CHAPTER 3 Understanding the Normal Distribution and the t- Distribution If you have ever watched someone sift flour in a kitchen or dig a hole and pile the dirt in the same place outside, you have seen a normal distribu- tion, albeit somewhat imperfect. It is bell shaped, symmetric, peaked in the center with tails that trail off rapidly the greater the distance from the center. While imperfect versions of the normal distribution are easily seen in ordinary life, its discovery is credited to the great German mathemati- cian Johann Carl Friedrich Gauss (1777­1855), who documented that errors of routine measurements often follow a normal distribution. Some normal distributions can be huge, as seen in a pile of hulls at the end of a dump chute from an almond hulling plant, for example, and others quite tiny in comparison. Normal distributions occur because the greatest numbers of elements in the "pile" fall straight down below the location of the end of the chute or the bottom of the sifter or the tip of the shovel. Some elements tumble off center, down the growing sides of the "pile." Unless otherwise constrained, occasionally an element will tumble rela- tively far away from the center. Because the height, , and breadth, , in different distributions can differ over many magnitudes, the standard normal distribution is introduced to standardize their discussion. The Standard Normal Distribution All normal distributions share the same shape, differing only by the location of the center and the degree of spread. The standard normal dis- tribution is the normal distribution that has a mean of 0 and a standard deviation of 1. The axis along the bottom of the distribution represents