Question

calculate the variance of the given data set \(4, 16, 22, 32, 11, 12

Answer 1

To calculate the variance of the data set \(4, 16, 22, 32, 11, 12\), we will follow these steps:

1. Calculate the mean of the data set.
2. Subtract the mean from each data point and square the result (these are the squared deviations).
3. Calculate the average of these squared deviations (this is the variance).

### Step 1: Calculate the mean

\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{4 + 16 + 22 + 32 + 11 + 12}{6}
\]

\[
\text{Sum} = 4 + 16 + 22 + 32 + 11 + 12 = 97
\]

\[
\text{Mean} = \frac{97}{6} \approx 16.17
\]

### Step 2: Calculate the squared deviations

Now we will calculate the squared deviations from the mean for each data point:

\[
(4 - 16.17)^2 \approx (-12.17)^2 \approx 148.1889
\]
\[
(16 - 16.17)^2 \approx (-0.17)^2 \approx 0.0289
\]
\[
(22 - 16.17)^2 \approx (5.83)^2 \approx 33.9289
\]
\[
(32 - 16.17)^2 \approx (15.83)^2 \approx 251.4289
\]
\[
(11 - 16.17)^2 \approx (-5.17)^2 \approx 26.7289
\]
\[
(12 - 16.17)^2 \approx (-4.17)^2 \approx 17.3889
\]

### Step 3: Calculate the variance

Next, we find the average of these squared deviations:

\[
\text{Variance} = \frac{\text{Sum of squared deviations}}{\text{Number of data points}}
\]
\[
\text{Sum of squared deviations} = 148.1889 + 0.0289 + 33.9289 + 251.4289 + 26.7289 + 17.3889 \approx 477.1934
\]
\[
\text{Variance} = \frac{477.1934}{6} \approx 79.5319
\]

### Final Answer

Thus, the variance of the given data set is approximately \(79.53\).