Mean Absolute Deviation (MAD)

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Mean Absolute Deviation (MAD): meaning, method and solved example (6th Grade) – Maths PDF

In short: Mean absolute deviation is the typical gap between the data values and their own average, computed by averaging each value’s distance from the mean.

6th GradeStatistics6.SP.B.5c

What is Mean Absolute Deviation?

The mean absolute deviation, or MAD, answers one question: if you pick a value at random, how far from the average will it land, give or take? Values that hug the average give a tiny answer, and scattered values give a bigger one.

With mean \(\bar{x}\) and \(n\) values, the formula is MAD \(= \frac{|x_1 – \bar{x}| + |x_2 – \bar{x}| + \cdots + |x_n – \bar{x}|}{n}\). The bars turn every gap into a positive distance, so values above and below the mean cannot cancel each other out.

How to spot it

  • A question asks how much the values typically differ from their average.
  • Two data sets share the same mean and you must say which one is more consistent.
  • A table has a column of distances from the mean, with a total at the bottom.
  • The letters MAD appear next to a small list of numbers.

The method

  1. Find the mean of the data first.
  2. Subtract the mean from each value and drop any minus sign, which gives that value’s distance.
  3. Add all the distances together.
  4. Divide the total by how many values there are; that quotient is the mean absolute deviation, so state the unit of the data.
4567891011mean 8distance 3distance 3Data 5, 8, 8, 11: (3 + 0 + 0 + 3) / 4 = 1.5
Two values sit 3 from the mean and two sit on it, so the average distance is 1.5.

Mean Absolute Deviation: solved example

Problem

A bakery in Providence sold 12, 15, 15, 18 and 20 loaves on five mornings. Find the MAD, and say what it means for the bakery.

Solution

Mean: \(\frac{12 + 15 + 15 + 18 + 20}{5} = \frac{80}{5} = 16\). The distances from 16 are 4, 1, 1, 2 and 4.

They total 12, and \(12 \div 5 = 2.4\). The MAD is 2.4 loaves: daily sales typically land about 2.4 loaves from 16.

Try it

Question 1

Find the MAD of 10, 14, 18.

Show the answer

Mean 14; distances 4, 0, 4; sum 8; \(8 \div 3 \approx 2.67\).

Question 2

Set A has a MAD of 1.2 and set B has a MAD of 5.8. Which set is more spread out?

Show the answer

Set B, since its values lie farther from the mean on average.

Mean Absolute Deviation vs. Range

The range looks only at the two extremes, so the middle of the data never matters. The MAD uses every single value, measuring each one’s distance from the mean, so it reflects how the whole list is spread and not just its ends.

Range (Statistics)

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