To compare the variability of the datasets using the standard deviation, we first need to calculate the standard deviation for each dataset.
Dataset A: 4, 5, 6, 12, 13
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Calculate the Mean of Dataset A: \[ \text{Mean} = \frac{4 + 5 + 6 + 12 + 13}{5} = \frac{40}{5} = 8 \]
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Calculate the Sum of Squared Deviations from the Mean: \[ (4 - 8)^2 = 16 \] \[ (5 - 8)^2 = 9 \] \[ (6 - 8)^2 = 4 \] \[ (12 - 8)^2 = 16 \] \[ (13 - 8)^2 = 25 \] \[ \text{Sum of Squared Deviations} = 16 + 9 + 4 + 16 + 25 = 70 \]
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Calculate Variance: \[ \text{Variance} = \frac{70}{5} = 14 \]
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Calculate Standard Deviation: \[ \text{Standard Deviation} = \sqrt{14} \approx 3.74 \]
Dataset B: 4, 5, 7, 9, 10
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Calculate the Mean of Dataset B: \[ \text{Mean} = \frac{4 + 5 + 7 + 9 + 10}{5} = \frac{35}{5} = 7 \]
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Calculate the Sum of Squared Deviations from the Mean: \[ (4 - 7)^2 = 9 \] \[ (5 - 7)^2 = 4 \] \[ (7 - 7)^2 = 0 \] \[ (9 - 7)^2 = 4 \] \[ (10 - 7)^2 = 9 \] \[ \text{Sum of Squared Deviations} = 9 + 4 + 0 + 4 + 9 = 26 \]
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Calculate Variance: \[ \text{Variance} = \frac{26}{5} = 5.2 \]
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Calculate Standard Deviation: \[ \text{Standard Deviation} = \sqrt{5.2} \approx 2.28 \]
Conclusion:
- The standard deviation of Dataset A is approximately 3.74.
- The standard deviation of Dataset B is approximately 2.28.
Thus, the correct response is:
1. The standard deviation of Dataset A is 3.74. The standard deviation of Dataset B is 2.28. The standard deviation of Dataset A is larger than that of Dataset B, so Dataset A has more variability.