# Chebyshev's inequalities

Hey guys ,

was doing a reading on statistics basics and I’m unable to understand what does chebyshev’s theory tell us ? what its used for ? and how to do it ?

It would be kind if anyone can explain me dispite all the measures of location and measures of dispersion what does chebyshev’s inequality have to offer ?

Hedgefudge

Chebychev’s inequality simply tells us the percentage of observations that fall within a certain number of standard deviations from the mean. The number of standard deviations is represented by the letter k in the formula 1-1/(k^2). The rule applies to any normal distribution.

For example if you want to know how many observations from sample population will fall within 2 standard deviations from the mean (k=2), we find that 1-1/(2^2) = 0.75 = 75%.

Therefore, for any normal distribution 75% of all observations will fall within 2 stadard deviations from the mean. Since this applies to all normal distributions Chbebychev’s inequality is more of a Rule than a formula really.

To get a better idea of what it means look at the bell curve when practicing with the formula to understand how it is applied.

While it is true that Chebychev’s inequality applies to a normal distribution, it would be worthless if that were the only distribution to which it applies. (Why use a very conservative estimate when you can calculate the exact number easily?)

The strength – and, indeed, the sheer beauty – of Chebychev’s inequality is that it applies not only to normal distributions, but to _ any probability distribution _. _ Every probability distribution imaginable _ satisfies Chebychev’s inequality. Normal distributions, uniform distributions, beta distributions, Poisson distributions, hypergeometric distributions, Weibull distributions, weird bimodal, trimodal, tetramodal, icosamodal, whatevermodal distributions, strange, one-off, empirical distributions: they all satisfy Chebychev’s inequality.

No matter what distribution you have, if you want to know what fraction of the distribution lies in the range of μ – _k_σ to μ + _k_σ, for any number k of standard deviations, Chebychev’s inequality tells you that it will be at least 1 – 1/_k_².

Note, however, that the number you get from Chebychev’s inequality is quite conservative. It tells you that at least:

• 1 – 1/1² = 0% of the distribution lies within one standard deviation of the mean; for a normal distribution, the actual number is 68.27% (which is certainly bigger than 0%)
• 1 – 1/2² = 75% of the distribution lies within two standard deviations of the mean; for a normal distribution, the actual number is 95.45%
• 1 – 1/3² = 88.89% of the distribution lies within three standard deviations of the mean; for a normal distribution, the actual number is 99.73%
• 1 – 1/4² = 93.75% of the distribution lies within four standard deviations of the mean; for a normal distribution, the actual number is 99.99%
• 1 – 1/5² = 96% of the distribution lies within five standard deviations of the mean; for a normal distribution, the actual number is 99.9999%

THink of Chebychev’s as a quick and dirty rule you can use to get a MINIMUM percentage of the distribution WITHOUT NEEDING ANY OTHER INFORMATION. If you know more about the distribution (like, it’s normally distributed) you can get a better estimate. But for Chebychev’s holds for ANY distribution.