Editing Chebyshev's inequality (section)

====Semivariances====
An alternative method of obtaining sharper bounds is through the use of [[Variance#Semivariance|semivariance]]s (partial variances). The upper (''σ''<sub>+</sub><sup>2</sup>) and lower (''σ''<sub>−</sub><sup>2</sup>) semivariances are defined as

: <math> \sigma_+^2 = \frac { \sum_{x>m} (x - m)^2 } { n - 1 } ,</math>

: <math> \sigma_-^2 = \frac { \sum_{x<m} (m - x)^2 } { n - 1 }, </math>

where ''m'' is the arithmetic mean of the sample and ''n'' is the number of elements in the sample.

The variance of the sample is the sum of the two semivariances:

: <math> \sigma^2 = \sigma_+^2 + \sigma_-^2. </math>

In terms of the lower semivariance Chebyshev's inequality can be written<ref name=Berck1982>{{cite journal|author-link1=Peter Berck |last1=Berck |first1=Peter |last2=Hihn |first2=Jairus M. |title=Using the Semivariance to Estimate Safety-First Rules |journal=American Journal of Agricultural Economics |date=May 1982 |volume=64 |issue=2 |pages=298–300 |doi=10.2307/1241139 |issn=0002-9092|jstor=1241139 |doi-access= }}</ref>

: <math> \Pr(x \le m - a \sigma_-) \le \frac { 1 } { a^2 }.</math>

Putting

: <math> a = \frac{ k \sigma } { \sigma_- }. </math>

Chebyshev's inequality can now be written

: <math> \Pr(x \le m - k \sigma) \le \frac { 1 } { k^2 } \frac { \sigma_-^2 } { \sigma^2 }.</math>

A similar result can also be derived for the upper semivariance.

If we put

: <math> \sigma_u^2 = \max(\sigma_-^2, \sigma_+^2) , </math>

Chebyshev's inequality can be written

: <math> \Pr(| x \le m - k \sigma |) \le \frac 1 {k^2} \frac { \sigma_u^2 } { \sigma^2 } .</math>

Because ''σ''<sub>u</sub><sup>2</sup> ≤ ''σ''<sup>2</sup>, use of the semivariance sharpens the original inequality.

If the distribution is known to be symmetric, then

: <math> \sigma_+^2 = \sigma_-^2  = \frac{ 1 } { 2 } \sigma^2 </math>

and

: <math> \Pr(x \le m - k \sigma) \le \frac 1 {2k^2} .</math>

This result agrees with that derived using standardised variables.

;Note: The inequality with the lower semivariance has been found to be of use in estimating downside risk in finance and agriculture.<ref name="Berck1982"/><ref name=Nantell1979>{{cite journal |last1=Nantell |first1=Timothy J. |last2=Price |first2=Barbara |title=An Analytical Comparison of Variance and Semivariance Capital Market Theories |journal=[[The Journal of Financial and Quantitative Analysis]] |date=June 1979 |volume=14 |issue=2 |pages=221–42 |doi=10.2307/2330500 |jstor=2330500  |s2cid=154652959 }}</ref><ref name=Neave2008>{{cite journal
  |title = Distinguishing upside potential from downside risk
  |last1 = Neave
  |first1 = Edwin H.
  |last2 = Ross
  |first2 = Michael N.
  |last3 = Yang
  |first3 = Jun
  |journal = [[Management Research News]]
  |issn = 0140-9174
  |year = 2009
  |volume = 32
  |issue = 1
  |pages = 26–36
  |doi = 10.1108/01409170910922005
  }}</ref>