how to find the mad

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. 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:

Formula for the mean absolute deviation.

Where:

X = the value of a data point
µ = mean
|X – µ| = absolute deviation
N = sample size

The calculations involve 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.

Let's see how that works on a number line!

Number Line 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.

Number line that displays a smaller mean absolute deviation.

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, the mean absolute deviation of this dataset is 1.5. The average distance between the data points and the mean is 1.5.

Number line that displays a broader range and a larger MAD.

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 for the mean absolute deviations in the worksheet involve the following:

Taking each observation.
Subtracting the sample average.
Calculating the difference.
Obtaining the absolute value.

Worksheet that displays at dataset and the calculations for the mean absolute deviation.

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:

${\displaystyle \frac {{\text{MAD}}}{{\text{SD}}}} = 0.79788$

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.

Graph that compares the mean absolute deviation to the standard deviation.

Graph that compares MAD to the standard deviation for a second dataset.

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!