To compare the variability of the datasets, we need to calculate the standard deviation for each dataset.
Dataset A: 4, 5, 6, 12, 13
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Calculate the mean (average): \[ \text{Mean} = \frac{4 + 5 + 6 + 12 + 13}{5} = \frac{40}{5} = 8 \]
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Calculate each deviation from the mean, square it, and find the average of these squared deviations: \[ (4 - 8)^2 = (-4)^2 = 16 \] \[ (5 - 8)^2 = (-3)^2 = 9 \] \[ (6 - 8)^2 = (-2)^2 = 4 \] \[ (12 - 8)^2 = (4)^2 = 16 \] \[ (13 - 8)^2 = (5)^2 = 25 \]
Now we sum these squares: \[ 16 + 9 + 4 + 16 + 25 = 70 \]
The variance (population variance formula) is: \[ \text{Variance} = \frac{70}{5} = 14 \]
Therefore, the standard deviation is: \[ \text{Standard Deviation} = \sqrt{14} \approx 3.74 \]
Dataset B: 4, 5, 7, 9, 10
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Calculate the mean (average): \[ \text{Mean} = \frac{4 + 5 + 7 + 9 + 10}{5} = \frac{35}{5} = 7 \]
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Calculate each deviation from the mean, square it, and find the average of these squared deviations: \[ (4 - 7)^2 = (-3)^2 = 9 \] \[ (5 - 7)^2 = (-2)^2 = 4 \] \[ (7 - 7)^2 = (0)^2 = 0 \] \[ (9 - 7)^2 = (2)^2 = 4 \] \[ (10 - 7)^2 = (3)^2 = 9 \]
Now we sum these squares: \[ 9 + 4 + 0 + 4 + 9 = 26 \]
The variance (population variance formula) is: \[ \text{Variance} = \frac{26}{5} = 5.2 \]
Therefore, the standard deviation is: \[ \text{Standard Deviation} = \sqrt{5.2} \approx 2.28 \]
Now to complete the sentences:
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.