To calculate the sample variance, we follow these steps:

Step 1: Find the mean (average) of the data.

The data points are: $12, 10, 13, 3, 4, 6$.

$\text{Mean} = \frac{12 + 10 + 13 + 3 + 4 + 6}{6} = \frac{48}{6} = 8$

Step 2: Calculate the squared differences from the mean.

For each data point $x_i$, we compute $(x_i - \text{mean})^2$:

$(12 - 8)^2 = 16, \quad (10 - 8)^2 = 4, \quad (13 - 8)^2 = 25$ $(3 - 8)^2 = 25, \quad (4 - 8)^2 = 16, \quad (6 - 8)^2 = 4$

Step 3: Find the sum of the squared differences.

$16 + 4 + 25 + 25 + 16 + 4 = 90$

Step 4: Divide the sum by $n - 1$ (where $n = 6$).

$\text{Sample Variance} = \frac{90}{6 - 1} = \frac{90}{5} = 18$

Thus, the sample variance is 18.0.

Would you like further details or have any questions?

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Here are five questions for further exploration:

1. How would the sample variance change if an additional data point is added?
2. What is the standard deviation of the given data?
3. How is sample variance different from population variance?
4. Can you calculate the mean absolute deviation of the data?
5. How do outliers affect the sample variance?

Tip: Sample variance is sensitive to extreme values (outliers), so be cautious when interpreting it for datasets with significant outliers.