Median Absolute Deviation: Formula & Examples Video

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  • 0:04 Median Absolute Deviation
  • 0:45 The MAD Formula
  • 3:50 MAD and Outliers
  • 5:23 MAD Practice
  • 6:18 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Measures of deviation are commonly quoted when referring to data. In this lesson, we look at the median absolute deviation, showing how it is calculated and why it is insensitive to outliers.

Median Absolute Deviation

Your friend Jay has taken a weekly botany quiz for 9 weeks and consistently received scores between 83 and 86. Her mean score is 84.6 with a very narrow spread. Jay's twin brother, Out, has the identical scores for the first 8 weeks. For some reason, maybe illness, the result of Out's ninth quiz is very low: a 45. This score throws off Out's performance report and produces a spread which misrepresents the actual consistency of his quiz scores. In this lesson, we describe a measure of the spread called the median absolute deviation, which ignores data outside of a trend.

The MAD Formula

The median absolute deviation is known by its acronym MAD. How cool is that?

The MAD formula is the following:

MAD = median(| x - median(x)|)

where x represents the collection of numbers. In other words, the formula says:

  • First, find the median of x
  • Then, subtract this median from each value in x
  • Then, take the absolute value of these differences
  • Find the median of these absolute differences

Let's return to Jay's quiz scores to make some sense out of this.

First, the 9 scores are 83, 83, 84, 85, 85, 86, 86, 83, and 86. The mean of these scores is the sum 761 divided by 9. The mean is 84.6.

The deviation is a measure of how each score differs from some average. This deviation is the spread we have alluded to. To calculate the standard deviation, we take each score and subtract the mean from it. This gives -1.5556, -1.5556, -0.5556, 0.4444, 0.4444, 1.4444, 1.4444, -1.5556, and 1.4444. Do you see how some of the differences are negative and some are positive? The next step in calculating the standard deviation is to square these differences: 2.4198, 2.4198, 0.3086, 0.1975, 0.1975, 2.0864, 2.0864, 2.4198, and 2.0864. The squaring turns the difference into a positive numbers. We add these positive numbers to get 14.2222. Then, we divide by 9 to get 1.5802. This is the mean of the squared differences.

The last step is to take the square root. This gives us 1.2571, which is the standard deviation. If your calculator or computer program computes the standard deviation, this answer can be arrived at much more quickly. There are two ways to calculate the standard deviation: one method divides by the number in the method known as the ' n of items' and the other method divides by n-1. We are using the divide by n method.

Okay, those 9 scores are really closely bunched together. A small calculated deviation of 1.2571 is a meaningful measure. Now, for some fun! If we compute the standard deviation for Out, we get a deviation of 12.4276. The outlier (the score that doesn't belong with the rest) has had a huge effect! The deviation appears to be nearly ten times larger than Jay's, and all because of one bad quiz result.

MAD and Outliers

This is where the MAD comes on the scene. The MAD measure of deviation handles outliers better.

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