To determine the standard deviation for both datasets, we will go through the calculations step by step.
-
Calculate the mean of each dataset.
- Dataset A:
- Mean = (11 + 7 + 12 + 8 + 9 + 5) / 6 = 52 / 6 = 8.67
- Dataset B:
- Mean = (10 + 8 + 10 + 8 + 8 + 4) / 6 = 58 / 6 = 9.67
- Dataset A:
-
Calculate the variance for each dataset.
- Dataset A:
- Deviations from the mean: (11 - 8.67), (7 - 8.67), (12 - 8.67), (8 - 8.67), (9 - 8.67), (5 - 8.67)
- Squared deviations: (2.33^2), (-1.67^2), (3.33^2), (-0.67^2), (0.33^2), (-3.67^2)
- = (5.4289), (2.7889), (11.0889), (0.4489), (0.1089), (13.4689)
- Sum of squared deviations = 5.4289 + 2.7889 + 11.0889 + 0.4489 + 0.1089 + 13.4689 = 33.3334
- Variance = Sum of squared deviations / (Number of observations - 1) = 33.3334 / 5 = 6.66668
- Dataset B:
- Deviations from the mean: (10 - 9.67), (8 - 9.67), (10 - 9.67), (8 - 9.67), (8 - 9.67), (4 - 9.67)
- Squared deviations: (0.33^2), (-1.67^2), (0.33^2), (-1.67^2), (-1.67^2), (-5.67^2)
- = (0.1089), (2.7889), (0.1089), (2.7889), (2.7889), (32.1289)
- Sum of squared deviations = 0.1089 + 2.7889 + 0.1089 + 2.7889 + 2.7889 + 32.1289 = 40.7034
- Variance = Sum of squared deviations / (Number of observations - 1) = 40.7034 / 5 = 8.14068
- Dataset A:
-
Calculate standard deviation by taking the square root of the variance.
- Dataset A:
- Standard Deviation = √6.66668 ≈ 2.58
- Dataset B:
- Standard Deviation = √8.14068 ≈ 2.85
- Dataset A:
Now, we can compare the results.
- The standard deviation of Dataset A is approximately 2.58.
- The standard deviation of Dataset B is approximately 2.85.
Thus, the correct conclusion is:
The standard deviation of Dataset A is 2.58. The standard deviation of Dataset B is 2.85. The standard deviation of Dataset B is larger than that of Dataset A, so Dataset B has more variability.