Question

Value 3 4 5
Total Observations 2 3 2
Charles records the number of miles he walks every day. The chart includes how many days he walked the given number of miles for one week. Calculate the standard deviation for the number of miles he walked. Round the answer to the nearest hundredth.

(1 point)
Responses

0.82
0.82

0.76
0.76

0.45
0.45

0.67

Answer 1

To calculate the standard deviation of the number of miles Charles walked, we first need to find the mean (average) of the observations and then use it to find the standard deviation.

1. **Calculate the total number of miles walked:**

We have the following data:
- 3 miles: 2 days
- 4 miles: 3 days
- 5 miles: 2 days

We can calculate the total miles walked for each category:
- Total miles for 3 miles = 3 * 2 = 6
- Total miles for 4 miles = 4 * 3 = 12
- Total miles for 5 miles = 5 * 2 = 10

Now sum those up for the total number of miles:
\[
\text{Total miles} = 6 + 12 + 10 = 28
\]

2. **Calculate the total number of days:**

Total observations = 2 + 3 + 2 = 7 days.

3. **Calculate the mean miles walked:**

\[
\text{Mean} = \frac{\text{Total miles}}{\text{Total observations}} = \frac{28}{7} = 4
\]

4. **Calculate the variance:**

Calculate the squared difference from the mean for each observation, weighted by the number of days:
- For 3 miles:
\[
(3 - 4)^2 \times 2 = 1 \times 2 = 2
\]
- For 4 miles:
\[
(4 - 4)^2 \times 3 = 0 \times 3 = 0
\]
- For 5 miles:
\[
(5 - 4)^2 \times 2 = 1 \times 2 = 2
\]

Now sum the squared differences:
\[
\text{Sum of squared differences} = 2 + 0 + 2 = 4
\]

Now divide by the total number of observations (N) to get the variance:
\[
\text{Variance} = \frac{\text{Sum of squared differences}}{N} = \frac{4}{7} \approx 0.5714
\]

5. **Calculate the standard deviation:**

The standard deviation is the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{0.5714} \approx 0.76
\]

So, the standard deviation for the number of miles he walked, rounded to the nearest hundredth, is **0.76**.

**Final Response:** 0.76