AVERAGE

In mathematics, an 'average', or 'central tendency' of a data set refers to a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency. The most common method is the arithmetic mean, but there are more than one type of average (median being another common example). Colloquially, people often use the term 'average' to refer to an intuitive 'central tendency' without having a specific measurement of central tendency in mind. Please see the table of mathematical symbols for explanations of the symbols used.
In statistics, the term ''central tendency'' is used in some fields of empirical research to refer to what statisticians sometimes call "location". A "measure of central tendency" is either a location parameter or a statistic used to estimate a location parameter.

Contents
Measures of central tendency
Other averages
Average applied to a Data Stream
Derivation of the name
External links

Measures of central tendency


There are several different kinds of calculations for central tendency, the kind of calculation that should be used depends on the type of data (level of measurement) and purpose for which the central tendency is being calculated:
{|class="wikitable" style="background: white;"
|-
! Name !! Equation or description
|-
| Arithmetic mean || ar{x} = rac{1}{n}sum_{i=1}^n x_i = rac{1}{n} (x_1+cdots+x_n)
|-
| Median || The middle value that separates the higher half from the lower half of the data set
|-
| Geometric median || A rotation invariant extension of the median for points in Rn
|-
| Mode || The most frequent value in the data set
|-
| Geometric mean || igg(prod_{i=1}^n x_i igg)^{1/n} = sqrt[n]{x_1 cdot x_2 dotsb x_n}
|-
| Harmonic mean || rac{n}{ rac{1}{x_1} + rac{1}{x_2} + cdots + rac{1}{x_n}}
|-
| Quadratic mean
(or RMS) || sqrt{ rac{1}{n} sum_{i=1}^{n} x_i^2} =
sqrt { rac{x_1^2 + x_2^2 + cdots + x_n^2}{n}}

|-
| Generalized mean || sqrt[p]{ rac{1}{n} cdot sum_{i=1}^n x_{i}^p}
|-
| Heronian mean || rac{2}{n(n+1)} cdot sum_{i=1}^n sum_{j=1}^n sqrt{x_i x_j}
|-
| Weighted mean || rac{ sum_{i=1}^n w_i x_i}{sum_{i=1}^n w_i} = rac{w_1 x_1 + w_2 x_2 + cdots + w_n x_n}{w_1 + w_2 + cdots + w_n}
|-
| Truncated mean || The arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded
|-
| Interquartile mean || A special case of the truncated mean, using the interquartile range
|-
| Midrange || rac{max x + min x}{2}
|-
| Winsorized mean || Similar to the truncated mean, but, rather than deleting the extreme values, they are set equal to the largest and smallest values that remain
|}

Other averages


Other more sophisticated averages are: trimean, trimedian, and normalized mean. These are usually more representative of the whole data set.
One can create one's own average metric using Generalized f-mean: y = f^{-1}left( rac{f(x_1)+f(x_2)+dots+f(x_n)}{n}
ight) where f is any invertible function. The harmonic mean is an example of this using f(x) = 1/x, and the geometric mean is another, using f(x) = log x. Another example, expmean (exponential mean) is a mean using the function f(x) = e^x, and it is inherently biased towards the higher values.

Average applied to a Data Stream


The concept of an average can be applied to a stream of data as well as a bounded set, the goal being to find a value about which recent data is in some way clustered. The stream may be distributed in time, as in samples taken by some data acquisition system from which we want to remove noise, or in space, as in pixels in an image from which we want to extract some property. An easy-to-understand and widely used application of average to a stream is the simple moving average in which we compute the arithmetic mean of the most recent N data items in the stream. To advance one position in the stream, we add 1/N times the new data item and subtract 1/N times the data item N places back in the stream.

Derivation of the name


The original meaning of the word is "damage sustained at sea": the same word is found in Arabic as ''awar'', in Italian as ''avaria'' and in French as ''avarie''. Hence an ''average adjuster'' is a person who assesses an insurable loss.
Marine damage is either ''particular average'', which is borne only by the owner of the damaged property, or general average, where the owner can claim a proportional contribution from all the parties to the marine venture. The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean".

External links



★ Median as a weighted arithmetic mean of all Sample Observations

★ Calculations and comparison between arithmetic and geometric mean of two values

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