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Chebyshev’s Inequality

A probability theory that guarantees only a definite fraction of values will be found within a specific distance from the mean of a distribution

What is Chebyshev’s Inequality?

Chebyshev’s inequality is a probability theory that guarantees that within a specified range or distance from the mean, for a large range of probability distributions, no more than a specific fraction of values will be present. In other words, only a definite fraction of values will be found within a specific distance from the mean of a distribution.

The formula for the fraction for which no more than a certain number of values can exceed is 1/K2; in other words, 1/K2 of a distribution’s values can be more than or equal to K standard deviations away from the mean of the distribution. Further, it also holds that 1–(1/K2) of a distribution’s values must be within, but not including, K standard deviations away from the mean of the distribution.

 

Summary

  • Chebyshev’s inequality is a probability theory that guarantees only a definite fraction of values will be found within a specific distance from the mean of a distribution.
  • The fraction for which no more than a certain number of values can exceed is represented by 1/K2.
  • Chebyshev’s inequality can be applied to a wide range of distributions so long as the distribution includes a defined mean and variance. It is similar to the 65-95-99.7 rule in practice.

 

Understanding Chebyshev’s Inequality

Chebyshev’s inequality is similar to the 68-95-99.7 rule; however, the latter rule only applies to normal distributions. Chebyshev’s inequality is broader; it can be applied to any distribution so long as the distribution includes a defined variance and mean.

Chebyshev’s inequality states that within two standard deviations away from the mean contains 75% of the values, and within three standard deviations away from the mean contains 88.9% of the values. It holds for a wide range of probability distributions, not only the normal distribution.

However, when applied to the normal distribution, Chebyshev’s inequality is less precise than the 65-95-99.7 rule; yet, it is important to keep in mind that the theory applies to a far broader range of distributions. It should be noted that standard deviations equal to or less than one are not valid for Chebyshev’s inequality formula.

 

Chebyshev’s Inequality History

Chebyshev’s inequality was proven by Pafnuty Chebyshev, a Russian mathematician, in 1867. It was stated earlier by French statistician Irénée-Jules Bienaymé in 1853; however, there was no proof for the theory made with the statement. After Pafnuty Chebyshev proved Chebyshev’s inequality, one of his students, Andrey Markov, provided another proof for the theory in 1884.

 

Chebyshev’s Inequality Statement

Let X be a random variable with a finite mean denoted as µ and a finite non-zero variance, which is denoted as σ2, for any real number, K>0.

 

Chebyshev’s Inequality

 

Practical Example

Assume that an asset is picked from a population of assets at random. The average return of the population of assets is 12%, and the standard deviation of the population of assets is 5%. To calculate the probability that an asset picked at random from this population, which has a return less than 4% or greater than 20%, Chebyshev’s inequality can be applied.

Since there is a limited amount of information and only the mean and standard deviation of a distribution is given, the exact probability of this scenario cannot be determined; thus, Chebyshev’s inequality is applied. Below is the application of the theory:

 

|X – µ| ≥ K

 

P(|X – µ|≥K) ≤ (σ2/ K2) = (5%2/8%2)
P(|X – µ|≥K) ≤ (σ2/ K2) = 39.06%

 

Where:

  • Standard deviation: 5%
  • Mean: 12%
  • K: 8%

 

Thus, the probability of an asset’s return to be less than 4% or greater than 20% from the population of assets, which has a mean return of 12% with a standard deviation of 5%, is less than 39.06%, according to Chebyshev’s inequality.

 

More Resources

CFI is the official provider of the global Business Intelligence & Data Analyst (BIDA)® certification program, designed to help anyone become a world-class financial analyst. To keep advancing your career, the additional CFI resources below will be useful:

  • Arithmetic Mean
  • Rate of Return
  • Sampling Errors
  • Unconditional Probability