To calculate the standard deviation of the miles Charles walked, we first need to determine the mean (average) of the data and then use it to find the variance and finally the standard deviation.
Step 1: Create a frequency distribution table
From the data provided:
- 3 miles: 2 days
- 4 miles: 3 days
- 5 miles: 2 days
To create a frequency distribution:
| Miles (x) | Frequency (f) | |-----------|---------------| | 3 | 2 | | 4 | 3 | | 5 | 2 |
Step 2: Calculate the mean (µ)
The mean is calculated using the formula:
\[ \text{Mean} (\mu) = \frac{\sum (f \cdot x)}{\sum f} \]
Where:
- \( f \) is the frequency
- \( x \) is the value (miles)
Calculating \( f \cdot x \):
\[ f \cdot x = (3 \cdot 2) + (4 \cdot 3) + (5 \cdot 2) = 6 + 12 + 10 = 28 \]
Total observations (sum of frequencies):
\[ \sum f = 2 + 3 + 2 = 7 \]
Now we can calculate the mean:
\[ \mu = \frac{28}{7} = 4 \]
Step 3: Calculate variance (σ²)
The variance is calculated using the formula:
\[ \sigma^2 = \frac{\sum f (x - \mu)^2}{\sum f} \]
Calculating \( (x - \mu)^2 \):
| Miles (x) | f | x - μ | (x - μ)² | f(x - μ)² | |-----------|---|--------|----------|-----------| | 3 | 2 | 3 - 4 = -1 | 1 | 2 | | 4 | 3 | 4 - 4 = 0 | 0 | 0 | | 5 | 2 | 5 - 4 = 1 | 1 | 2 |
Now sum \( f(x - \mu)² \):
\[ \sum f(x - \mu)² = 2 + 0 + 2 = 4 \]
Now we can calculate the variance:
\[ \sigma^2 = \frac{4}{7} \approx 0.57142857 \]
Step 4: Calculate standard deviation (σ)
The standard deviation is the square root of the variance:
\[ \sigma = \sqrt{0.57142857} \approx 0.756 \]
Step 5: Round to the nearest hundredth
Rounding 0.756 to the nearest hundredth gives:
\[ \text{Standard Deviation} \approx 0.76 \]
Thus, the standard deviation for the number of miles Charles walked is 0.76.