To solve the problem, let's calculate the sample variance and sample standard deviation for the given dataset $[3, 7, 4, 1, 10, 5]$. Here's the step-by-step solution:

1. Formula for Sample Variance

The sample variance $s^2$ is calculated as: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$ where:

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

2. Calculate the Mean ($\bar{x}$)

$\bar{x} = \frac{\sum x_i}{n} = \frac{3 + 7 + 4 + 1 + 10 + 5}{6} = \frac{30}{6} = 5$

3. Subtract the Mean and Square the Differences

For each data point, calculate $(x_i - \bar{x})^2$:

• $(3 - 5)^2 = (-2)^2 = 4$
• $(7 - 5)^2 = (2)^2 = 4$
• $(4 - 5)^2 = (-1)^2 = 1$
• $(1 - 5)^2 = (-4)^2 = 16$
• $(10 - 5)^2 = (5)^2 = 25$
• $(5 - 5)^2 = (0)^2 = 0$

4. Sum of Squared Differences

$\sum (x_i - \bar{x})^2 = 4 + 4 + 1 + 16 + 25 + 0 = 50$

5. Calculate the Variance

$s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{50}{6 - 1} = \frac{50}{5} = 10$

6. Calculate the Standard Deviation

The sample standard deviation $s$ is the square root of the variance: $s = \sqrt{s^2} = \sqrt{10} \approx 3.2$

Final Results:

• Sample Variance: $10.0$
• Sample Standard Deviation: $3.2$

Let me know if you would like more details or have additional questions!

Here are five related questions to deepen your understanding:

1. How does sample variance differ from population variance?
2. Why do we divide by $n-1$ instead of $n$ when calculating sample variance?
3. Can you find the range of this dataset? How is it different from standard deviation?
4. What would the variance be if all data points were identical?
5. How is standard deviation useful in interpreting real-world data?

Tip: Always double-check calculations involving rounding when solving problems with specific decimal requirements.