Statistical Inference: A Short Course

Appendix 7.B Assessing Normality Via the Normal probability Plot

How can we determine if a simple random sample taken from some unknown population is normally distributed (or approximately so)? One way to answer this question is to see if the sample data values plot out as a frequency distribution that is bell-shaped and symmetrical. As one might expect, for large data sets, a distinct normal pattern may emerge that would lead one to conclude that the underlying population is indeed approximately normally distributed. However, for small data sets, simple graphics might not work very well; the shape of the population might not be readily discernable from only a small subset of observations taken from it.

A convenient tool for assessing the normality of a data set is the normal probability plot—a graph involving the observed data values and their normal scores in the sample or “expected” Z-scores. (A more formal test of normality appears in Appendix 10.C.) The steps involved in obtaining a normal probability plot are:

1. Arrange the sample data values X_i, i = 1, . . . , n, in an increasing (ordered) sequence.

2. Determine the probability level

(7.B.1)

where j serves to index the position of the data point X_i in the ordered sample data set.

3. Find the expected Z-score or normal score for p_j, using the N (0, 1) area table.

Given that the population variable X is N(μ, σ), we may ...