Question

Calculate the variance for the data set 4, 16, 21, 32, 11, 12

Answer 1

To calculate the variance of the dataset \(4, 16, 21, 32, 11, 12\), follow these steps:

1. **Calculate the Mean (\(\mu\))**:
\[
\mu = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{4 + 16 + 21 + 32 + 11 + 12}{6}
\]
\[
\mu = \frac{96}{6} = 16
\]

2. **Calculate the squared differences from the Mean**:
\[
(4 - 16)^2 = (-12)^2 = 144
\]
\[
(16 - 16)^2 = 0^2 = 0
\]
\[
(21 - 16)^2 = 5^2 = 25
\]
\[
(32 - 16)^2 = 16^2 = 256
\]
\[
(11 - 16)^2 = (-5)^2 = 25
\]
\[
(12 - 16)^2 = (-4)^2 = 16
\]

3. **Sum of Squared Differences**:
\[
\text{Sum} = 144 + 0 + 25 + 256 + 25 + 16 = 466
\]

4. **Calculate the Variance (\(\sigma^2\))**:
For a sample variance, you would divide by \(n - 1\) (where \(n\) is the number of data points). However, since we are looking at the population variance, we’ll divide by \(n\):
\[
\sigma^2 = \frac{\text{Sum of squared differences}}{n} = \frac{466}{6} \approx 77.67
\]

Thus, the variance of the dataset \(4, 16, 21, 32, 11, 12\) is approximately \(77.67\).