Calculating the Mean Absolute Deviation

Calculating the Mean Absolute Deviation There are many measurements of spread or dispersion in statistics. Although the range and standard deviation are most commonly used, there are other ways to quantify dispersion.  We will look at how to calculate the mean absolute deviation for a data set.   Definition We begin with the definition of the mean absolute deviation, which is also referred to as the average absolute deviation. The formula displayed with this article is the formal definition of the mean absolute deviation. It may make more sense to consider this formula as a process, or series of steps, that we can use to obtain our statistic. We start with an average, or measurement of the center, of a data set, which we will denote by m.  Next, we find how much each of the data values deviates from m.  This means that we take the difference between each of the data values and m.  After this, we take the absolute value of each of the difference from the previous step. In other words, we drop any negative signs for any of the differences.  The reason for doing this is that there are positive and negative deviations from m.  If we do not figure out a way to eliminate the negative signs, all of the deviations will cancel one another out if we add them together.Now we add together all of these absolute values.Finally, we divide this sum by n, which is the total number of data values.  The result is the mean absolute deviation. Variations There are several variations for the above process.  Note that we did not specify exactly what m is. The reason for this is that we could use a variety of statistics for m.  Typically this is the center of our data set, and so any of the measurements of central tendency can be used. The most common statistical measurements of the center of a data set are the mean, median and the mode.  Thus any of these could be used as m in the calculation of the mean absolute deviation. This is why it is common to refer to the mean absolute deviation about the mean or the mean absolute deviation about the median. We will see several examples of this. Example:  Mean Absolute Deviation About the Mean Suppose that we start with the following data set: 1, 2, 2, 3, 5, 7, 7, 7, 7, 9. The mean of this data set is 5.  The following table will organize our work in calculating the mean absolute deviation about the mean.   Data Value Deviation from mean Absolute Value of Deviation 1 1 - 5 = -4 |-4| = 4 2 2 - 5 = -3 |-3| = 3 2 2 - 5 = -3 |-3| = 3 3 3 - 5 = -2 |-2| = 2 5 5 - 5 = 0 |0| = 0 7 7 - 5 = 2 |2| = 2 7 7 - 5 = 2 |2| = 2 7 7 - 5 = 2 |2| = 2 7 7 - 5 = 2 |2| = 2 9 9 - 5 = 4 |4| = 4 Total of Absolute Deviations: 24 We now divide this sum by 10, since there are a total of ten data values.  The mean absolute deviation about the mean is 24/10 2.4. Example:  Mean Absolute Deviation About the Mean Now we start with a different data set: 1, 1, 4, 5, 5, 5, 5, 7, 7, 10. Just like the previous data set, the mean of this data set is 5.   Data Value Deviation from mean Absolute Value of Deviation 1 1 - 5 = -4 |-4| = 4 1 1 - 5 = -4 |-4| = 4 4 4 - 5 = -1 |-1| = 1 5 5 - 5 = 0 |0| = 0 5 5 - 5 = 0 |0| = 0 5 5 - 5 = 0 |0| = 0 5 5 - 5 = 0 |0| = 0 7 7 - 5 = 2 |2| = 2 7 7 - 5 = 2 |2| = 2 10 10 - 5 = 5 |5| = 5 Total of Absolute Deviations: 18 Thus the mean absolute deviation about the mean is 18/10 1.8.  We compare this result to the first example.  Although the mean was identical for each of these examples, the data in the first example was more spread out. We see from these two examples that the mean absolute deviation from the first example is greater than the mean absolute deviation from the second example. The greater the mean absolute deviation, the greater the dispersion of our data. Example:  Mean Absolute Deviation About the Median Start with the same data set as the first example: 1, 2, 2, 3, 5, 7, 7, 7, 7, 9. The median of the data set is 6.  In the following table,  we show the details of the calculation of the mean absolute deviation about the median. Data Value Deviation from median Absolute Value of Deviation 1 1 - 6 = -5 |-5| = 5 2 2 - 6 = -4 |-4| = 4 2 2 - 6 = -4 |-4| = 4 3 3 - 6 = -3 |-3| = 3 5 5 - 6 = -1 |-1| = 1 7 7 - 6 = 1 |1| = 1 7 7 - 6 = 1 |1| = 1 7 7 - 6 = 1 |1| = 1 7 7 - 6 = 1 |1| = 1 9 9 - 6 = 3 |3| = 3 Total of Absolute Deviations: 24 Again we divide the total by 10 and obtain a mean average deviation about the median as 24/10 2.4. Example:  Mean Absolute Deviation About the Median Start with the same data set as before: 1, 2, 2, 3, 5, 7, 7, 7, 7, 9. This time we find the mode of this data set to be 7.  In the following table,  we show the details of the calculation of the mean absolute deviation about the mode. Data Deviation from mode Absolute Value of Deviation 1 1 - 7 = -6 |-5| = 6 2 2 - 7 = -5 |-5| = 5 2 2 - 7 = -5 |-5| = 5 3 3 - 7 = -4 |-4| = 4 5 5 - 7 = -2 |-2| = 2 7 7 - 7 = 0 |0| = 0 7 7 - 7 = 0 |0| = 0 7 7 - 7 = 0 |0| = 0 7 7 - 7 = 0 |0| = 0 9 9 - 7 = 2 |2| = 2 Total of Absolute Deviations: 22 We divide the sum of the absolute deviations and see that we have a mean absolute deviation about the mode of 22/10 2.2. Fast Facts There are a few basic properties concerning mean absolute deviations The mean absolute deviation about the median is always less than or equal to the mean absolute deviation about the mean.The standard deviation is greater than or equal to the mean absolute deviation about the mean.The mean absolute deviation is sometimes abbreviated by MAD.  Unfortunately, this can be ambiguous as MAD may alternately refer to the median absolute deviation.The mean absolute deviation for a normal distribution is approximately 0.8 times the size of the standard deviation. Common Uses The mean absolute deviation has a few applications.  The first application is that this statistic may be used to teach some of the ideas behind the standard deviation. The mean absolute deviation about the mean is much easier to calculate than the standard deviation. It does not require us to square the deviations, and we do not need to find a square root at the end of our calculation. Furthermore, the mean absolute deviation is more intuitively connected to the spread of the data set than what the standard deviation is. This is why the mean absolute deviation is sometimes taught first, before introducing the standard deviation. Some have gone so far as to argue that the standard deviation should be replaced by the mean absolute deviation.  Although the standard deviation is important for scientific and mathematical applications, it is not as intuitive as the mean absolute deviation. For day-to-day applications, the mean absolute deviation is a more tangible way to measure how spread out data are.