Among all the laws in probability theory, the normal distribution law occurs most often, including more often than the uniform one. 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 a range of random variables, each of which affects in its own way. Normal (or Gaussian) distribution in this case is obtained by adding different distributions. It is thanks to the wide distribution that the normal distribution law got its name.
Whenever we talk about an average, whether it's monthly rainfall, per capita income, or class performance, the normal distribution is usually used to calculate its value. This average value is called the mathematical expectation and corresponds to the maximum on the graph (usually denoted as M). With a correct distribution, the curve is symmetrical about the maximum, but in reality this is not always the case, and thisallowed.
To describe the normal law of distribution of a random variable, it is also necessary to know the standard deviation (denoted σ - sigma). It sets the shape of the curve on the graph. The larger σ, the flatter the curve will be. On the other hand, the smaller σ, the more accurately the average value of the quantity in the sample is determined. Therefore, with large standard deviations, one has 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 shows itself the better, the larger the sample, i.e. the number of objects that participate in the measurements. However, another effect is manifested here: with a large sample, the probability of meeting a certain value of a quantity, including the mean, becomes very small. Values are only grouped around the average. Therefore, it is more correct to say that a random variable will be close to a certain value with such and such a degree of probability.
Determine how high the probability is and the standard deviation helps. In the interval "three sigma", i.e. M +/- 3σ, fits 97.3% of all values in the sample, and about 99% fits into the five sigma interval. These intervals are usually used to determine, when necessary, the maximum and minimum values of the values in the sample. The probability that the value of the quantity will come out offive sigma interval is negligible. In practice, three sigma intervals are usually used.
The normal distribution law can be multidimensional. In this case, it is assumed that an object has several independent parameters expressed in one unit of measurement. For example, the deviation of a bullet from the center of the target vertically and horizontally when 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.
