Understanding Mean Absolute Deviation Quiz

1) Data set A (daily quiz scores out of 10): 7, 9, 8, 8, 8, 7, 9, 8 What is the mean of Data set A?

7.5

7.75

8.0

8.25

Mean = (7 + 9 + 8 + 8 + 8 + 7 + 9 + 8) / 8 = 64 / 8 = 8.0

So, the mean of Data set A is 8.0.

Explanation

Mean = (7 + 9 + 8 + 8 + 8 + 7 + 9 + 8) / 8 = 64 / 8 = 8.0

So, the mean of Data set A is 8.0.

2) What does deviation from the mean represent for a single score?

The average of all scores

The difference between the score and the highest score

The distance a score is from the mean

The number of points above 5

Deviation shows how far a score is from the mean. For example, if the mean = 8 and a score = 7, deviation = -1 → distance from mean = 1.

Explanation

Deviation shows how far a score is from the mean. For example, if the mean = 8 and a score = 7, deviation = -1 → distance from mean = 1.

3) Why do we use absolute value when calculating MAD?

To make larger numbers smaller

To avoid negative deviations canceling positive ones

To ensure the mean is always an integer

To remove outliers

Using absolute value turns all deviations into positive distances so that large and small scores don’t cancel each other out.

Explanation

Using absolute value turns all deviations into positive distances so that large and small scores don’t cancel each other out.

4) Which list shows the absolute deviations from the mean for Data set A?

1, 1, 0, 0, 0, 1, 1, 0

1, 1, 0, 0, 2, 1, 1, 0

2, 1, 0, 0, 0, 2, 1, 0

1, 0, 0, 0, 1, 2, 2, 0

Mean = 8

Deviations = (7−8)=−1, (9−8)=1, (8−8)=0...

Absolute deviations = 1, 1, 0, 0, 0, 1, 1, 0

Explanation

Mean = 8

Deviations = (7−8)=−1, (9−8)=1, (8−8)=0...

Absolute deviations = 1, 1, 0, 0, 0, 1, 1, 0

5) What is the MAD of Data set A?

0.25

0.5

0.75

1.0

MAD = (1 + 1 + 0 + 0 + 0 + 1 + 1 + 0)/8 = 4/8 = 0.5

Explanation

MAD = (1 + 1 + 0 + 0 + 0 + 1 + 1 + 0)/8 = 4/8 = 0.5

6) What does the MAD in question 5 tell you about consistency?

Scores are very spread out

Scores are perfectly consistent

Scores are fairly consistent around the mean

Scores are inconsistent with many outliers

MAD = 0.5 means most scores are about half a point from the mean, showing fair consistency.

Explanation

MAD = 0.5 means most scores are about half a point from the mean, showing fair consistency.

7) Which data set has a MAD of 0? (Data set B: 3, 7, 11, 5, 9; Data set C: 7, 7, 7, 7, 7)

Data set B only

Data set C only

Both B and C

Neither B nor C

All values in Data set C are 7, so all deviations = 0 → MAD = 0. Data set B has variation, so MAD > 0.

Explanation

All values in Data set C are 7, so all deviations = 0 → MAD = 0. Data set B has variation, so MAD > 0.

8) Which statement best compares the spreads of Data sets B and C?

B and C have the same spread

C has more spread than B

B has more spread than C

Neither set has any spread

Data set B values range from 3 to 11 (spread out), while Data set C’s values are all the same (no spread).

Explanation

Data set B values range from 3 to 11 (spread out), while Data set C’s values are all the same (no spread).

9) If the mean of Data set B is 7, what is the absolute deviation of the value 11?

2

3

4

5

Deviation = 11 − 7 = 4 → absolute deviation = |4| = 4.

Explanation

Deviation = 11 − 7 = 4 → absolute deviation = |4| = 4.

10) Which interpretation of MAD is most accurate?

The typical distance of data values from the mean

The median value of the data

The difference between the max and min

The proportion of outliers in the set

MAD measures how far data values usually are from the mean — it’s the average of absolute deviations.

Explanation

MAD measures how far data values usually are from the mean — it’s the average of absolute deviations.

11) Who is more consistent based on MAD? (Runner D: 8, 9, 8, 10, 7, 8; Runner E: 8, 8, 8, 8, 8, 8)

Runner D

Runner E

They are equally consistent

Not enough information

Runner E’s times are all the same → deviations = 0 → MAD = 0 → perfectly consistent.

Explanation

Runner E’s times are all the same → deviations = 0 → MAD = 0 → perfectly consistent.

12) If both runners have the same mean, what does a larger MAD imply?

Faster average speed

Slower average speed

More variable day-to-day performance

Fewer data points

A larger MAD means times fluctuate more from the mean → less consistent performance.

Explanation

A larger MAD means times fluctuate more from the mean → less consistent performance.

13) Which change would decrease Runner D’s MAD the most?

Replace 10 with 8

Replace 7 with 9

Replace 8 with 11

Replace 9 with 12

10 is far from the mean (~8). Changing it to 8 removes a large deviation, reducing the MAD the most.

Explanation

10 is far from the mean (~8). Changing it to 8 removes a large deviation, reducing the MAD the most.

14) If Runner D’s MAD is approximately 1, what does that mean?

Times are usually within about 1 minute of the mean

The mean is exactly 1 minute

The range is 1 minute

There are exactly 1 outlier

MAD ≈ 1 means each time is typically about 1 minute away from the average time.

Explanation

MAD ≈ 1 means each time is typically about 1 minute away from the average time.

15) Which class likely has the larger MAD? (Class F: 6,7,7,8,8,9; Class G: 2,5,8,10,10,10)

Class F

Class G

Both the same

Not enough data

Class G’s scores vary from 2 to 10 — much wider spread → larger MAD.

Explanation

Class G’s scores vary from 2 to 10 — much wider spread → larger MAD.

16) Which measure would best show that Class G has more inconsistent scores?

Higher mean

Lower mean

Higher MAD

Lower MAD

A higher MAD indicates greater variation and less consistency.

Explanation

A higher MAD indicates greater variation and less consistency.

17) If both classes have the same mean, which statement could still be true?

They must have the same MAD

Class F could have a smaller MAD

Class G must have a smaller MAD

Their ranges must be equal

Even if two classes share the same mean, one can still have scores closer to that mean (smaller MAD).

Explanation

Even if two classes share the same mean, one can still have scores closer to that mean (smaller MAD).

18) Which data set has the greatest MAD?

4, 4, 4, 4, 4

3, 5, 7, 9, 11

6, 6, 6, 7, 6

10, 9, 10, 9, 10

Data set B spans from 3 to 11 — largest range → greatest average distance from mean → highest MAD.

Explanation

Data set B spans from 3 to 11 — largest range → greatest average distance from mean → highest MAD.

19) What happens to the MAD if each data value increases by 5?

MAD increases by 5

MAD decreases by 5

MAD stays the same

MAD becomes 0

Adding 5 to every value shifts the data, but the distances from the mean remain unchanged → MAD unchanged.

Explanation

Adding 5 to every value shifts the data, but the distances from the mean remain unchanged → MAD unchanged.

20) Which single statement best captures what MAD represents?

The average of the deviations from the median

The typical distance of values from the mean using absolute value

The sum of squared deviations from the mean

The proportion of values equal to the mean

MAD = the mean of absolute deviations → the average distance each value is from the mean.

Explanation

MAD = the mean of absolute deviations → the average distance each value is from the mean.