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In colloquial language, an **average** is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an **average value**.

In mathematics and statistics, the **arithmetic mean**, or simply the mean or **average** when the context is clear, is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

The **median** is the value separating the higher half from the lower half of a data sample. For a data set, it may be thought of as the "middle" value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fifth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.

The **mode** of a set of data values is the value that appears most often. If * X* is a discrete random variable, the mode is the value

The most common type of average is the arithmetic mean. If * n * numbers are given, each number denoted by *a _{i}* (where

In mathematics, **summation** is the addition of a sequence of any kind of numbers, called *addends* or *summands*; the result is their *sum* or *total*. Besides numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any types of mathematical objects on which an operation denoted "+" is defined.

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 to 2, 8, and 11, the arithmetic mean 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.

Along with the arithmetic mean above, the geometric mean and the harmonic mean are known collectively as the Pythagorean means.

The geometric mean of *n* positive numbers is obtained by multiplying them all together and then taking the *n*th root. In algebraic terms, the geometric mean of *a*_{1}, *a*_{2}, ..., *a*_{n} is defined as

In mathematics, the **geometric mean** is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the *n*th root of the product of n numbers, i.e., for a set of numbers *x*_{1}, *x*_{2}, ..., *x _{n}*, the geometric mean is defined as

Geometric mean can be thought of as the antilog of the arithmetic mean of the logs of the numbers.

In mathematics, the **logarithm** is the inverse function to exponentiation (it is an example of a concave function). That means the logarithm of a given number x is the exponent to which another fixed number, the *base* b, must be raised, to produce that number x. In the simplest case, the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 10^{3}, the "logarithm to base 10" of 1000 is 3. The logarithm of x to *base*b is denoted as log_{b} (*x*) (or, without parentheses, as log_{b} *x*, or even without explicit base as log *x*, when no confusion is possible). More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:

Example: Geometric mean of 2 and 8 is

Harmonic mean for a non-empty collection of numbers *a*_{1}, *a*_{2}, ..., *a*_{n}, all different from 0, is defined as the reciprocal of the arithmetic mean of the reciprocals of the *a*_{i}'s:

In mathematics, the **harmonic mean** is one of several kinds of average, and in particular one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired.

In mathematics, a **multiplicative inverse** or **reciprocal** for a number *x*, denoted by 1/*x* or *x*^{−1}, is a number which when multiplied by *x* yields the multiplicative identity, 1. The multiplicative inverse of a fraction *a*/*b* is *b*/*a*. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The **reciprocal function**, the function *f*(*x*) that maps *x* to 1/*x*, is one of the simplest examples of a function which is its own inverse.

One example where the harmonic mean is useful is when examining the 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 harmonic mean speed is given by

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.

In mathematics, the **inequality of arithmetic and geometric means**, or more briefly the **AM–GM inequality**, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.

Thus for the above harmonic mean example: AM = 50, GM ≈ 49, and HM = 48 km/h.

The mode, the median, and the mid-range are often used in addition to the mean as estimates of central tendency in descriptive statistics. These can all be seen as minimizing variation by some measure; see Central tendency § Solutions to variational problems.

Type | Description | Example | Result |
---|---|---|---|

Arithmetic mean | Sum of values of a data set divided by number of values: | (1+2+2+3+4+7+9) / 7 | 4 |

Median | Middle value separating the greater and lesser halves of a data set | 1, 2, 2, 3, 4, 7, 9 | 3 |

Mode | Most frequent value in a data set | 1, 2, 2, 3, 4, 7, 9 | 2 |

Mid-range | The arithmetic mean of the highest and lowest values of a set | (1+9) / 2 | 5 |

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. It may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there is no agreed definition of mode. Some authors say they are all modes and some say there is no mode.

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.

The mid-range is the arithmetic mean of the highest and lowest values of a set.

Name | Equation or description |
---|---|

Arithmetic mean | |

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 R^{n} |

Mode | The most frequent value in the data set |

Geometric mean | |

Harmonic mean | |

Quadratic mean (or RMS) | |

Cubic mean | |

Generalized mean | |

Weighted mean | |

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. A special case of the inter-quantile truncated mean, which operates on quantiles (often deciles or percentiles) that are equidistant but on opposite sides of the median. |

Midrange | |

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 |

The table of mathematical symbols explains the symbols used below.

