The standard deviation of the past five monthly returns for PG Company are 2.75 percent, −0.75 percent, 4.15 percent, 6.29 percent, and 3.84 percent. What is the standard deviation?
A) 2.309 percent
B) 2.581 percent
C) 3.256 percent
D) 3.406 percent

Question

Quizplus · Accepted Answer

To calculate the standard deviation of these monthly returns, we first find the mean (average) of these returns, then calculate the variance by finding the squared differences from the mean, average those squared differences, and finally take the square root of that average to get the standard deviation.1. Calculate the mean:$$ \text{Mean} = \frac{2.75 + (-0.75) + 4.15 + 6.29 + 3.84}{5} = \frac{16.28}{5} = 3.256\% $$2. Calculate each squared difference from the mean, then the variance:$$ \text{Variance} = \frac{(2.75-3.256)^2 + (-0.75-3.256)^2 + (4.15-3.256)^2 + (6.29-3.256)^2 + (3.84-3.256)^2}{5} $$$$ \text{Variance} = \frac{(0.506)^2 + (-4.006)^2 + (0.894)^2 + (3.034)^2 + (0.584)^2}{5} $$$$ \text{Variance} = \frac{0.256 + 16.048 + 0.799 + 9.205 + 0.341}{5} $$$$ \text{Variance} = \frac{26.649}{5} = 5.3298 $$3. Calculate the standard deviation:$$ \text{Standard Deviation} = \sqrt{5.3298} \approx 2.309\% $$It appears there was a mistake in my calculation. Let's correct the process:1. **Mean (Average) Calculation**:$$ \text{Mean} = \frac{2.75 + (-0.75) + 4.15 + 6.29 + 3.84}{5} = 3.256\% $$2. **Variance Calculation**:$$ \text{Variance} = \frac{(2.75-3.256)^2 + (-0.75-3.256)^2 + (4.15-3.256)^2 + (6.29-3.256)^2 + (3.84-3.256)^2}{4} $$ (Note: The denominator is 4, not 5, because for sample variance, we divide by $n-1$, where $n$ is the number of observations.)$$ \text{Variance} = \frac{(-0.506)^2 + (-4.006)^2 + (0.894)^2 + (3.034)^2 + (0.584)^2}{4} $$$$ \text{Variance} \approx \frac{0.256 + 16.048 + 0.799 + 9.205 + 0.341}{4} $$$$ \text{Variance} \approx \frac{26.649}{4} = 6.66225 $$3. **Standard Deviation Calculation**:$$ \text{Standard Deviation} = \sqrt{6.66225} \approx 2.581\% $$The correct answer is B) 2.581 percent, acknowledging the error in the initial explanation.

The Standard Deviation of the Past Five Monthly Returns for PG