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Current member states of the United Nations |
Monarchies of Europe |
Prince-bishoprics |
Principalities |
Pyrenees |
Southern European countries |
Southwestern European countries |
Spanish-speaking countries and territories |
Special economic zones |
States and territories established in 1278 In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or simply just the mean or the average (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results o... |
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. |
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). For skewed distributions, such as the distribution of income for which a few people's incomes are s... |
Definition |
Given a data set , the arithmetic mean (or mean or average), denoted (read bar), is the mean of the values . |
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 ... |
(for an explanation of the summation operator, see summation.) |
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, and denoted by the Greek letter . If the data set is a statistical sample (a subset of the population), then we call the statistic res... |
The arithmetic mean can be similarly defined for vectors in multiple dimension, not only scalar values; this is often referred to as a centroid. More generally, because the arithmetic mean is a convex combination (coefficients sum to 1), it can be defined on a convex space, not only a vector space. |
Motivating properties |
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)... |
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 sens... |
Contrast with median |
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 examp... |
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. |
Generalizations |
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. 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 muc... |
Continuous probability distributions |
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... |
Angles |
Particular care must be taken when using cyclic data, such as phases or angles. Naively 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°, since each one of them 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 di... |
Symbols and encoding |
The arithmetic mean is often denoted by a bar, (a.k.a vinculum or macron), for example as in (read bar). |
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 ¯). |
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. |
See also |
Fréchet mean |
Generalized mean |
Geometric mean |
Harmonic mean |
Inequality of arithmetic and geometric means |
Mode |
Sample mean and covariance |
Standard deviation |
Standard error of the mean |
Summary statistics |
References |
Further reading |
External links |
Calculations and comparisons between arithmetic mean and geometric mean of two numbers |
Calculate the arithmetic mean of a series of numbers on fxSolver |
Means The American Football Conference (AFC) is one of the two conferences of the National Football League (NFL), the highest professional level of American football in the United States. This conference currently contains 16 teams organized into 4 divisions, as does its counterpart, the National Football Conference (N... |
Teams |
Like the NFC, the conference has 16 teams organized into four divisions each with four teams: East, North, South and West. |
Season structure |
This chart of the 2021 season standings displays an application of the NFL scheduling formula. The Bengals in 2021 (highlighted in green) finished in first place in the AFC North. Thus, in 2021, the Bengals are scheduled to play two games against each of its division rivals (highlighted in light blue), one game against... |
Currently, the fourteen opponents each team faces over the 17-game regular season schedule are set using a pre-determined formula: |
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