Question

Calculate the variance for the data set. Show all of your steps.

{4, 16, 21, 32, 11, 12}

Answer 1

To calculate the variance for the data set {4, 16, 21, 32, 11, 12}, follow these steps:

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

2. **Calculate the squared deviations from the mean** for each data point:
- For 4: \((4 - 16)^2 = (-12)^2 = 144\)
- For 16: \((16 - 16)^2 = (0)^2 = 0\)
- For 21: \((21 - 16)^2 = (5)^2 = 25\)
- For 32: \((32 - 16)^2 = (16)^2 = 256\)
- For 11: \((11 - 16)^2 = (-5)^2 = 25\)
- For 12: \((12 - 16)^2 = (-4)^2 = 16\)

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

4. **Divide by the number of data points minus 1** (n-1) to get the sample variance:
\[
\text{Sample Variance} = \frac{\text{Sum of squared deviations}}{n-1} = \frac{466}{6-1} = \frac{466}{5} = 93.2
\]

Thus, the variance of the data set {4, 16, 21, 32, 11, 12} is \( \boxed{93.2} \).