Quadratic deviation
quadratic variance, standard deviation, of quantities 
from 
The square root of the expression
 | (*) |
The quadratic deviation takes its smallest value when 
, where 
is the arithmetic mean of 
:
 |
In this case the quadratic deviation serves as a measure of the variance (cf. Dispersion) of the quantities 
. Also used is the more general concept of a weighted quadratic deviation:
 |
where the 
are the so-called weights associated with 
. The weighted quadratic deviation attains its smallest value when 
is the weighted mean:
 |
In probability theory, the quadratic deviation 
of a random variable 
(from its mathematical expectation) refers to the square root of its variance: 
.
The quadratic deviation is taken as a measure of the quality of statistical estimators and in this case is referred to as the quadratic error.
Comments
The expression (*) itself is sometimes referred to as the mean-squared error or mean-square error, and its root as the root mean-square error. Similarly one has a weighted mean-square error, etc.
References
| [a1] | K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. 1318 |
| [a2] | A.M. Mood, F.A. Graybill, "Introduction to the theory of statistics" , McGraw-Hill (1963) pp. 166, 176 |
Quadratic deviation. BSE-3 (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Quadratic_deviation&oldid=14081