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Encyclopedia > Arithmetic 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 by the number of items in the list. The arithmetic mean is what students are taught very early to call the "average". If the list is a statistical population, then the mean of that population is called a population mean. If the list is a statistical sample, we call the resulting statistic a sample mean. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article is about the field of statistics. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ... In statistics, a statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. ... Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. ... A statistic (singular) is the result of applying a statistical algorithm to a set of data. ...


When the mean is not an accurate estimate of the median, the list of numbers, or frequency distribution, is said to be skewed. In probability theory and statistics, a median is a type of average that is described as the number dividing the higher half of a sample, a population, or a probability distribution, from the lower half. ... In statistics, a frequency distribution is a list of the values that a variable takes in a sample. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ...

Contents

Introduction

If we denote a set of data by X = (x1, x2, ..., xn), then the sample mean is typically denoted with a horizontal bar over the variable (bar{x} ,, enunciated "x bar").


The symbol μ (Greek: mu) is used to denote the arithmetic mean of an entire population. Or, for a random number that has a defined mean, μ is the probabilistic mean or expected value of the random number. If the set X is a collection of random numbers with probabilistic mean of μ, then for any individual sample, xi, from that collection, μ = E{xi} is the expected value of that sample. Random number may refer to: A number generated for or part of a set exhibiting statistical randomness. ... 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... 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...


In practice, the difference between μ and bar{x} , is that μ is typically unobservable because one observes only a sample rather than the whole population, and if the sample is drawn randomly, then one may treat bar{x} ,, but not μ, as a random variable, attributing a probability distribution to it (the sampling distribution of the mean). In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data values in a sample). ...


Both are computed in the same way:

bar{x} = frac{1}{n}sum_{i=1}^n x_i = frac{1}{n} (x_1+cdots+x_n).

If X is a random variable, then the expected value of X can be seen as the long-term arithmetic mean that occurs on repeated measurements of X. This is the content of the law of large numbers. As a result, the sample mean is used to estimate unknown expected values. In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... 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... // The law of large numbers (LLN) is any of several theorems in probability. ...


Note that several other "means" have been defined, including the generalized mean, the generalized f-mean, the harmonic mean, the arithmetic-geometric mean, and various weighted means. A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means. ... 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). ... In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ... In mathematics, the arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a1, i. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ...


Examples

  • If you have 3 numbers then add them and divide them by 3: frac{x_1 + x_2 + x_3}{3}
  • If you have 4 numbers add them and divide by 4: frac{x_1 + x_2 + x_3 + x_4}{4}

Formulation as an optimization problem

The arithmetic mean is the value with minimal quadratical distance from the given values:

bar{x} = arg min_{yinR} {sum_{k=1}^n (y-x_k)^2}.

Equivalently, the mean is the best least squares fit of a constant function to the given data. In regression analysis, least squares, also known as ordinary least squares analysis, is a method for linear regression that determines the values of unknown quantities in a statistical model by minimizing the sum of the residuals (the difference between the predicted and observed values) squared. ...


Problems with some uses of the mean

While the mean is often used to report central tendency, it may not be appropriate for describing skewed distributions, because it is easily misinterpreted. The arithmetic mean is greatly influenced by outliers. These distortions can occur when the mean is different from the median. When this happens the median may be a better description of central tendency. 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. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... Figure 1. ... In probability theory and statistics, a median is a type of average that is described as the number dividing the higher half of a sample, a population, or a probability distribution, from the lower half. ...


A classic example is average income. The arithmetic mean may be misinterpreted to imply that most people's incomes are higher than is in fact the case. When presented with an "average" one may be led to believe that most people's incomes are near this number. 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). However, this "average" says nothing about the number of people near the median income (nor does it say anything about the modal income that most people are near). Nevertheless, because one might carelessly relate "average" and "most people" one might incorrectly assume that most people's incomes would be higher (nearer this inflated "average") than they are. For instance, reporting the "average" net worth in Medina, Washington as the arithmetic mean of all annual net worths would yield a surprisingly high number because of Bill Gates. Consider the scores (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five out of six scores are below this. Net worth (sometimes net assets) is the total assets minus total liabilities of an individual or company. ... Medina is a city located in King County, Washington, on the eastern shore of Lake Washington opposite Seattle. ... For other persons named Bill Gates, see Bill Gates (disambiguation). ...


In certain situations, the arithmetic mean is the wrong measure of central tendency altogether. For example, if a stock fell 10 % in the first year, and rose 30 % in the second year, then it would be incorrect to report its "average" increase per year over this two year period as the arithmetic mean (−10 % + 30 %)/2 = 10 %; the correct average in this case is the geometric mean which yields an average increase per year of only 8.2 %. The reason for this is that each of those percents have different starting points. If the stock starts at $30 and falls 10 %, it is now at $27. If the stock then rises 30 %, it is now $35.1. The arithmetic mean of those rises is 10 %, but since the stock rose by $5.1 in 2 years, an average of 8.2 % would result in the final $35.1 figure [$30(1-10 %)(1+30 %) = $30(1+8.2 %)(1+8.2 %) = $35.1]. If one used the arithmetic mean 10 % in the same way, onewould not get the actual increase [$30(1+10 %)(1+10 %) = $36.3]. 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. ... In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ...


Particular care must be taken when using cyclic data such as phases or angles. Taking the arithmetic mean of 1 degree and 359 degrees yields a result of 180 degrees, whereas 1 and 359 are both adjacent to 360 degrees which may be a more correct average value. In general application such an oversight will lead to the average value artificially moving towards the middle of the numerical range.


See also

In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, a median is a type of average that is described as the number dividing the higher half of a sample, a population, or a probability distribution, from the lower half. ... In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate as much as possible as simply as possible. ... 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 generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means. ... 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. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... 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. ... 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... In mathematics, Muirheads inequality, also known as the bunching method, generalizes the inequality of arithmetic and geometric means. ... The sample size of a statistical sample is the number of repeated measurements that constitute it. ... Sample mean and covariance are statistics computed from a collection of data, thought of as being random. ... In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables. ...

Further reading

  • Darrell Huff, How to lie with statistics, Victor Gollancz, 1954 (ISBN 0-393-31072-8).

Darrell Huff (July 15, 1913 - June 27, 2001) was an American writer, and is best known as the author of How to Lie with Statistics (1954), a brief, breezy, illustrated volume which is the best-selling statistics book of all time. ...

External links

  • Calculations and comparisons between arithmetic and geometric mean between two numbers
  • Mean or Average
  • Rational Mean

  Results from FactBites:
 
PlanetMath: arithmetic mean (181 words)
The arithmetic mean is what is commonly called the average of the numbers.
See Also: geometric mean, harmonic mean, arithmetic-geometric-harmonic means inequality, general means inequality, weighted power mean, power mean, geometric distribution, root-mean-square, proof of general means inequality, proof of arithmetic-geometric-harmonic means inequality, derivation of geometric mean as the limit of the power mean, mean, a prime theorem of a convergent sequence
This is version 8 of arithmetic mean, born on 2001-10-20, modified 2006-11-11.
  More results at FactBites »

 
 

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