Questions: The following data are the distances from the workplace (in miles) for the 5 employees of a small business 4, 4, 19, 4, 19. Assuming that these distances constitute an entire population, find the standard deviation of the population. Round your answer to two decimal places.

Solution Steps

To find the mean \( \mu \) of the distances, we use the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} \]

Given the distances \( x_1 = 4, x_2 = 4, x_3 = 19, x_4 = 4, x_5 = 19 \), we calculate:

\[ \sum x_i = 4 + 4 + 19 + 4 + 19 = 50 \]

The number of employees \( N = 5 \). Thus, the mean is:

\[ \mu = \frac{50}{5} = 10.0 \]

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

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \]

First, we find \( (x_i - \mu)^2 \) for each distance:

• For \( x_1 = 4 \): \( (4 - 10)^2 = 36 \)
• For \( x_2 = 4 \): \( (4 - 10)^2 = 36 \)
• For \( x_3 = 19 \): \( (19 - 10)^2 = 81 \)
• For \( x_4 = 4 \): \( (4 - 10)^2 = 36 \)
• For \( x_5 = 19 \): \( (19 - 10)^2 = 81 \)

Now, summing these values:

\[ \sum (x_i - \mu)^2 = 36 + 36 + 81 + 36 + 81 = 270 \]

Thus, the variance is:

\[ \sigma^2 = \frac{270}{5} = 54.0 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{54.0} \approx 7.35 \]

Final Answer