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In the chapter on probability, we saw that the binomial distribution could be used to solve problems such as "If afair 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: $$P(x)=\frac{N!}{x!(N-x)!}\pi ^{x}(1-\pi )^{(N-x)}$$ where $x$ is the number of heads (60), $N$ is the number of flips (100), and $\pi $ is the probability of a head (0.5). Therefore, to solve this problem, you computethe probability of 60 heads, then the probability of 61 heads, 62 heads, etc, and add up all these probabilities. Imagine how longit 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 lengthycomputations. de Moivre noted that when the number of events (coin flips) increased, the shape of the binomial distributionapproached a very smooth curve. Binomial distributions for 2, 4, and 12 flips are shown in .
de Moivre reasoned that if he could find a mathematical expression for this curve, he would be able to solve problemssuch as finding the probability of 60 or more heads out of 100 coin flips much more easily. This is exactly what he did, andthe curve he discovered is now called the normal curve .The importance of the normal curve stems primarily from the fact that the distribution of many natural phenomena are at leastapproximately normally distributed. One of the first applications of the normal distribution was to the analysis oferrors of measurement made in astronomical observations, errors that occurred because of imperfect instruments and imperfectobservers. Galileo in the 17th century noted that these errors were symmetric and that small errors occurred more frequentlythan large errors. This led to several hypothesized distributions of errors, but it was not until the early 19thcentury that it was discovered that these errors followed a normal distribution. Independently the mathematicians Adrian in1808 and Gauss in 1809 developed the formula for the normal distribution and showed that errors were fit well by thisdistribution.
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 repeatedsamples from the distribution would be very nearly normal, and that the the larger the sample size, the closer the distributionwould be to a normal distribution. Most statistical procedures for testing differences between means assume normaldistributions. Because the distribution of means is very close to normal, these tests work well even if the distribution itselfis only roughly normal.
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