To find the Mean Absolute Deviation (MAD) of the dataset \(99, 45, 23, 67, 45, 91, 82, 78, 62, 51\), follow these steps:
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Calculate the Mean: \[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \] \[ \text{Mean} = \frac{99 + 45 + 23 + 67 + 45 + 91 + 82 + 78 + 62 + 51}{10} = \frac{ 99 + 45 + 23 + 67 + 45 + 91 + 82 + 78 + 62 + 51}{10} = \frac{ 99 + 45 + 23 + 67 + 45 + 91 + 82 + 78 + 62 + 51}{10} = \frac{ 45 + 67 + 99 + 23 + 55 + 78 + 91 + 82 + 62 + 51}{10}= \frac{651}{10} = 65.1 \]
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Calculate the Absolute Deviations: Subtract the mean from each data point and take the absolute value: \[ |99 - 65.1| = 33.9 \] \[ |45 - 65.1| = 20.1 \] \[ |23 - 65.1| = 42.1 \] \[ |67 - 65.1| = 1.9 \] \[ |45 - 65.1| = 20.1 \] \[ |91 - 65.1| = 25.9 \] \[ |82 - 65.1| = 16.9 \] \[ |78 - 65.1| = 12.9 \] \[ |62 - 65.1| = 3.1 \] \[ |51 - 65.1| = 14.1 \]
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Calculate the Mean Absolute Deviation: Now sum these absolute deviations and divide by the number of data points: \[ \text{MAD} = \frac{|99 - 65.1| + |45 - 65.1| + |23 - 65.1| + |67 - 65.1| + |45 - 65.1| + |91 - 65.1| + |82 - 65.1| + |78 - 65.1| + |62 - 65.1| + |51 - 65.1|}{10} \] \[ = \frac{33.9 + 20.1 + 42.1 + 1.9 + 20.1 + 25.9 + 16.9 + 12.9 + 3.1 + 14.1}{10} \] \[ = \frac{ %Data%}{10} \] \[ = \frac{ 33.9 + 20.1 + 42.1 + 1.9 + 20.1 + 25.9 + 16.9 + 12.9 + 3.1 + 14.1}{10.} = \frac{ 77. 13.4 + + %Data% + 1.9 + 20.1 + 25.9 + 16.9 + 12.9 + 3.1 + 14.1}{10} ≈ 18.51 \]
Hence, the Mean Absolute Deviation (MAD) of the dataset is approximately \( 18.51 \).
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