To calculate the standard deviation of the data set \(7, 8, 1, 8, 8\), we'll follow these steps:
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Calculate the mean (average): \[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{7 + 8 + 1 + 8 + 8}{5} = \frac{32}{5} = 6.4 \]
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Calculate the squared deviations from the mean: \[ (7 - 6.4)^2 = (0.6)^2 = 0.36 \] \[ (8 - 6.4)^2 = (1.6)^2 = 2.56 \] \[ (1 - 6.4)^2 = (-5.4)^2 = 29.16 \] \[ (8 - 6.4)^2 = (1.6)^2 = 2.56 \] \[ (8 - 6.4)^2 = (1.6)^2 = 2.56 \]
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Sum of the squared deviations: \[ 0.36 + 2.56 + 29.16 + 2.56 + 2.56 = 37.2 \]
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Calculate the variance: Since this is a sample standard deviation, we'll divide by \(n-1\) (4 in this case): \[ \text{Variance} = \frac{37.2}{5 - 1} = \frac{37.2}{4} = 9.3 \]
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Calculate the standard deviation: \[ \text{Standard deviation} (σ) = \sqrt{9.3} \approx 3.0 \]
Thus, the standard deviation rounded to the nearest tenth is: \[ \text{Standard deviation (σ)}: 3.0 \]