1. An investor records annual returns of 8%, 12%, and 10% over three years.
- Compute the average return over these years.
- Calculate the sample variance, variance, and standard deviation of the returns.

Solve this using math with detailed explanation

Let's tackle the problem step by step. We need to compute the average return, the sample variance, the variance, and the standard deviation of the returns. 

### 1. Average Return
The average return, often called the arithmetic mean, is calculated by summing the returns and dividing by the number of years.

Given returns:
- Year 1: \( r_1 = 8\% \)
- Year 2: \( r_2 = 12\% \)
- Year 3: \( r_3 = 10\% \)

The formula for the average return is:

\[
\text{Average Return} = \frac{r_1 + r_2 + r_3}{n}
\]

where \( n \) is the number of years.

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

### 2. Sample Variance
Sample variance is a measure of how much the returns vary from the average return. The formula for sample variance \( S^2 \) is:

\[
S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (r_i - \bar{r})^2
\]

where:
- \( \bar{r} \) is the average return (which we calculated as 10%),
- \( r_i \) is each individual return,
- \( n \) is the number of years.

First, let's calculate the deviations from the mean for each year:

- \( r_1 - \bar{r} = 8\% - 10\% = -2\% \)
- \( r_2 - \bar{r} = 12\% - 10\% = 2\% \)
- \( r_3 - \bar{r} = 10\% - 10\% = 0\% \)

Now, square each of these deviations:

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

Sum these squared deviations:

\[
\sum (r_i - \bar{r})^2 = 4\% + 4\% + 0\% = 8\%
\]

Finally, divide by \( n-1 = 3-1 = 2 \):

\[
S^2 = \frac{8\%}{2} = 4\%
\]

### 3. Variance
In this case, variance and sample variance would typically be the same unless there is a reason to distinguish between population variance and sample variance (which is the case here). Since we only have three data points and treat them as a sample, the variance \( S^2 \) we calculated is the sample variance.

However, if we treated these three data points as the entire population, we would calculate the population variance by dividing by \( n \) instead of \( n-1 \):

\[
\sigma^2 = \frac{8\%}{3} = \frac{8\%}{3} \approx 2.67\%
\]

### 4. Standard Deviation
The standard deviation is the square root of the variance. We need to compute both the sample standard deviation and the population standard deviation.

- **Sample Standard Deviation \( S \)**:
\[
S = \sqrt{S^2} = \sqrt{4\%} = 2\%
\]

- **Population Standard Deviation \( \sigma \)**:
\[
\sigma = \sqrt{\sigma^2} = \sqrt{2.67\%} \approx 1.63\%
\]

### Summary of Results
- **Average Return**: \( 10\% \)
- **Sample Variance**: \( 4\% \)
- **Population Variance**: \( 2.67\% \)
- **Sample Standard Deviation**: \( 2\% \)
- **Population Standard Deviation**: \( 1.63\% \)

Would you like any more details or have any further questions?

### Relative Questions:
1. How does sample variance differ from population variance in statistical analysis?
2. What are the implications of high variance in investment returns?
3. How would the average return change if the returns for each year were different?
4. How do standard deviation and variance help in assessing investment risk?
5. Can you calculate the geometric mean return for these investment returns?

### Tip:
When comparing different investments, understanding both the average return and the variability of returns (variance and standard deviation) is crucial to gauge the risk-return profile.

Learn how to calculate the average return, sample variance, population variance, sample standard deviation, and population standard deviation from investment returns of 8%, 12%, and 10% over three years. Understand the statistical measures crucial for assessing investment performance.

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