Mean Absolute Deviation Quiz: Finding Mean Absolute Deviation

MAD measures how far data values are from the mean on average. It uses absolute distances so that values above and below the mean do not cancel each other out. A larger MAD indicates greater spread while a smaller MAD indicates values are clustered closely around the mean. MAD is always expressed in the same units as the original data.

The answer is True. MAD equals 0 only when every absolute deviation equals 0. Each absolute deviation equals 0 only when each value equals the mean. If every value equals the mean, then all values in the data set are identical. For example {5, 5, 5, 5} has mean 5, all absolute deviations are 0, and MAD equals 0. It is impossible for MAD to equal 0 when any value differs from the mean.

Computing each absolute deviation: the absolute value of 4 minus 7 equals 3, the absolute value of 6 minus 7 equals 1, the absolute value of 8 minus 7 equals 1, and the absolute value of 10 minus 7 equals 3. The absolute deviations are 3, 1, 1, and 3. Taking absolute values ensures all results are non-negative regardless of whether the value is above or below the mean.

The mean equals (4 plus 6 plus 8 plus 10) divided by 4 equals 7. The absolute deviations are 3, 1, 1, and 3. The sum of absolute deviations is 3 plus 1 plus 1 plus 3 equals 8. MAD equals 8 divided by 4 equals 2. This means the values in the data set are on average 2 units away from the mean of 7.

Every value in this data set equals the mean of 10. Since every value equals the mean, every deviation is 0, every absolute deviation is 0, and the sum of absolute deviations is 0. MAD equals 0 divided by 4 equals 0. A MAD of 0 means there is no spread at all — every value is identical and sits exactly at the mean.

The mean equals (2 plus 4 plus 8 plus 10) divided by 4 equals 6. The absolute deviations are 4, 2, 2, and 4. The sum is 4 plus 2 plus 2 plus 4 equals 12. MAD equals 12 divided by 4 equals 3.

Computing each absolute deviation: the absolute value of 1 minus 7 equals 6, the absolute value of 5 minus 7 equals 2, the absolute value of 9 minus 7 equals 2, and the absolute value of 13 minus 7 equals 6. The absolute deviations are 6, 2, 2, and 6. Values equally spaced from the mean produce matching pairs of absolute deviations.

The mean equals (1 plus 5 plus 9 plus 13) divided by 4 equals 7. The absolute deviations are 6, 2, 2, and 6. The sum is 6 plus 2 plus 2 plus 6 equals 16. MAD equals 16 divided by 4 equals 4. This means the values are on average 4 units away from the mean of 7, reflecting the relatively wide spread of this data set.

The mean equals (5 plus 8 plus 11) divided by 3 equals 8. The absolute deviations are the absolute value of 5 minus 8 equals 3, the absolute value of 8 minus 8 equals 0, and the absolute value of 11 minus 8 equals 3. The sum is 3 plus 0 plus 3 equals 6. MAD equals 6 divided by 3 equals 2.

Computing each absolute deviation: the absolute value of 3 minus 6 equals 3, the absolute value of 3 minus 6 equals 3, the absolute value of 9 minus 6 equals 3, and the absolute value of 9 minus 6 equals 3. All four absolute deviations equal 3 because the data has only two distinct values, each equally far from the mean. The sum is 3 plus 3 plus 3 plus 3 equals 12.

The answer is True. Adding a constant shifts every data value and the mean by the same amount. Each deviation (value minus mean) stays exactly the same as before, so each absolute deviation stays the same, and the average of the absolute deviations remains unchanged. For example {2, 4, 6} and {12, 14, 16} (adding 10 to each value) have the same MAD of 1.33 because the spread between the values has not changed.

The mean of {2, 7, 12} is 7. The absolute deviations are 5, 0, and 5. The sum is 10 and MAD equals 10 divided by 3 equals 3.33. The student reported 5, which is one of the individual absolute deviations rather than the average of all three. This is a common error — computing a single absolute deviation and treating it as the MAD without completing the averaging step.

The mean equals (2 plus 7 plus 12) divided by 3 equals 7. The absolute deviations are the absolute value of 2 minus 7 equals 5, the absolute value of 7 minus 7 equals 0, and the absolute value of 12 minus 7 equals 5. The sum is 5 plus 0 plus 5 equals 10. MAD equals 10 divided by 3 equals approximately 3.33.

The answer is False. Variance squares each deviation before averaging while MAD takes absolute values. Squaring amplifies larger deviations significantly, so variance is often much larger than MAD for the same data set. For example the data set {1, 5, 9, 13} has MAD equals 4 but population variance equals 20. There is no general rule that MAD is always larger than variance.

The mean equals (0 plus 4 plus 8) divided by 3 equals 4. The absolute deviations are the absolute value of 0 minus 4 equals 4, the absolute value of 4 minus 4 equals 0, and the absolute value of 8 minus 4 equals 4. The sum is 4 plus 0 plus 4 equals 8. MAD equals 8 divided by 3 equals approximately 2.67.

Option A is correct: MAD takes the absolute value of each deviation while variance squares each deviation. Option C is correct: a larger MAD means values are farther from the mean on average, indicating more spread. Option B is incorrect: MAD is in the same units as the data but variance is in squared units. For example if data is in centimetres, MAD is in centimetres but variance is in centimetres squared. Option D is incorrect: because absolute values and squares produce different results, MAD and variance nearly always differ numerically.

The mean equals (3 plus 3 plus 9 plus 9) divided by 4 equals 6. The sum of absolute deviations is 12, as computed above. MAD equals 12 divided by 4 equals 3. This means each value is on average 3 units away from the mean of 6, which makes sense since the two distinct values (3 and 9) are each exactly 3 units from the mean.

The answer is False. MAD and variance both measure spread but use different approaches. MAD takes the absolute value of each deviation before averaging, keeping results in the original units of the data. Variance squares each deviation before averaging, producing results in squared units. Squaring gives more weight to larger deviations than taking absolute values does, which is why variance and MAD produce different numerical results for the same data set.

Options A, B, and D are all correct steps. Finding the mean comes first. Finding the absolute deviation of each value by subtracting the mean and taking the absolute value removes negative signs so that values above and below the mean both contribute positively. Dividing the sum of absolute deviations by n gives the MAD. Option C is incorrect: squaring each deviation is the step used in computing variance, not MAD. Using squares instead of absolute values produces a different and larger result and changes the units of measurement to squared units.

The mean equals (1 plus 3 plus 5 plus 9) divided by 4 equals 4.5. The absolute deviations are 3.5, 1.5, 0.5, and 4.5. The sum of absolute deviations is 3.5 plus 1.5 plus 0.5 plus 4.5 equals 10. MAD equals 10 divided by 4 equals 2.5.