To calculate the Mean Absolute Deviation (MAD), you first need to find the mean of the dataset, then calculate the absolute deviations from the mean, and finally find the average of those absolute deviations.
Here’s how to do it step by step:
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Calculate the mean: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{12 + 47 + 26 + 25 + 38 + 45 + 35 + 35 + 41 + 39 + 33 + 25 + 18 + 30}{14} \]
\[ \text{Sum} = 12 + 47 + 26 + 25 + 38 + 45 + 35 + 35 + 41 + 39 + 33 + 25 + 18 + 30 = 474 \]
\[ \text{Mean} = \frac{474}{14} = 33.8571 \]
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Calculate the absolute deviations from the mean:
- For each value in the dataset, calculate the absolute difference from the mean:
\[ |12 - 33.8571| = 21.8571 \] \[ |47 - 33.8571| = 13.1429 \] \[ |26 - 33.8571| = 7.8571 \] \[ |25 - 33.8571| = 8.8571 \] \[ |38 - 33.8571| = 4.1429 \] \[ |45 - 33.8571| = 11.1429 \] \[ |35 - 33.8571| = 1.1429 \] \[ |35 - 33.8571| = 1.1429 \] \[ |41 - 33.8571| = 7.1429 \] \[ |39 - 33.8571| = 5.1429 \] \[ |33 - 33.8571| = 0.8571 \] \[ |25 - 33.8571| = 8.8571 \] \[ |18 - 33.8571| = 15.8571 \] \[ |30 - 33.8571| = 3.8571 \]
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Sum of absolute deviations: \[ \text{Sum of absolute deviations} = 21.8571 + 13.1429 + 7.8571 + 8.8571 + 4.1429 + 11.1429 + 1.1429 + 1.1429 + 7.1429 + 5.1429 + 0.8571 + 8.8571 + 15.8571 + 3.8571 = 109.1429 \]
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Calculate the Mean Absolute Deviation (MAD): \[ \text{MAD} = \frac{\text{Sum of absolute deviations}}{\text{Number of values}} = \frac{109.1429}{14} \approx 7.7957 \]
Thus, the Mean Absolute Deviation (MAD) of the dataset is approximately 7.80.
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