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In colloquial language average usually means the sum of a list of numbers divided by the size of the list, in other words the arithmetic mean. However it can alternatively mean the median, the mode, or some other central or typical value. In statistics, these are all known as measures of central tendency. The concept of an average can be extended in various ways in mathematics, but in those contexts it is usually referred to as a mean (for example the mean of a function)
Pythagorean means 
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The three most common averages are the Pythagorean means – the arithmetic mean, the geometric mean, and the harmonic mean.
Arithmetic mean 
If n numbers are given, each number denoted by ai, where i = 1, ..., n, the arithmetic mean is the [sum] of the ai's divided by n or
The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. One may find that A = (2 + 8)/2 = 5. Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list for which we want an average, we get, for example, that the arithmetic mean of 2, 8, and 11 is found by solving for the value of A in the equation 2 + 8 + 11 = A + A + A. One finds that A = (2 + 8 + 11)/3 = 7.
Geometric mean 
The geometric mean of n non-negative numbers is obtained by multiplying them all together and then taking the nth root. In algebraic terms, the geometric mean of a1, a2, ..., an is defined as
Example: Geometric mean of 2 and 8 is
Harmonic mean 
Harmonic mean for a non-empty collection of numbers a1, a2, ..., an, all different from 0, is defined as the reciprocal of the arithmetic mean of the reciprocals of the ai's:
One example where it is useful is calculating the average speed for a number of fixed-distance trips. For example, if the speed for going from point A to B was 60 km/h, and the speed for returning from B to A was 40 km/h, then the average speed is given by
Inequality concerning AM, GM, and HM 
A well known inequality concerning arithmetic, geometric, and harmonic means for any set of positive numbers is
It is easy to remember noting that the alphabetical order of the letters A, G, and H is preserved in the inequality. See Inequality of arithmetic and geometric means.
The most frequently occurring number in a list is called the mode. For example, the mode of the list (1, 2, 2, 3, 3, 3, 4) is 3. The mode is not necessarily uniquely defined; for example, the list (1, 2, 2, 3, 3, 5) has the two modes 2 and 3. The mode is more meaningful and potentially useful if there are many numbers in the list, and the frequency of the numbers progresses smoothly (e.g., if out of a group of 1000 people, 30 people weigh 61 kg, 32 weigh 62 kg, 29 weigh 63 kg, and all the other possible weights occur less frequently, then 62 kg is the mode).
The mode has the advantage that it can be used with non-numerical data (e.g., red cars are most frequent), while other averages cannot.
The median is the middle number of the group when they are ranked in order. (If there are an even number of numbers, the mean of the middle two is taken.)
Thus to find the median, order the list according to its elements' magnitude and then repeatedly remove the pair consisting of the highest and lowest values until either one or two values are left. If exactly one value is left, it is the median; if two values, the median is the arithmetic mean of these two. This method takes the list 1, 7, 3, 13 and orders it to read 1, 3, 7, 13. Then the 1 and 13 are removed to obtain the list 3, 7. Since there are two elements in this remaining list, the median is their arithmetic mean, (3 + 7)/2 = 5.
Summary of Types 
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The table of mathematical symbols explains the symbols used below.
|Name||Equation or description|
|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|
|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|
|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|
Miscellaneous types 
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One can create one's own average metric using the generalized f-mean:
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.
However, this method for generating means is not general enough to capture all averages. A more general method for defining an average takes any function g(x1, x2, ..., xn) of a list of arguments that is continuous, strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average y is then the value that, when replacing each member of the list, results in the same function value: g(y, y, ..., y) = g(x1, x2, ..., xn). This most general definition still captures the important property of all averages that the average of a list of identical elements is that element itself. The function g(x1, x2, ..., xn) = x1+x2+ ··· + xn provides the arithmetic mean. The function g(x1, x2, ..., xn) = x1x2···xn (where the list elements are positive numbers) provides the geometric mean. The function g(x1, x2, ..., xn) = −(x1−1+x2−1+ ··· + xn−1) (where the list elements are positive numbers) provides the harmonic mean.
Average Percentage Return and CAGR 
A type of average used in finance is the average percentage return. It is an example of a geometric mean. When the returns are annual, it is called the Compound Annual Growth Rate (CAGR). For example, if we are considering a period of two years, and the investment return in the first year is −10% and the return in the second year is +60%, then the average percentage return or CAGR, R, can be obtained by solving the equation: (1 − 10%) × (1 + 60%) = (1 − 0.1) × (1 + 0.6) = (1 + R) × (1 + R). The value of R that makes this equation true is 0.2, or 20%. This means that the total return over the 2-year period is the same as if there had been 20% growth each year. Note that the order of the years makes no difference – the average percentage returns of +60% and −10% is the same result as that for −10% and +60%.
This method can be generalized to examples in which the periods are not equal. For example, consider a period of a half of a year for which the return is −23% and a period of two and a half years for which the return is +13%. The average percentage return for the combined period is the single year return, R, that is the solution of the following equation: (1 − 0.23)0.5 × (1 + 0.13)2.5 = (1 + R)0.5+2.5, giving an average percentage return R of 0.0600 or 6.00%.
In data streams 
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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 that recent data clusters about in some way. 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.
- Update rule for a window of size upon seeing new element :
"Few words have received more etymological investigation."  In the 16th century average meant a customs duty, or the like, and was used in the Mediterranean area. It came to mean the cost of damage sustained at sea. From that came an "average adjuster" who decided how to apportion a loss between the owners and insurers of a ship and cargo.
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".
The root is found in Arabic as awar, in Italian as avaria, in French as avarie and in Dutch as averij. It is unclear in which language the word first appeared.
There is earlier (from at least the 11th century), unrelated use of the word. It appears to be an old legal term for a tenant's day labour obligation to a sheriff, probably anglicised from "avera" found in the English Domesday Book (1085).
See also 
- Algorithms for calculating variance
- Law of averages
- Spherical mean
- Measures of statistical dispersion
- Merigo, Jose M.; Cananovas, Montserrat (2009). "The Generalized Hybrid Averaging Operator and its Application in Decision Making". Journal of Quantitative Methods for Economics and Business Administration 9: 69–84. ISSN 1886-516X.
- John Bibby (1974). “Axiomatisations of the average and a further generalisation of monotonic sequences”. Glasgow Mathematical Journal, vol. 15, pp. 63–65.
- Oxford English Dictionary
- Hardy, G.H.; Littlewood, J.E.; Pólya, G. (1988). Inequalities (2nd ed.). Cambridge University Press. ISBN 978-0-521-35880-4
|Look up average in Wiktionary, the free dictionary.|