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Chebyshev’s Inequality: What is it used for?

Statistics Definitions > Chebyshev’s Inequality

Chebyshev’s theorem refers to several theorems, all proven by Russian mathematician Pafnuty Chebyshev. They include: Chebyshev’s inequality, Bertrand’s postulate, Chebyshev’s sum inequality and Chebyshev’s equioscillation theorem.
Chebyshev’s inequality is the theorem most often used in stats. It states that no more than 1/k2 of a distribution’s values are more than “k” standard deviations away from the mean. With a normal distribution, standard deviations tell you how much of that distribution’s data are within k standard deviations from the mean. If you have a distribution that isn’t normal, you can use Chebyshev’s to help you find out what percentage of the data is clustered around the mean.

Chebyshev’s Inequality relates to the distribution of numbers in a set. The formula was originally developed by Chebyschev’s friend, Irénée-Jules Bienaymé. In layman’s terms, the formula helps you figure out the number of values that are inside and outside the standard deviation. The standard deviation tells you how far away values are from the average of the set. Roughly two-thirds of the values should fall within one standard deviation either side of mean in a normal distribution. In statistics, it’s often referred to as Chebyshev’s Theorem (as opposed to Chebyshev’s Inequality). This can be a little confusing, but it’s just semantics; if you’re asked to calculate Chebyshev’s Inequality in Statistics, watch this video for the steps.

See: Chebyshev’s Theorem, How to Calculate it

Chebyshev’s Inequality Definition

Chebyshev's Inequality

Pafnuty Chebyshev

Chebyschev’s Inequality formula is able to prove (with little information given on your part) the probability of outliers existing at a certain interval. Given X is a random variable, A stands for the mean of the set, K is the number of standard deviations, and Y is the value of the standard deviation, the formula reads as follows: Pr(|X-A|=>KY)<=1/K2, The absolute value of the difference of X minus A is greater than or equal to the K times Y has the probability of less than or equal to one divided by K squared. You can learn how to calculate Chebyshev’s here.

Chebyshev’s Inequality Uses

The formula was used with calculus to develop the weak version of the law of large numbers. This law states that as a sample set increases in size, the closer it should be to its theoretical mean. A simple example is that when rolling a six-sided die, the probable average is 3.5. A sample size of 5 rolls may result in drastically different results. Roll the die 20 times; The average should begin approaching 3.5. As you add more and more rolls, the average should continue to near 3.5 until reaching it. Or, it becomes so close that they are pretty much equal.

Another application is in finding the difference between the mean and median of a set of numbers. Using a one-sided version of Chebyshev’s Inequality theorem, also known as Cantelli’s theorem, you can prove the absolute value of the difference between the median and the mean will always be less than or equal to the standard deviation. This is handy in determining if a median you derived is plausible.

Other Theorems

Bertrand’s postulate
Bertand’s Postulate is used in number theory. It has very few applications to stats and you probably won’t come across it in an elementary stats course. According to the University of Tennessee, it states that if n is an integer greater than 3, there is at least one prime between n and 2n-2. It can also be stated as “If n is a positive integer, then there is a prime p with n < p < 2n." Chebyshev’s sum inequality
Used in calculus. It states that:
Chebyshev's Inequality 2
Chebyshev's Inequality 3

You won’t encounter this theorem in an elementary stats course. However, if you take advanced stats courses that includes calculus, you might use this particular version.

Chebyshev’s equioscillation theorem
The Chebyshev equioscillation theorem shows the pattern of a continuous function on a closed interval. You won’t come across this theory in regular stats courses; It is used in numerical analysis courses at the graduate level and involves a somewhat complicated proof.


If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Chebyshev’s Inequality: What is it used for? was last modified: March 17th, 2018 by Stephanie

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