Other more sophisticated averages are: trimean, trimedian, and normalized mean, with their generalizations.^{ [1] }

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^{ [2] } for defining an average takes any function *g*(*x*_{1}, *x*_{2}, ..., *x*_{n}) 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*(*x*_{1}, *x*_{2}, ..., *x*_{n}). 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*(*x*_{1}, *x*_{2}, ..., *x*_{n}) =*x*_{1}+*x*_{2}+ ··· + *x*_{n} provides the arithmetic mean. The function *g*(*x*_{1}, *x*_{2}, ..., *x*_{n}) =*x*_{1}*x*_{2}···*x*_{n} (where the list elements are positive numbers) provides the geometric mean. The function *g*(*x*_{1}, *x*_{2}, ..., *x*_{n}) =−(*x*_{1}^{−1}+*x*_{2}^{−1}+ ··· + *x*_{n}^{−1}) (where the list elements are positive numbers) provides the harmonic mean.^{ [2] }

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 return *R* of 0.0600 or 6.00%.

Given a time series such as daily stock market prices or yearly temperatures people often want to create a smoother series.^{ [3] } This helps to show underlying trends or perhaps periodic behavior. An easy way to do this is the *moving average*: one chooses a number *n* and creates a new series by taking the arithmetic mean of the first *n* values, then moving forward one place by dropping the oldest value and introducing a new value at the other end of the list, and so on. This is the simplest form of moving average. More complicated forms involve using a weighted average. The weighting can be used to enhance or suppress various periodic behavior and there is very extensive analysis of what weightings to use in the literature on filtering. In digital signal processing the term "moving average" is used even when the sum of the weights is not 1.0 (so the output series is a scaled version of the averages).^{ [4] } The reason for this is that the analyst is usually interested only in the trend or the periodic behavior.

The first recorded time that the arithmetic mean was extended from 2 to n cases for the use of estimation was in the sixteenth century. From the late sixteenth century onwards, it gradually became a common method to use for reducing errors of measurement in various areas.^{ [5] }^{ [6] } At the time, astronomers wanted to know a real value from noisy measurement, such as the position of a planet or the diameter of the moon. Using the mean of several measured values, scientists assumed that the errors add up to a relatively small number when compared to the total of all measured values. The method of taking the mean for reducing observation errors was indeed mainly developed in astronomy.^{ [5] }^{ [7] } A possible precursor to the arithmetic mean is the mid-range (the mean of the two extreme values), used for example in Arabian astronomy of the ninth to eleventh centuries, but also in metallurgy and navigation.^{ [6] }

However, there are various older vague references to the use of the arithmetic mean (which are not as clear, but might reasonably have to do with our modern definition of the mean). In a text from the 4th century, it was written that (text in square brackets is a possible missing text that might clarify the meaning):^{ [8] }

- In the first place, we must set out in a row the sequence of numbers from the monad up to nine: 1, 2, 3, 4, 5, 6, 7, 8, 9. Then we must add up the amount of all of them together, and since the row contains nine terms, we must look for the ninth part of the total to see if it is already naturally present among the numbers in the row; and we will find that the property of being [one] ninth [of the sum] only belongs to the [arithmetic] mean itself...

Even older potential references exist. There are records that from about 700 BC, merchants and shippers agreed that damage to the cargo and ship (their "contribution" in case of damage by the sea) should be shared equally among themselves.^{ [7] } This might have been calculated using the average, although there seem to be no direct record of the calculation.

