To compare the variability of the two datasets using standard deviation, we first need to calculate the standard deviation for each dataset.
Dataset A: 11, 7, 12, 8, 9, 5
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Find the mean (average): \[ \text{Mean} = \frac{11 + 7 + 12 + 8 + 9 + 5}{6} = \frac{52}{6} \approx 8.67 \]
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Calculate the squared differences from the mean:
- \( (11 - 8.67)^2 = (2.33)^2 \approx 5.43 \)
- \( (7 - 8.67)^2 = (-1.67)^2 \approx 2.78 \)
- \( (12 - 8.67)^2 = (3.33)^2 \approx 11.09 \)
- \( (8 - 8.67)^2 = (-0.67)^2 \approx 0.45 \)
- \( (9 - 8.67)^2 = (0.33)^2 \approx 0.11 \)
- \( (5 - 8.67)^2 = (-3.67)^2 \approx 13.51 \)
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Find the variance (average of the squared differences): \[ \text{Variance} = \frac{5.43 + 2.78 + 11.09 + 0.45 + 0.11 + 13.51}{6} \approx \frac{33.37}{6} \approx 5.56 \]
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Calculate the standard deviation: \[ \text{Standard Deviation} = \sqrt{5.56} \approx 2.36 \]
Dataset B: 10, 8, 10, 8, 8, 4
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Find the mean: \[ \text{Mean} = \frac{10 + 8 + 10 + 8 + 8 + 4}{6} = \frac{58}{6} \approx 9.67 \]
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Calculate the squared differences from the mean:
- \( (10 - 9.67)^2 = (0.33)^2 \approx 0.11 \)
- \( (8 - 9.67)^2 = (-1.67)^2 \approx 2.78 \)
- \( (10 - 9.67)^2 = (0.33)^2 \approx 0.11 \)
- \( (8 - 9.67)^2 = (-1.67)^2 \approx 2.78 \)
- \( (8 - 9.67)^2 = (-1.67)^2 \approx 2.78 \)
- \( (4 - 9.67)^2 = (-5.67)^2 \approx 32.11 \)
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Find the variance: \[ \text{Variance} = \frac{0.11 + 2.78 + 0.11 + 2.78 + 2.78 + 32.11}{6} \approx \frac{40.67}{6} \approx 6.78 \]
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Calculate the standard deviation: \[ \text{Standard Deviation} = \sqrt{6.78} \approx 2.60 \]
Now we can summarize:
- The standard deviation of Dataset A is approximately 2.36.
- The standard deviation of Dataset B is approximately 2.60.
Conclusion:
The standard deviation of Dataset A is 2.36 and the standard deviation of Dataset B is 2.60. Since Dataset B has a higher standard deviation, it indicates that Dataset B has more variability compared to Dataset A.
None of the provided response options seem to match the calculated values, but the appropriate choice based on our calculations should state that Dataset B has more variability.
If you need to make a choice among the options provided:
"The standard deviation of Dataset A is 2.58. The standard deviation of Dataset B is 2.19. The standard deviation of Dataset B is larger than that of Dataset A, so Dataset B has more variability.”
This option might have some errors in the values but infers the correct direction about variability.