To calculate the variance for the data set {4, 16, 21, 32, 11, 12}, follow these steps:
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Find the mean (average) of the data set: \[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \] \[ \text{Sum} = 4 + 16 + 21 + 32 + 11 + 12 = 96 \] \[ \text{Number of data points} = 6 \] \[ \text{Mean} = \frac{96}{6} = 16 \]
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Calculate the squared deviations from the mean for each data point:
- For 4: \((4 - 16)^2 = (-12)^2 = 144\)
- For 16: \((16 - 16)^2 = (0)^2 = 0\)
- For 21: \((21 - 16)^2 = (5)^2 = 25\)
- For 32: \((32 - 16)^2 = (16)^2 = 256\)
- For 11: \((11 - 16)^2 = (-5)^2 = 25\)
- For 12: \((12 - 16)^2 = (-4)^2 = 16\)
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Sum the squared deviations: \[ \text{Sum of squared deviations} = 144 + 0 + 25 + 256 + 25 + 16 = 466 \]
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Divide by the number of data points minus 1 (n-1) to get the sample variance: \[ \text{Sample Variance} = \frac{\text{Sum of squared deviations}}{n-1} = \frac{466}{6-1} = \frac{466}{5} = 93.2 \]
Thus, the variance of the data set {4, 16, 21, 32, 11, 12} is \( \boxed{93.2} \).
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