To find the Mean Absolute Deviation (MAD) of the dataset, we can follow these steps:
- Calculate the Mean: Find the average of the numbers.
- Calculate the Absolute Deviations: Find the absolute difference between each data point and the mean.
- Calculate the Mean Absolute Deviation: Take the average of the absolute deviations.
Let's break it down:
Step 1: Calculate the Mean
The given dataset is:
55, 63, 88, 97, 58, 90, 88, 71, 65, 77, 75, 88, 95, 86
To calculate the mean: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{55 + 63 + 88 + 97 + 58 + 90 + 88 + 71 + 65 + 77 + 75 + 88 + 95 + 86}{14} \]
Calculating the sum: \[ \text{Sum} = 55 + 63 + 88 + 97 + 58 + 90 + 88 + 71 + 65 + 77 + 75 + 88 + 95 + 86 = 1228 \]
Now, calculating the mean: \[ \text{Mean} = \frac{1228}{14} \approx 87.7143 \]
Step 2: Calculate Absolute Deviations
Now we find the absolute deviations from the mean:
- |55 - 87.7143| ≈ 32.7143
- |63 - 87.7143| ≈ 24.7143
- |88 - 87.7143| ≈ 0.2857
- |97 - 87.7143| ≈ 9.2857
- |58 - 87.7143| ≈ 29.7143
- |90 - 87.7143| ≈ 2.2857
- |88 - 87.7143| ≈ 0.2857
- |71 - 87.7143| ≈ 16.7143
- |65 - 87.7143| ≈ 22.7143
- |77 - 87.7143| ≈ 10.7143
- |75 - 87.7143| ≈ 12.7143
- |88 - 87.7143| ≈ 0.2857
- |95 - 87.7143| ≈ 7.2857
- |86 - 87.7143| ≈ 1.7143
Step 3: Calculate the Mean Absolute Deviation
Now, let's sum the absolute deviations: \[ \text{Sum of absolute deviations} \approx 32.7143 + 24.7143 + 0.2857 + 9.2857 + 29.7143 + 2.2857 + 0.2857 + 16.7143 + 22.7143 + 10.7143 + 12.7143 + 0.2857 + 7.2857 + 1.7143 \approx 221.4286 \]
Finally, the Mean Absolute Deviation is: \[ \text{MAD} = \frac{\text{Sum of absolute deviations}}{\text{Number of values}} \approx \frac{221.4286}{14} \approx 15.6 \]
Conclusion
The Mean Absolute Deviation (MAD) of the dataset is approximately 15.6.
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