differences between each data value and the mean of the data set, that is, $\mathrm{MAD}=\frac{\sum \left|x_i-\mu \right|}{n}$.
• B. The mean absolute deviation is the same as the median of the data set.
• C. The mean absolute deviation is always equal to the standard deviation ($\sigma$) for any data set.
• D. The mean absolute deviation is calculated by squaring the differences between each data value and the mean.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand what the mean absolute deviation (MAD) measures. It quantifies the average distance between each data point and the mean of the data set, using absolute values to avoid negative differences.
2. Step 2: Recall the formula for MAD, which is given by: \(\text{MAD} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|\), where \(x_i\) represents each data value, \(\bar{x}\) is the mean of the data set, and \(n\) is the number of data points.
3. Step 3: Compare MAD to other measures mentioned: it is not the median of the data set, so the statement claiming MAD equals the median is incorrect.
4. Step 4: Understand that MAD is generally different from the standard deviation because standard deviation involves squaring the differences before averaging, while MAD uses absolute values. Therefore, MAD is not always equal to the standard deviation.
5. Step 5: Recognize that the calculation of MAD does not involve squaring the differences; that process is part of the variance and standard deviation calculations, so the statement about squaring differences for MAD is incorrect.