To calculate the standard deviation for the number of miles Charles walked based on the provided data, we first need to find the mean, variance, and then take the square root of the variance.
Data Summary
- For 3 miles: 2 days
- For 4 miles: 3 days
- For 5 miles: 2 days
Step 1: Calculate the Mean
The formula for the mean (μ) is:
\[ \mu = \frac{\sum (x \cdot f)}{N} \]
where:
- \( x \) is the value (miles walked)
- \( f \) is the frequency (number of days)
- \( N \) is the total number of observations
Calculating total observations (N):
\[ N = 2 + 3 + 2 = 7 \text{ days} \]
Calculating \( \sum (x \cdot f) \):
\[ \sum (x \cdot f) = (3 \cdot 2) + (4 \cdot 3) + (5 \cdot 2) = 6 + 12 + 10 = 28 \]
Now calculate the mean:
\[ \mu = \frac{28}{7} = 4 \]
Step 2: Calculate the Variance
The formula for variance (\( \sigma^2 \)) is:
\[ \sigma^2 = \frac{\sum f \cdot (x - \mu)^2}{N} \]
Calculating \( (x - \mu)^2 \) for each value:
-
For 3 miles: \[ (3 - 4)^2 = (-1)^2 = 1 \] Contribution to the variance: \[ 2 \cdot 1 = 2 \]
-
For 4 miles: \[ (4 - 4)^2 = (0)^2 = 0 \] Contribution to the variance: \[ 3 \cdot 0 = 0 \]
-
For 5 miles: \[ (5 - 4)^2 = (1)^2 = 1 \] Contribution to the variance: \[ 2 \cdot 1 = 2 \]
Now, summing these contributions:
\[ \sum f \cdot (x - \mu)^2 = 2 + 0 + 2 = 4 \]
Finally, calculate the variance:
\[ \sigma^2 = \frac{4}{7} \approx 0.57 \]
Step 3: Calculate the Standard Deviation
The standard deviation (\( \sigma \)) is the square root of the variance:
\[ \sigma = \sqrt{0.57} \approx 0.76 \]
Final Answer
The standard deviation for the number of miles he walked, rounded to the nearest hundredth, is:
0.76
So, the correct response is:
0.76