To calculate the variance of a data set, we follow these steps:
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Calculate the Mean: The mean (average) is calculated by summing all the values and then dividing by the number of values.
\( \text{Mean} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \)
For the data set (4, 16, 22, 32, 11, 12): \[ \text{Mean} = \frac{4 + 16 + 22 + 32 + 11 + 12}{6} = \frac{97}{6} \approx 16.33 \]
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Calculate the Deviations from the Mean: For each data point, subtract the mean and then square the result.
\[ \begin{align*} (4 - 16.33)^2 & = (-12.33)^2 \approx 152.1289 \ (16 - 16.33)^2 & = (-0.33)^2 \approx 0.1089 \ (22 - 16.33)^2 & = (5.67)^2 \approx 32.1489 \ (32 - 16.33)^2 & = (15.67)^2 \approx 245.5089 \ (11 - 16.33)^2 & = (-5.33)^2 \approx 28.4089 \ (12 - 16.33)^2 & = (-4.33)^2 \approx 18.7489 \ \end{align*} \]
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Sum the Squared Deviations:
\[ \text{Sum of Squared Deviations} \approx 152.1289 + 0.1089 + 32.1489 + 245.5089 + 28.4089 + 18.7489 \approx 477.6534 \]
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Calculate the Variance: For a sample, we divide by \( n - 1 \) (degrees of freedom). For a population, we divide by \( n \). Here, we'll assume it's a sample variance calculation.
\[ \text{Variance} = \frac{\text{Sum of Squared Deviations}}{n - 1} = \frac{477.6534}{6 - 1} = \frac{477.6534}{5} \approx 95.53068 \]
Thus, the variance of the data set is approximately 95.53.