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Arithmetic mean

In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk ˈmn/ arr-ith-MET-ik), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection.[1] The collection is often a set of results from an experiment, an observational study, or a survey. The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic.

In addition to mathematics and statistics, the arithmetic mean is frequently used in economics, anthropology, history, and almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population.

While the arithmetic mean is often used to report central tendencies, it is not a robust statistic: it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions, such as the distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median, may provide a better description of central tendency.

Definition edit

The arithmetic mean of a set of observed data is equal to the sum of the numerical values of each observation, divided by the total number of observations. Symbolically, for a data set consisting of the values  , the arithmetic mean is defined by the formula:

 [2]

(For an explanation of the summation operator, see summation.)

For example, if the monthly salaries of   employees are  , then the arithmetic mean is:

 

If the data set is a statistical population (i.e., consists of every possible observation and not just a subset of them), then the mean of that population is called the population mean and denoted by the Greek letter  . If the data set is a statistical sample (a subset of the population), it is called the sample mean (which for a data set   is denoted as  ).

The arithmetic mean can be similarly defined for vectors in multiple dimensions, not only scalar values; this is often referred to as a centroid. More generally, because the arithmetic mean is a convex combination (meaning its coefficients sum to  ), it can be defined on a convex space, not only a vector space.

Motivating properties edit

The arithmetic mean has several properties that make it interesting, especially as a measure of central tendency. These include:

  • If numbers   have mean  , then  . Since   is the distance from a given number to the mean, one way to interpret this property is by saying that the numbers to the left of the mean are balanced by the numbers to the right. The mean is the only number for which the residuals (deviations from the estimate) sum to zero. This can also be interpreted as saying that the mean is translationally invariant in the sense that for any real number  ,  .
  • If it is required to use a single number as a "typical" value for a set of known numbers  , then the arithmetic mean of the numbers does this best since it minimizes the sum of squared deviations from the typical value: the sum of  . The sample mean is also the best single predictor because it has the lowest root mean squared error.[3] If the arithmetic mean of a population of numbers is desired, then the estimate of it that is unbiased is the arithmetic mean of a sample drawn from the population.
  • The arithmetic mean is independent of scale of the units of measurement, in the sense that   So, for example, calculating a mean of liters and then converting to gallons is the same as converting to gallons first and then calculating the mean. This is also called first order homogeneity.

Additional properties edit

  • The arithmetic mean of a sample is always between the largest and smallest values in that sample.
  • The arithmetic mean of any amount of equal-sized number groups together is the arithmetic mean of the arithmetic means of each group.

Contrast with median edit

The arithmetic mean may be contrasted with the median. The median is defined such that no more than half the values are larger, and no more than half are smaller than it. If elements in the data increase arithmetically when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample  . The mean is  , as is the median. However, when we consider a sample that cannot be arranged to increase arithmetically, such as  , the median and arithmetic average can differ significantly. In this case, the arithmetic average is  , while the median is  . The average value can vary considerably from most values in the sample and can be larger or smaller than most.

There are applications of this phenomenon in many fields. For example, since the 1980s, the median income in the United States has increased more slowly than the arithmetic average of income.[4]

Generalizations edit

Weighted average edit

A weighted average, or weighted mean, is an average in which some data points count more heavily than others in that they are given more weight in the calculation.[5] For example, the arithmetic mean of   and   is  , or equivalently  . In contrast, a weighted mean in which the first number receives, for example, twice as much weight as the second (perhaps because it is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as  . Here the weights, which necessarily sum to one, are   and  , the former being twice the latter. The arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all weights are equal to the same number (  in the above example and   in a situation with   numbers being averaged).

Continuous probability distributions edit

 
Comparison of two log-normal distributions with equal median, but different skewness, resulting in various means and modes

If a numerical property, and any sample of data from it, can take on any value from a continuous range instead of, for example, just integers, then the probability of a number falling into some range of possible values can be described by integrating a continuous probability distribution across this range, even when the naive probability for a sample number taking one certain value from infinitely many is zero. In this context, the analog of a weighted average, in which there are infinitely many possibilities for the precise value of the variable in each range, is called the mean of the probability distribution. The most widely encountered probability distribution is called the normal distribution; it has the property that all measures of its central tendency, including not just the mean but also the median mentioned above and the mode (the three Ms[6]), are equal. This equality does not hold for other probability distributions, as illustrated for the log-normal distribution here.

Angles edit

Particular care is needed when using cyclic data, such as phases or angles. Taking the arithmetic mean of 1° and 359° yields a result of 180°. This is incorrect for two reasons:

  • Firstly, angle measurements are only defined up to an additive constant of 360° (  or  , if measuring in radians). Thus, these could easily be called 1° and -1°, or 361° and 719°, since each one of them produces a different average.
  • Secondly, in this situation, 0° (or 360°) is geometrically a better average value: there is lower dispersion about it (the points are both 1° from it and 179° from 180°, the putative average).

