Average

In mathematics, an average, or central tendency^[1] of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average.

There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. These include arithmetic mean, the median and the mode. Other statistical measures such as the standard deviation and the range are called measures of spread and describe how spread out the data is.

An average is a single value that is meant to typify a list of values. If all the numbers in the list are the same, then this number should be used. If the numbers are not the same, the average is calculated by combining the values from the set in a specific way and computing a single number as being the average of the set.

The most common method is the arithmetic mean but there are many other types of central tendency, such as median (which is used most often when the distribution of the values is skewed with some small numbers of very high values, as seen with house prices or incomes).^[2]

1 Calculation
2 Types
3 Solutions to variational problems
4 Miscellaneous types
5 In data streams
6 Averages of functions
7 Etymology
8 Notes
9 See also
10 References
11 External links

Calculation

The three most common averages are the Pythagorean means -- the arithmetic mean, the geometric mean, and the harmonic mean.

Arithmetic mean

Main article: Arithmetic mean

If n numbers are given, each number denoted by a_i, where i = 1, ..., n, the arithmetic mean is the [sum] of the a_i's divided by n or

$AM=\frac{1}{n}\sum_{i=1}^na_i=\frac{a_1%2Ba_2%2B\cdots%2Ba_n}{n}.$

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 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

Main article: Geometric mean

The geometric mean of n numbers is obtained by multiplying them all together and then taking the nth root. In algebraic terms, the geometric mean of a₁, a₂, ..., a_n is defined as

$\text{GM=} \sqrt[n]{\prod_{i=1}^n a_i}=\sqrt[n]{a_1 a_2 \cdots a_n}.$

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 $GM = \sqrt{2 \cdot 8} = 4.$

Harmonic mean

Main article: Harmonic mean

Harmonic mean for a set of numbers a₁, a₂, ..., a_n is defined as the reciprocal of the arithmetic mean of the reciprocals of a_i's:

$HM = \frac{1}{\frac{1}{n}\sum_{i=1}^n \frac{1}{a_i}}=\frac{n}{\frac{1}{a_1}%2B\frac{1}{a_2}%2B\cdots%2B\frac{1}{a_n}}.$

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

$\frac{2}{1/60%2B1/40}=48.$

Inequality concerning AM, GM, and HM

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

$AM \ge GM \ge HM. \,$

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.

Mode and median

Main articles: Mode (statistics) and Median

The most frequently occurring number in a list is called the mode. The mode of the list (1, 2, 2, 3, 3, 3, 4) is 3. The mode is not necessarily well defined, the list (1, 2, 2, 3, 3, 5) has the two modes 2 and 3. The mode can be subsumed under the general method of defining averages by understanding it as taking the list and setting each member of the list equal to the most common value in the list if there is a most common value. This list is then equated to the resulting list with all values replaced by the same value. Since they are already all the same, this does not require any change. 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.

Average Percentage Return

The average percentage return is a type of average used in finance. It is an example of a geometric mean. 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, 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%. Note that changing the order to find the average percentage returns of +60% and −10% gives the same result as the average percentage returns of −10% and +60%.

This method can be generalized to examples in which the periods are not all of one-year duration. Average percentage of a set of returns is a variation on the geometric average that provides the intensive property of a return per year corresponding to a list of percentage returns. For example, consider a period of a half of a year for which the return is −23% and a period of two and one 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%.

Types

The table of mathematical symbols explains the symbols used below.

Name	Equation or description
Arithmetic mean	$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i = \frac{1}{n} (x_1%2B\cdots%2Bx_n)$
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	$\bigg(\prod_{i=1}^n x_i \bigg)^{\frac{1}{n}} = \sqrt[n]{x_1 \cdot x_2 \dotsb x_n}$
Harmonic mean	$\frac{n}{\frac{1}{x_1} %2B \frac{1}{x_2} %2B \cdots %2B \frac{1}{x_n}}$
Quadratic mean (or RMS)	$\sqrt{\frac{1}{n} \sum_{i=1}^{n} x_i^2} = \sqrt {\frac{x_1^2 %2B x_2^2 %2B \cdots %2B x_n^2}{n}}$
Generalized mean	$\sqrt[p]{\frac{1}{n} \cdot \sum_{i=1}^n x_{i}^p}$
Weighted mean	$\frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} = \frac{w_1 x_1 %2B w_2 x_2 %2B \cdots %2B w_n x_n}{w_1 %2B w_2 %2B \cdots %2B w_n}$
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
Midrange	$\frac{\max x %2B \min x}{2}$
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
Annualization	${\left[ \prod (1%2BR_i )^{t_i} \right] }^{1/\sum t_i} -1$

