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.


Your thoughts and comments on this ME-P are appreciated. Feel free to review our top-left column, and top-right sidebar materials, links, URLs and related websites, too. Then, subscribe to the ME-P. It is fast, free and secure.

Speaker: If you need a moderator or speaker for an upcoming event, Dr. David E. Marcinko; MBA – Publisher-in-Chief of the Medical Executive-Post – is available for seminar or speaking engagements.

Book Marcinko: https://medicalexecutivepost.com/dr-david-marcinkos-bookings/

Subscribe: MEDICAL EXECUTIVE POST for curated news, essays, opinions and analysis from the public health, economics, finance, marketing, IT, business and policy management ecosystem


“Insurance & Risk Management Strategies for Doctors” https://tinyurl.com/ydx9kd93

“Fiduciary Financial Planning for Physicians” https://tinyurl.com/y7f5pnox

“Business of Medical Practice 2.0” https://tinyurl.com/yb3x6wr8

Risk Management, Liability Insurance, and Asset Protection Strategies for Doctors and Advisors: Best Practices from Leading Consultants and Certified Medical Planners™8Comprehensive Financial Planning Strategies for Doctors and Advisors: Best Practices from Leading Consultants and Certified Medical Planners™


%d bloggers like this: