Samuelson's inequality [SEP] named after
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 "x", ..., "x", 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 formula_4
for finite samples. Samuelson's inequality states that all values of a sample will lie within standard deviations of the mean. Chebyshev's bound improves as the sample size increases.
When "N" = 10, Samuelson's inequality states that all members of the sample lie within 3 standard deviations of the mean: in contrast Chebyshev's states that 99.5% of the sample lies within 13.5789 standard deviations of the mean.
When "N" = 100, Samuelson's inequality states that all members of the sample lie within