To calculate the standard deviation of the dataset, follow these steps:
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Find the Mean (Average): First, we need to calculate the mean of the dataset.
The values and their frequencies are:
- Value = -3, Frequency = 1
- Value = -2, Frequency = 2
- Value = -1, Frequency = 3
- Value = 0, Frequency = 2
- Value = 1, Frequency = 1
The total number of observations (N) = 1 + 2 + 3 + 2 + 1 = 9.
Now we find the sum of the values weighted by their frequencies: \[ \text{Sum} = (-3 \times 1) + (-2 \times 2) + (-1 \times 3) + (0 \times 2) + (1 \times 1) = -3 - 4 - 3 + 0 + 1 = -9 \]
Now calculate the mean: \[ \mu = \frac{\text{Sum}}{N} = \frac{-9}{9} = -1 \]
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Calculate the Variance: The variance is calculated as follows: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2 \cdot f_i}{N} \] Where \(f_i\) is the frequency of each value \(x_i\).
Calculate \((x_i - \mu)^2\) for each value:
- For x = -3: \((-3 - (-1))^2 = (-2)^2 = 4\), Contribution to variance = \(4 \cdot 1 = 4\)
- For x = -2: \((-2 - (-1))^2 = (-1)^2 = 1\), Contribution to variance = \(1 \cdot 2 = 2\)
- For x = -1: \((-1 - (-1))^2 = (0)^2 = 0\), Contribution to variance = \(0 \cdot 3 = 0\)
- For x = 0: \((0 - (-1))^2 = (1)^2 = 1\), Contribution to variance = \(1 \cdot 2 = 2\)
- For x = 1: \((1 - (-1))^2 = (2)^2 = 4\), Contribution to variance = \(4 \cdot 1 = 4\)
Now add these contributions: \[ \text{Total} = 4 + 2 + 0 + 2 + 4 = 12 \]
Now calculate the variance: \[ \sigma^2 = \frac{12}{9} = \frac{4}{3} \approx 1.3333 \]
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Calculate the Standard Deviation: The standard deviation is the square root of the variance: \[ \sigma = \sqrt{1.3333} \approx 1.1547 \]
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Round to the nearest hundredth: Finally, rounding this value to the nearest hundredth gives us: \[ \sigma \approx 1.15 \]
Thus, the standard deviation of the dataset is 1.15.