The root is found in Arabic as عوار *awar*, a defect, or anything defective or damaged, including partially spoiled merchandise; and عواري *ʿawārī* (also عوارة *ʿawāra*) = "of or relating to *ʿawār*, a state of partial damage".^{ [9] } Within the Western languages the word's history begins in medieval sea-commerce on the Mediterranean. 12th and 13th century Genoa Latin *avaria* meant "damage, loss and non-normal expenses arising in connection with a merchant sea voyage"; and the same meaning for *avaria* is in Marseille in 1210, Barcelona in 1258 and Florence in the late 13th.^{ [10] } 15th-century French *avarie* had the same meaning, and it begot English "averay" (1491) and English "average" (1502) with the same meaning. Today, Italian *avaria*, Catalan *avaria* and French *avarie* still have the primary meaning of "damage". The huge transformation of the meaning in English began with the practice in later medieval and early modern Western merchant-marine law contracts under which if the ship met a bad storm and some of the goods had to be thrown overboard to make the ship lighter and safer, then all merchants whose goods were on the ship were to suffer proportionately (and not whoever's goods were thrown overboard); and more generally there was to be proportionate distribution of any *avaria*. From there the word was adopted by British insurers, creditors, and merchants for talking about their losses as being spread across their whole portfolio of assets and having a mean proportion. Today's meaning developed out of that, and started in the mid-18th century, and started in English.^{ [10] } .

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".

A second English usage, documented as early as 1674 and sometimes spelled "averish", is as the residue and second growth of field crops, which were considered suited to consumption by draught animals ("avers").^{ [11] }

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).

In mathematics, the **arithmetic–geometric mean** (**AGM**) of two positive real numbers *x* and *y* is defined as follows:

In statistics, a **central tendency** is a central or typical value for a probability distribution. It may also be called a **center** or **location** of the distribution. Colloquially, measures of central tendency are often called *averages.* The term *central tendency* dates from the late 1920s.

In mathematics, **generalized means** are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means. The generalized mean is also known as **power mean** or **Hölder mean**.

There are several kinds of **means** in various branches of mathematics.

In mathematics, a **square root** of a number *a* is a number *y* such that *y*^{2} = *a*; in other words, a number *y* whose *square* (the result of multiplying the number by itself, or *y* ⋅ *y*) is *a*. For example, 4 and −4 are square roots of 16 because 4^{2} = (−4)^{2} = 16. Every nonnegative real number *a* has a unique nonnegative square root, called the *principal square root*, which is denoted by √*a*, where √ is called the *radical sign* or *radix*. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 3^{2} = 3 · 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the *radicand*. The radicand is the number or expression underneath the radical sign, in this example 9.

In mathematics, the **geometric–harmonic mean** M(*x*, *y*) of two positive real numbers *x* and *y* is defined as follows: we form the geometric mean of *g*_{0} = *x* and *h*_{0} = *y* and call it *g*_{1}, i.e. *g*_{1} is the square root of *xy*. We also form the harmonic mean of *x* and *y* and call it *h*_{1}, i.e. *h*_{1} is the reciprocal of the arithmetic mean of the reciprocals of *x* and *y*. These may be done sequentially or simultaneously.

In statistics, given a set of data,

In mathematics, the three classical **Pythagorean means** are the arithmetic mean (*AM*), the geometric mean (*GM*), and the harmonic mean (*HM*). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.

In mathematics, the **logarithmic mean** is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer.

In mathematics, a **contraharmonic mean** is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, , where p = 2.

In statistics, the **median absolute deviation** (**MAD**) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.

In mathematics, a **geometric progression**, also known as a **geometric sequence**, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the *common ratio*. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.

In mathematics and statistics, the **Fréchet mean** is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. **Karcher mean** is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.

In calculus, and especially multivariable calculus, the **mean of a function** is loosely defined as the average value of the function over its domain. In one variable, the mean of a function *f*(*x*) over the interval (*a,b*) is defined by

