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In mathematics and statistics, the **arithmetic mean** ( /ˌærɪθˈmɛtɪk**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 of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

**Statistics** is the discipline that concerns the collection, organization, displaying, analysis, interpretation and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

There are several kinds of **means** in various branches of mathematics.

- Definition
- Motivating properties
- Contrast with median
- Generalizations
- Weighted average
- Continuous probability distributions
- Angles
- Symbols and encoding
- See also
- References
- Further reading
- External links

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

**Economics** is the social science that studies the production, distribution, and consumption of goods and services.

**Anthropology** is the scientific study of humans, human behavior and societies in the past and present. Social anthropology studies patterns of behaviour and cultural anthropology studies cultural meaning, including norms and values. Linguistic anthropology studies how language influences social life. Biological or physical anthropology studies the biological development of humans.

**History** is the past as it is described in written documents, and the study thereof. Events occurring before written records are considered prehistory. "History" is an umbrella term that relates to past events as well as the memory, discovery, collection, organization, presentation, and interpretation of information about these events. Scholars who write about history are called historians.

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

In statistics, a **central tendency** is a central or typical value for a probability distribution. It may also be called a **center** or **location** of the distribution. Colloquially, measures of central tendency are often called *averages.* The term *central tendency* dates from the late 1920s.

In statistics, an **outlier** is a data point that differs significantly from other observations. An outlier may be due to variability in the measurement or it may indicate experimental error; the latter are sometimes excluded from the data set. An outlier can cause serious problems in statistical analyses.

The **median** is the value separating the higher half from the lower half of a data sample. For a data set, it may be thought of as the "middle" value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.

The **arithmetic mean** (or **mean** or **average**), (read *bar*), is the mean of the values .^{ [2] }

The arithmetic mean is the most commonly used and readily understood measure of central tendency in a data set. In statistics, the term average refers to any of the measures of central tendency. The arithmetic mean of a set of observed data is defined as being equal to the sum of the numerical values of each and every observation divided by the total number of observations. Symbolically, if we have a data set consisting of the values , then the arithmetic mean is defined by the formula:

A **data set** is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the data set in question. The data set lists values for each of the variables, such as height and weight of an object, for each member of the data set. Each value is known as a datum. Data sets can also consist of a collection of documents or files.

In colloquial language, an **average** is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an **average value**.

(See summation for an explanation of the summation operator).

In mathematics, **summation** is the addition of a sequence of any kind of numbers, called *addends* or *summands*; the result is their *sum* or *total*. Besides numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any types of mathematical objects on which an operation denoted "+" is defined.

For example, consider the monthly salary of 10 employees of a firm: 2500, 2700, 2400, 2300, 2550, 2650, 2750, 2450, 2600, 2400. 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**. If the data set is a statistical sample (a subset of the population), we call the statistic resulting from this calculation a **sample mean**.

The arithmetic mean has several properties that make it useful, 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 as saying that the numbers to the left of the mean are balanced by the numbers to the right of the mean. The mean is the only single number for which the residuals (deviations from the estimate) sum to zero.
- 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, in the sense of minimizing the sum of squared deviations from the typical value: the sum of . (It follows that the sample mean is also the best single predictor in the sense of having the lowest root mean squared error.)
^{ [2] }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 may be contrasted with the median. The median is defined such that no more than half the values are larger than, and no more than half are smaller than, the median. 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 average is , as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as , the median and arithmetic average can differ significantly. In this case, the arithmetic average is 6.2 and the median is 4. In general, the average value can vary significantly from most values in the sample, and can be larger or smaller than most of them.

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.^{ [3] }

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. 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 the value 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 the weights are equal to each other (equal to in the above example, and equal to in a situation with numbers being averaged).

If a numerical property, and any sample of data from it, could 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. The analog of a weighted average in this context, in which there are an infinite number of possibilities for the precise value of the variable in each range, is called the *mean of the probability distribution*. A 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 aforementioned median and the mode (the three M's^{ [4] }), are equal to each other. This equality does not hold for other probability distributions, as illustrated for the lognormal distribution here.

Particular care must be taken when using cyclic data, such as phases or angles. Naïvely 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 2π, if measuring in radians). Thus one could as easily call these 1° and −1°, or 361° and 719°, each of which gives a different average.
- Secondly, in this situation, 0° (equivalently, 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 (viz., 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°).

The arithmetic mean is often denoted by a bar, for example as in (read *bar*).^{ [2] }

Some software (text processors, web browsers) may not display the x̄ symbol properly. For example, the x̄ symbol in HTML is actually a combination of two codes - the base letter x plus a code for the line above (̄: or ¯).^{ [5] }

In some texts, such as pdfs, the x̄ symbol may be replaced by a cent (¢) symbol (Unicode ¢) when copied to text processor such as Microsoft Word.

In probability theory, the **expected value** of a random variable is a key aspect of its probability distribution. Intuitively, a random variable's expected value represents the average of a large number of independent realizations of the random variable. For example, the expected value of rolling a six-sided die is 3.5, because the average of all the numbers that come up converges to 3.5 as the number of rolls approaches infinity. The expected value is also known as the **expectation**, **mathematical expectation**, **mean**, or **first moment**.

In mathematics, the **geometric mean** is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the *n*th root of the product of n numbers, i.e., for a set of numbers *x*_{1}, *x*_{2}, ..., *x _{n}*, the geometric mean is defined as

In mathematics, the **harmonic mean** is one of several kinds of average, and in particular one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired.

In probability theory, the **normal****distribution** is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be **normally distributed** and is called a **normal deviate**.

In statistics, the **standard deviation** is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

The **weighted arithmetic mean** is similar to an ordinary arithmetic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

In statistics, the **Pearson correlation coefficient**, also referred to as **Pearson's r**, the

The **standard error** (**SE**) of a statistic is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the parameter or the statistic is the mean, it is called the **standard error of the mean** (**SEM**).

In statistics, a **moving average** is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a **moving mean** (**MM**) or **rolling mean** and is a type of finite impulse response filter. Variations include: simple, and cumulative, or weighted forms.

The **Kruskal–Wallis test** by ranks, **Kruskal–Wallis H test**, or

*Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.*

In mathematics, a **contraharmonic mean** is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, , where p = 2.

In statistics, the **bias** of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called **unbiased**. In statistics, "bias" is an **objective** property of an estimator. Unlike the *ordinary English use* of the term "bias", it is not pejorative even though it's not a desired property.

In mathematics, a **mean of circular quantities** is a mean which is sometimes better-suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on circular quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because for most purposes 360° is the same thing as 0°. As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. This is one of the simplest examples of statistics of non-Euclidean spaces.

The **sample mean** or **empirical mean** and the **sample covariance** are statistics computed from a collection of data on one or more random variables. The sample mean and sample covariance are estimators of the population mean and population covariance, where the term *population* refers to the set from which the sample was taken.

In statistics, the **reduced chi-squared statistic** is used extensively in goodness of fit testing. It is also known as **mean square weighted deviation** (**MSWD**) in isotopic dating and **variance of unit weight** in the context of weighted least squares.

- ↑ Jacobs, Harold R. (1994).
*Mathematics: A Human Endeavor*(Third ed.). W. H. Freeman. p. 547. ISBN 0-7167-2426-X. - 1 2 3 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*. - ↑ 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*. Retrieved 14 October 2018.

- Huff, Darrell (1993).
*How to Lie with Statistics*. W. W. Norton. ISBN 978-0-393-31072-6.

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