Question

Calculate the variance

Given the data: 4, 16, 22, 32, 11, 12

Answer 1

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

1. **Calculate the mean** (average):
\[
\text{Mean} = \frac{\sum x_i}{n} = \frac{4 + 16 + 22 + 32 + 11 + 12}{6} = \frac{97}{6} \approx 16.33
\]

2. **Calculate the squared differences from the mean**:
\[
(x_i - \text{Mean})^2
\]
- For \(4\): \( (4 - 16.33)^2 \approx ( -12.33)^2 \approx 152.11 \)
- For \(16\): \( (16 - 16.33)^2 \approx ( -0.33)^2 \approx 0.11 \)
- For \(22\): \( (22 - 16.33)^2 \approx (5.67)^2 \approx 32.11 \)
- For \(32\): \( (32 - 16.33)^2 \approx (15.67)^2 \approx 246.11 \)
- For \(11\): \( (11 - 16.33)^2 \approx ( -5.33)^2 \approx 28.41 \)
- For \(12\): \( (12 - 16.33)^2 \approx ( -4.33)^2 \approx 18.75 \)

3. **Sum the squared differences**:
\[
152.11 + 0.11 + 32.11 + 246.11 + 28.41 + 18.75 \approx 477.70
\]

4. **Calculate the variance**:
Since this is a sample, the formula for variance \(s^2\) is:
\[
s^2 = \frac{\sum (x_i - \text{Mean})^2}{n - 1}
\]
where \(n\) is the number of observations (6 in this case):
\[
s^2 = \frac{477.70}{6 - 1} = \frac{477.70}{5} \approx 95.54
\]

The variance of the given data set is approximately **95.54**.