The standard deviation is one of the most common ways to measure the spread of a dataset.
It is calculated as:
Standard Deviation = √( Σ(xi – x)2 / n )
An alternative way to measure the spread of observations in a dataset is the mean absolute deviation.
It is calculated as:
Mean Absolute Deviation = Σ|xi – x| / n
This tutorial explains the differences between these two metrics along with examples of how to calculate each.
Similarities & Differences
As the names imply, both the standard deviation and mean absolute deviation attempt to quantify the typical deviation of observations from the mean in a given dataset.
However, the method that each metric uses is different.
Standard Deviation
The standard deviation finds the squared difference between each observation and the mean of a dataset. It then takes the average of these squared differences and takes the square root.
This leaves us with a number that represents the “standard” or typical deviation of an observation from the mean.
Mean Absolute Deviation
Conversely, the mean absolute deviation finds the absolute deviation between each observation and the mean of the dataset. It then finds the average of these deviations.
This leaves us with a number that represents the average deviation of observations from the mean.
Because the standard deviation finds the squared differences, it will always be equal to or larger than the mean absolute deviation.
When extreme outliers are present, the standard deviation will be considerably larger than the mean absolute deviation. The following example illustrates this point.
Example: Mean Absolute Deviation vs. Standard Deviation
Suppose we have the following dataset of 8 values:
The mean turns out to be 11.
Thus, we would calculate the mean absolute deviation as:
Mean Absolute Deviation = (|3-11| + |5-11| + |6-11| + |8-11| + |11-11| + |14-11| + |17-11| + |24-11|) / 8 = 5.5.
And we would calculate the standard deviation as:
Standard Deviation = √((3-11)2 + (5-11)2 + (6-11)2 + (8-11)2 + (11-11)2 + (14-11)2 + (17-11)2 + (24-11)2) / 8) = 6.595.
As mentioned earlier, the standard deviation will always be equal to or larger than the mean absolute deviation.
However, the difference between the standard deviation and the mean absolute deviation will be particularly large if there are extreme outliers in the dataset.
For example, consider the following dataset with an extreme outlier for the last value:
It turns out that the standard deviation for this dataset is 63.27 while the mean absolute deviation is 41.75.
The extreme outlier causes the standard deviation to be much larger than the mean absolute deviation.