To compare the variability of the two datasets using standard deviation, we first need to calculate the standard deviation of each dataset.
Dataset A: 4, 5, 6, 12, 13
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Calculate the Mean: \[ \text{Mean} = \frac{4 + 5 + 6 + 12 + 13}{5} = \frac{40}{5} = 8 \]
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Calculate the Variance: \[ \text{Variance} = \frac{(4-8)^2 + (5-8)^2 + (6-8)^2 + (12-8)^2 + (13-8)^2}{5} \] \[ = \frac{(-4)^2 + (-3)^2 + (-2)^2 + (4)^2 + (5)^2}{5} \] \[ = \frac{16 + 9 + 4 + 16 + 25}{5} = \frac{70}{5} = 14 \]
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Calculate the Standard Deviation: \[ \text{Standard Deviation} = \sqrt{14} \approx 3.74 \]
Dataset B: 4, 5, 7, 9, 10
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Calculate the Mean: \[ \text{Mean} = \frac{4 + 5 + 7 + 9 + 10}{5} = \frac{35}{5} = 7 \]
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Calculate the Variance: \[ \text{Variance} = \frac{(4-7)^2 + (5-7)^2 + (7-7)^2 + (9-7)^2 + (10-7)^2}{5} \] \[ = \frac{(-3)^2 + (-2)^2 + (0)^2 + (2)^2 + (3)^2}{5} \] \[ = \frac{9 + 4 + 0 + 4 + 9}{5} = \frac{26}{5} = 5.2 \]
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Calculate the Standard Deviation: \[ \text{Standard Deviation} = \sqrt{5.2} \approx 2.28 \]
Summary of Results:
- Standard Deviation of Dataset A = 3.74
- Standard Deviation of Dataset B = 2.28
Thus, the higher standard deviation in Dataset A indicates greater variability compared to Dataset B.