To compare the variability of the two data sets using the standard deviation, we need to calculate the standard deviation for each data set.
Data Set 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 (SD): \[ \text{SD} = \sqrt{14} \approx 3.74 \]
Data Set 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 (SD): \[ \text{SD} = \sqrt{5.2} \approx 2.28 \]
Conclusion
- Standard Deviation of Data Set A: Approximately 3.74
- Standard Deviation of Data Set B: Approximately 2.28
Rounding to the nearest hundredths:
- SD of Data Set A: 3.74
- SD of Data Set B: 2.28
Variability Comparison
Data Set A has a higher standard deviation of 3.74, indicating greater variability in the data compared to Data Set B, which has a standard deviation of 2.28.