We can find such a formula because our definition gives us the following differential equation. However, in order to use calculus to find these areas, we need a formula for the curve. In a probability distribution, this is the area under the curve over the interval: a typical calculus problem. 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. This gives us the followingĭefinition: 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. 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. 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. 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 express this by saying that the rate at which the frequencies fall off is proportional to the distance from the true value. As we get farther from the true value, the chances of landing there gets less and less. we are more likely to land on or near the target. 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. These data are approximately normally distributed. If we make up a histogram for this data, we get,Īt this point we can see that the true value would be 135.675m.

Gauss would have them measure it about 15 times, and they would get

Which one is it? Each time they meausre they have gotten a different answer. The next time, they might get an answer of 135.675m. or 135.677m.? They would have to measure it again. The second time, they might get an answer of 135.677m. To tell if that is the correct distance, they would check their work by measuring it again. For instance, a survey crew may measure a distance to be 135.674m. Gauss was working as the royal surveyor for the king of Prussia. The bell shaped curve was discovered by Carl Friedrich Gauss (1777-1855), whom many mathematical historians consider to have been the greatest mathematician of all time. In practice, one can tell by looking at a histogram if the data are normally distributed. Wilson PhD.ĭata are said to be normally distributed if their frequency histogram is apporximated by a bell shaped curve.

The Normal Distribution A Derivation of the Normal Distribution Robert S.

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