Among all the laws in probability theory, the normal distribution law is most often encountered, including more often than uniform. Perhaps this phenomenon has a deep fundamental nature. After all, this type of distribution is also observed when several factors participate in the presentation of the range of random variables, each of which affects in its own way. The normal (or Gaussian) distribution in this case is obtained due to the addition of different distributions. It is thanks to the wide distribution that the normal distribution law got its name.
Whenever we talk about an average value, be it the monthly rainfall rate, per capita income or class performance, when calculating its value, as a rule, the normal distribution law is used. This average value is called the mathematical expectation and on the graph corresponds to the maximum (usually denoted as M). With the correct distribution, the curve is symmetrical with respect to the maximum, but in reality this is not always the case, and this is permissible.
To describe the normal distribution law of a random variable, it is also necessary to know the standard deviation (denoted by ฯ - sigma). It sets the shape of the curve on the graph. The greater ฯ, the more gentle the curve will be. On the other hand, the smaller ฯ, the more accurately the average value of the quantity in the sample is determined. Therefore, for large standard deviations, we have to say that the average value lies in a certain range of numbers, and does not correspond to any number.
Like other laws of statistics, the normal law of probability distribution manifests itself the better, the larger the sample, i.e. the number of objects that participate in the measurements. However, one more effect is manifested here: with a large sample, it becomes very unlikely to meet a certain value, including the average. Values โโare only grouped near the average. Therefore, it is more correct to say that a random variable will be close to a certain value with such a certain probability.
Determining how high the probability is, and the standard deviation helps. In the โthree sigmaโ interval, i.e. M +/- 3 * ฯ, fits 97.3% of all values โโin the sample, and about 99% in the โfive sigmaโ interval. These intervals are usually used to determine, when necessary, the maximum and minimum value of the values โโin the sample. The probability that the value of the value goes out of the interval of five sigma is negligible. In practice, an interval of three sigma is usually used.
The normal distribution law can be multidimensional. Moreover, it is assumed that a certain object has several independent parameters expressed in one unit of measurement. For example, the deviation of the bullet from the center of the target vertically and horizontally when shooting will be described by a two-dimensional normal distribution. The graph of such a distribution in the ideal case is similar to the figure of rotation of a plane curve (Gaussian), which was mentioned above.