Average

In mathematics, an average or central tendency of a set (list) of data refers to a measure of the "middle" 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.

A list of 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:

• Arithmetic mean - the sum of all measurements divided by the number of observations in the data set
• Median - the middle value that separates the higher half from the lower half of the data set
• Mode - the most frequent value in the data set
• Geometric mean - the nth root of the product of n data values
• Harmonic mean - the reciprocal of the arithmetic mean of the reciprocals of the data values
• Quadratic mean or root mean square (RMS) - the square root of the arithmetic mean of the squares of the data values
• Generalized mean - generalizing the above, the nth root of the arithmetic mean of the nth powers of the data values
• Weighted mean - an arithmetic mean that incorporates weighting to certain data elements
• 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
• Midrange - the arithmetic mean of the highest and lowest values of the data or distribution.
• 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 normalised mean, to name a few. These are usually more representative of the whole dataset.

One can create one's own average metric using Generalised f-mean: [itex]y = f^{-1}((f(x_1)+f(x_2)+\dots+f(x_n))/n)[/itex] where [itex]f[/itex] is any invertible function. The harmonic mean is an example of this using [itex]f(x) = 1/x[/itex], and the geometric mean is another, using [itex]f(x) = \log x[/itex]. Another example, expmean (exponential mean) is a mean using the function [itex]f(x) = e^x[/itex], 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.

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".