What is the Normal Distribution of Mathematics
We now come to the most important distribution in statistics—the normal distribution. The formula for this distribution was first published by Abraham de Moivre (1667-1754) in 1733. Other mathematicians linked with the history of the normal distribution are Pierre Simon, Marquis de Laplace (1667-1754) and Carl Friedrich Gauss (1777-1855), in whose honor it is sometimes called the Gaussian distribution. The distribution has two parameters: JJL, the mean, and CT, the standard deviation. The graph of the normal distribution is the familiar bell-shaped curve.Characteristics of the Normal Distribution
The following are some important characteristics of the normal distribution:
It is symmetrical about its mean,u,. The curve on either side of u, is a mirror image of the other side.
The mean, the median, and the mode are all equal.
The total area under the curve above the x axis is equal to 1.Because of the symmetry of the normal curve, 50% of the area is to the right of a perpendicular line erected at the mean, and 50% is to the left.
Suppose that we erect vertical lines one standard deviation from the mean in each direction. The area enclosed by these lines, the x axis, and the curve will be equal to approximately 68% of the total area. If we erect these lateral boundaries two standard deviations from the mean in each direction, they will enclose approximately 95% of the area. Perpendiculars erected three standard deviations on either side of the mean will enclose approximately 99. 7% of the total area. If we know that a random variable is normally distributed, we can make more powerful probability statements than we could fuse Chebyshev’s theorem.
The normal distribution is completely determined by its parameters JJL and CT. That is, each different value of JJL or a specifies a different normal distribution.
The Standard Normal Distribution
The normal distribution is really a family of distributions in which one member is distinguished from another on the basis of the values of (x and a. In other words, as already indicated, there is a different normal distribution for each different value of either u, or a.
The most important member of this family of distributions is the standard normal distribution, which has a mean of 0 and a standard deviation of 1. We usually use the letter z for the random variable that results from the standard normal distribution. The probability that z lies between any two points on the z axis is determined by the area bounded by perpendiculars erected at each of these points, the curve, and the horizontal axis. We find areas under the curve of a continuous distribution by integrating the function between two values of the variable. There are tables that give the results of integrations in which we might be interested. The table of the standard normal distribution may be presented in many different forms
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