To compare the variability of the datasets using standard deviation, we first need to calculate the standard deviation for each dataset.
Calculating Standard Deviation for Dataset A:
-
Calculate the mean: \[ \text{Mean} = \frac{11 + 7 + 12 + 8 + 9 + 5}{6} = \frac{52}{6} \approx 8.67 \]
-
Calculate the squared differences from the mean: \[ (11 - 8.67)^2 \approx 5.56 \ (7 - 8.67)^2 \approx 2.78 \ (12 - 8.67)^2 \approx 11.11 \ (8 - 8.67)^2 \approx 0.44 \ (9 - 8.67)^2 \approx 0.11 \ (5 - 8.67)^2 \approx 13.49 \ \]
-
Calculate the variance (average of squared differences): \[ \text{Variance} = \frac{5.56 + 2.78 + 11.11 + 0.44 + 0.11 + 13.49}{6} \approx \frac{33.49}{6} \approx 5.58 \]
-
Calculate the standard deviation (square root of variance): \[ \text{Standard Deviation A} \approx \sqrt{5.58} \approx 2.36 \]
Calculating Standard Deviation for Dataset B:
-
Calculate the mean: \[ \text{Mean} = \frac{10 + 8 + 10 + 8 + 8 + 4}{6} = \frac{58}{6} \approx 9.67 \]
-
Calculate the squared differences from the mean: \[ (10 - 9.67)^2 \approx 0.11 \ (8 - 9.67)^2 \approx 2.78 \ (10 - 9.67)^2 \approx 0.11 \ (8 - 9.67)^2 \approx 2.78 \ (8 - 9.67)^2 \approx 2.78 \ (4 - 9.67)^2 \approx 32.11 \ \]
-
Calculate the variance: \[ \text{Variance} = \frac{0.11 + 2.78 + 0.11 + 2.78 + 2.78 + 32.11}{6} \approx \frac{40.67}{6} \approx 6.78 \]
-
Calculate the standard deviation: \[ \text{Standard Deviation B} \approx \sqrt{6.78} \approx 2.60 \]
Based on the calculations:
- The standard deviation of Dataset A is approximately 2.36.
- The standard deviation of Dataset B is approximately 2.60.
Since 2.60 (Dataset B) > 2.36 (Dataset A), Dataset B has more variability.
From the options provided, the correct interpretation based on our calculated standard deviations is:
The standard deviation of Dataset A is 2.36. The standard deviation of Dataset B is 2.60. The standard deviation of Dataset B is larger than that of Dataset A, so Dataset B has more variability. (Note: Since our calculations are slightly different due to rounding, please ensure to use the exact numbers from your calculations.)