The Normal Distribution
Carl Friedrich Gauss, a genius mathematician among the genius mathematicians, discovered the Normal distribution, which is sometimes called the Gauss distribution, but is familiar to most people as the bell-shaped curve. The distribution is perfectly symmetrical, and reaches from minus infinity to plus infinity, though the area under the curve beyond four standard deviations from the middle is negligible. Because the Normal curve is a continuous distribution, calculus is needed to compute probabilities. However, there tables and computer programs are readily available to allow ordinary people to use this distribution.
Although the distribution of many things may be approximately normally distributed, there is no reason to assume that everyday things are normally distributed. The importance of this distribution is not that many things are normally distributed. Rather its importance comes from the magic of the Central Limit Theorem, discussed in the next section.
Generations of students have had to find probabilities using a Normal table, which shows the probabilities for a Normal distribution with a mean of zero and a standard deviation of 1 or N(0,1). The key to solving these problems is to use a bit of thought mixed with a bit of algebra and geometry. Drawing a picture is often helpful. One decides what area (because area is the same as probability in the table) one wants to find, and then figures out how to use the information one has to find that area. For example, in the picture below, if one wants to find area Y but can only find areas X and X+Y in the table, one can find area Y by subtracting X from X+Y.
Many times, however, one is not working with a N(0,1) distribution. One starts with a distribution with a non-zero mean and a standard deviation different from one. In order to solve these problems, one must convert the problem to a N(0,1) problem by finding z-scores for everything, which converts the problem into a N(0,1) problem.
Normal curve problems come in two forms. One can begin with observations and compute probabilities. Or one can begin with probabilities and work backward to compute observations.
Many students find using the tables a chore, in part because they want to memorize rather than solve problems. Dealing with this bit of the statistics course does not involve statistics. If is about using some reasoning with a bit of basic algebra and some elementary geometry.