Samuelson's inequality


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 x1, ..., xn, 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 for if and only if all the n − 1 s other than are equal to each other and greater than

Comparison to 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 of polynomials.

Consider a polynomial with all roots real:

Without loss of generality let and let

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 unless the constant term is also zero.