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 bellshaped 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 nonzero
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 zscores 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.
