2. A portfolio manager tracks the return on investment (ROI) for three different assets as 6%,

8%, and 10%.

- Compute the average ROI.

- Calculate the sample variance, variance, and standard deviation of the ROI.

To solve the problem, we need to compute the following:

1. **Average ROI** (Mean)
2. **Sample Variance**
3. **Population Variance**
4. **Standard Deviation**

Given the ROIs for three assets: 6%, 8%, and 10%.

### 1. Compute the Average ROI (Mean)

The average ROI is calculated as:

\[
\text{Average ROI} = \frac{\text{Sum of all ROIs}}{\text{Number of ROIs}} = \frac{6\% + 8\% + 10\%}{3}
\]

\[
\text{Average ROI} = \frac{24\%}{3} = 8\%
\]

### 2. Calculate the Sample Variance

The sample variance (\(s^2\)) is calculated using the formula:

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

Where:
- \(x_i\) are the individual ROI values,
- \(\bar{x}\) is the average ROI,
- \(n\) is the number of observations.

For our dataTo solve the problem, we need to compute the following:

1. **Average ROI** (Mean)
2. **Sample Variance**
3. **Population Variance**
4. **Standard Deviation**

Given the ROIs for three assets: 6%, 8%, and 10%.

### 1. Compute the Average ROI (Mean)

The average ROI is calculated as:

\[
\text{Average ROI} = \frac{\text{Sum of all ROIs}}{\text{Number of ROIs}} = \frac{6\% + 8\% + 10\%}{3}
\]

\[
\text{Average ROI} = \frac{24\%}{3} = 8\%
\]

### 2. Calculate the Sample Variance

The sample variance (\(s^2\)) is calculated using the formula:

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

Where:
- \(x_i\) are the individual ROI values,
- \(\bar{x}\) is the average ROI,
- \(n\) is the number of observations.

For our 
- \(x_1 = 6\%\), \(x_2 = 8\%\), \(x_3 = 10\%\)
- \(\bar{x} = 8\%\)
- \(n = 3\)

Now, let's calculate each term:

\[
(6\% - 8\%)^2 = (-2\%)^2 = 4\%
\]
\[
(8\% - 8\%)^2 = (0\%)^2 = 0\%
\]
\[
(10\% - 8\%)^2 = (2\%)^2 = 4\%
\]

Now, sum these squared deviations:

\[
\sum_{i=1}^{3} (x_i - \bar{x})^2 = 4\% + 0\% + 4\% = 8\%
\]

Finally, calculate the sample variance:

\[
s^2 = \frac{8\%}{3 - 1} = \frac{8\%}{2} = 4\%
\]

### 3. Calculate the Population Variance

The population variance (\(\sigma^2\)) is calculated using the formula:

\[
\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2
\]

Using the sum of squared deviations:

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

### 4. Calculate the Standard Deviation

The standard deviation is the square root of the variance.

For the sample standard deviation:

\[
s = \sqrt{s^2} = \sqrt{4\%} = 2\%
\]

For the population standard deviation:

\[
\sigma = \sqrt{\sigma^2} = \sqrt{2.67\%} \approx 1.63\%
\]

### Summary of Results:
- **Average ROI**: 8%
- **Sample Variance**: 4%
- **Population Variance**: 2.67%
- **Sample Standard Deviation**: 2%
- **Population Standard Deviation**: 1.63%

Would you like a more detailed explanation of any part of this solution, or do you have any questions?

#### Relative Questions:
1. How does the sample size affect the difference between sample variance and population variance?
2. What would happen to the standard deviation if we added a fourth ROI of 12%?
3. How can you interpret the standard deviation in the context of investment risk?
4. How do you calculate the weighted average ROI if each asset has a different investment amount?
5. What is the significance of using sample variance versus population variance in financial analysis?

#### Tip:
Understanding the relationship between variance and standard deviation is crucial for assessing the risk associated with an investment portfolio. Lower variance and standard deviation indicate more consistent returns, while higher values suggest greater risk.

Learn how to calculate the average ROI, sample variance, population variance, and standard deviation from ROI data. This problem involves statistical concepts such as variance and standard deviation, essential for analyzing investment returns. Suitable for advanced high school students and anyone interested in financial analysis.

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