To calculate the standard deviation for the number of miles Charles walked, we first need to determine the mean (average) number of miles walked based on the observations given.
Here are the values and their total observations:
- 3 miles: 2 days
- 4 miles: 3 days
- 5 miles: 2 days
Step 1: Calculate the Mean (μ)
The total number of miles walked can be calculated as follows:
\[ \text{Total Miles} = (3 \times 2) + (4 \times 3) + (5 \times 2) = 6 + 12 + 10 = 28 \]
Next, we calculate the total number of observations (days):
\[ \text{Total Days} = 2 + 3 + 2 = 7 \]
Now, we find the mean:
\[ \mu = \frac{\text{Total Miles}}{\text{Total Days}} = \frac{28}{7} = 4 \]
Step 2: Calculate the Variance (σ²)
Now, we need to calculate the variance using the formula:
\[ \sigma^2 = \frac{\sum (x_i - \mu)^2 \cdot f_i}{N} \]
Where:
- \(x_i\) = each value (3, 4, 5)
- \(f_i\) = frequency of each value (2, 3, 2)
- \(N\) = total observations (7)
Calculating each term:
-
For \(x_1 = 3\): \[ (3 - 4)^2 \cdot 2 = (-1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
-
For \(x_2 = 4\): \[ (4 - 4)^2 \cdot 3 = (0)^2 \cdot 3 = 0 \]
-
For \(x_3 = 5\): \[ (5 - 4)^2 \cdot 2 = (1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
Now, sum these up:
\[ \text{Sum} = 2 + 0 + 2 = 4 \]
Then, we find the variance:
\[ \sigma^2 = \frac{4}{7} \approx 0.5714 \]
Step 3: Calculate the Standard Deviation (σ)
Now we take the square root of the variance to get the standard deviation:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{0.5714} \approx 0.756 \]
When rounded to the nearest hundredth:
\[ \sigma \approx 0.76 \]
Final Answer:
The standard deviation for the number of miles Charles walked is 0.76.