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Encyclopedia > Average

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. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. Batting average is a statistic in both cricket and baseball measuring the performance of cricket batsmen and baseball hitters, respectively. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ... Image File history File links Question_book-3. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In statistics, central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. ... A data set (or dataset) is a collection of data, usually presented in tabular form. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... Descriptive statistics are used to describe the basic features of the data in a study. ...

The most common method is the arithmetic mean, but there are many other types of averages.[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. Look up common in Wiktionary, the free dictionary. ... 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 by the number of items in the list. ...

### 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. 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. 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 by the number of items in the list. ...

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. Thus, the geometric mean of 2 and 8 is obtained by solving for G in the following equation: 2 * 8 = G * G. Thus, the geometric mean of 2 and 8 is G = sqrt(2 * 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 * 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. The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ...

### Mode and median

The most frequently occurring number in a list of numbers is called the mode. 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 a the resulting list with all values replaced by the same value. Since they are already all the same, this does not require any change. In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...

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. 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. This article is about the statistical concept. ...

### Annualized return

Th 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. The following table lists many specialized symbols commonly used in 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 $frac{n}{frac{1}{x_1} + frac{1}{x_2} + cdots + frac{1}{x_n}}$
(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. 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 central tendency". In the sense of Lp spaces, the correspondence is: Calculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. ... In descriptive statistics, statistical dispersion (also called statistical variability) is quantifiable variation of measurements of differing members of a population within the scale on which they are measured. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

Lp dispersion central tendency
L1 average absolute deviation median
L2 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 L2 and $L^infty$ norms are convex functions. 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 absolute deviation of an element of a data set is the absolute difference between that element and a given point. ... This article is about the statistical concept. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... This article is about mathematical mean. ... The midrange of a set of statistical data values is the arithmetic mean of the smallest and largest values in the set. ... Convex optimization is a subfield of mathematical optimization. ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...

Similarly, the mode minimizes qualitative variation.[citation needed] In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... Qualitative variation (QV) allows us to assess the degree of statistical dispersion in nominal distributions. ...

## Miscellaneous types

Other more sophisticated averages are: trimean, trimedian, and normalized mean. These are usually more representative of the whole data set.[citation needed]

One can create one's own average metric using generalized f-mean: In mathematics and statistics, the generalised f-mean is the natural generalisation of the more familar means such as the arithmetic mean and the geometric mean, using a function f(x). ...

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

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) = ex, 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, ..., xn), 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, ..., xn) = 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, ..., xn) =x1+x2+ ...+ xn provides the arithmetic mean. The function g(x1, x2, ..., xn) =x1·x2· ...· xn provides the geometric mean. The function g(x1, x2, ..., xn) =x1−1+x2−1+ ...+ xn−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. The term moving average is used in different contexts. ...

## 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 type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean". 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 of part of the ship or cargo to save the whole in an emergency. ...

Results from FactBites:

 Average - Wikipedia, the free encyclopedia (665 words) This "average" (arithmetic mean) income is higher than most people's incomes, because high income outliers skew the result higher (in contrast, the median income "resists" such skew). Other more sophisticated averages, usually more representative of the whole dataset are: trimean, trimedian, and normalised mean, to name a few. The only significant reason why the arithmetic mean (classical average) is generally used in scientific papers is that there are various (statistical) tests which can be applied to test the statistical significance of the results, as well as the correlations that are explored through these metrics.
 Mean - Wikipedia, the free encyclopedia (1058 words) The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or most likely (mode). The geometric mean is an average that is useful for sets of numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean). The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).
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