To calculate the standard deviation of the dataset given in the table, we will follow these steps:
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List the values and their frequencies:
- Value \( -3 \): 1 observation
- Value \( -2 \): 2 observations
- Value \( -1 \): 3 observations
- Value \( 0 \): 2 observations
- Value \( 1 \): 1 observation
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Expand the dataset based on the frequency:
- \( -3, -2, -2, -1, -1, -1, 0, 0, 1 \)
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Calculate the mean (average) of the dataset: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of observations}} = \frac{-3 + (-2) + (-2) + (-1) + (-1) + (-1) + 0 + 0 + 1}{10} \] \[ = \frac{-3 - 2 - 2 - 1 - 1 - 1 + 0 + 0 + 1}{10} = \frac{-9}{10} = -0.9 \]
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Calculate the squared deviations from the mean:
- For \( -3 \): \( (-3 - (-0.9))^2 = (-3 + 0.9)^2 = (-2.1)^2 = 4.41 \)
- For \( -2 \): \( (-2 - (-0.9))^2 = (-2 + 0.9)^2 = (-1.1)^2 = 1.21 \) (this occurs twice, so multiply by 2)
- For \( -1 \): \( (-1 - (-0.9))^2 = (-1 + 0.9)^2 = (-0.1)^2 = 0.01 \) (this occurs three times, so multiply by 3)
- For \( 0 \): \( (0 - (-0.9))^2 = (0 + 0.9)^2 = (0.9)^2 = 0.81 \) (this occurs twice, so multiply by 2)
- For \( 1 \): \( (1 - (-0.9))^2 = (1 + 0.9)^2 = (1.9)^2 = 3.61 \)
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Sum the squared deviations: \[ 4.41 + 2 \times 1.21 + 3 \times 0.01 + 2 \times 0.81 + 3.61 \] \[ = 4.41 + 2.42 + 0.03 + 1.62 + 3.61 = 12.09 \]
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Calculate the variance: Since we are calculating the sample standard deviation, we use \( n - 1 \) (where \( n \) is the total number of observations, which is 10 here) to find the variance. \[ \text{Variance} = \frac{\text{Sum of squared deviations}}{n - 1} = \frac{12.09}{10 - 1} = \frac{12.09}{9} = 1.343333 \]
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Calculate the standard deviation: \[ \text{Standard Deviation} = \sqrt{1.343333} \approx 1.16 \]
Rounding to the nearest hundredth, the standard deviation is 1.16.