The normal distribution law, or the Gaussian distribution

Among all the laws in probability theory, the normal distribution law is most often encountered, including more often than the uniform distribution law. Perhaps, this phenomenon has a deep fundamental nature. After all, this type of distribution is also observed when several factors participate in the representation 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 due to the wide distribution of the normal distribution law and got its name.

Whenever we talk about some average value, whether it's the monthly rainfall norm, per capita income or grade 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). If the distribution is correct, the curve is symmetrical with respect to the maximum, but in reality this does not always happen, and this is permissible.

To describe the normal distribution law of a random variable, it is also necessary to know the root-mean-square deviation (denoted by σ-sigma). It sets the shape of the curve on the graph. The larger the σ, the flatter the curve will be. On the other hand, the smaller σ, the more accurately the average value of the value in the sample is determined. Therefore, for large root-mean-square 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 statistics laws, the normal law of probability distribution shows itself better the larger the sample, i.e. Number of objects that participate in the measurements. However, another effect is manifested here: with a large sample, it is very unlikely to meet a certain value of the value, including the mean. Values are only grouped near the middle. Therefore, it is more correct to say that a random variable will be close to a certain value with such a share of probability.

Determine how high the probability is, and the root-mean-square deviation helps. In the interval "three sigma", i.e. M +/- 3 * σ, 97.3% of all values fit into the sample, and in the interval "five sigma" - about 99%. These intervals are usually used to determine, when necessary, the maximum and minimum value of the values in the sample. The probability that the value of the value will leave the interval of five sigma is negligible. In practice, usually use an interval of three sigma.

The normal distribution law can be multidimensional. It is assumed that an 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 during firing 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 flat curve (gaussian), which was mentioned above.