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.

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.

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.

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.

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.

Cite this Article

Format

Your Citation

Taylor, Courtney. "Calculating the Mean Absolute Deviation." ThoughtCo, Apr. 14, 2025, thoughtco.com/what-is-the-mean-absolute-deviation-4120569. Taylor, Courtney. (2025, April 14). Calculating the Mean Absolute Deviation. Retrieved from https://www.thoughtco.com/what-is-the-mean-absolute-deviation-4120569 Taylor, Courtney. "Calculating the Mean Absolute Deviation." ThoughtCo. https://www.thoughtco.com/what-is-the-mean-absolute-deviation-4120569 (accessed July 21, 2025).