To calculate the sample variance, we will use the following formula:

$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$

Where:

• $s^2$ is the sample variance,
• $n$ is the number of data points,
• $x_i$ are the data points,
• $\bar{x}$ is the sample mean.

Step-by-step process:

1. Find the sample mean $\bar{x}$:

$\bar{x} = \frac{\sum x_i}{n} = \frac{9 + (-4) + 11 + 11 + 9 + 9 + 11}{7}$

$\bar{x} = \frac{56}{7} = 8$

2. Subtract the mean from each data point and square the result:

For each data point $x_i$:

$(9 - 8)^2 = 1$ $(-4 - 8)^2 = 144$ $(11 - 8)^2 = 9$ $(11 - 8)^2 = 9$ $(9 - 8)^2 = 1$ $(9 - 8)^2 = 1$ $(11 - 8)^2 = 9$

3. Sum of the squared differences:

$1 + 144 + 9 + 9 + 1 + 1 + 9 = 174$

4. Divide by $n-1$ (degrees of freedom):

Since $n = 7$, we divide by $n-1 = 6$:

$s^2 = \frac{174}{6} = 29$

Final answer:

The sample variance is $\boxed{29.0}$.

Would you like more details, or do you have any questions?

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Here are five related questions to explore:

1. How does the sample variance differ from population variance?
2. What is the standard deviation based on this sample variance?
3. How would the variance change if we added another data point?
4. Can we calculate the variance of a population with fewer than two data points?
5. How does variance help in understanding data spread?

Tip: Always remember to use $n-1$ for the sample variance to account for degrees of freedom.