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 √n − 1 uncorrected sample standard deviations of their sample mean.
Statement of the inequality If we let
x ¯ = x 1 + ⋯ + x n n {\displaystyle {\overline {x}}={\frac {x_{1}+\cdots +x_{n}}{n}}}
be the sample mean and
s = 1 n ∑ i = 1 n ( x i − x ¯ ) 2 {\displaystyle s={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}}
be the standard deviation of the sample, then
x ¯ − s n − 1 ≤ x j ≤ x ¯ + s n − 1 for j = 1 , … , n . {\displaystyle {\overline {x}}-s{\sqrt {n-1}}\leq x_{j}\leq {\overline {x}}+s{\sqrt {n-1}}\qquad {\text{for }}j=1,\dots ,n.}
Equality holds on the left (or right) for x j {\displaystyle x_{j}} if and only if all the n − 1 x i {\displaystyle x_{i}} s other than x j {\displaystyle x_{j}} are equal to each other and greater (smaller) than x j . {\displaystyle x_{j}.}
If you instead define s = 1 n − 1 ∑ i = 1 n ( x i − x ¯ ) 2 {\displaystyle s={\sqrt {{\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}} then the inequality x ¯ − s n − 1 ≤ x j ≤ x ¯ + s n − 1 {\displaystyle {\overline {x}}-s{\sqrt {n-1}}\leq x_{j}\leq {\overline {x}}+s{\sqrt {n-1}}} still applies and can be slightly tightened to x ¯ − s n − 1 n ≤ x j ≤ x ¯ + s n − 1 n . {\displaystyle {\overline {x}}-s{\tfrac {n-1}{\sqrt {n}}}\leq x_{j}\leq {\overline {x}}+s{\tfrac {n-1}{\sqrt {n}}}.}
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, Chebyshev's inequality is more useful.
Applications
Samuelson’s inequality has several applications in statistics and mathematics. It is useful in the 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 theory, Samuelson’s inequality is used to locate the eigenvalues of certain matrices and tensors. Furthermore, generalizations of this inequality apply to complex data and random variables in a probability space.
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:
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