To calculate the standard deviation of the dataset represented in the table, we first need to organize the data based on the values and their total observations.
The data from the table can be expressed as follows:
- Value 4 occurs 1 time
- Value 5 occurs 3 times
- Value 6 occurs 1 time
This gives us the dataset: \( 4, 5, 5, 5, 6 \).
Step 1: Calculate the mean (average)
To compute the mean, we sum all the values and divide by the total number of observations.
\[ \text{Mean} = \frac{(4 \times 1) + (5 \times 3) + (6 \times 1)}{1 + 3 + 1} = \frac{4 + 15 + 6}{5} = \frac{25}{5} = 5 \]
Step 2: Calculate the variance
The variance is calculated by taking the average of the squared differences from the mean.
- Calculate each squared difference from the mean:
\[ (4 - 5)^2 = (-1)^2 = 1 \] \[ (5 - 5)^2 = (0)^2 = 0 \quad \text{(for 3 occurrences)} \] \[ (6 - 5)^2 = (1)^2 = 1 \]
- We weigh these by their frequencies:
\[ \text{Variance} = \frac{(1 \times 1) + (3 \times 0) + (1 \times 1)}{5} \] \[ = \frac{1 + 0 + 1}{5} = \frac{2}{5} = 0.4 \]
Step 3: Calculate the standard deviation
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{0.4} \approx 0.6325 \]
Rounding
Rounding to the nearest hundredth, we get:
\[ \text{Standard Deviation} \approx 0.63 \]
Conclusion
Based on the provided options, this standard deviation does not exactly match the given multiple-choice answers. However, since the given numbers do not include 0.63, the closest match based on our calculation is not available in the responses.
Please double-check the calculations or values provided in the table to ensure accuracy.