To calculate the standard deviation of the given data set, we first need to identify the values along with their frequencies.
The data set is as follows:
- Value 4 occurs 1 time.
- Value 5 occurs 3 times.
- Value 6 occurs 1 time.
We can summarize this data set as:
- \(4, 5, 5, 5, 6\) (expanding the values based on their frequency).
Step 1: Calculate the mean (average).
\[ \text{Mean} = \frac{\text{Total Sum of Values}}{\text{Total Count of Values}} \] \[ \text{Total Sum} = (4 \times 1) + (5 \times 3) + (6 \times 1) = 4 + 15 + 6 = 25 \] \[ \text{Total Count} = 1 + 3 + 1 = 5 \] \[ \text{Mean} = \frac{25}{5} = 5 \]
Step 2: Calculate the variance.
The variance is calculated using the formula:
\[ \text{Variance} = \frac{\sum (x_i - \mu)^2}{N} \] where \(x_i\) is each value, \(\mu\) is the mean, and \(N\) is the number of observations.
Calculating the squared differences from the mean:
- For 4: \((4 - 5)^2 = (-1)^2 = 1\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\) (3 times, so add 0 three times)
- For 6: \((6 - 5)^2 = (1)^2 = 1\)
Thus, we have: \[ \sum (x_i - \mu)^2 = 1 + 0 + 0 + 0 + 1 = 2 \] Now, we can compute the variance: \[ \text{Variance} = \frac{2}{5} = 0.4 \]
Step 3: Calculate the standard deviation.
Standard deviation is the square root of variance: \[ \text{Standard Deviation} = \sqrt{0.4} \approx 0.6325 \]
Rounding this to the nearest hundredth gives: \[ \text{Standard Deviation} \approx 0.63 \]
Among the provided options, 0.71 is the closest representation, but a value of 0.63 does not appear in your responses. Based on the computation, the nearest hundredth is indeed 0.63 which may imply a slight error in available options.
The answer is not explicitly listed among your provided options, and the calculated result is approximately 0.63.
1 answer