In general application, such an oversight will lead to the average value artificially moving towards the middle of the numerical range. A solution to this problem is to use the optimization formulation (that is, define the mean as the central point: the point about which one has the lowest dispersion) and redefine the difference as a modular distance (i.e., the distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°).

 
Proof without words of the inequality of arithmetic and geometric means:
  is the diameter of a circle centered on  ; its radius   is the arithmetic mean of   and  . Using the geometric mean theorem, triangle  's altitude   is the geometric mean. For any ratio  ,  .

Symbols and encoding edit

The arithmetic mean is often denoted by a bar (vinculum or macron), as in  .[3]

Some software (text processors, web browsers) may not display the "x̄" symbol correctly. For example, the HTML symbol "x̄" combines two codes — the base letter "x" plus a code for the line above (̄ or ¯).[7]

In some document formats (such as PDF), the symbol may be replaced by a "¢" (cent) symbol when copied to a text processor such as Microsoft Word.

See also edit

 
Geometric proof without words that max (a,b) > root mean square (RMS) or quadratic mean (QM) > arithmetic mean (AM) > geometric mean (GM) > harmonic mean (HM) > min (a,b) of two distinct positive numbers a and b [note 1]

Notes edit

  1. ^ If AC = a and BC = b. OC = AM of a and b, and radius r = QO = OG.
    Using Pythagoras' theorem, QC² = QO² + OC² ∴ QC = √QO² + OC² = QM.
    Using Pythagoras' theorem, OC² = OG² + GC² ∴ GC = √OC² − OG² = GM.
    Using similar triangles, HC/GC = GC/OC ∴ HC = GC²/OC = HM.

References edit

  1. ^ Jacobs, Harold R. (1994). Mathematics: A Human Endeavor (Third ed.). W. H. Freeman. p. 547. ISBN 0-7167-2426-X.
  2. ^ Weisstein, Eric W. "Arithmetic Mean". mathworld.wolfram.com. Retrieved 21 August 2020.
  3. ^ a b Medhi, Jyotiprasad (1992). Statistical Methods: An Introductory Text. New Age International. pp. 53–58. ISBN 9788122404197.
  4. ^ Krugman, Paul (4 June 2014) [Fall 1992]. "The Rich, the Right, and the Facts: Deconstructing the Income Distribution Debate". The American Prospect.
  5. ^ "Mean | mathematics". Encyclopedia Britannica. Retrieved 21 August 2020.
  6. ^ Thinkmap Visual Thesaurus (30 June 2010). "The Three M's of Statistics: Mode, Median, Mean June 30, 2010". www.visualthesaurus.com. Retrieved 3 December 2018.
  7. ^ . www.personal.psu.edu. Archived from the original on 31 March 2022. Retrieved 14 October 2018.

Further reading edit

External links edit

  • Calculations and comparisons between arithmetic mean and geometric mean of two numbers
  • Calculate the arithmetic mean of a series of numbers on fxSolver