Solutions to variational problems

Several measures of central tendency can be characterized as solving a variational problem, in the sense of the calculus of variations, namely minimizing variation from the center. That is, given a measure of statistical dispersion, one asks for a measure of central tendency that minimizes variation: such that variation from the center is minimal among all choices of center. In a quip, "dispersion precedes location". In the sense of L^p spaces, the correspondence is:

L^p	dispersion	central tendency
L¹	average absolute deviation	median
L²	standard deviation	mean
L^∞	maximum deviation	midrange

Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point. The uniqueness of this characterization of mean follows from convex optimization. Indeed, for a given (fixed) data set x, the function

$f_2(c) = \|x-c\|_2$

represents the dispersion about a constant value c relative to the L² norm. Because the function ƒ₂ is a strictly convex coercive function, the minimizer exists and is unique.

Note that the median in this sense is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation. The dispersion in the L¹ norm, given by

$f_1(c) = \|x-c\|_1$

is not strictly convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. In spite of this, the minimizer is unique for the L^∞ norm.

Miscellaneous types

Other more sophisticated averages are: trimean, trimedian, and normalized mean, with their generalizations.^[3]

One can create one's own average metric using generalized f-mean:

$y = f^{-1}\left(\frac{f(x_1)%2Bf(x_2)%2B\cdots%2Bf(x_n)}{n}\right),$

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. Another example, expmean (exponential mean) is a mean using the function f(x) = e^x, and it is inherently biased towards the higher values. However, this method for generating means is not general enough to capture all averages. A more general method for defining an average, y, takes any function of a list g(x₁, x₂, ..., x_n), which is symmetric under permutation of the members of the list, and equates it to the same function with the value of the average replacing each member of the list: g(x₁, x₂, ..., x_n) = g(y, y, ..., y). 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, ..., x_n) =x₁+x₂+ ...+ x_n provides the arithmetic mean. The function g(x₁, x₂, ..., x_n) =x₁·x₂· ...· x_n provides the geometric mean. The function g(x₁, x₂, ..., x_n) =x₁⁻¹+x₂⁻¹+ ...+ x_n⁻¹ provides the harmonic mean. (See John Bibby (1974) “Axiomatisations of the average and a further generalisation of monotonic sequences,” Glasgow Mathematical Journal, vol. 15, pp. 63–65.)

In data streams

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 about which recent data is in some way clustered. 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 $k$ upon seeing new element $x_n$ :

$\mu_{n,{n-k}} = \mu_{n-1,n-k-1} %2B \frac{x_n}{k} - \frac{x_{n-k-1}}{k}$

Averages of functions

The concept of average can be extended to functions.^[4] In calculus, the average value of an integrable function ƒ on an interval [a,b] is defined by

$\overline{f} = \frac{1}{b-a}\int_a^bf(x)\,dx.$

Etymology

An early meaning (c. 1500) of the word average is "damage sustained at sea". The root is found in Arabic as awar, in Italian as avaria, in French as avarie and in Dutch as averij. 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".

However, according to the Oxford English Dictionary, the earliest usage in English (1489 or earlier) 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). This pre-existing term thus lay to hand when an equivalent for avarie was wanted.

Notes

^ In statistics, the term central tendency is used in some fields of empirical research to refer to what statisticians sometimes call "location".
^ An axiomatic approach to averages is provided by John Bibby (1974) "Axiomatisations of the average and a further generalization of monotonic sequences", Glasgow Mathematical Journal, vol. 15, pp. 63–65.
^ 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. http://www.upo.es/RevMetCuant/art.php?id=38.
^ G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities (2nd ed.), Cambridge University Press, ISBN 978-0521358804, 1988.

References

Hardy, G.H.; Littlewood, J.E.; Pólya, G. (1988). Inequalities (2nd ed.). Cambridge University Press. ISBN 978-0521358804