/kloc-in the 9th century, the Russian mathematician Chebyshev studied statistical laws and demonstrated and expressed an inequality with standard deviation, which is of universal significance and is called Chebyshev theorem. Its main idea is:
In any data set, the proportion (or part) within m standard deviations of its average is always at least 1- 1/m2, where m is any positive number greater than 1. For m=2, m=3 and m=5, the following results are obtained:
In all data, at least 3/4 (or 75%) is within 2 standard deviations of the average.
In all data, at least 8/9 (or 88.9%) is within 3 standard deviations of the average.
In all data, at least 24/25 (or 96%) is within 5 standard deviations of the average.
Chebyshev inequality, which is suitable for almost infinite types of probability distribution, works under looser assumptions than normal.
Extended data:
Chebyshev (182 1~ 1894), formerly known as пант in Russian? тий Льво? виччебышёв, Russian mathematician and mechanic. 1821was born in okatovo, kaluga province on may 26th, and died in Petersburg on February 8th.
He published more than 70 scientific papers in his life, covering number theory, probability theory, function approximation theory, integral calculus and so on. He proved beltran formula, prime number distribution theorem of natural sequence, general formula of law of large numbers and central limit theorem. He not only attaches importance to pure mathematics, but also attaches great importance to the application of mathematics.
Regarding the great significance of the methodological change introduced by Chebyshev in probability theory, Andrey Kolmogorov, a famous Soviet mathematician, wrote in The Development of Russian Probability Science (рольсусскойн).
"From the perspective of methodology, the main significance of the fundamental change brought by Chebyshev is not that he is the first mathematician who insists on absolute accuracy in limit theory (the proofs of A. de Moivre, P-S. Laplace and Poisson are not in harmony with the background of formal logic, and it is also different from Jakob Bernoulli's proof of his limit theorem with detailed arithmetic accuracy).
The main significance of Chebyshev's work is that he is always eager to accurately estimate the possible deviation from the limit law in any experiment and express it with effective inequalities. In addition, Chebyshev was the first person to clearly foresee the value of concepts such as' random variable' and its' expected (average) value' and apply them.
Baidu encyclopedia-Chebyshev theorem