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:
\[ \boxed{0.76} \]
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