Mean

In mathematics, mean has several different definitions depending on the context.

In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving . An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite: for example, when the probability of the value is for n = 1, 2, 3, ....

For a data set, the terms arithmetic mean, mathematical expectation, and sometimes average are used synonymously to refer to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is typically denoted by , pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean (denoted ) to distinguish it from the population mean (denoted or ).

Mean (song)

"Mean" is a song written and recorded by American country pop singer-songwriter Taylor Swift for her third studio album, Speak Now (2010). Produced by Swift alongside Nathan Chapman, the song was sent to country radio in the United States on March 13, 2011, as the third single from Speak Now. "Mean" garnered mixed to positive reviews from critics for its lyrical detail and profound country sound. The song received commercial success in the United States and Canada, debuting at number 11 on the Billboard Hot 100 and number ten on the Canadian Hot 100. The song also appeared on the Australian Singles Chart at number 45.

The song's accompanying music video was directed by Declan Whitebloom, who developed the concept together with Swift. It received mixed reviews from critics who perceived ambivalent messages in the video, despite the prevalent self-empowerment and anti-bullying themes. "Mean" was performed for the first time by Swift at the 46th Annual Academy of Country Music Awards on April 3, 2011. The song won the Grammy Awards for Best Country Song and Best Country Solo Performance at the 54th Grammy Awards.Rolling Stone Magazine ranked Mean as #24 on their list of the 100 greatest country songs of all time. Taylor Swift was tied up in the music video for mean.

Arithmetic mean

In mathematics and statistics, the arithmetic mean (pronunciation: /ˌærɪθˈmɛtɪk ˈmiːn/, stress on third syllable of "arithmetic"), or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection. The collection is often a set of results of an experiment, or 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.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, 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). 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 accord with one's notion of "middle", and robust statistics, such as the median, may be a better description of central tendency.