Question

Calculate the variance for the data set. 4, 26, 21, 32, 11, 12

Answer 1

To calculate the variance of the given data set \(4, 26, 21, 32, 11, 12\), follow these steps:

1. **Calculate the mean**.
\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{4 + 26 + 21 + 32 + 11 + 12}{6}
\]
\[
= \frac{106}{6} \approx 17.67
\]

2. **Calculate each data point's deviation from the mean** and square it:
- For 4: \((4 - 17.67)^2 \approx ( -13.67)^2 \approx 186.91\)
- For 26: \((26 - 17.67)^2 \approx (8.33)^2 \approx 69.39\)
- For 21: \((21 - 17.67)^2 \approx (3.33)^2 \approx 11.09\)
- For 32: \((32 - 17.67)^2 \approx (14.33)^2 \approx 205.69\)
- For 11: \((11 - 17.67)^2 \approx (-6.67)^2 \approx 44.49\)
- For 12: \((12 - 17.67)^2 \approx (-5.67)^2 \approx 32.11\)

3. **Sum all squared deviations**:
\[
\text{Sum} = 186.91 + 69.39 + 11.09 + 205.69 + 44.49 + 32.11 \approx 549.68
\]

4. **Calculate the variance**. Since this is a sample data set, use the sample variance formula; if you consider it a whole population, divide by \(n\); if it's a sample, divide by \(n - 1\).
\[
\text{Variance} = \frac{\text{Sum of squared deviations}}{n - 1} \quad \text{(for sample)}
\]

Here, \(n = 6\):

Sample variance:
\[
\text{Variance} = \frac{549.68}{6 - 1} \approx \frac{549.68}{5} \approx 109.94
\]

Therefore, the variance of the data set \(4, 26, 21, 32, 11, 12\) is approximately **109.94**.