To find the Mean Absolute Deviation (MAD) of the given data set (23, 56, 45, 65, 59, 55, 61, 54, 85, 25), follow these steps:
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Calculate the Mean (Average): \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]
Sum = 23 + 56 + 45 + 65 + 59 + 55 + 61 + 54 + 85 + 25 = 499
Number of values = 10
\[ \text{Mean} = \frac{499}{10} = 49.9 \]
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Calculate the Absolute Deviations from the Mean: \[ |x_i - \text{Mean}| \]
- For 23: \( |23 - 49.9| = 26.9 \)
- For 56: \( |56 - 49.9| = 6.1 \)
- For 45: \( |45 - 49.9| = 4.9 \)
- For 65: \( |65 - 49.9| = 15.1 \)
- For 59: \( |59 - 49.9| = 9.1 \)
- For 55: \( |55 - 49.9| = 5.1 \)
- For 61: \( |61 - 49.9| = 11.1 \)
- For 54: \( |54 - 49.9| = 4.1 \)
- For 85: \( |85 - 49.9| = 35.1 \)
- For 25: \( |25 - 49.9| = 24.9 \)
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Sum the Absolute Deviations: \[ \text{Sum of Absolute Deviations} = 26.9 + 6.1 + 4.9 + 15.1 + 9.1 + 5.1 + 11.1 + 4.1 + 35.1 + 24.9 = 113.3 \]
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Calculate the Mean Absolute Deviation (MAD): \[ \text{MAD} = \frac{\text{Sum of Absolute Deviations}}{\text{Number of Values}} = \frac{113.3}{10} = 11.33 \]
Thus, the Mean Absolute Deviation (MAD) for the given data set is approximately 11.33.
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