# Arithmetic mean

**meanaveragearithmeticmean averagesample meanmeansaveragedaverage ratingarithmetical meansample means**

In mathematics and statistics, the arithmetic mean (, 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 count of numbers in the collection.wikipedia

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### Mean

**mean valuepopulation meanaverage**

In mathematics and statistics, the arithmetic mean (, 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 count of numbers in the collection.

For a data set, the arithmetic mean, also called the mathematical expectation or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values.

### Average

**Rushing averageReceiving averagemean**

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.

Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged.

### Geometric mean

**geometric averagegeometricmean**

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 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 (as opposed to the arithmetic mean which uses their sum).

### Harmonic mean

**harmonicweighted harmonic meanharmonic average**

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.

The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations.

### Central tendency

**Localitycentral locationcentral point**

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).

The most common measures of central tendency are the arithmetic mean, the median and the mode.

### Median

**averagesample medianmedian-unbiased estimator**

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.

The basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by extremely large or small values, and so it may give a better idea of a "typical" value.

### Outlier

**outliersconservative estimateirregularities**

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).

In most larger samplings of data, some data points will be further away from the sample mean than what is deemed reasonable.

### Robust statistics

**robustbreakdown pointrobustness**

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).

The median is a robust measure of central tendency, while the mean is not. The median has a breakdown point of 50%, while the mean has a breakdown point of 0% (a single large observation can throw it off).

### Errors and residuals

**residualserror termerror**

If numbers have mean \bar{x}, then . Since x_i-\bar{x} 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.

The expected value, being the mean of the entire population, is typically unobservable, and hence the statistical error cannot be observed either.

### Mode (statistics)

**modemodalmodes**

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 ), are equal to each other.

### Generalized mean

**power meangeneralisedHölder generalized mean**

Generalized mean

In mathematics, generalized means are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).

### Summary statistics

**summary statisticSummarizationdata summarization**

Summary statistics

a measure of location, or central tendency, such as the arithmetic mean

### Fréchet mean

**Fréchet sense**

Fréchet mean

On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.

### Mathematics

**mathematicalmathmathematician**

In mathematics and statistics, the arithmetic mean (, 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 count of numbers in the collection.

### Statistics

**statisticalstatistical analysisstatistician**

In mathematics and statistics, the arithmetic mean (, 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 count of numbers in the collection.

### Experiment

**experimentalexperimentationexperiments**

The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey.

### Observational study

**observational studiesobservationalobservational data**

The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey.

### Survey methodology

**surveysurveysstatistical survey**

The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey.

### Economics

**economiceconomisteconomic theory**

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.

### Anthropology

**anthropologistanthropologicalanthropologists**

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.

### History

**historical recordshistoricalhistoric**

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.

### Per capita income

**per-capita incomeper capitaincome per capita**

For example, per capita income is the arithmetic average income of a nation's population.

### Skewness

**skewedskewskewed distribution**

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.

### Income distribution

**distribution of incomeincomedistribution**

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.

### Data set

**datasetdatasetsdata**

The arithmetic mean is the most commonly used and readily understood measure of central tendency in a data set.