Question

Use the table to answer the 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

0.67
0.67

0.45
0.45

0.76

Answer 1

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

### Step 1: Calculate the Mean

The values and their corresponding frequencies from the table are:

- Miles walked = 3, Frequency = 2
- Miles walked = 4, Frequency = 3
- Miles walked = 5, Frequency = 2

To find the mean, we can use the formula:

\[
\text{Mean} = \frac{\sum (x_i \cdot f_i)}{N}
\]

where \( x_i \) is the number of miles and \( f_i \) is the frequency, and \( N \) is the total number of observations.

Calculating \( \sum (x_i \cdot f_i) \):

\[
(3 \cdot 2) + (4 \cdot 3) + (5 \cdot 2) = 6 + 12 + 10 = 28
\]

Total observations \( N \):

\[
N = 2 + 3 + 2 = 7
\]

Now, calculate the mean:

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

### Step 2: Calculate the Variance

Next, we calculate the variance using the formula:

\[
\text{Variance} = \frac{\sum f_i (x_i - \text{Mean})^2}{N}
\]

Calculating each \( (x_i - \text{Mean})^2 \):

For \( x_1 = 3 \):

\[
(3 - 4)^2 = (-1)^2 = 1
\]

For \( x_2 = 4 \):

\[
(4 - 4)^2 = (0)^2 = 0
\]

For \( x_3 = 5 \):

\[
(5 - 4)^2 = (1)^2 = 1
\]

Now, multiply by the frequencies:

For \( x_1 = 3, f_1 = 2\):

\[
2 \cdot 1 = 2
\]

For \( x_2 = 4, f_2 = 3\):

\[
3 \cdot 0 = 0
\]

For \( x_3 = 5, f_3 = 2\):

\[
2 \cdot 1 = 2
\]

Summing these values gives:

\[
2 + 0 + 2 = 4
\]

Now we can calculate the variance:

\[
\text{Variance} = \frac{4}{7} \approx 0.5714
\]

### Step 3: Calculate the Standard Deviation

Taking the square root of the variance gives us the standard deviation:

\[
\text{Standard Deviation} = \sqrt{0.5714} \approx 0.76 \text{ (rounded to two decimal places)}
\]

### Conclusion

Thus, the standard deviation for the number of miles Charles walked, rounded to the nearest hundredth, is:

\[
\boxed{0.76}
\]