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X redirects here For the character see macron diacritic For broader coverage of this topic see Mean In mathematics and statistics the arithmetic mean ˌ ae r ɪ 8 ˈ m ɛ t ɪ k ˈ m iː n arr ith MET ik arithmetic average or just the mean or average when the context is clear is the sum of a collection of numbers divided by the count of numbers in the collection 1 The collection is often a set of results from an experiment an observational study or a survey The term arithmetic mean is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means such as geometric and harmonic In addition to mathematics and statistics the arithmetic mean is frequently used in economics anthropology history and almost every academic field to some extent For example per capita income is the arithmetic average income of a nation s population While the arithmetic mean is often used to report central tendencies it is not a robust statistic it is greatly influenced by outliers values much larger or smaller than most others For skewed distributions such as the distribution of income for which a few people s incomes are substantially higher than most people s the arithmetic mean may not coincide with one s notion of middle In that case robust statistics such as the median may provide a better description of central tendency Contents 1 Definition 2 Motivating properties 2 1 Additional properties 3 Contrast with median 4 Generalizations 4 1 Weighted average 4 2 Continuous probability distributions 4 3 Angles 5 Symbols and encoding 6 See also 7 Notes 8 References 9 Further reading 10 External linksDefinition editThe arithmetic mean of a set of observed data is equal to the sum of the numerical values of each observation divided by the total number of observations Symbolically for a data set consisting of the values x 1 x n displaystyle x 1 dots x n nbsp the arithmetic mean is defined by the formula x 1 n i 1 n x i x 1 x 2 x n n displaystyle bar x frac 1 n left sum i 1 n x i right frac x 1 x 2 dots x n n nbsp 2 For an explanation of the summation operator see summation For example if the monthly salaries of 10 displaystyle 10 nbsp employees are 2500 2700 2400 2300 2550 2650 2750 2450 2600 2400 displaystyle 2500 2700 2400 2300 2550 2650 2750 2450 2600 2400 nbsp then the arithmetic mean is 2500 2700 2400 2300 2550 2650 2750 2450 2600 2400 10 2530 displaystyle frac 2500 2700 2400 2300 2550 2650 2750 2450 2600 2400 10 2530 nbsp If the data set is a statistical population i e consists of every possible observation and not just a subset of them then the mean of that population is called the population mean and denoted by the Greek letter m displaystyle mu nbsp If the data set is a statistical sample a subset of the population it is called the sample mean which for a data set X displaystyle X nbsp is denoted as X displaystyle overline X nbsp The arithmetic mean can be similarly defined for vectors in multiple dimensions not only scalar values this is often referred to as a centroid More generally because the arithmetic mean is a convex combination meaning its coefficients sum to 1 displaystyle 1 nbsp it can be defined on a convex space not only a vector space Motivating properties editThe arithmetic mean has several properties that make it interesting especially as a measure of central tendency These include If numbers x 1 x n displaystyle x 1 dotsc x n nbsp have mean x displaystyle bar x nbsp then x 1 x x n x 0 displaystyle x 1 bar x dotsb x n bar x 0 nbsp Since x i x displaystyle x i bar x nbsp is the distance from a given number to the mean one way to interpret this property is by saying that the numbers to the left of the mean are balanced by the numbers to the right The mean is the only number for which the residuals deviations from the estimate sum to zero This can also be interpreted as saying that the mean is translationally invariant in the sense that for any real number a displaystyle a nbsp x a x a displaystyle overline x a bar x a nbsp If it is required to use a single number as a typical value for a set of known numbers x 1 x n displaystyle x 1 dotsc x n nbsp then the arithmetic mean of the numbers does this best since it minimizes the sum of squared deviations from the typical value the sum of x i x 2 displaystyle x i bar x 2 nbsp The sample mean is also the best single predictor because it has the lowest root mean squared error 3 If the arithmetic mean of a population of numbers is desired then the estimate of it that is unbiased is the arithmetic mean of a sample drawn from the population The arithmetic mean is independent of scale of the units of measurement in the sense that avg c a 1 c a n c avg a 1 a n displaystyle text avg ca 1 cdots ca n c cdot text avg a 1 cdots a n nbsp So for example calculating a mean of liters and then converting to gallons is the same as converting to gallons first and then calculating the mean This is also called first order homogeneity Additional properties edit The arithmetic mean of a sample is always between the largest and smallest values in that sample The arithmetic mean of any amount of equal sized number groups together is the arithmetic mean of the arithmetic means of each group Contrast with median editMain article Median The arithmetic mean may be contrasted with the median The median is defined such that no more than half the values are larger and no more than half are smaller than it If elements in the data increase arithmetically when placed in some order then the median and arithmetic average are equal For example consider the data sample 1 2 3 4 displaystyle 1 2 3 4 nbsp The mean is 2 5 displaystyle 2 5 nbsp as is the median However when we consider a sample that cannot be arranged to increase arithmetically such as 1 2 4 8 16 displaystyle 1 2 4 8 16 nbsp the median and arithmetic average can differ significantly In this case the arithmetic average is 6 2 displaystyle 6 2 nbsp while the median is 4 displaystyle 4 nbsp The average value can vary considerably from most values in the sample and can be larger or smaller than most There are applications of this phenomenon in many fields For example since the 1980s the median income in the United States has increased more slowly than the arithmetic average of