To calculate the standard deviation for the number of miles Charles walked, we first need to calculate the mean (average) of the data. We will use the given values and the corresponding total observations to compute the mean and then the standard deviation.
Step 1: Calculate the Mean
The values and their frequencies are as follows:
- Value 3 (2 days)
- Value 4 (3 days)
- Value 5 (2 days)
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Calculate the total number of observations: \[ \text{Total Observations} = 2 + 3 + 2 = 7 \]
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Calculate the total value of miles walked: \[ \text{Total Value} = (3 \times 2) + (4 \times 3) + (5 \times 2) = 6 + 12 + 10 = 28 \]
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Calculate the mean: \[ \text{Mean} = \frac{\text{Total Value}}{\text{Total Observations}} = \frac{28}{7} = 4 \]
Step 2: Calculate the Variance
Now we will calculate the variance using the formula: \[ \text{Variance} = \frac{\sum (x_i - \text{mean})^2 \cdot f_i}{N} \] Where \(x_i\) is each value, \(f_i\) is the frequency of each value, and \(N\) is the total number of observations.
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Calculate the sum of squared differences multiplied by frequency:
- For value 3: \[ (3 - 4)^2 \cdot 2 = (-1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
- For value 4: \[ (4 - 4)^2 \cdot 3 = (0)^2 \cdot 3 = 0 \cdot 3 = 0 \]
- For value 5: \[ (5 - 4)^2 \cdot 2 = (1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
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Summing these values gives: \[ 2 + 0 + 2 = 4 \]
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Now, calculate the variance: \[ \text{Variance} = \frac{4}{7} \approx 0.5714 \]
Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.5714} \approx 0.76 \]
Conclusion
Rounding to the nearest hundredth, the standard deviation is: \[ \boxed{0.76} \]
Thus, the answer is 0.76.