Editing Arithmetic mean (section)

==Motivating properties==
The arithmetic mean has several properties that make it interesting, especially as a measure of central tendency. These include:

*If numbers <math>x_1,\dotsc,x_n</math> have mean <math>\bar{x}</math>, then <math>(x_1-\bar{x})+\dotsb+(x_n-\bar{x})=0</math>. Since <math>x_i-\bar{x}</math> is the distance from a given number to the mean, one way to interpret this property is by saying that the numbers to the left of the mean are balanced by the numbers to the right. The mean is the only number for which the [[Errors and residuals in statistics|residuals]] (deviations from the estimate) sum to zero.  This can also be interpreted as saying that the mean is [[Translational symmetry|translationally invariant]] in the sense that for any real number <math>a</math>, <math>\overline{x + a} = \bar{x} + a</math>.
*If it is required to use a single number as a "typical" value for a set of known numbers <math>x_1,\dotsc,x_n</math>, then the arithmetic mean of the numbers does this best since it minimizes the sum of squared deviations from the typical value: the sum of <math>(x_i-\bar{x})^2</math>. The sample mean is also the best single predictor because it has the lowest [[root mean squared error]].<ref name="JM">{{cite book|last=Medhi|first=Jyotiprasad|title=Statistical Methods: An Introductory Text|url=https://books.google.com/books?id=bRUwgf_q5RsC|year=1992|publisher=New Age International|isbn=9788122404197|pages=53–58}}</ref> If the arithmetic mean of a population of numbers is desired, then the estimate of it that is [[Unbiased estimate|unbiased]] is the arithmetic mean of a sample drawn from the population.

*The arithmetic mean is independent of scale of the units of measurement, in the sense that <math>\text{avg}(ca_{1},\cdots,ca_{n})=c\cdot\text{avg}(a_{1},\cdots,a_{n}).</math>  So, for example, calculating a mean of liters and then converting to gallons is the same as converting to gallons first and then calculating the mean.  This is also called [[Homogeneous function|first order homogeneity]].

===Additional properties===
* The arithmetic mean of a sample is always between the largest and smallest values in that sample.
*The arithmetic mean of any amount of equal-sized number groups together is the arithmetic mean of the arithmetic means of each group.