Understanding Chebyshev’s Theorem?

What it – How it works

By Dr. David Edward Marcinko MBA CMP™


The rule is often called Chebyshev’s theorem, about the range of standard deviations around the mean, in statistics www.HealthDictionarySeries.org

The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. But first, let us review the Empirical Rule.

Empirical Rule

In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying


Central Limit Theorem



What is a Galton Board?


Chebyshev’s Theorem

The Russian mathematician P.L. Chebyshev (1821-1894) proved a theorem which is valid for any distribution of data:

For any number k greater than 1, at least 1-1/k2 of the data will fall within k standard deviations of the mean. This theorem produces a few useful rules:

  • no information can be obtained on the fraction of values falling within 1 standard deviation of the mean
  • at least 75% will fall within 2 standard deviations
  • at least 88.8 % will fall within 3 standard deviations

The following figure is a graphical presentation of these rules.


Chebyshev’s Theorem: The proportion of any distribution that lies within k standard deviations of the mean is at least 1 – (1/k2), where k is any positive number larger than 1. This theorem applies to all distributions of data.


OK; so basically Chebyshev’s theorem states that 75% of your data will lie within 2 standard deviations of the mean and that 89% of your data will lie within 3 standard deviations of the mean.

And, I believe that this theorem is much more precise than the Empirical Rule, which assumes normality and can be off.


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One Response

  1. It’s the Context

    This was so cool and the context was awesome; very helpful.




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