What is the Mean Absolute Deviation?
The mean absolute deviation (MAD) is a measure of variability that indicates the average distance between observations and their mean. MAD uses the original units of the data, which simplifies interpretation. Larger values signify that the data points spread out further from the average. Conversely, lower values correspond to data points bunching closer to it. The mean absolute deviation is also known as the mean deviation and average absolute deviation.
This definition of the mean absolute deviation sounds similar to the standard deviation (SD). While both measure variability, they have different calculations. In recent years, some proponents of MAD have suggested that it replace the SD as the primary measure because it is a simpler concept that better fits real life.
In this post, you’ll learn how to find and interpret the mean absolute deviation and understand its formula. I’ll close by comparing it to the standard deviation.
Related post: Measures of Variability
How to Find the Mean Absolute Deviation
The process for finding the mean absolute deviation involves the following three steps.
- Calculate the sample average by summing all observations and dividing by the sample size.
- Find the absolute deviation of all data points from the mean. Take the observed values and subtract them from the mean and then disregard negative signs when they occur.
- Calculate the average of the absolute deviations. Sum the values in step #2 and divide it by the sample size.
The formula for the mean absolute deviation is the following:
Where:
- X = the value of a data point
- µ = mean
- |X – µ| = absolute deviation
- N = sample size
The formula involves absolute deviations. A deviation is the difference between a data point and the mean. The absolute value of the deviation simply tosses out any minus signs that occur. If we didn’t use the absolute value, the pluses and minuses would cancel each other out! Large distances from the mean, whether they’re positive or negative, now all have positive values.
Next, see how finding the mean absolute deviation works on a number line!
Number Line Examples
Let’s bring the mean absolute deviation formula to life by working through some examples. Imagine we have the following two datasets with four data points in each. They both have an average of ten, but their variability is different. The number lines show where each data point falls relative to the average and the distance from it. The mean absolute deviation simply takes those distances and averages them.
In the above dataset, we have absolute deviations of 1, 1, 2, 2. We sum those to obtain 6. Then divide by four to get the mean, which is 1.5. Therefore, we find that the mean absolute deviation of this dataset is 1.5. The average distance between the data points and the mean is 1.5.
In the second dataset, the absolute differences are larger because the data points spread out further. We have distances of 2, 2, 4, 4. They sum to 12. Dividing by 4, the mean is 3. Consequently, MAD is 3 for this dataset. The average distance between the data points and the mean is 3. The larger value indicates the spread of the second dataset is greater than the first.
Worked Example of Finding the Mean Absolute Deviation by Hand
In the worksheet below, we’ll assume that we’ve already calculated the sample average, which is 32. For comparison purposes, I employ the same dataset that I use in my post about SD calculations.
The calculations in the worksheet involve applying the mean absolute deviation formula:
- Taking each observation.
- Subtracting the sample average.
- Calculating the difference.
- Obtaining the absolute value.
Then, at the bottom, we sum the column of absolute deviations and divide it by the sample size of 17. The MAD of this dataset is 11.647. In contrast, the SD is 14.177.
Mean Absolute Deviation vs. Standard Deviation
These two statistics have similarities. They both are measures of variability that use the original data units, and they compare the data points to the mean.
However, there are differences. MAD uses the average distances of the data points from the mean. On the other hand, the SD squares the difference between each data point and the mean, sums the squared differences, divides by the degrees of freedom, and then takes the square root of that sum.
Because the standard deviation squares the differences, outliers have a larger impact on it than on MAD.
According to Geary (1935), the ratio between the two statistics in a normal distribution is the following:
In other words, the mean absolute deviation is approximately 80% the value of the SD in a normal distribution.
Related post: Standard Deviation
Benefits of the Standard Deviation
Clearly, finding and interpreting MAD are more intuitive than they are for the SD. This intuitive nature is why some have made a call to retire the standard deviation as the principal measure of variability.
I’d also guess that the mean absolute deviation is closer to how people think of differences from the mean. Consequently, MAD is a good entry point for understanding the concept of variability. That’s why some high schools are incorporating it into their statistical curriculum.
While the mean absolute deviation is easier to calculate and interpret, the standard deviation has some benefits that MAD can’t match.
For instance, the SD has a special place in normal distributions. For starters, it is one of the distribution’s parameters. Additionally, the empirical rule uses it to estimate the frequency of values in ranges of normal distributions. For nonnormal distributions, you can use the SD with Chebyshev’s theorem for similar reasons. None of that exists for the mean absolute value.
Standard deviations are also baked into many of the formulas for hypothesis tests and confidence intervals.
Finally, the SD can better reflect differences in variability in some cases.
In the graphs below, dataset 2 has more variability than dataset 1.
Despite dataset 2 having more variability, both datasets have a mean absolute deviation of two. However, the SD accurately indicates that dataset 2 has more variability, 2.24 vs. 2.
The SD squares the differences, which gives extra weight to the values further away from the mean. This additional impact reflects the properties of the normal distribution where outliers are substantially less likely to occur. Extreme values do not taper off linearly as the mean absolute deviation implies.
While MAD is easier to calculate and interpret, the standard deviation won’t disappear anytime soon. I’d love to hear about your thoughts on these two statistics in the comment section!
Reference
Geary, R. C. (1935). The ratio of the mean deviation to the SD as a test of normality. Biometrika, 27(3/4), 310–332.
Technically, the two statistics are not typically freely interchangeable. The MAD is the proper descriptive statistic when the population is known, and there is no need to account for un-sampled cohort. This is why the SD weights the outliers heavier than typically ranged data, it is an attempt to make up for the missing data under the presumption that the data is normally distributed. It is inappropriate to use the SD when the population is fully accounted for, using it this way gives an overly rigid impression that is less capable of transparently reflecting the data and revealing variation.
The allegation and example claiming the SD to be more sensitive to variability is flawed. Search the “Datasaurus Dozen” or “Anscombe’s quartet,” these are data sets designed so that evidently different distributions produce the same or nearly the same mean variance and SD, but in all of the cases, the MAD properly distinguishes different looking data (though this goes unmentioned in most presentations). In fact, the SD is not the correct statistic for these examples because, as explained, all the data is presently accounted. Nevertheless, the mathematical point is made, and that is that the SD is NOT as sensitive to the actual data, I suspect that there is actually just as much variability in both of the examples presented here with MAD = 2, that a formal definition of “variability” would reveal, which is why the MAD returns the same value. The intuition likely is missing a dimension of relevant regularity that balances out whatever is salient that leads you to think the second sample has more variability.
SD is here to stay because it has a different role to play than MAD. The controversy misses the fact that they are not competing.
The MAD certainly is more intuitive. It should be possible to modify the empirical rule, hypothesis tests and the like to accommodate the MAD in the future.
When do we use coefficient of variantion
Hi Susan,
Please see my post about the coefficient of variation for details about that.