Math Core

The IQR measures spread using the median. When you describe the center with the mean, there's a matching measure of spread: the mean absolute deviation, or MAD. It answers the question "On average, how far are the data values from the mean?"

Distance from the mean

Look at these two sets of quiz scores. Both have a mean of 14.

Team A: $10, \ 12, \ 14, \ 16, \ 18$
Team B: $6, \ 8, \ 14, \ 20, \ 22$

Team A

Team B

Team B's scores are clearly more spread out. To measure how much more, find how far each score is from the mean. A distance is never negative: 10 is 4 away from 14, and 18 is also 4 away from 14. The distance between a value and the mean is called the value's absolute deviation.

Team A score	10	12	14	16	18
Distance from 14	4	2	0	2	4

Team B score	6	8	14	20	22
Distance from 14	8	6	0	6	8

Now find the mean of the distances:

Team A: $\dfrac{4 + 2 + 0 + 2 + 4}{5} = \dfrac{12}{5} = 2.4$
Team B: $\dfrac{8 + 6 + 0 + 6 + 8}{5} = \dfrac{28}{5} = 5.6$

On average, Team A's scores are 2.4 points from the mean, and Team B's are 5.6 points from the mean. Team B has more variability.

Definition

Mean absolute deviation (MAD)

The mean absolute deviation of a data set is the mean of the distances between each data value and the mean of the data set.

How to find the MAD

Find the mean of the data.
Find the distance from each value to the mean. (Subtract the smaller number from the larger, so every distance is positive or zero.)
Find the mean of those distances. That is the MAD.

Finding the MAD

Worked example: A small data set

Find the MAD of $3, \ 5, \ 6, \ 8, \ 13$ .

Step 1: mean. $\dfrac{3 + 5 + 6 + 8 + 13}{5} = \dfrac{35}{5} = 7$ .

Step 2: distances from 7.

Value	3	5	6	8	13
Distance from 7	4	2	1	1	6

Step 3: mean of the distances. $\dfrac{4 + 2 + 1 + 1 + 6}{5} = \dfrac{14}{5} = 2.8$ .

The MAD is 2.8. On average, the values are 2.8 away from the mean.

Common mistake

Don't subtract "value minus mean" and keep the negative signs. If you add $-4, -2, -1, 1, 6$ , you get $0$ . In fact, the signed differences always add up to 0, because the mean is the balance point. That's exactly why the MAD uses distances, which are never negative.

Worked example: A MAD that needs rounding

The number of text messages Sam sent on six days was $7, \ 9, \ 10, \ 14, \ 15, \ 17$ . Find the MAD, rounded to the nearest tenth.

Step 1: mean. $\dfrac{7 + 9 + 10 + 14 + 15 + 17}{6} = \dfrac{72}{6} = 12$ .

Step 2: distances from 12.

Value	7	9	10	14	15	17
Distance from 12	5	3	2	2	3	5

Step 3: mean of the distances. $\dfrac{5 + 3 + 2 + 2 + 3 + 5}{6} = \dfrac{20}{6} \approx 3.33$ .

Rounded to the nearest tenth, the MAD is about 3.3 messages.

What the MAD tells you

Worked example: Comparing two cities

Over one week, the daily high temperatures in two cities had these summaries:

Seaside: mean $70^\circ$ F, MAD $1.5^\circ$ F
Valley Town: mean $70^\circ$ F, MAD $8^\circ$ F

Both cities have the same mean. But in Seaside, a typical day was only about $1.5^\circ$ away from $70^\circ$ , while in Valley Town a typical day was about $8^\circ$ away. If you want to pack the same clothes every day, Seaside is the more predictable city.

Tip

A MAD of 0 means every value is exactly the same. The bigger the MAD, the more spread out the data. Also, the MAD can never be bigger than the range, so if yours is, check your work.

Practice

Practice 1

Find the MAD of $2, \ 4, \ 6, \ 8$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Find the MAD of $10, \ 12, \ 15, \ 17, \ 21$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

The dot plot shows how many glasses of water 10 students drank on Monday. Find the MAD.

Glasses of water

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Find the MAD of $4, \ 6, \ 7, \ 9, \ 10, \ 12$ . Round to the nearest tenth.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Each data set below has a mean of 50. Which one has the greatest MAD?

Practice 6

Two machines fill bags of rice. Both put a mean of 5 pounds in each bag. Machine 1 has a MAD of 0.4 pound. Machine 2 has a MAD of 0.1 pound. Which statement is correct?

Practice 7

The rainfall, in inches, in five months was $1.5, \ 2.5, \ 3, \ 4, \ 6$ . Find the MAD, to the nearest hundredth.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.