Arithmetic mean

In

In addition to mathematics and statistics, the arithmetic mean is frequently used in economics, anthropology, history, and 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

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

(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

${\displaystyle \mu }$. If the data set is a

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.

Motivating properties

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
  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.

Additional properties

• The arithmetic mean of a sample is always between the largest and smallest values in that sample.
• The arithmetic mean of any amount of equal-sized number groups together is the arithmetic mean of the arithmetic means of each group.

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, 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]

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.^[5] For example, the arithmetic mean of ${\displaystyle 3}$ and ${\displaystyle 5}$ is ${\displaystyle {\frac {3+5}{2}}=4}$, or equivalently ${\displaystyle 3{\frac {1}{2}}+5{\frac {1}{2}}=4}$. 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 ${\displaystyle 3{\frac {2}{3}}+5{\frac {1}{3}}={\frac {11}{3}}}$. Here the weights, which necessarily sum to one, are ${\displaystyle {\frac {2}{3}}}$ and ${\displaystyle {\frac {1}{3}}}$, 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 weights are equal to the same number (${\displaystyle {\frac {1}{2}}}$ in the above example and ${\displaystyle {\frac {1}{n}}}$ in a situation with ${\displaystyle n}$ numbers being averaged).

Continuous probability distributions

If a numerical property, and any sample of data from it, can take on any value from a continuous range instead of, for example, just integers, then the

Particular care is needed when using cyclic data, such as phases or angles. 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° (${\displaystyle 2\pi }$ or ${\displaystyle \tau }$, if measuring in radians). Thus, these could easily be called 1° and -1°, or 361° and 719°, since each one of them produces a different average.
• Secondly, in this situation, 0° (or 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 (that is, 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°).

Symbols and encoding

The arithmetic mean is often denoted by a bar (vinculum or macron), as in ${\displaystyle {\bar {x}}}$.^[3]

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.

See also

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

Notes