Samuelson's inequality

If we let

${\displaystyle {\overline {x}}={\frac {x_{1}+\cdots +x_{n}}{n}}}$

be the sample mean and

${\displaystyle s={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}}$

be the standard deviation of the sample, then

${\displaystyle {\overline {x}}-s{\sqrt {n-1}}\leq x_{j}\leq {\overline {x}}+s{\sqrt {n-1}}\qquad {\text{for }}j=1,\dots ,n.}$[4]

Equality holds on the left (or right) for ${\displaystyle x_{j}}$ if and only if all the n − 1 ${\displaystyle x_{i}}$s other than ${\displaystyle x_{j}}$ are equal to each other and greater (smaller) than ${\displaystyle x_{j}.}$[2]

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.

Samuelson's inequality may be considered a reason why studentization of residuals should be done externally.

Samuelson was not the first to describe this relationship: the first was probably Laguerre in 1880 while investigating the roots (zeros) of polynomials.[2][5]

Consider a polynomial with all roots real:

${\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}=0}$

Without loss of generality let ${\displaystyle a_{0}=1}$ and let

${\displaystyle t_{1}=\sum x_{i}}$ and ${\displaystyle t_{2}=\sum x_{i}^{2}}$

Then

${\displaystyle a_{1}=-\sum x_{i}=-t_{1}}$

and

${\displaystyle a_{2}=\sum x_{i}x_{j}={\frac {t_{1}^{2}-t_{2}}{2}}\qquad {\text{ where }}i<j}$

In terms of the coefficients

${\displaystyle t_{2}=a_{1}^{2}-2a_{2}}$

Laguerre showed that the roots of this polynomial were bounded by

${\displaystyle -a_{1}/n\pm b{\sqrt {n-1}}}$

where

${\displaystyle b={\frac {\sqrt {nt_{2}-t_{1}^{2}}}{n}}={\frac {\sqrt {na_{1}^{2}-a_{1}^{2}-2na_{2}}}{n}}}$

Inspection shows that ${\displaystyle -{\tfrac {a_{1}}{n}}}$ 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 ${\displaystyle a_{1}}$ and ${\displaystyle a_{2}}$ 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.