Question

Use the table to answer the question. Value 4 5 6 Total Observations 1 3 1 Calculate the standard deviation of the data set. Round the answer to the nearest hundredth. (1 point) Responses 5.00 5.00 0.25 0.25 0.71 0.71 0.50 0.50 Skip to navigation

Answer 1

To calculate the standard deviation of the data set given the values and their corresponding observations, we can follow these steps:

1. **Construct the Data Set**: From the table, we can represent the data set based on the values and their frequency:
- Value 4 → 1 observation
- Value 5 → 3 observations
- Value 6 → 1 observation

Therefore, the data set is:
- \(4, 5, 5, 5, 6\)

2. **Calculate the Mean**:
\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of observations}} = \frac{4 + 5 + 5 + 5 + 6}{5} = \frac{25}{5} = 5
\]

3. **Calculate the Variance**:
- Find the squared differences from the mean:
- For \(4\): \((4 - 5)^2 = (-1)^2 = 1\)
- For \(5\): \((5 - 5)^2 = (0)^2 = 0\) (this happens for the three 5's)
- For \(6\): \((6 - 5)^2 = (1)^2 = 1\)

- Now calculate the variance:
\[
\text{Variance} = \frac{\text{Sum of squared differences}}{\text{Total number of observations}} = \frac{1 + 0 + 0 + 0 + 1}{5} = \frac{2}{5} = 0.4
\]

4. **Calculate the Standard Deviation**:
\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.4} \approx 0.6325
\]

5. **Round to the Nearest Hundredth**:
When rounded to the nearest hundredth:
\[
0.6325 \approx 0.63
\]

Thus, the standard deviation of the data set is approximately **0.63**.