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

0.50
0.50

5.00
5.00

0.71
0.71

0.25

Answer 1

To calculate the standard deviation, we first need to compile the data set based on the values and their corresponding total observations.

Here, the data points are:
- One instance of 4
- Three instances of 5
- One instance of 6

This gives us the following data points:
- 4, 5, 5, 5, 6

Next, we can calculate the mean (average) of the data set:

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

2. Next, we find the squared differences from the mean for each data point:
- For 4: \((4 - 5)^2 = (-1)^2 = 1\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\)
- For 6: \((6 - 5)^2 = (1)^2 = 1\)

3. Calculate the sum of the squared differences:
\[
\text{Sum of squared differences} = 1 + 0 + 0 + 0 + 1 = 2
\]

4. Divide the sum of squared differences by the number of observations to find the variance:
\[
\text{Variance} = \frac{\text{Sum of squared differences}}{\text{Total observations}} = \frac{2}{5} = 0.4
\]

5. Finally, take the square root of the variance to find the standard deviation:
\[
\text{Standard deviation} = \sqrt{0.4} \approx 0.6325
\]

Rounding to the nearest hundredth gives us:
\[
\text{Standard deviation} \approx 0.63
\]

However, looking at the available responses:
- 0.50
- 5.00
- 0.71
- 0.25

None of these options match the calculated standard deviation. If only rounded values are considered, the closest value provided is **0.71**.

So based on the choices given, the answer would be:
**0.71**.