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Mean absolute deviation

The mean distance from the mean; comparing spreads · Foundations · 22 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Find the mean
  2. Distance from the mean
  3. Mean of the distances
  4. Check

The steps for MAD of 2, 4, 6, 8, 10. Every solution names its steps, so you always know what comes next.

Worked examples

MAD of 2, 4, 6, 8, 10 Easy

The question

Question: find the mean absolute deviation, the MAD, of 2, 4, 6, 8, 10.

2, 4, 6, 8, 10

The MAD says how spread out the numbers are: on average, how far each one is from the mean.

MAD = the mean distance from the mean

1 Find the mean

The mean is the sum divided by how many.

The sum is 30.

There are 5 numbers.

30 divided by 5 is 6.

= mean = 2 + 4 + 6 + 8 + 105 = 305 = 6

2 Distance from the mean

We make a table: each number, and how far it is from the mean.

2 is 4 away from 6: 6 minus 2.

4

4 is 2 away.

2

And so on.

0
2
4

Each time, bigger minus smaller.

So a distance is never negative.

= distance = bigger − smaller

Careful!

If we used minus signs, the numbers below the mean would cancel the ones above.

We would always get 0.

Careful: a distance is never negative

3 Mean of the distances

Now the mean of the distances.

They add up to 12.

12 divided by 5 is 2.4.

= MAD = 4 + 2 + 0 + 2 + 45 = 125 = 2.4

So on average, the numbers are 2.4 away from the mean.

4 Check

Let's check: the distances below the mean add up to 6.

Above the mean: 6.

The same, because the mean is the balance point.

So the distances are right!

below: 4 + 2 = 6, above: 2 + 4 = 6

Let's look back.

First: find the mean.

Then: distance from the mean.

Then: mean of the distances.

Last: check.

Answer
= MAD = 4 + 2 + 0 + 2 + 45 = 125 = 2.4

▶ Watch it solved

MAD of 1, 2, 4, 7, 11 Medium

The question

Question: find the mean absolute deviation, the MAD, of 1, 2, 4, 7, 11.

1, 2, 4, 7, 11

The MAD says how spread out the numbers are: on average, how far each one is from the mean.

MAD = the mean distance from the mean

1 Find the mean

The mean is the sum divided by how many.

The sum is 25.

There are 5 numbers.

25 divided by 5 is 5.

= mean = 1 + 2 + 4 + 7 + 115 = 255 = 5

2 Distance from the mean

We make a table: each number, and how far it is from the mean.

1 is 4 away from 5: 5 minus 1.

4

2 is 3 away.

3

And so on.

1
2
6

Each time, bigger minus smaller.

So a distance is never negative.

= distance = bigger − smaller

Careful!

If we used minus signs, the numbers below the mean would cancel the ones above.

We would always get 0.

Careful: a distance is never negative

3 Mean of the distances

Now the mean of the distances.

They add up to 16.

16 divided by 5 is 3.2.

= MAD = 4 + 3 + 1 + 2 + 65 = 165 = 3.2

So on average, the numbers are 3.2 away from the mean.

4 Check

Let's check: the distances below the mean add up to 8.

Above the mean: 8.

The same, because the mean is the balance point.

So the distances are right!

below: 4 + 3 + 1 = 8, above: 2 + 6 = 8

Let's look back.

First: find the mean.

Then: distance from the mean.

Then: mean of the distances.

Last: check.

Answer
= MAD = 4 + 3 + 1 + 2 + 65 = 165 = 3.2

▶ Watch it solved

Compare MAD: team A 2, 4, 6, 8, 10; team B 5, 6, 6, 7, 6 Harder

The question

Question: compare the MAD of team A, 2, 4, 6, 8, 10, and team B, 5, 6, 6, 7, 6.

team A: 2, 4, 6, 8, 10
= team B: 5, 6, 6, 7, 6

The MAD says how spread out the numbers are.

A smaller MAD means the numbers are closer together.

smaller MAD = closer together

1 MAD of team A

Team A: the mean is the sum divided by how many.

The sum is 30.

There are 5 numbers.

30 divided by 5 is 6.

= mean of team A = 2 + 4 + 6 + 8 + 105 = 305 = 6

The distances from 6: 4, 2, 0, 2, 4.

= distances: 4, 2, 0, 2, 4

They add up to 12.

