To calculate the sample variance, we can follow these steps:

Step 1: Calculate the Mean (Average)

The mean is calculated by summing all data values and then dividing by the number of values.

$\text{Mean} = \frac{27 + 28 + 20 + 15 + 30 + 34 + 25 + 28}{8} = \frac{207}{8} = 25.875$

Step 2: Calculate the Squared Differences from the Mean

For each data value, subtract the mean and then square the result:

$(27 - 25.875)^2 = 1.2656$ $(28 - 25.875)^2 = 4.5156$ $(20 - 25.875)^2 = 34.5156$ $(15 - 25.875)^2 = 119.3906$ $(30 - 25.875)^2 = 17.0156$ $(34 - 25.875)^2 = 66.3906$ $(25 - 25.875)^2 = 0.7656$ $(28 - 25.875)^2 = 4.5156$

Step 3: Calculate the Sample Variance

The sample variance is the average of the squared differences from the mean, divided by $n - 1$ (where $n$ is the sample size).

$\text{Sample Variance} = \frac{1.2656 + 4.5156 + 34.5156 + 119.3906 + 17.0156 + 66.3906 + 0.7656 + 4.5156}{7}$

$\text{Sample Variance} = \frac{248.375}{7} = 35.4821$

Thus, the sample variance is approximately 35.48.

Would you like any further details?

---

Here are 5 follow-up questions for deeper exploration:

1. How would you calculate the sample standard deviation based on this variance?
2. What is the difference between population variance and sample variance?
3. How would adding another data point affect the sample variance?
4. Can you explain why we divide by $n - 1$ for sample variance instead of $n$?
5. How do you interpret a high or low variance in a dataset?

Tip: When working with small sample sizes, sample variance provides a better estimate of population variance than just using the formula for population variance.