Editing Arithmetic mean (section)

==Generalizations==

===Weighted average===
{{main|Weighted average}}

A weighted average, or weighted mean, is an average in which some data points count more heavily than others in that they are given more weight in the calculation.<ref>{{Cite web|title=Mean {{!}} mathematics|url=https://www.britannica.com/science/mean|access-date=2020-08-21|website=Encyclopedia Britannica|language=en}}</ref> For example, the arithmetic mean of <math>3</math> and <math>5</math> is <math>\frac{3+5}{2}=4</math>, or equivalently <math>3 \cdot \frac{1}{2}+5 \cdot \frac{1}{2}=4</math>. In contrast, a ''weighted'' mean in which the first number receives, for example, twice as much weight as the second (perhaps because it is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as <math>3 \cdot \frac{2}{3}+5 \cdot \frac{1}{3}=\frac{11}{3}</math>. Here the weights, which necessarily sum to one, are <math>\frac{2}{3}</math> and <math>\frac{1}{3}</math>, the former being twice the latter. The arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all weights are equal to the same number (<math>\frac{1}{2}</math> in the above example and <math>\frac{1}{n}</math> in a situation with <math>n</math> numbers being averaged).

===Functions===
{{excerpt|Arithmetic mean of a function}}

===Continuous probability distributions===
[[File:Comparison mean median mode.svg|thumb|300px|Comparison of two [[log-normal distribution]]s with equal median, but different [[skewness]], resulting in various means and [[mode (statistics)|mode]]s]]

If a numerical property, and any sample of data from it, can take on any value from a continuous range instead of, for example, just integers, then the [[probability]] of a number falling into some range of possible values can be described by integrating a [[continuous probability distribution]] across this range, even when the naive probability for a sample number taking one certain value from infinitely many is zero. In this context, the analog of a weighted average, in which there are infinitely many possibilities for the precise value of the variable in each range, is called the ''mean of the [[probability distribution]]''. The most widely encountered probability distribution is called the [[normal distribution]]; it has the property that all measures of its central tendency, including not just the mean but also the median mentioned above and the mode (the three Ms<ref name=ThreeMs>{{cite web|url=https://www.visualthesaurus.com/cm/lessons/the-three-ms-of-statistics-mode-median-mean/|title=The Three M's of Statistics: Mode, Median, Mean June 30, 2010|website=www.visualthesaurus.com|author=Thinkmap Visual Thesaurus|date=2010-06-30|access-date=2018-12-03}}</ref>), are equal. This equality does not hold for other probability distributions, as illustrated for the log-normal distribution here.

===Angles===
{{main|Mean of circular quantities}}

Particular care is needed when using cyclic data, such as phases or [[angle]]s. Taking the arithmetic mean of 1° and 359° yields a result of 180[[degree (angle)|°]].
This is incorrect for two reasons:
*Firstly, angle measurements are only defined up to an additive constant of 360° (<math>2\pi</math> or <math>\tau</math>, if measuring in [[radian]]s). Thus, these could easily be called 1° and -1°, or 361° and 719°, since each one of them produces a different average.
*Secondly, in this situation, 0° (or 360°) is geometrically a better ''average'' value: there is lower [[Statistical dispersion|dispersion]] about it (the points are both 1° from it and 179° from 180°, the putative average).

In general application, such an oversight will lead to the average value artificially moving towards the middle of the numerical range. A solution to this problem is to use the optimization formulation (that is, define the mean as the central point: the point about which one has the lowest dispersion) and redefine the difference as a modular distance (i.e., the distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°).

{{AM_GM_inequality_visual_proof.svg}}