Question

find the mean absolute deviation
10, 7, 13, 10, 8
table with data, mean, difference, positive value columns; sum, count, mean absolute deviation section
the average of the \positive value\ column

find the mean absolute deviation
110, 114, 104, 108, 106
table with data, mean, difference, positive value columns; sum, count, mean absolute deviation section
the average of the \positive value\ column

find the mean absolute deviation
87, 75, 85, 77, 74, 82, 90, 88, 79, 81
table with data, mean, difference, positive value columns; sum, count, mean absolute deviation section
the average of the \positive value\ column

find the mean absolute deviation
15, 17, 15, 17, 21, 17, 15, 23, 20, 18
table with data, mean, difference, positive value columns; sum, count, mean absolute deviation section
the average of the \positive value\ column

Explanation:

First Dataset: 10, 7, 13, 10, 8

Step1: Calculate the mean

The mean ($\bar{x}$) is calculated as the sum of the data divided by the number of data points.
Sum of data: $10 + 7 + 13 + 10 + 8 = 48$
Number of data points ($n$): $5$
Mean: $\bar{x} = \frac{48}{5} = 9.6$

Step2: Find the differences from the mean

For each data point $x_i$, calculate $x_i - \bar{x}$:

• $10 - 9.6 = 0.4$
• $7 - 9.6 = -2.6$
• $13 - 9.6 = 3.4$
• $10 - 9.6 = 0.4$
• $8 - 9.6 = -1.6$

Step3: Take absolute values (positive values)

Absolute differences:

• $|0.4| = 0.4$
• $|-2.6| = 2.6$
• $|3.4| = 3.4$
• $|0.4| = 0.4$
• $|-1.6| = 1.6$

Step4: Calculate the mean of these absolute differences

Sum of absolute differences: $0.4 + 2.6 + 3.4 + 0.4 + 1.6 = 8.4$
Mean Absolute Deviation (MAD): $\frac{8.4}{5} = 1.68$

Step1: Calculate the mean

Sum of data: $110 + 114 + 104 + 108 + 106 = 542$
Number of data points: $5$
Mean: $\bar{x} = \frac{542}{5} = 108.4$

Step2: Find the differences from the mean

• $110 - 108.4 = 1.6$
• $114 - 108.4 = 5.6$
• $104 - 108.4 = -4.4$
• $108 - 108.4 = -0.4$
• $106 - 108.4 = -2.4$

Step3: Take absolute values

Absolute differences:

• $|1.6| = 1.6$
• $|5.6| = 5.6$
• $|-4.4| = 4.4$
• $|-0.4| = 0.4$
• $|-2.4| = 2.4$

Step4: Calculate the mean of absolute differences

Sum of absolute differences: $1.6 + 5.6 + 4.4 + 0.4 + 2.4 = 14.4$
MAD: $\frac{14.4}{5} = 2.88$

Step1: Calculate the mean

Sum of data: $87 + 75 + 85 + 77 + 74 + 82 + 90 + 88 + 79 + 81 = 828$
Number of data points: $10$
Mean: $\bar{x} = \frac{828}{10} = 82.8$

Step2: Find the differences from the mean

• $87 - 82.8 = 4.2$
• $75 - 82.8 = -7.8$
• $85 - 82.8 = 2.2$
• $77 - 82.8 = -5.8$
• $74 - 82.8 = -8.8$
• $82 - 82.8 = -0.8$
• $90 - 82.8 = 7.2$
• $88 - 82.8 = 5.2$
• $79 - 82.8 = -3.8$
• $81 - 82.8 = -1.8$

Step3: Take absolute values

Absolute differences:

• $|4.2| = 4.2$
• $|-7.8| = 7.8$
• $|2.2| = 2.2$
• $|-5.8| = 5.8$
• $|-8.8| = 8.8$
• $|-0.8| = 0.8$
• $|7.2| = 7.2$
• $|5.2| = 5.2$
• $|-3.8| = 3.8$
• $|-1.8| = 1.8$

Step4: Calculate the mean of absolute differences

Sum of absolute differences: $4.2 + 7.8 + 2.2 + 5.8 + 8.8 + 0.8 + 7.2 + 5.2 + 3.8 + 1.8 = 47.6$
MAD: $\frac{47.6}{10} = 4.76$

Answer:

1.68

Second Dataset: 110, 114, 104, 108, 106