# Average

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In colloquial language, an average is a single number taken as representative of a non-empty 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.

## General properties

If all numbers in a list are the same number, then their average is also equal to this number. This property is shared by each of the many types of average.

Another universal property is monotonicity: if two lists of numbers A and B have the same length, and each entry of list A is at least as large as the corresponding entry on list B, then the average of list A is at least as that of list B. Also, all averages satisfy linear homogeneity: if all numbers of a list are multiplied by the same positive number, then its average changes by the same factor.

In some types of average, the items in the list are assigned different weights before the average is determined. These include the weighted arithmetic mean, the weighted geometric mean and the weighted median. Also, for some types of moving average, the weight of an item depends on its position in the list. Most types of average, however, satisfy permutation-insensitivity: all items count equally in determining their average value and their positions in the list are irrelevant; the average of (1, 2, 3, 4, 5) is the same as that of (3, 2, 5, 4, 1).

## Types

### Pythagorean means

The arithmetic mean, the geometric mean and the harmonic mean are known collectively as the Pythagorean means.

#### Arithmetic mean

The most common type of average is the arithmetic mean. If n numbers are given, each number denoted by ai (where i = 1,2, ..., n), the arithmetic mean is the sum of the as divided by n or

${\displaystyle {\text{AM}}={\frac {1}{n}}\sum _{i=1}^{n}a_{i}={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}}$

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.

#### Geometric mean

The geometric mean of n positive 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

${\displaystyle {\text{GM}}={\sqrt[{n}]{\prod _{i=1}^{n}a_{i}}}={\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}}}}$

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

Example: Geometric mean of 2 and 8 is ${\displaystyle {\text{GM}}={\sqrt {2\cdot 8}}=4}$

#### 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:

${\displaystyle {\text{HM}}={\frac {1}{{\dfrac {1}{n}}\displaystyle \sum \limits _{i=1}^{n}{\frac {1}{a_{i}}}}}={\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}}$

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

${\displaystyle {\frac {2}{{\frac {1}{60}}+{\frac {1}{40}}}}=48}$

#### Inequality concerning AM, GM, and HM

A well known inequality concerning arithmetic, geometric, and harmonic means for any set of positive numbers is

${\displaystyle {\text{AM}}\geq {\text{GM}}\geq {\text{HM}}}$

(The alphabetical order of the letters A, G, and H is preserved in the inequality.) See Inequality of arithmetic and geometric means.

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

## Statistical location

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.

Comparison of common averages of values { 1, 2, 2, 3, 4, 7, 9 }
TypeDescriptionExampleResult
Arithmetic mean Sum of values of a data set divided by number of values: ${\displaystyle \scriptstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$(1+2+2+3+4+7+9) / 74
Median Middle value separating the greater and lesser halves of a data set1, 2, 2, 3, 4, 7, 93
Mode Most frequent value in a data set1, 2, 2, 3, 4, 7, 92
Mid-range The arithmetic mean of the highest and lowest values of a set(1+9) / 25

### Mode

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.

### Median

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.

### Mid-range

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

## Summary of types

NameEquation or description
Arithmetic mean ${\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {1}{n}}(x_{1}+\cdots +x_{n})}$
Median The middle value that separates the higher half from the lower half of the data set
Geometric median A rotation invariant extension of the median for points in Rn
Mode The most frequent value in the data set
Geometric mean ${\displaystyle {\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}={\sqrt[{n}]{x_{1}\cdot x_{2}\dotsb x_{n}}}}$
Harmonic mean ${\displaystyle {\frac {n}{{\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}}}}$
(or RMS)
${\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}}$
Cubic mean ${\displaystyle {\sqrt[{3}]{{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{3}}}={\sqrt[{3}]{{\frac {1}{n}}\left(x_{1}^{3}+x_{2}^{3}+\cdots +x_{n}^{3}\right)}}}$
Generalized mean ${\displaystyle {\sqrt[{p}]{{\frac {1}{n}}\cdot \sum _{i=1}^{n}x_{i}^{p}}}}$
Weighted mean ${\displaystyle {\frac {\sum _{i=1}^{n}w_{i}x_{i}}{\sum _{i=1}^{n}w_{i}}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}}$
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. 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 ${\displaystyle {\frac {1}{2}}\left(\max x+\min x\right)}$
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.

## Miscellaneous types

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:

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

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 [2] 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. [2]

### 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. 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%.

## Moving average

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.

## History

### 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. [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.

### 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". [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).

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. [12]

## Averages as a rhetorical tool

Due to the aforementioned colloquial nature of the term "average", the term can be used to obfuscate the true meaning of data and suggest varying answers to questions based on the averaging method (most frequently arithmetic mean, median, or mode) used. In his article "Framed for Lying: Statistics as In/Artistic Proof", University of Pittsburgh faculty member Daniel Libertz comments that statistical information is frequently dismissed from rhetorical arguments for this reason. [13] However, due to their persuasive power, averages and other statistical values should not be discarded completely, but instead used and interpreted with caution. Libertz invites us to engage critically not only with statistical information such as averages, but also with the language used to describe the data and its uses, saying: "If statistics rely on interpretation, rhetors should invite their audience to interpret rather than insist on an interpretation." [13] In many cases, data and specific calculations are provided to help facilitate this audience-based interpretation.

## Related Research Articles

In mathematics and statistics, the arithmetic mean, or simply the mean or the average, 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.

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, 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 nth root of the product of n numbers, i.e., for a set of numbers x1, x2, ..., xn, the geometric mean is defined as

In mathematics, generalized means are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means.

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 statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result.

There are several kinds of mean in mathematics, especially in statistics:

In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function . It is also called Kolmogorov mean after Russian mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.

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.

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 g0 = x and h0 = y and call it g1, i.e. g1 is the square root of xy. We also form the harmonic mean of x and y and call it h1, i.e. h1 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 statistical analysis of binary classification, the F-score or F-measure is a measure of a test's accuracy. It is calculated from the precision and recall of the test, where the precision is the number of true positive results divided by the number of all positive results, including those not identified correctly, and the recall is the number of true positive results divided by the number of all samples that should have been identified as positive. Precision is also known as positive predictive value, and recall is also known as sensitivity in diagnostic binary classification.

In mathematics, the Lehmer mean of a tuple of positive real numbers, named after Derrick Henry Lehmer, is defined as:

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 mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero 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

## References

1. 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.
2. Bibby, John (1974). "Axiomatisations of the average and a further generalisation of monotonic sequences". Glasgow Mathematical Journal . 15: 63–65. doi:.
3. Box, George E.P.; Jenkins, Gwilym M. (1976). Time Series Analysis: Forecasting and Control (revised ed.). Holden-Day. ISBN   0816211043.
4. Haykin, Simon (1986). Adaptive Filter Theory. Prentice-Hall. ISBN   0130040525.
5. Plackett, R. L. (1958). "Studies in the History of Probability and Statistics: VII. The Principle of the Arithmetic Mean". Biometrika. 45 (1/2): 130–135. doi:10.2307/2333051. JSTOR   2333051.
6. "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.
7. 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 Archived 2013-10-29 at the Wayback Machine , 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).