To solve the problem, let's break it down into two parts: calculating the average ROI and then computing the variance, sample variance, and standard deviation.

Part 1: Compute the Average ROI

The average ROI is calculated by taking the sum of all the ROIs and dividing by the number of assets.

Given ROIs: 6%, 8%, 10%

$\text{Average ROI} = \frac{6\% + 8\% + 10\%}{3} = \frac{24\%}{3} = 8\%$

Part 2: Calculate the Variance, Sample Variance, and Standard Deviation

Let's first calculate the variance, then the sample variance, and finally the standard deviation.

1. Variance

Variance ($\sigma^2$) measures the dispersion of the ROIs from the mean.

1. Calculate the deviations from the mean:

   • $6\% - 8\% = -2\%$
   • $8\% - 8\% = 0\%$
   • $10\% - 8\% = 2\%$

2. Square the deviations:

   • $(-2\%)^2 = 4\%$
   • $(0\%)^2 = 0\%$
   • $(2\%)^2 = 4\%$

3. Calculate the average of the squared deviations: $\text{Variance} = \frac{4\% + 0\% + 4\%}{3} = \frac{8\%}{3} \approx 2.67\%$

2. Sample Variance

Sample variance ($s^2$) is calculated similarly to variance, but we divide by $n-1$ instead of $n$, where $n$ is the number of observations.

$\text{Sample Variance} = \frac{4\% + 0\% + 4\%}{2} = \frac{8\%}{2} = 4\%$

3. Standard Deviation

Standard deviation ($\sigma$) is the square root of the variance.

$\text{Standard Deviation} = \sqrt{2.67\%} \approx 1.63\%$

For the sample standard deviation, take the square root of the sample variance:

$\text{Sample Standard Deviation} = \sqrt{4\%} = 2\%$

Summary of Results

• Average ROI: 8%
• Variance: 2.67%
• Sample Variance: 4%
• Standard Deviation: 1.63%
• Sample Standard Deviation: 2%

Would you like any further details or have any questions?

Here are five related questions you might find helpful:

1. How does the standard deviation reflect the risk in an investment portfolio?
2. What is the difference between variance and sample variance?
3. How would the ROI statistics change if another asset with a different ROI was added?
4. Why is the sample variance larger than the population variance?
5. How can the standard deviation be used to assess the stability of asset returns?

Tip: When calculating sample variance, always remember to divide by $n-1$ to account for the smaller sample size, providing an unbiased estimate of the population variance.