Question

Calculate the variance for this data. 4, 16, 22, 32, 11, 12

Answer 1

To calculate the variance, follow these steps:

1. **Calculate the mean (average)** of the data set.
2. **Calculate the squared differences** from the mean for each data point.
3. **Calculate the average of those squared differences**.

### Step 1: Calculate the mean

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

\[
\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}} = \frac{4 + 16 + 22 + 32 + 11 + 12}{6} = \frac{97}{6} \approx 16.33
\]

### Step 2: Calculate the squared differences from the mean

Now, calculate each squared difference from the mean \( \mu \approx 16.33 \):

- For 4: \( (4 - 16.33)^2 \approx ( -12.33)^2 \approx 152.1889 \)
- For 16: \( (16 - 16.33)^2 \approx ( -0.33)^2 \approx 0.1089 \)
- For 22: \( (22 - 16.33)^2 \approx (5.67)^2 \approx 32.1489 \)
- For 32: \( (32 - 16.33)^2 \approx (15.67)^2 \approx 245.0889 \)
- For 11: \( (11 - 16.33)^2 \approx ( -5.33)^2 \approx 28.4089 \)
- For 12: \( (12 - 16.33)^2 \approx ( -4.33)^2 \approx 18.7489 \)

### Step 3: Calculate the average of squared differences

Now, we sum the squared differences:

\[
152.1889 + 0.1089 + 32.1489 + 245.0889 + 28.4089 + 18.7489 \approx 476.2344
\]

Now, divide by the number of data points (use \( n \) for population variance or \( n-1 \) for sample variance):

- For population variance:

\[
\text{Variance} = \frac{476.2344}{6} \approx 79.3724
\]

- For sample variance:

\[
\text{Variance} = \frac{476.2344}{5} \approx 95.24688
\]

Thus, the population variance is approximately **79.37**, while the sample variance is approximately **95.25**.