Key takeaway: The mean absolute deviation (MAD) is the average distance between each data value and the mean, so it tells how far a typical value lies from the center of the data.
Grade 66.SP.B.5cStatistics and Data
Definition of Mean Absolute Deviation
The mean absolute deviation, or MAD, measures how spread out a data set is by averaging the distances of its values from the mean. A small MAD means the values hug the mean; a large one means they are scattered.
The calculation has four steps: find the mean, subtract it from each value, take the absolute value of each difference so that every distance is positive, and then find the mean of those distances. Taking absolute values is the key: without it, differences above and below the mean cancel to zero. MAD uses the same units as the data.
Formula and notation
With values \(x_1, x_2, \text{…}, x_n\) and mean \(\,\overline{x}\):
\[\text{MAD} = \frac{|x_1 – \,\overline{x}| + |x_2 – \,\overline{x}| + \text{…} + |x_n – \,\overline{x}|}{n}\]
Each bar pair turns a difference into a distance. The sum of the distances is divided by the number of values.
Key facts
- A MAD of zero means every value equals the mean.
- Plain differences from the mean always add up to zero, which is why absolute values are used.
- It describes spread around the mean, while the IQR describes spread around the median.
- Compare two sets by comparing their MADs: the smaller one is more consistent.
Mean Absolute Deviation example
Jaden made 4, 8, 6, 5, 3 and 10 free throws in six practices. Find the mean absolute deviation.
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Mean: \(\frac{4 + 8 + 6 + 5 + 3 + 10}{6} = \frac{36}{6} = 6\).
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Distances from 6: \(|4-6| = 2\), \(|8-6| = 2\), \(|6-6| = 0\), \(|5-6| = 1\), \(|3-6| = 3\), \(|10-6| = 4\).
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Add them: \(2 + 2 + 0 + 1 + 3 + 4 = 12\).
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Average: \(12 : 6 = 2\).
The MAD is 2 free throws: on a typical day Jaden is 2 shots away from his mean of 6.
Common mistake
Watch out
Kira skips the absolute values and adds the signed differences \(-2 + 2 + 0 + (-1) + (-3) + 4\), getting 0.
Why it is wrong
Signed differences cancel, so the sum of deviations from the mean is always 0. Take the absolute value of each difference first, so every distance counts as positive, and then average them.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Statistics and Data