Average

From Wikipedia, the free encyclopedia

In mathematics, an average, mean, 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, and the one generally referred to simply as the average, is the arithmetic mean. 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.

1 Measures of central tendency
2 Other averages
3 Average applied to a Data Stream
4 Derivation of the name
5 See also
6 External links

[edit] Measures of central tendency

See also Category:Means

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:

Name	Equation or description
Arithmetic mean	$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i = \frac{1}{n} (x_1+\cdots+x_n)$
Median	The middle value that separates the higher half from the lower half of the data set
Geometric median	An rotation invariant extension of the median for points in Rⁿ
Mode	The most frequent value in the data set
Geometric mean	$\bigg(\prod_{i=1}^n x_i \bigg)^{1/n} = \sqrt[n]{x_1 \cdot x_2 \dotsb x_n}$
Harmonic mean	$\frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}$
Quadratic mean (or RMS)	$\sqrt{\frac{1}{n} \sum_{i=1}^{n} x_i^2} = \sqrt {\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}}$
Generalized mean	$\sqrt[p]{\frac{1}{n} \cdot \sum_{i=1}^n x_{i}^p}$
Heronian mean	$\frac{2}{n(n+1)} \cdot \sum_{i=1}^n \sum_{j=1}^n \sqrt{x_i x_j}$
Weighted mean	$\frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}$
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	$\frac{\max x + \min x}{2}$
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

[edit] Other averages

Other more sophisticated averages are: trimean, trimedian, and normalized mean. These are usually more representative of the whole data set.^{[citation needed]}

One can create one's own average metric using Generalised f-mean: $y = f^{-1}\left(\frac{f(x_1)+f(x_2)+\dots+f(x_n)}{n}\right)$ 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.

The only significant reason why the arithmetic mean (classical average) is generally used in scientific papers is that there are various (statistical) tests which can be applied to test the statistical significance of the results, as well as the correlations that are explored through these metrics.^{[citation needed]}

[edit] 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.

[edit] 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".