Samuelson's inequality
Introduction In statistics, '''Samuelson's inequality''', named after the economist Paul Samuelson, also called the '''Laguerre–Samuelson inequality''', after the mathematician Edmond Laguerre, states that every one of any collection ''x''1, ..., ''x''''n'', is within  uncorrected sample standard deviations of their sample mean.
Statement of the inequality If we let: be the sample mean and: be the standard deviation of the sample, then: Equality holds on the left (or right) for  if and only if all the ''n'' − 1 s other than  are equal to each other and greater (smaller) than If you instead define  then the inequality  still applies and can be slightly tightened to
Comparison to Chebyshev's inequality Chebyshev's inequality locates a certain fraction of the data within certain bounds, while Samuelson's inequality locates ''all'' the data points within certain bounds.The bounds given by Chebyshev's inequality are unaffected by the number of data points, while for Samuelson's inequality the bounds loosen as the sample size increases. Thus for large enough data sets, Chebychev's inequality is more useful.
Applications Samuelson's inequality may be considered a reason why studentization of residuals should be done externally.
Relationship to polynomials Samuelson was not the first to describe this relationship: the first  was probably Laguerre in 1880 while investigating the roots (zeros) of polynomials. Consider a polynomial with all roots real: :  	Without loss of generality let  and let:  and  	Then: and: In terms of the coefficients: Laguerre showed that the roots of this polynomial were bounded by: where: Inspection shows that  is the mean of the roots and that ''b'' is the standard deviation of the roots.Laguerre failed to notice this relationship with the means and standard deviations of the roots, being more interested in the bounds themselves. This relationship permits a rapid estimate of the bounds of the roots and may be of use in their location.When the coefficients  and  are both zero no information can be obtained about the location of the roots, because not all roots are real (as can be seen from Descartes' rule of signs) unless the constant term is also zero.
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