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{{Short description|Concept in statistics}}
[[File:Paul Samuelson.jpg|alt=Samuelson Inequality|thumb]]
In [[statistics]], '''Samuelson's inequality''', named after the economist [[Paul Samuelson]],<ref>{{cite journal |first=Paul |last=Samuelson |title=How Deviant Can You Be? |journal=[[Journal of the American Statistical Association]] |volume=63 |issue=324 |year=1968 |pages=1522–1525 |jstor=2285901 |doi=10.2307/2285901 }}</ref> also called the '''Laguerre–Samuelson inequality''',<ref name=Jensen>{{cite thesis |type=MSc |last=Jensen |first=Shane Tyler |date=1999 |title=The Laguerre–Samuelson Inequality with Extensions and Applications in Statistics and Matrix Theory |publisher=Department of Mathematics and Statistics, [[McGill University]] |url=http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape10/PQDD_0027/MQ50799.pdf }}</ref><ref>{{cite book |first1=Shane T. |last1=Jensen |first2=George P. H. |last2=Styan |year=1999 |chapter=Some Comments and a Bibliography on the Laguerre-Samuelson Inequality with Extensions and Applications in Statistics and Matrix Theory |title=Analytic and Geometric Inequalities and Applications |pages=151–181 |doi=10.1007/978-94-011-4577-0_10 |isbn=978-94-010-5938-1 }}</ref> after the mathematician [[Edmond Laguerre]], states that every one of any collection ''x''1, ..., ''x''''n'', is within {{radic|''n'' − 1}} uncorrected sample [[standard deviation]]s of their sample mean.

==Statement of the inequality==
If we let

: <math> \overline{x} = \frac{x_1+\cdots+x_n}{n} </math>

be the sample [[mean]] and

: <math> s = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \overline{x})^2 } </math>

be the standard deviation of the sample, then

: <math> \overline{x} - s\sqrt{n-1} \le x_j \le \overline{x} + s\sqrt{n-1}\qquad \text{for } j = 1,\dots,n. </math><ref>{{cite book |title=Advances in Inequalities from Probability Theory and Statistics |first1=Neil S. |last1=Barnett |first2=Sever Silvestru |last2=Dragomir |publisher=Nova Publishers |year=2008 |isbn=978-1-60021-943-6 |page=164 }}</ref>

Equality holds on the left (or right) for <math>x_j</math> [[if and only if]] all the ''n'' − 1 <math>x_i</math>s other than <math>x_j</math> are equal to each other and greater (smaller) than <math>x_j.</math><ref name=Jensen/>

If you instead define <math> s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2 } </math> then the inequality <math> \overline{x} - s\sqrt{n-1} \le x_j \le \overline{x} + s\sqrt{n-1}</math> still applies and can be slightly tightened to <math> \overline{x} - s\tfrac{n-1}{\sqrt{n}} \le x_j \le \overline{x} + s\tfrac{n-1}{\sqrt{n}}. </math>

==Comparison to Chebyshev's inequality==
{{Main|Chebyshev's inequality#Samuelson'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, Chebyshev's inequality is more useful.

==Applications==
{{expand section|date=July 2017}}

Samuelson’s inequality has several applications in [[statistics]] and [[mathematics]]. It is useful in the [[Studentized residual|studentization of residuals]] which shows a rationale for why this process should be done externally to better understand the spread of residuals in [[regression analysis]].

In [[Matrix (mathematics)|matrix theory]], Samuelson’s inequality is used to locate the [[Eigenvalues and eigenvectors|eigenvalues]] of certain matrices and tensors.

Furthermore, generalizations of this inequality apply to complex data and random variables in a [[probability space]].<ref>{{Cite journal |last1=JIN |first1=HONGWEI |last2=BEN´ITEZ |first2=JULIO |title=Some generalizations and probability versions of Samuelson's inequality |url=https://files.ele-math.com/articles/mia-20-01.pdf |access-date=4 September 2024 |journal=Mathematical Inequalities & Applications |date=2017 |pages=1–12 |language=English |doi=10.7153/mia-20-01}}</ref><ref>{{Cite book |title= Paul Samuelson|chapter=Samuelson's Approach to Revealed Preference Theory: Some Recent Advances |chapter-url=https://dipot.ulb.ac.be/dspace/bitstream/2013/314701/3/working_paper.pdf |series=Remaking Economics: Eminent Post-War Economists |date=2019 |doi=10.1057/978-1-137-56812-0_9 |last1=Demuynck |first1=Thomas |last2=Hjertstrand |first2=Per |pages=193–227 |isbn=978-1-137-56811-3 }}</ref>

==Relationship to polynomials==

Samuelson was not the first to describe this relationship: the first was probably [[Laguerre]] in 1880 while investigating the [[Equation solving|root]]s (zeros) of [[polynomial]]s.<ref name=Jensen/><ref name=Laguerre1880>Laguerre E. (1880) Mémoire pour obtenir par approximation les racines d'une équation algébrique qui a toutes les racines réelles. Nouv Ann Math 2e série, 19, 161–172, 193–202</ref>

Consider a polynomial with all roots real:

: <math> a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n = 0 </math>

Without loss of generality let <math>a_0 = 1</math> and let

: <math> t_1 = \sum x_i </math> and <math> t_2 = \sum x_i^2 </math>

Then

: <math> a_1 = - \sum x_i = -t_1 </math>

and

: <math> a_2 = \sum x_ix_j = \frac{t_1^2 - t_2}{2} \qquad \text{ where } i < j </math>

In terms of the coefficients

: <math> t_2 = a_1^2 - 2a_2 </math>

Laguerre showed that the roots of this polynomial were bounded by

: <math> -a_1 / n \pm b \sqrt{n - 1} </math>

where

: <math> b = \frac{\sqrt{nt_2 - t_1^2}}{n} = \frac{\sqrt{na_1^2 - a_1^2 - 2na_2}}{n} </math>

Inspection shows that <math>-\tfrac{a_1}{n}</math> 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 <math> a_1 </math> and <math> a_2 </math> 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.

== References ==
{{reflist}}

[[Category:Statistical inequalities]]

Samuelson's inequality - Revision history

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