Question

evaluating an outliers effect on mean absolute deviation
tammy’s last four homework scores were 68, 76, 70, and 74. the mean absolute deviation of her scores is
tammy worked hard on yesterday’s assignment and feels confident that this homework score will be at least a 90. predict what will happen to her mean absolute deviation if she is correct:
tammy was correct. she earned a 92! the new mean absolute deviation of her homework scores now that they are 68, 76, 70, 74, and 92 is:
2.4
3
6.4
32

Explanation:

First Part: Mean Absolute Deviation of 68, 76, 70, 74

Step1: Calculate the mean

The mean ($\mu$) of a set of numbers is the sum of the numbers divided by the count of numbers. For the scores 68, 76, 70, 74:
$$\mu = \frac{68 + 76 + 70 + 74}{4} = \frac{288}{4} = 72$$

Step2: Calculate absolute deviations

Find the absolute difference between each score and the mean:

• $|68 - 72| = 4$
• $|76 - 72| = 4$
• $|70 - 72| = 2$
• $|74 - 72| = 2$

Step3: Calculate mean absolute deviation (MAD)

The MAD is the mean of these absolute deviations:
$$MAD = \frac{4 + 4 + 2 + 2}{4} = \frac{12}{4} = 3$$

Second Part: Predict the effect of adding a score of at least 90

An outlier (a score much higher or lower than the rest) will increase the spread of the data. The mean absolute deviation measures the average distance from the mean. Adding a score (90 or higher) that is far from the original mean (72) will increase the average distance of the scores from the new mean (which will also increase, but the deviation of the new score from the mean will be large), so the MAD will increase.

Third Part: Mean Absolute Deviation of 68, 76, 70, 74, 92

Step1: Calculate the new mean

For the scores 68, 76, 70, 74, 92:
$$\mu = \frac{68 + 76 + 70 + 74 + 92}{5} = \frac{380}{5} = 76$$

Step2: Calculate absolute deviations

Find the absolute difference between each score and the new mean:

• $|68 - 76| = 8$
• $|76 - 76| = 0$
• $|70 - 76| = 6$
• $|74 - 76| = 2$
• $|92 - 76| = 16$

Step3: Calculate new MAD

The MAD is the mean of these absolute deviations:
$$MAD = \frac{8 + 0 + 6 + 2 + 16}{5} = \frac{32}{5} = 6.4$$

Answer:

s:

• First blank (MAD of 68,76,70,74): 3
• Second blank (effect on MAD): It will increase
• Third blank (new MAD with 92): 6.4