To calculate the standard deviation of the data set given the values and their corresponding observations, we can follow these steps:
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Construct the Data Set: From the table, we can represent the data set based on the values and their frequency:
- Value 4 → 1 observation
- Value 5 → 3 observations
- Value 6 → 1 observation
Therefore, the data set is:
- \(4, 5, 5, 5, 6\)
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Calculate the Mean: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of observations}} = \frac{4 + 5 + 5 + 5 + 6}{5} = \frac{25}{5} = 5 \]
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Calculate the Variance:
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Find the squared differences from the mean:
- For \(4\): \((4 - 5)^2 = (-1)^2 = 1\)
- For \(5\): \((5 - 5)^2 = (0)^2 = 0\) (this happens for the three 5's)
- For \(6\): \((6 - 5)^2 = (1)^2 = 1\)
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Now calculate the variance: \[ \text{Variance} = \frac{\text{Sum of squared differences}}{\text{Total number of observations}} = \frac{1 + 0 + 0 + 0 + 1}{5} = \frac{2}{5} = 0.4 \]
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Calculate the Standard Deviation: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.4} \approx 0.6325 \]
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Round to the Nearest Hundredth: When rounded to the nearest hundredth: \[ 0.6325 \approx 0.63 \]
Thus, the standard deviation of the data set is approximately 0.63.