To calculate the standard deviation of the given data set (1, 1, 2, 3, 3), we will follow these steps:
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Calculate the mean (average): \[ \text{Mean} = \frac{\text{Sum of all responses}}{\text{Number of responses}} = \frac{1 + 1 + 2 + 3 + 3}{5} = \frac{10}{5} = 2 \]
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Calculate the variance: The variance is calculated as the average of the squared differences from the mean. \[ \text{Variance} = \frac{\sum (x_i - \text{mean})^2}{N} \] Where \(x_i\) are the individual responses and \(N\) is the number of responses.
Calculating each squared difference:
- For \(1\): \((1 - 2)^2 = (-1)^2 = 1\)
- For \(1\): \((1 - 2)^2 = (-1)^2 = 1\)
- For \(2\): \((2 - 2)^2 = (0)^2 = 0\)
- For \(3\): \((3 - 2)^2 = (1)^2 = 1\)
- For \(3\): \((3 - 2)^2 = (1)^2 = 1\)
Now summing these squared differences: \[ 1 + 1 + 0 + 1 + 1 = 4 \]
Now divide by the number of responses to get the variance: \[ \text{Variance} = \frac{4}{5} = 0.8 \]
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Calculate the standard deviation: The standard deviation is the square root of the variance. \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.8} \approx 0.8944 \]
Thus, the standard deviation of the data is approximately 0.894.