Chebyshev's theorem is any of several theorems proven by Russian mathematician Pafnuty Chebyshev. Bertrand's postulate, that for every n there is a prime between n and 2 n. has a limit at infinity, then the limit is 1 (where π is the prime-counting function). This result has been superseded by the prime number theorem.

In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) provides an upper bound on the probability of deviation of a random variable (with finite variance) from its mean.

设X是一个随机变数取区间（0，∞）上的值，F (x)是它的分布函数，设Xα（α >0）的数学期望M (Xα )存在，a>0，则不等式成立。 这叫做切比雪夫定理，或者切比雪夫不等式。 [1] 任意一个数据集中，位于其平均数±m个标准差范围内的比例（或部分）总是至少为1－1/m2，其中m为大于1的任意 正数。 对于m=2，m=3和m=5有如下结果： 所有数据中，至少有3/4（或75%）的数据位于 平均数 2个标准差范围内。 所有数据中，至少有8/9（或88.9%）的数据位于平均数3个标准差范 …

2019年12月3日 · 切比雪夫定理（Chebyshev's theorem）：适用于任何数据集，而不论数据的分布情况如何。 与平均数的距离在z个标准差之内的数值所占的比例至少为(1-1/z2)，其中z是大于1的任意实数。

Chebyshev’s theorem is used to find the minimum proportion of numerical data that occur within a certain number of standard deviations from the mean. In normally-distributed numerical data: 68% of the data are within 1 standard deviation from the mean. 95% of the data are within 2 standard deviations from the mean.

2021年4月19日 · Chebyshev’s Theorem estimates the minimum proportion of observations that fall within a specified number of standard deviations from the mean. This theorem applies to a broad range of probability distributions. Chebyshev’s Theorem is …

2018年10月18日 · 切比雪夫定理：设 \ (\int x^m (a+bx^n)^p\mathrm {d}x\) 为一个二项微分式积分，其中 \ (p,m,n\) 均为有理数，则其可表示为初等函数的充要条件是 \ (p,\frac {m+1} {n},\frac {m+1} {n}+p\) 中至少一个为 整数。 这篇文章来自前苏联盖·伊·德林费尔特所著的《普通数学分析教程补篇》第六章，原书的中译本出版于1960年，后来一直未尝再版。 （网上可下载到电子书） 积分的初等可积性是分析内容中一个比较古典的课题，不过随着一些现代元素的引入（例如微分代数之 …

2020年5月26日 · This statistics video provides a basic introduction into Chebyshev's theorem which states that the minimum percentage of distribution values that lie within k standard deviations of the mean is...

This relationship is described by Chebyshev's Theorem: For every population of $n$ values and real value $k \gt 1$, the proportion of values within $k$ standard deviations of the mean is at least $$1 - \frac{1}{k^2}$$

2023年6月1日 · Chebyshev’s Theorem is also known as Chebyshev’s inequality, and it’s a fundamental concept in probability theory and statistics. It provides a way to estimate the proportion of data that falls within a certain range around the mean, regardless of the shape of the probability distribution.