Editing Chebyshev's inequality (section)

==Related inequalities==
Several other related inequalities are also known.

===Paley–Zygmund inequality===
{{main|Paley–Zygmund inequality}}

The Paley–Zygmund inequality gives a lower bound on tail probabilities, as opposed to Chebyshev's inequality which gives an upper bound.<ref name=Godwin1964a>Godwin H. J. (1964) Inequalities on distribution functions. (Chapter 3) New York, Hafner Pub. Co.</ref> Applying it to the square of a random variable, we get

: <math> \Pr( | Z | > \theta \sqrt{E[Z^2]} ) \ge \frac{ ( 1 - \theta^2 )^2 E[Z^2]^2 }{E[Z^4]}.</math>

===Haldane's transformation===
One use of Chebyshev's inequality in applications is to create confidence intervals for variates with an unknown distribution. [[J. B. S. Haldane|Haldane]] noted,<ref name=Haldane1952>{{cite journal | last1 = Haldane | first1 = J. B.|author-link=J. B. S. Haldane | year = 1952 | title = Simple tests for bimodality and bitangentiality | journal = [[Annals of Eugenics]] | volume = 16 | issue = 4| pages = 359–364 | doi = 10.1111/j.1469-1809.1951.tb02488.x | pmid = 14953132}}</ref> using an equation derived by [[Maurice Kendall|Kendall]],<ref name=Kendall1943>Kendall M. G. (1943) The Advanced Theory of Statistics, 1. London</ref> that if a variate (''x'') has a zero mean, unit variance and both finite [[skewness]] (''γ'') and [[kurtosis]] (''κ'') then the variate can be converted to a normally distributed [[standard score]] (''z''):

: <math> z = x - \frac{\gamma}{6} (x^2 - 1) + \frac{ x }{ 72 } [ 2 \gamma^2 (4 x^2 - 7) - 3 \kappa (x^2 - 3) ] + \cdots </math>

This transformation may be useful as an alternative to Chebyshev's inequality or as an adjunct to it for deriving confidence intervals for variates with unknown distributions.

While this transformation may be useful for moderately skewed and/or kurtotic distributions, it performs poorly when the distribution is markedly skewed and/or kurtotic.

===He, Zhang and Zhang's inequality===
For any collection of {{mvar|n}} non-negative independent random variables {{mvar|X<sub>i</sub>}} with expectation 1 <ref name=He2010>{{cite journal
| last1=He | first1=Simai
| last2=Zhang | first2=Jiawei
| last3=Zhang | first3=Shuzhong
| s2cid=11298475
| date=2010
| title=Bounding probability of small deviation: a fourth moment approach
| journal=[[Mathematics of Operations Research]]
| volume=35
| issue=1
| pages=208–232
| doi=10.1287/moor.1090.0438}}</ref>

: <math> \Pr\left ( \frac{\sum_{i=1}^n X_i }{n} - 1 \ge \frac{1}{n} \right) \le \frac{ 7 }{ 8 }. </math>