Box plots and the standard normal distribution. What relationship exists between the standard normal distribution and the box-plot methodology (Section 2.7) for describing distributions of data by means of quartiles?

The answer depends on the true underlying probability distribution of the data. Assume for the remainder of this exercise that the distribution is normal.

a. Calculate the values zL and zU of the standard normal random variable z that correspond, respectively, to the hinges of the box plot (i.e., the lower and upper quartiles QL and QU) of the probability distribution.

b. Calculate the z values that correspond to the inner fences of the box plot for a normal probability distribution.

c. Calculate the z values that correspond to the outer fences of the box plot for a normal probability distribution.

d. What is the probability that an observation lies beyond the inner fences of a normal probability distribution?

The outer fences?

e. Can you now better understand why the inner and outer fences of a box plot are used to detect outliers in a distribution? Explain.