Let's tackle the problem step by step by calculating the average return, sample variance, variance, and standard deviation.

Part 1: Compute the Average Return

The average return is found by taking the sum of all the returns and dividing by the number of years.

Given annual returns: 8%, 12%, 10%

$\text{Average Return} = \frac{8\% + 12\% + 10\%}{3} = \frac{30\%}{3} = 10\%$

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

Next, we calculate the variance, sample variance, and standard deviation.

1. Variance

Variance ($\sigma^2$) is the average of the squared deviations from the mean.

1. Calculate the deviations from the mean:

   • $8\% - 10\% = -2\%$
   • $12\% - 10\% = 2\%$
   • $10\% - 10\% = 0\%$

2. Square the deviations:

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

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

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.

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

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 Return: 10%
• 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 sample variance differ if more years of returns were recorded?
2. What does the standard deviation tell us about the volatility of the investment returns?
3. How can these statistics help in comparing different investment opportunities?
4. What impact does an outlier in the data set have on the variance and standard deviation?
5. Why is it important to distinguish between sample and population variance?

Tip: When dealing with financial data, always check whether you're working with a sample or a full population, as this will affect whether you use variance or sample variance in your calculations.