What is the Gaussian distribution curve primarily used to represent?

How does the population standard deviation (Sigma) influence the shape of the Gaussian distribution curve?

A Gaussian distribution has a mean (mu) of 50 and a standard deviation (Sigma) of 5. If the mean is increased to 60 and the standard deviation is decreased to 3, how will the new curve compare to the original?

In a quality control process, a manufacturer finds that the diameter of produced bolts follows a Gaussian distribution with a mean of 10 mm and a standard deviation of 0.1 mm. If the acceptable range is 9.8 mm to 10.2 mm, what percentage of bolts are expected to be within the acceptable range?

Given a data set with values: 4, 6, 8, 10, 12, calculate the mean and standard deviation.