To find the variance of the data set \(-4, 4, 4, 4\), we can follow these steps:
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Calculate the Mean: \[ \text{Mean} = \frac{(-4) + 4 + 4 + 4}{4} = \frac{8}{4} = 2 \]
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Calculate the Squared Differences from the Mean:
- For \(-4\): \((-4 - 2)^2 = (-6)^2 = 36\)
- For \(4\): \((4 - 2)^2 = (2)^2 = 4\)
- For \(4\): \((4 - 2)^2 = (2)^2 = 4\)
- For \(4\): \((4 - 2)^2 = (2)^2 = 4\)
Now we have the squared differences: \(36, 4, 4, 4\).
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Calculate the Variance: The variance is the average of the squared differences.
First, find the sum of the squared differences: \[ 36 + 4 + 4 + 4 = 48 \]
Then divide by the number of data points (4): \[ \text{Variance} = \frac{48}{4} = 12 \]
Thus, the variance of the data set \(-4, 4, 4, 4\) is \(\boxed{12}\).