Arithmetic mean

Given a data set ${\displaystyle X=\{x_{1},\ldots ,x_{n}\}}$, the arithmetic mean (also mean or average), denoted ${\displaystyle {\bar {x}}}$ (read ${\displaystyle x}$ bar), is the mean of the ${\displaystyle n}$ values ${\displaystyle x_{1},\ldots ,x_{n}}$.[2]

The arithmetic mean is a data set's most commonly used and readily understood measure of central tendency. In statistics, the term average refers to any measurement of central tendency. The arithmetic mean of a set of observed data is equal to the sum of the numerical values of each observation, divided by the total number of observations. Symbolically, for a data set consisting of the values ${\displaystyle x_{1},\dots ,x_{n}}$, the arithmetic mean is defined by the formula:

${\displaystyle {\bar {x}}={\frac {1}{n}}\left(\sum _{i=1}^{n}{x_{i}}\right)={\frac {x_{1}+x_{2}+\dots +x_{n}}{n}}}$[3]

(For an explanation of the summation operator, see summation.)

For example, if the monthly salaries of ${\displaystyle 10}$ employees are ${\displaystyle \{2500,2700,2400,2300,2550,2650,2750,2450,2600,2400\}}$, then the arithmetic mean is:

${\displaystyle {\frac {2500+2700+2400+2300+2550+2650+2750+2450+2600+2400}{10}}=2530}$

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 ${\displaystyle \mu }$. If the data set is a statistical sample (a subset of the population), it is called the sample mean (which for a data set ${\displaystyle X}$ is denoted as ${\displaystyle {\overline {X}}}$).

The arithmetic mean can be similarly defined for vectors in multiple dimensions, not only scalar values; this is often referred to as a centroid. More generally, because the arithmetic mean is a convex combination (meaning its coefficients sum to ${\displaystyle 1}$), it can be defined on a convex space, not only a vector space.

The arithmetic mean has several properties that make it interesting, especially as a measure of central tendency. These include:

• If numbers ${\displaystyle x_{1},\dotsc ,x_{n}}$ have mean ${\displaystyle {\bar {x}}}$, then ${\displaystyle (x_{1}-{\bar {x}})+\dotsb +(x_{n}-{\bar {x}})=0}$. Since ${\displaystyle x_{i}-{\bar {x}}}$ is the distance from a given number to the mean, one way to interpret this property is by saying that the numbers to the left of the mean are balanced by the numbers to the right. The mean is the only number for which the residuals (deviations from the estimate) sum to zero. This can also be interpreted as saying that the mean is translationally invariant in the sense that for any real number ${\displaystyle a}$, ${\displaystyle {\overline {x+a}}={\bar {x}}+a}$.
• If it is required to use a single number as a "typical" value for a set of known numbers ${\displaystyle x_{1},\dotsc ,x_{n}}$, then the arithmetic mean of the numbers does this best since it minimizes the sum of squared deviations from the typical value: the sum of ${\displaystyle (x_{i}-{\bar {x}})^{2}}$. The sample mean is also the best single predictor because it has 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 is independent of scale of the units of measurement, in the sense that ${\displaystyle {\text{avg}}(ca_{1},\cdots ,ca_{n})=c\cdot {\text{avg}}(a_{1},\cdots ,a_{n}).}$ So, for example, calculating a mean of liters and then converting to gallons is the same as converting to gallons first and then calculating the mean. This is also called first order homogeneity.

The arithmetic mean may be contrasted with the median. The median is defined such that no more than half the values are larger, and no more than half are smaller than it. 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 ${\displaystyle \{1,2,3,4\}}$. The mean is ${\displaystyle 2.5}$, as is the median. However, when we consider a sample that cannot be arranged to increase arithmetically, such as ${\displaystyle \{1,2,4,8,16\}}$, the median and arithmetic average can differ significantly. In this case, the arithmetic average is ${\displaystyle 6.2}$, while the median is ${\displaystyle 4}$. The average value can vary considerably from most values in the sample and can be larger or smaller than most.

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.[4]

The arithmetic mean is often denoted by a bar (vinculum or macron), as in ${\displaystyle {\bar {x}}}$.[2]

Some software (text processors, web browsers) may not display the "x̄" symbol correctly. For example, the HTML symbol "x̄" combines two codes — the base letter "x" plus a code for the line above (̄ or ¯).[7]

In some document formats (such as PDF), the symbol may be replaced by a "¢" (cent) symbol when copied to a text processor such as Microsoft Word.

Geometric proof without words that max (a,b) > root mean square (RMS) or quadratic mean (QM) > arithmetic mean (AM) > geometric mean (GM) > harmonic mean (HM) > min (a,b) of two distinct positive numbers a and b [note 1]

• Fréchet mean
• Generalized mean
• Inequality of arithmetic and geometric means
• Sample mean and covariance
• Standard deviation
• Standard error of the mean
• Summary statistics

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

• Calculations and comparisons between arithmetic mean and geometric mean of two numbers
• Calculate the arithmetic mean of a series of numbers on fxSolver