High Quality Content by WIKIPEDIA articles! 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) standard deviations of their mean. In other words, if we let overline{x} = frac{x_1+cdots+x_n}{n} be the mean and s = sqrt{fr ...Full description
High Quality Content by WIKIPEDIA articles! 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) standard deviations of their mean. In other words, if we let overline{x} = frac{x_1+cdots+x_n}{n} be the mean and s = sqrt{frac{1}{n} sum_{i=1}^n (x_i - overline{x})^2 } be the standard deviation, then overline{x} - ssqrt{n-1} le x_i le overline{x} + ssqrt{n-1}qquad text{for }i = 1,dots,n. Equality holds on the left if and only if the n ¿ 1 smallest of the n numbers are equal to each other, and on the right iff the n ¿ 1 largest ones are equal. Samuelson's inequality may be considered a reason why studentization of residuals should be done externally.