The rates of return on Cherry Jalopies, Incorporated, stock over the last five years were 18 percent, 11 percent, −2 percent, 7 percent, and 10 percent. Over the same period, the returns on Straw Construction Company’s stock were 16 percent, 21 percent, −2 percent, 4 percent, and 13 percent.

Calculate the variances and the standard deviations for Cherry and Straw.

To calculate the variances and standard deviations for the returns of Cherry Jalopies, Incorporated, and Straw Construction Company, we need to follow these steps:

### Step 1: Calculate the Average (Mean) Return
For each company, calculate the mean return over the five years.

### Cherry Jalopies, Incorporated
Returns: 18%, 11%, −2%, 7%, 10%

\[
\text{Mean} = \frac{18 + 11 + (-2) + 7 + 10}{5} = \frac{44}{5} = 8.8\%
\]

### Straw Construction Company
Returns: 16%, 21%, −2%, 4%, 13%

\[
\text{Mean} = \frac{16 + 21 + (-2) + 4 + 13}{5} = \frac{52}{5} = 10.4\%
\]

### Step 2: Calculate the Variance
Variance is calculated by taking the average of the squared differences from the Mean.

### Cherry Jalopies, Incorporated

\[
\text{Variance} = \frac{(18 - 8.8)^2 + (11 - 8.8)^2 + (-2 - 8.8)^2 + (7 - 8.8)^2 + (10 - 8.8)^2}{5}
\]

Let's calculate each squared difference:

\[
(18 - 8.8)^2 = 84.64
\]
\[
(11 - 8.8)^2 = 4.84
\]
\[
(-2 - 8.8)^2 = 118.81
\]
\[
(7 - 8.8)^2 = 3.24
\]
\[
(10 - 8.8)^2 = 1.44
\]

Now, sum these squared differences:

\[
\text{Sum of Squared Differences} = 84.64 + 4.84 + 118.81 + 3.24 + 1.44 = 212.97
\]

Finally, divide by the number of observations (5) to get the variance:

\[
\text{Variance} = \frac{212.97}{5} = 42.594
\]

### Straw Construction Company

\[
\text{Variance} = \frac{(16 - 10.4)^2 + (21 - 10.4)^2 + (-2 - 10.4)^2 + (4 - 10.4)^2 + (13 - 10.4)^2}{5}
\]

Let's calculate each squared difference:

\[
(16 - 10.4)^2 = 31.36
\]
\[
(21 - 10.4)^2 = 112.36
\]
\[
(-2 - 10.4)^2 = 153.76
\]
\[
(4 - 10.4)^2 = 40.96
\]
\[
(13 - 10.4)^2 = 6.76
\]

Now, sum these squared differences:

\[
\text{Sum of Squared Differences} = 31.36 + 112.36 + 153.76 + 40.96 + 6.76 = 345.20
\]

Finally, divide by the number of observations (5) to get the variance:

\[
\text{Variance} = \frac{345.20}{5} = 69.04
\]

### Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance.

### Cherry Jalopies, Incorporated

\[
\text{Standard Deviation} = \sqrt{42.594} \approx 6.525\%
\]

### Straw Construction Company

\[
\text{Standard Deviation} = \sqrt{69.04} \approx 8.308\%
\]

### Summary of Results:
- **Cherry Jalopies, Incorporated:**
  - Variance: 42.594
  - Standard Deviation: 6.525%

- **Straw Construction Company:**
  - Variance: 69.04
  - Standard Deviation: 8.308%

Would you like further details or have any questions?

### Relative Questions:
1. How does the standard deviation relate to investment risk?
2. What does a higher variance indicate about a stock's returns?
3. How would adding more years of data affect the calculated variance and standard deviation?
4. Why is it important to calculate the standard deviation in finance?
5. Can we compare the risk of two stocks only by looking at their standard deviations?

### Tip:
In finance, the standard deviation is a key measure of risk, as it quantifies the volatility of returns.

The rates of return on Cherry Jalopies, Incorporated, stock over the last five years were 18 percent, 11 percent, −2 percent, 7 percent, and 10 percent. Over the same period, the returns on Straw Construction Company’s stock were 16 percent, 21 percent, −2 percent, 4 percent, and 13 percent. Calculate the variances and the standard deviations for Cherry and Straw.

Calculate the variance and standard deviation for Cherry Jalopies, Incorporated, and Straw Construction Company stock returns over the past five years. Understand the statistical formulas used to assess stock volatility and risk, including key steps in calculating the mean, variance, and standard deviation.

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