What is the Median Absolute Deviation?
The median absolute deviation is a measure of variability that indicates the typical distance between observations and the median. Unlike the mean absolute deviation, which uses the average, this method centers on the median, making it more resistant to outliers. The result uses the same units as the data, which helps with interpretation. Larger values signify that the data points spread further from the median, while lower values mean they cluster more tightly around it. Statisticians frequently abbreviate it as MAD, but sometimes use MADM to avoid confusion with the mean absolute deviation.
The median absolute deviation definition sounds similar to both the mean AD and standard deviation (SD). All these statistics measure variability using the natural data units, but they use different calculations. While MAD is less common in introductory courses, it is a standard tool in robust statistics because of its resistance to outliers. Some advanced classes or textbooks introduce it alongside the standard deviation to highlight this robustness.
The median absolute deviation belongs to the family of robust statistics, which are methods designed to work reliably even when data include outliers or come from skewed distributions. Unlike traditional measures, where a few extreme values can distort the results, robust methods give results that better reflect the bulk of the data. This property makes the MAD especially useful when you want a measure of spread that isn’t overly influenced by unusual observations.
An unusual feature of the median absolute deviation is that it can equal zero when at least half of the data values are the same. That doesn’t occur with most other measures of dispersion, making it a distinctive property of this statistic.
In this post, you’ll learn what the median absolute deviation is, how to find it, the formula, how it compares to the standard deviation, and using it to find outliers.
Related post: Measures of Variability
Example
Psychologists often use the median absolute deviation when analyzing data like reaction times. In these studies, a few participants might respond extremely slowly or quickly, creating outliers. Researchers can’t simply remove those unusual values because they are legitimate data points. If they used the standard deviation, those values would inflate the variability, making the data look more variable than it really is. By using MAD instead, analysts get a measure of spread that reflects the typical differences in response times without being distorted by a handful of extreme cases (Leys, Ley, Klein, Bernard, & Licata, 2013).
How to Find the Median Absolute Deviation
The process for finding the median absolute deviation involves three steps:
- Calculate the median of the dataset.
- Find the absolute deviation of each data point from the median. Subtract the median from each value and disregard any negative signs.
- Find the median of those absolute deviations.
The formula for the median absolute deviation is the following:
Where:
- Xi = a data point
- M = the sample median
- |Xi – M| = absolute deviation from the median
This formula uses absolute deviations just like the mean absolute deviation. However, instead of averaging them, you take their median. That shift gives the statistic its strong resistance to outliers.
We’ll go through this calculation process in the worked example later.
Learn about Median Definition and Uses.
Number Line Examples
Let’s illustrate the median absolute deviation with some simple examples. Suppose we have two datasets, each with four data points. Both have a median of ten, but their spreads differ. You can see how far each point lies from the median on the number lines below.
In the first dataset, the absolute deviations are 1, 1, 2, 2. The median of those deviations is 1.5. So MAD is 1.5.
In the second dataset, the deviations are 2, 2, 4, 4. The median of those is 3. Consequently, the MAD is 3 for this dataset.
These results tell us the typical distance of a data point from the median is larger in the second dataset, reflecting greater spread.
Worked Example of Finding the Median Absolute Deviation by Hand
Let’s calculate the median absolute deviation for a larger dataset. In the worksheet below, we’ll assume that we’ve already calculated the sample median, which is 29. For comparison purposes, I employ the same dataset that I use in my post about SD calculations. Here is the CSV dataset to try it yourself: VariabilityExample.
Alternatively, use my free Median Absolute Deviation Calculator! It calculates the results, finds outliers, and creates a histogram of your data.
The calculations in the worksheet involve applying the mean absolute deviation formula. The steps are the following:
- Taking each observation.
- Subtracting the sample median.
- Calculating the difference.
- Obtaining the absolute value.
- Find the median of the absolute deviations.
After you calculate all the absolute deviations from the sample median in Step 4, the final step is to take the median of those absolute deviations. This process involves sorting the absolute values from smallest to largest and then identifying the middle value. It’s easy to forget this second median step and mistakenly average the deviations instead.
For Step 5, the sorted absolute deviations (ADs) are below:
0, 3, 4, 4, 5, 7, 8, 9, 9, 10, 10, 13, 17, 18, 23, 26, 29
Following this process, the MAD of this dataset is 9 because there are eight values below it and above it the list of sorted ADs. Conversely, the standard deviation is 14.2.
Median Absolute Deviation vs. Standard Deviation
Both the median absolute deviation and the standard deviation are measures of variability. Each uses the original data units and compares data points to a central tendency.
However, there are crucial differences:
- Resistance to outliers: MAD is highly robust because it relies on the median. A single extreme value won’t change it much. In contrast, the standard deviation can grow substantially with outliers because it squares the differences.
- Interpretation: MAD is more intuitive in some situations, because it shows the typical distance from the center without being distorted by a few extreme values.
- Mathematical properties: The standard deviation is deeply tied to the normal distribution. It appears as a parameter in its definition, powers the empirical rule, and is central to hypothesis tests and confidence intervals. MAD doesn’t have those same special roles.
Interestingly, for a normal distribution, MAD relates to the standard deviation through a constant factor:
SD ≈ 1.4826 × MAD
This scaling factor allows statisticians to use the MAD as a robust estimate of the standard deviation when data may contain outliers.
Related post: Standard Deviation
Benefits of the Median Absolute Deviation
The median absolute deviation is straightforward, robust, and easy to interpret. It shows how tightly data points cluster around the median without being thrown off by extreme values.
Because of these advantages, analysts frequently use MAD in robust statistics, where outliers and skewed distributions make other measures less reliable. It’s also an excellent teaching tool for illustrating dispersion in a way that resists distortion from unusual values.
For example, researchers use the median absolute deviation as a practical tool for detecting outliers. Leys et al. (2013) recommend identifying outliers as values that fall outside the range of the median plus or minus 2.5 times the MAD. They show that the common mean ± SD rule can completely miss obvious outliers because both the mean and standard deviation are distorted by extreme values. In contrast, the MAD-based rule resists those distortions and gives a clearer boundary, allowing analysts to detect truly unusual values even in small or skewed samples. Learn more about 5 Ways to Find Outliers.
Still, the standard deviation won’t disappear anytime soon. Its mathematical connections to probability distributions and statistical methods give it a role that the MAD can’t replace.
I’d love to hear your thoughts about the median absolute deviation in the comments below!
References
Croux, C., & Dehon, C. (2010). Robust estimation of location and scale. KU Leuven / Université Libre de Bruxelles. Retrieved from https://feb.kuleuven.be/public/u0017833/PDF-FILES/Croux_Dehon5.pdf
Leys, C., Ley, C., Klein, O., Bernard, P., & Licata, L. (2013). Detecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median. Journal of Experimental Social Psychology, 49(4), 764–766. https://doi.org/10.1016/j.jesp.2013.03.013
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