Divided by 5: 2.4.

= MAD = 125 = 2.4

2 MAD of team B

Team B: the mean is the sum divided by how many.

The sum is 30.

There are 5 numbers.

30 divided by 5 is 6.

= mean of team B = 5 + 6 + 6 + 7 + 65 = 305 = 6

The distances from 6: 1, 0, 0, 1, 0.

= distances: 1, 0, 0, 1, 0

They add up to 2.

Divided by 5: 0.4.

= MAD = 25 = 0.4

Careful!

The question is about spread.

So we compare the MADs, not the means.

Careful: compare the spread, not the mean

3 Compare

0.4 is smaller.

So team B is closer together: more consistent.

= 2.4 > 0.4 → team B is closer together

4 Check

Let's check with the lists: team A goes from 2 to 10.

team B goes from 5 to 7.

team B is more bunched up.

So the answer is right!

team A: 2 to 10, team B: 5 to 7

Let's look back.

First: mAD of team A.

Then: mAD of team B.

Then: compare.

Last: check.

Answer
= 2.4 > 0.4 → team B is closer together

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. MAD of 2, 4, 6, 8, 10
    Answer
    = MAD = 4 + 2 + 0 + 2 + 45 = 125 = 2.4
  2. MAD of 1, 3, 5
    Answer
    = MAD = 2 + 0 + 23 = 43 ≈ 1.3
  3. MAD of 4, 6, 8
    Answer
    = MAD = 2 + 0 + 23 = 43 ≈ 1.3
  4. MAD of 3, 5, 6, 9, 12
    Answer
    = MAD = 4 + 2 + 1 + 2 + 55 = 145 = 2.8
  5. MAD of 10, 12, 14, 16
    Answer
    = MAD = 3 + 1 + 1 + 34 = 84 = 2
  6. MAD of 5, 5, 5, 5
    Answer
    = MAD = 0 + 0 + 0 + 04 = 04 = 0
  7. MAD of 2, 8, 2, 8
    Answer
    = MAD = 3 + 3 + 3 + 34 = 124 = 3
  8. MAD of 6, 9, 12
    Answer
    = MAD = 3 + 0 + 33 = 63 = 2

Medium

  1. MAD of 1, 2, 4, 7, 11
    Answer
    = MAD = 4 + 3 + 1 + 2 + 65 = 165 = 3.2
  2. MAD of 12, 15, 15, 18, 20
    Answer
    = MAD = 4 + 1 + 1 + 2 + 45 = 125 = 2.4
  3. MAD of 3, 4, 8, 9
    Answer
    = MAD = 3 + 2 + 2 + 34 = 104 = 2.5
  4. MAD of 20, 25, 30, 45
    Answer
    = MAD = 10 + 5 + 0 + 154 = 304 = 7.5
  5. MAD of 7, 8, 10, 11, 14
    Answer
    = MAD = 3 + 2 + 0 + 1 + 45 = 105 = 2
  6. MAD of 2.5, 3.5, 4, 6
    Answer
    = MAD = 1.5 + 0.5 + 0 + 24 = 44 = 1
  7. MAD of 0, 4, 5, 6, 10
    Answer
    = MAD = 5 + 1 + 0 + 1 + 55 = 125 = 2.4
  8. MAD of 30, 32, 35, 38, 40
    Answer
    = MAD = 5 + 3 + 0 + 3 + 55 = 165 = 3.2

Harder

  1. compare MAD: team A 2, 4, 6, 8, 10; team B 5, 6, 6, 7, 6
    Answer
    = 2.4 > 0.4 → team B is closer together
  2. compare MAD: class A 70, 80, 90; class B 78, 80, 82
    Answer
    = 6.7 > 1.3 → class B is closer together
  3. compare MAD: Monday 3, 9, 6, 6; Tuesday 1, 11, 4, 8
    Answer
    = 1.5 < 3.5 → Monday is closer together
  4. MAD of 1, 1, 2, 6, 10
    Answer
    = MAD = 3 + 3 + 2 + 2 + 65 = 165 = 3.2
  5. MAD of 15, 18, 21, 22, 24
    Answer
    = MAD = 5 + 2 + 1 + 2 + 45 = 145 = 2.8
  6. compare MAD: shop A 10, 12, 14; shop B 8, 12, 16
    Answer
    = 1.3 < 2.7 → shop A is closer together