To calculate the standard deviation of the dataset, we first need to find the mean and then use that to find the variance and the standard deviation.
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List the data points based on the values and their frequencies:
- \(-3\) appears \(1\) time
- \(-2\) appears \(2\) times
- \(-1\) appears \(3\) times
- \(0\) appears \(2\) times
- \(1\) appears \(1\) time
Thus, the dataset is: \[ -3, -2, -2, -1, -1, -1, 0, 0, 1 \]
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Calculate the mean (\(\mu\)) of the dataset:
\[ \text{Sum of all observations} = -3 + (-2) + (-2) + (-1) + (-1) + (-1) + 0 + 0 + 1 = -9 \]
\[ \text{Total number of observations} = 1 + 2 + 3 + 2 + 1 = 9 \]
\[ \mu = \frac{\text{Sum of all observations}}{\text{Total number of observations}} = \frac{-9}{9} = -1 \]
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Calculate the variance (\(\sigma^2\)):
\[ \text{Variance} = \frac{\sum (x_i - \mu)^2}{N} \] where \(x_i\) are the data points, \(\mu\) is the mean, and \(N\) is the number of observations.
Now, compute \((x_i - \mu)^2\) for each observation:
- \( (-3 - (-1))^2 = (-3 + 1)^2 = (-2)^2 = 4 \)
- \( (-2 - (-1))^2 = (-2 + 1)^2 = (-1)^2 = 1 \) (counted twice, so \(1 \times 2\))
- \( (-1 - (-1))^2 = (-1 + 1)^2 = (0)^2 = 0 \) (counted three times, so \(0 \times 3\))
- \( (0 - (-1))^2 = (0 + 1)^2 = (1)^2 = 1 \) (counted twice, so \(1 \times 2\))
- \( (1 - (-1))^2 = (1 + 1)^2 = (2)^2 = 4 \)
Now, sum them up: \[ \text{Sum of squared differences} = 4 + (1 \times 2) + (0 \times 3) + (1 \times 2) + 4 = 4 + 2 + 0 + 2 + 4 = 12 \]
Now divide by \(N = 9\): \[ \text{Variance} = \frac{12}{9} \approx 1.3333 \]
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Calculate the standard deviation (\(\sigma\)):
\[ \sigma = \sqrt{\text{Variance}} = \sqrt{\frac{12}{9}} \approx \sqrt{1.3333} \approx 1.1547 \]
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Round to the nearest hundredth: \[ \sigma \approx 1.15 \]
Thus, the standard deviation is \( \boxed{1.15} \).
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