Questions: The following data are the distances from the workplace (in miles) for the 12,14,20,7,2

Solution Steps

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

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

where \( N \) is the number of data points and \( x_i \) are the individual distances. For our data:

\[ \mu = \frac{12 + 14 + 20 + 7 + 2}{5} = \frac{55}{5} = 11.0 \]

Thus, the mean of the distances is \( 11.0 \).

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

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

Substituting the values, we first find \( (x_i - \mu)^2 \) for each distance:

• \( (12 - 11)^2 = 1 \)
• \( (14 - 11)^2 = 9 \)
• \( (20 - 11)^2 = 81 \)
• \( (7 - 11)^2 = 16 \)
• \( (2 - 11)^2 = 81 \)

Now, summing these values:

\[ \sum (x_i - \mu)^2 = 1 + 9 + 81 + 16 + 81 = 188 \]

Now, we calculate the variance:

\[ \sigma^2 = \frac{188}{5} = 37.6 \]

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{37.6} \approx 6.13 \]

Final Answer