income 4 Generalizations editWeighted average edit Main article Weighted average A weighted average or weighted mean is an average in which some data points count more heavily than others in that they are given more weight in the calculation 5 For example the arithmetic mean of 3 displaystyle 3 nbsp and 5 displaystyle 5 nbsp is 3 5 2 4 displaystyle frac 3 5 2 4 nbsp or equivalently 3 1 2 5 1 2 4 displaystyle 3 frac 1 2 5 frac 1 2 4 nbsp In contrast a weighted mean in which the first number receives for example twice as much weight as the second perhaps because it is assumed to appear twice as often in the general population from which these numbers were sampled would be calculated as 3 2 3 5 1 3 11 3 displaystyle 3 frac 2 3 5 frac 1 3 frac 11 3 nbsp Here the weights which necessarily sum to one are 2 3 displaystyle frac 2 3 nbsp and 1 3 displaystyle frac 1 3 nbsp the former being twice the latter The arithmetic mean sometimes called the unweighted average or equally weighted average can be interpreted as a special case of a weighted average in which all weights are equal to the same number 1 2 displaystyle frac 1 2 nbsp in the above example and 1 n displaystyle frac 1 n nbsp in a situation with n displaystyle n nbsp numbers being averaged Continuous probability distributions edit nbsp Comparison of two log normal distributions with equal median but different skewness resulting in various means and modesIf a numerical property and any sample of data from it can take on any value from a continuous range instead of for example just integers then the probability of a number falling into some range of possible values can be described by integrating a continuous probability distribution across this range even when the naive probability for a sample number taking one certain value from infinitely many is zero In this context the analog of a weighted average in which there are infinitely many possibilities for the precise value of the variable in each range is called the mean of the probability distribution The most widely encountered probability distribution is called the normal distribution it has the property that all measures of its central tendency including not just the mean but also the median mentioned above and the mode the three Ms 6 are equal This equality does not hold for other probability distributions as illustrated for the log normal distribution here Angles edit Main article Mean of circular quantities Particular care is needed when using cyclic data such as phases or angles Taking the arithmetic mean of 1 and 359 yields a result of 180 This is incorrect for two reasons Firstly angle measurements are only defined up to an additive constant of 360 2 p displaystyle 2 pi nbsp or t displaystyle tau nbsp if measuring in radians Thus these could easily be called 1 and 1 or 361 and 719 since each one of them produces a different average Secondly in this situation 0 or 360 is geometrically a better average value there is lower dispersion about it the points are both 1 from it and 179 from 180 the putative average In general application such an oversight will lead to the average value artificially moving towards the middle of the numerical range A solution to this problem is to use the optimization formulation that is define the mean as the central point the point about which one has the lowest dispersion and redefine the difference as a modular distance i e the distance on the circle so the modular distance between 1 and 359 is 2 not 358 nbsp Proof without words of the inequality of arithmetic and geometric means P R displaystyle PR nbsp is the diameter of a circle centered on O displaystyle O nbsp its radius A O displaystyle AO nbsp is the arithmetic mean of a displaystyle a nbsp and b displaystyle b nbsp Using the geometric mean theorem triangle P G R displaystyle PGR nbsp s altitude G Q displaystyle GQ nbsp is the geometric mean For any ratio a b displaystyle a b nbsp A O G Q displaystyle AO geq GQ nbsp Symbols and encoding editThe arithmetic mean is often denoted by a bar vinculum or macron as in x displaystyle bar x nbsp 3 Some software text processors web browsers may not display the x symbol correctly For example the HTML symbol x combines two codes the base letter x plus a code for the line above or 7 In some document formats such as PDF the symbol may be replaced by a cent symbol when copied to a text processor such as Microsoft Word See also edit nbsp Geometric proof without words that max a b gt root mean square RMS or quadratic mean QM gt arithmetic mean AM gt geometric mean GM gt harmonic mean HM gt min a b of two distinct positive numbers a and b note 1 Frechet mean Generalized mean Inequality of arithmetic and geometric means Sample mean and covariance Standard deviation Standard error of the mean Summary statisticsNotes edit If AC a and BC b OC AM of a and b and radius r QO OG Using Pythagoras theorem QC QO OC QC QO OC QM Using Pythagoras theorem OC OG GC GC OC OG GM Using similar triangles HC GC GC OC HC GC OC HM References edit Jacobs Harold R 1994 Mathematics A Human Endeavor Third ed W H Freeman p 547 ISBN 0 7167 2426 X Weisstein Eric W Arithmetic Mean mathworld wolfram com Retrieved 21 August 2020 a b Medhi Jyotiprasad 1992 Statistical Methods An Introductory Text New Age International pp 53 58 ISBN 9788122404197 Krugman Paul 4 June 2014 Fall 1992 The Rich the Right and the Facts Deconstructing the Income Distribution Debate The American Prospect Mean mathematics Encyclopedia Britannica Retrieved 21 August 2020 Thinkmap Visual Thesaurus 30 June 2010 The Three M s of Statistics Mode Median Mean June 30 2010 www visualthesaurus com Retrieved 3 December 2018 Notes on Unicode for Stat Symbols www personal psu edu Archived from the original on 31 March 2022 Retrieved 14 October 2018 Further reading editHuff Darrell 1993 How to Lie with Statistics W W Norton ISBN 978 0 393 31072 6 External links editCalculations and comparisons between arithmetic mean and geometric mean of two numbers Calculate the arithmetic mean of a series of numbers on fxSolver Portal nbsp Mathematics Retrieved from https en wikipedia org w index php title Arithmetic mean amp oldid 1187886757, wikipedia, wiki, book, books, library,

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