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The most commonly used definition of the average is the arithmetic mean, also known as "arithmetic average" i.e. the sum divided by the count, so the "average" of the list of numbers [2, 3, 4, 7, 9] is generally considered to be (2+3+4+7+9)/5 = 25/5 = 5. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted using an overhead bar,

{\displaystyle {\bar {x}}}

. If the numbers are from observing a sample of a larger group, the arithmetic mean is termed the sample mean (

{\displaystyle {\bar {x}}}

) to distinguish it from the group mean (or expected value) of the underlying distribution, denoted

{\displaystyle \mu }

{\displaystyle \mu _{x}}

However, other meanings are sometimes used depending on the context, which can lead to confusion; for instance, in teaching, "average" sometimes refers to "the three Ms": mean, median, and mode.

The median, defined as the value in the center after sorting the group, is usually used as the average in situations where the data is skewed or has outliers, in order to focus on the main part of the group rather than the long tail. For example, the average personal income is usually given as the median income, so that it represents the majority of the population rather than being overly influenced by the much higher incomes of the few rich people.

The harmonic mean, defined as the reciprocal of the mean of the reciprocals, is used in a variety of situations involving rates or ratios, such as computing the average speed from multiple measurements taken over the same distance. Indeed, unlike an arithmetic mean or median of speeds, a harmonic mean of speeds will give the value of the constant speed that would cause one to travel the same distance in the same amount of time.

The mode represents the most common value found in the group. It can be used when the data is categorical rather than numeric, when the frequency of each value is relevant (such as where a histogram, bar chart, or probability density function is being referenced), or to find a value that represents the majority of the group.

Other statistics that can be used as an average include the mid-range, the quadratic mean or the geometric mean, but they are rarely referred to as "the average".

Statistical location

In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may colloquially be called an "average" (more formally, a measure of central tendency). The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions.

The mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. If the random variable is denoted by

{\displaystyle X}

, then the mean is also known as the expected value of

{\displaystyle X}

(denoted

(

)

{\displaystyle E(X)}

). For a discrete probability distribution, the mean is given by

∑

(

)

{\displaystyle \textstyle \sum xP(x)}

, where the sum is taken over all possible values of the random variable and

(

)

{\displaystyle P(x)}

is the probability mass function. For a continuous distribution, the mean is

∫

−

∞

(

)

{\displaystyle \textstyle \int _{-\infty }^{\infty }xf(x)\,dx}

, where

(

)

{\displaystyle f(x)}

is the probability density function. In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure. The mean need not exist or be finite; for some probability distributions the mean is infinite (+∞ or −∞), while for others the mean is undefined.

Possible averages

Even though perhaps not an average, the

{\displaystyle \tau }

th quantile (another summary statistic that generalizes the median) can similarly be expressed as a solution to the optimization problem

argmin

∈

∑

max

(

−

)

(

−

)

(

−

)

argmin

∈

∑

(

−

(

−

)

{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}\max {\big (}(1-\tau )(x_{i}-x),\,\tau (x-x_{i}){\big )}={\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}{\big (}|x-x_{i}|+(1-2\tau )\,x{\big )}}

which aims to minimize the total tilted absolute value loss (or quantile loss or pinball loss).

Other more sophisticated averages are: trimmore sophistiean, trimedian, and normalized mean, with their generalizations.

In a more general fashion, one can create their own average metric using the generalized f-mean:

−

(

[

(

)

(

)

⋯

(

)

]

)

{\displaystyle y=f^{-1}\left({\frac {1}{n}}\left[f(x_{1})+f(x_{2})+\cdots +f(x_{n})\right]\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.

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)−1) (where the list elements are positive numbers) provides the harmonic mean.

Origin

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. 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 mainly developed in astronomy. 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.

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):

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. This might have been calculated using the average, although there seem to be no direct record of the calculation.

Etymology

The root is found in Arabic as عوار ʿawār, 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". 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. 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 transformation of the meaning in English began 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.

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").

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).

The Oxford English Dictionary, however, says that derivations from German hafen haven, and Arabic ʿawâr loss, damage, have been "quite disposed of" and the word has a Romance origin.

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Definitions

Central tendency

General properties

Statistical location

Relationships between the mean, median and mode

Possible averages

Pythagorean means

Power mean

Quasi-arithmetic mean

Weighted arithmetic mean

Truncated mean

Mean of a function

Mean of angles and cyclical quantities

Fréchet mean

Center of a triangle

Swanson's rule

Moving average

Solutions to variational problems

Uniqueness

Clustering

Information geometry

History

Origin

Etymology

Averages as a rhetorical tool

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