Average

"Averages" redirects here. For the baseball and cricket terminology, see Batting average. Batting average is a Statistic in both Cricket and Baseball measuring the performance of cricket batsmen and baseball hitters, respectively

"Mean value" redirects here. For the theorem in calculus, see Mean value theorem. In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative

In mathematics, an average, or central tendency ^[1] of a data set refers to a measure of the "middle" or "expected" value of the data set. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A data set (or dataset) is a collection of Data, usually presented in tabular form There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. Descriptive Statistics are used to describe the basic features of the Data gathered from an experimental study in various ways

The most common method is the arithmetic mean, but there are many other types of averages. In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided ^[2]The average is calculated by combining the measurements related to a group of people or objects, to compute a number as being the average of the group.

1 Calculation
2 Types
3 Solutions to variational problems
4 Miscellaneous types
5 In data streams
6 Etymology
7 Footnotes
8 External links

Calculation

Arithmetic mean

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 all the same, an easy way to get a representative value from a list is to randomly pick any number from the list. However, the word 'average' is usually reserved for more sophisticated methods that are generally found to be more useful.

The most common type of average is the arithmetic mean, often simply called the mean. In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided The arithmetic mean of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. It is then simple to 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. It is simple to find that A = (2 + 8 + 11)/3 = 7.

Again, changing the order of the three members of the list does not change the result: A = (8 + 11 + 2)/3 = 7, and that 7 is between 2 and 11. This summation method is easily generalized for lists with any number of elements. However, the mean of a list of integers is not necessarily an integer. "The average family has 1. 7 children" is a jarring way of making a statement that is more appropriately expressed by "the average number of children in the collection of families examined is 1. 7".

Geometric mean

With geometric mean, instead of adding numbers, the numbers are multiplied. The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers Thus, the geometric mean of 2 and 8 is obtained by solving for G in the following equation: $2 \cdot 8 = G \cdot G$ . Thus, the geometric mean of 2 and 8 is $G = \sqrt{2 \cdot 8} = 4$ . And again it is seen that changing the order of the members of the list to be averaged does not change the result: $G = \sqrt{8 \cdot 2} = 4$ . In order to make sense of the requirement that the mean must be at least as big as the smallest member of the list and no bigger than the largest, the geometric mean is usually only applied to lists of positive numbers, not to lists that can include negative numbers.

Mode and median

The most frequently occurring number in a list of numbers is called the mode. In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution. So 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.

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. In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution 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. Now do the same for the equal-sized list consisting of all the same value M: M, M, M, M. It is already ordered. We remove the two end values to get M, M. We take their arithmetic mean to get M. Finally, set this result equal to our previous result to get M = 5.

Annualized return

The annualized return is a type of average used in finance. For example, if there are two years in which the return in the first year is -10% and the return in the second year is +60%, then the annualized return, R, can be obtained by solving the equation: (1 - 10%) × (1 + 60%) = (1 + R) × (1 + R). The value of R that makes this equation true is 20%. Note that changing the order to find the annualized returns of +60% and -10% gives the same result as the annualized returns of -10% and +60%.

This method can be generalized to examples in which the periods are not all of one-year duration. Annualization 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 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 annualized return for the combined period is the single year return, R, that is the solution of the following equation: (1 - 23%)^{0. 5} × (1 + 13%)^{2. 5} = (1 + R)^{0. 5+2. 5}, giving an annualized return R of 6. 00%.

Types

The table of mathematical symbols explains the symbols used below. This is a listing of common symbols found within all branches of the science of Mathematics.

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

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. In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution The geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution. The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average. In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric The weighted mean is similar to an Arithmetic mean (the most common type of Average) where instead of each of the data points contributing equally to the final average A truncated mean or trimmed mean is a Statistical Measure of central tendency, much like the Mean and Median. The interquartile mean (IQM is a statistical measure of central tendency, much like the Mean (in more popular terms called the Average) the In Descriptive statistics, the interquartile range (IQR, also called the midspread, middle fifty and middle of the #s, is a measure of In Statistics, the mid-range or mid-extreme of a set of statistical data values is the Arithmetic mean of the maximum and minimum values in a Data A Winsorized mean is a Winsorized Statistical Measure of central tendency, much like the Mean and Median, and even more similar to Compound Annual Growth Rate (CAGR is a business and investing specific term for the geometric mean growth rate on an annualized basis Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. 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 Statistics, (statistical dispersion (also called statistical variability or variation) is variability or spread in a Variable or a Probability In a quip, "dispersion precedes central tendency". In the sense of $L p$ spaces, the correspondence is:

$L p$	dispersion	central tendency
$L 1$	average absolute deviation	median
$L 2$	standard deviation	mean
$L^\infty$	maximum deviation	midrange

Thus standard deviation about the mean is lower than standard deviation about any other point; the uniqueness of this characterization of mean and midrange follows from convex optimization, as the $L 2$ and $L^\infty$ norms are convex functions. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding In Statistics, the absolute deviation of an element of a Data set is the absolute difference between that element and a given point In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Statistics, the mid-range or mid-extreme of a set of statistical data values is the Arithmetic mean of the maximum and minimum values in a Data Convex optimization is a subfield of mathematical optimization. In Mathematics, a real-valued function f defined on an interval (or on any Convex subset of some Vector space) is called convex 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.

Similarly, the mode minimizes qualitative variation. In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution. An index of qualitative variation (IQV is a measure of Statistical dispersion in Nominal distributions There are a variety of these but they have been relatively

Miscellaneous types

Other more sophisticated averages are: trimean, trimedian, and normalized mean. In Statistics, the trimean (TM is a measure of a Probability distribution 's location, defined as a Weighted average of the distribution's These are usually more representative of the whole data set.

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

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

where f is any invertible function. In Mathematics and Statistics, the quasi-arithmetic mean or generalised f -mean is one generalisation of the more familiar Means such 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. In Statistics, a moving average, also called a rolling average and sometimes a running average, refers to a statistical technique used to analyze a 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.

Etymology

The original meaning of the word average is "damage sustained at sea": the same word is found in Arabic as awar, in Italian as avaria and in French as avarie. 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 law of general average is a legal principle of Maritime law according to which all parties in a sea venture proportionally share any losses resulting from a voluntary sacrifice The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean".

Footnotes

^ In statistics, the term central tendency is used in some fields of empirical research to refer to what statisticians sometimes call "location". Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Empirical research is any research that bases its findings on direct or indirect Observation as its test of Reality.
^ 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.

External links

Dictionary

-noun

(arithmetic) The arithmetic mean.
(arithmetic) Any measure of central tendency, especially the mean, median or mode (see Usage notes below).
In various sports, an indication of a player's ability calculated from his scoring record, etc.

-adjective

(not comparable) Constituting or relating to the average.
Neither very good nor very bad; rated somewhere in the middle of all others in the same category.
Typical.

-verb

(transitive, informal) To compute the arithmetic mean of.
(transitive) Over a period of time or across members of a population, to have or generate a mean value of.

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