Question

calculate the variance of the data set (4, 16, 22, 32, 11, 12 show steps

Answer 1

To calculate the variance of a data set, we follow these steps:

1. **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):
\[
\text{Mean} = \frac{4 + 16 + 22 + 32 + 11 + 12}{6} = \frac{97}{6} \approx 16.33
\]

2. **Calculate the Deviations from the Mean**:
For each data point, subtract the mean and then square the result.

\[
\begin{align*}
(4 - 16.33)^2 & = (-12.33)^2 \approx 152.1289 \\
(16 - 16.33)^2 & = (-0.33)^2 \approx 0.1089 \\
(22 - 16.33)^2 & = (5.67)^2 \approx 32.1489 \\
(32 - 16.33)^2 & = (15.67)^2 \approx 245.5089 \\
(11 - 16.33)^2 & = (-5.33)^2 \approx 28.4089 \\
(12 - 16.33)^2 & = (-4.33)^2 \approx 18.7489 \\
\end{align*}
\]

3. **Sum the Squared Deviations**:

\[
\text{Sum of Squared Deviations} \approx 152.1289 + 0.1089 + 32.1489 + 245.5089 + 28.4089 + 18.7489 \approx 477.6534
\]

4. **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**.