When was the normal distribution invented

The introductory section defines what it means for a distribution to be normal and presents some important properties of normal distributions. The interesting history of the discovery of the normal distribution is described in the second section. Methods for calculating probabilities based on the normal distribution are described in Areas of Normal Distributions. The Varieties of Normal Distribution Demo allows you to enter values for the mean and standard deviation of a normal distribution and see a graph of the resulting distribution.

A frequently used normal distribution is called the Standard Normal distribution and is described in the section with that name. The binomial distribution can be approximated by a normal distribution. The section Normal Approximation to the Binomial shows this approximation. The bell shaped curve was discovered by Carl Friedrich Gauss , whom many mathematical historians consider to have been the greatest mathematician of all time.

Gauss was working as the royal surveyor for the king of Prussia. Surveyors maesure distances. For instance, a survey crew may measure a distance to be To tell if that is the correct distance, they would check their work by measuring it again. The second time, they might get an answer of So is it They would have to measure it again.

The next time, they might get an answer of Which one is it? Each time they meausre they have gotten a different answer.

Gauss would have them measure it about 15 times, and they would get. At this point we can see that the true value would be These data are approximately normally distributed.

We will get a normal distribution if there is a true answer for the distance, but as we shoot for this distance, since, to err is human, we are likely to miss the target. As we get farther from the true value, the chances of landing there gets less and less. We can express this by saying that the rate at which the frequencies fall off is proportional to the distance from the true value.

If this were the end of the story, the histogram would be parabolically shaped, and as you got farther and farther from the true value, the frequencies would eventually become negative, and we can't have negative frequencies. We can get the frequencies to level off as they asymptotically approach zero by further requiring that the rate at which the frequencies fall off is also proportional to the frequencies themselves. Then as the frequencies approach zero, slope of the histogram also approaches zero, and the curve levels off as we get into the tail end of the curve.

This gives us the following. Definition : Data are said to be normally distributed if the rate at which the frequencies fall off is proportional to the distance of the score from the mean, and to the frequencies themselves. In practice, the value of the bell shaped curve is that we can find the proportion of the scores which lie over a certain interval.

In a probability distribution, this is the area under the curve over the interval: a typical calculus problem. Independently, the mathematicians Adrain in and Gauss in 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 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 normally distributed, and that the larger the sample size, the closer the distribution of means 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 original distribution is only roughly normal.

He noted that characteristics such as height, weight, and strength were normally distributed. Figure 1. Examples of binomial distributions.