History of Normal Distribution2.52003/05/142007/06/01 11:59:53.581 GMT-5DavidLanelane@rice.eduDavidLanelane@rice.eduElizabethGregoryelizabeth.gregory@gmail.combinomial distributionhistoryMoivrenormal curvenormal distributionThis module introduces the history of the normal curve.In the chapter on probability, we saw that the binomial
distribution could be used to solve problems such as "If a
fair coin is flipped 100 times, what is the probability of
getting 60 or more heads?" The probability of exactly
x heads out of
N flips is computed using the
formula:
PxNxNxx1Nx
where x is the number of heads
(60), N is the number of flips
(100), and is the probability
of a head (0.5). Therefore, to solve this problem, you compute
the probability of 60 heads, then the probability of 61 heads,
62 heads, etc, and add up all these probabilities. Imagine how long
it must have taken to compute binomial probabilities before the
advent of calculators and computers.
Abraham de Moivre, an 18th century statistician and consultant
to gamblers was often called upon to make these lengthy
computations. de Moivre noted that when the number of events
(coin flips) increased, the shape of the binomial distribution
approached a very smooth curve. Binomial distributions for 2, 4,
and 12 flips are shown in .
Examples of binomial distributions. The heights of the blue
bars represent the probabilities.
de Moivre reasoned that if he could find a mathematical
expression for this curve, he would be able to solve problems
such as finding the probability of 60 or more heads out of 100
coin flips much more easily. This is exactly what he did, and
the curve he discovered is now called the normal
curve.
The normal approximation to the binomial distribution for 12
coin flips. The smooth curve is the normal distribution. Note
how well it approximates the binomial probabilities
represented by the heights of the blue lines.
The importance of the normal curve stems primarily from the fact
that the distribution of many natural phenomena are at least
approximately normally distributed. One of the first
applications of the normal distribution was to the analysis of
errors of measurement made in astronomical observations, errors
that occurred because of imperfect instruments and imperfect
observers. Galileo in the 17th century noted that these errors
were symmetric and that small errors occurred more frequently
than large errors. This led to several hypothesized
distributions of errors, but it was not until the early 19th
century that it was discovered that these errors followed a
normal distribution. Independently the mathematicians Adrian in
1808 and Gauss in 1809 developed the formula for the normal
distribution and showed that errors were fit well by this
distribution.
This same distribution had been discovered by Laplace in 1778
when he derived the extremely important central limit
theorem, the topic of a later
section of this chapter. Laplace showed that even if a
distribution is not normally distributed, the means of repeated
samples from the distribution would be very nearly normal, and
that the the larger the sample size, the closer the distribution
would be to a normal distribution. Most statistical procedures
for testing differences between means assume normal
distributions. Because the distribution of means is very close
to normal, these tests work well even if the distribution itself
is only roughly normal.
Biographical Information for:
de
Moivre
Gauss
Laplace