- ↑ 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.^{[ permanent dead link ]} - 1 2 Bibby, John (1974). "Axiomatisations of the average and a further generalisation of monotonic sequences".
*Glasgow Mathematical Journal*.**15**: 63–65. doi:10.1017/s0017089500002135. - ↑ Box, George E.P.; Jenkins, Gwilym M. (1976).
*Time Series Analysis: Forecasting and Control*(revised ed.). Holden-Day. ISBN 0816211043. - ↑ Haykin, Simon (1986).
*Adaptive Filter Theory*. Prentice-Hall. ISBN 0130040525. - 1 2 "Studies in the History of Probability and Statistics: VII. The Principle of the Arithmetic Mean".
*Biometrika*.**45**: 130. doi:10.2307/2333051. - 1 2 Eisenhart, Churchill. "The development of the concept of the best mean of a set of measurements from antiquity to the present day." Unpublished presidential address, American Statistical Association, 131st Annual Meeting, Fort Collins, Colorado. 1971.
- 1 2 Bakker, Arthur. "The early history of average values and implications for education." Journal of Statistics Education 11.1 (2003): 17-26.
- ↑ "Waterfield, Robin. "The theology of arithmetic." On the Mystical, mathematical and Cosmological Symbolism of the First Ten Number (1988). page 70" (PDF). Archived from the original (PDF) on 2016-03-04. Retrieved 2018-11-27.
- ↑ Medieval Arabic had عور
*ʿawr*meaning "blind in one eye" and عوار*ʿawār*meant "any defect, or anything defective or damaged". Some medieval Arabic dictionaries are at Baheth.info, and some translation to English of what's in the medieval Arabic dictionaries is in Lane's*Arabic-English Lexicon*, pages 2193 and 2195. The medieval dictionaries do not list the word-form عوارية*ʿawārīa*.*ʿAwārīa*can be naturally formed in Arabic grammar to refer to things that have*ʿawār*, but in practice in medieval Arabic texts*ʿawārīa*is a rarity or non-existent, while the forms عواري*ʿawārī*and عوارة*ʿawāra*are frequently used when referring to things that have*ʿawār*or damage – this can be seen in the searchable collection of medieval texts at AlWaraq.net (book links are clickable on righthand side). - 1 2 The Arabic origin of
*avaria*was first reported by Reinhart Dozy in the 19th century. Dozy's original summary is in his 1869 book*Glossaire*. Summary information about the word's early records in Italian-Latin, Italian, Catalan, and French is at*avarie*@ CNRTL.fr. The seaport of Genoa is the location of the earliest-known record in European languages, year 1157. A set of medieval Latin records of*avaria*at Genoa is in the downloadable lexicon*Vocabolario Ligure*, by Sergio Aprosio, year 2001,*avaria*in Volume 1 pages 115-116. Many more records in medieval Latin at Genoa are at StoriaPatriaGenova.it, usually in the plurals*avariis*and*avarias*. At the port of Marseille in the 1st half of the 13th century notarized commercial contracts have dozens of instances of Latin*avariis*(ablative plural of*avaria*), as published in Blancard year 1884. Some information about the English word over the centuries is at NED (year 1888). See also the definition of English "average" in English dictionaries published in the early 18th century, i.e., in the time period just before the big transformation of the meaning: Kersey-Phillips' dictionary (1706), Blount's dictionary (1707 edition), Hatton's dictionary (1712), Bailey's dictionary (1726), Martin's dictionary (1749). Some complexities surrounding the English word's history are discussed in Hensleigh Wedgwood year 1882 page 11 and Walter Skeat year 1888 page 781. Today there is consensus that: (#1) today's English "average" descends from medieval Italian*avaria*, Catalan*avaria*, and (#2) among the Latins the word*avaria*started in the 12th century and it started as a term of Mediterranean sea-commerce, and (#3) there is no root for*avaria*to be found in Latin, and (#4) a substantial number of Arabic words entered Italian, Catalan and Provençal in the 12th and 13th centuries starting as terms of Mediterranean sea-commerce, and (#5) the Arabic*ʿawār | ʿawārī*is phonetically a good match for*avaria*, as conversion of w to v was regular in Latin and Italian, and*-ia*is a suffix in Italian, and the Western word's earliest records are in Italian-speaking locales (writing in Latin). And most commentators agree that (#6) the Arabic*ʿawār | ʿawārī*= "damage | relating to damage" is semantically a good match for*avaria*= "damage or damage expenses". A minority of commentators have been dubious about this on the grounds that the early records of Italian-Latin*avaria*have, in some cases, a meaning of "an expense" in a more general sense – see TLIO (in Italian). The majority view is that the meaning of "an expense" was an expansion from "damage and damage expense", and the chronological order of the meanings in the records supports this view, and the broad meaning "an expense" was never the most commonly used meaning. On the basis of the above points, the inferential step is made that the Latinate word came or probably came from the Arabic word. - ↑ Ray, John (1674).
*A Collection of English Words Not Generally Used*. London: H. Bruges. Retrieved 18 May 2015.

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