To calculate the variance of the data set \({4, 16, 21, 32, 11, 12}\), we’ll follow these steps:
Step 1: Calculate the Mean
The mean is calculated by summing all the values and dividing by the number of values.
\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]
\[ \text{Mean} = \frac{4 + 16 + 21 + 32 + 11 + 12}{6} \]
Calculating the sum:
\[ 4 + 16 + 21 + 32 + 11 + 12 = 96 \]
Now divide by the number of values (which is 6):
\[ \text{Mean} = \frac{96}{6} = 16 \]
Step 2: Calculate the Squared Deviations from the Mean
Next, we calculate the squared deviations from the mean for each data point, which is done by subtracting the mean from each data point and squaring the result.
\[ \text{Squared deviation} = (x_i - \text{Mean})^2 \]
Calculating for each \(x_i\):
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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 \(12\): \[ (12 - 16)^2 = (-4)^2 = 16 \]
Step 3: Sum of Squared Deviations
Now we sum the squared deviations calculated in the previous step:
\[ \text{Sum of squared deviations} = 144 + 0 + 25 + 256 + 25 + 16 = 466 \]
Step 4: Calculate the Variance
Since we have a sample, we’ll use the sample variance formula, which divides by \(n-1\) (where \(n\) is the number of observations):
\[ \text{Variance} = \frac{\text{Sum of squared deviations}}{n-1} \]
\[ \text{Variance} = \frac{466}{6-1} = \frac{466}{5} = 93.2 \]
Final Result
The variance of the data set \({4, 16, 21, 32, 11, 12}\) is \(93.2\).
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