To calculate the variance of a data set, we can follow these steps:
- Find the mean (average) of the data set.
- Subtract the mean from each data point and square the result (this is called the squared deviation).
- Find the average of these squared deviations.
- The result is the variance.
Let's perform the calculations step by step for the data set: {4, 16, 21, 32, 11, 12}.
Step 1: Calculate the Mean
The mean is calculated as follows: \[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \]
Calculating the sum of the data points: \[ 4 + 16 + 21 + 32 + 11 + 12 = 96 \]
Now, dividing by the number of data points (which is 6): \[ \text{Mean} = \frac{96}{6} = 16 \]
Step 2: Calculate the Squared Deviations
Next, we subtract the mean from each data point and square the result:
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For 4: \((4 - 16)^2 = (-12)^2 = 144\)
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For 16: \((16 - 16)^2 = (0)^2 = 0\)
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For 21: \((21 - 16)^2 = (5)^2 = 25\)
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For 32: \((32 - 16)^2 = (16)^2 = 256\)
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For 11: \((11 - 16)^2 = (-5)^2 = 25\)
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For 12: \((12 - 16)^2 = (-4)^2 = 16\)
Now compile the squared deviations: \[ 144, 0, 25, 256, 25, 16 \]
Step 3: Calculate the Average of the Squared Deviations
Now, we find the mean of these squared deviations: \[ \text{Sum of squared deviations} = 144 + 0 + 25 + 256 + 25 + 16 = 466 \] \[ \text{Variance} = \frac{\text{Sum of squared deviations}}{\text{Number of data points}} \]
Since this is a sample variance, we need to divide by \(n - 1\) where \(n\) is the number of data points. Here \(n = 6\).
Calculating the variance: \[ \text{Variance} = \frac{466}{6 - 1} = \frac{466}{5} = 93.2 \]
Final Result
Thus, the variance of the data set {4, 16, 21, 32, 11, 12} is: \[ \boxed{93.2} \]