Arithmetic mean

The arithmetic mean (or mean or average), ${\displaystyle {\bar {x}}}$ (read ${\displaystyle x}$ bar), is the mean of the ${\displaystyle n}$ values ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$.[2]

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 observation divided by the total number of observations. Symbolically, if we have a data set consisting of the values ${\displaystyle a_{1},a_{2},\ldots ,a_{n}}$, then the arithmetic mean ${\displaystyle A}$ is defined by the formula:

${\displaystyle A={\frac {1}{n}}\sum _{i=1}^{n}a_{i}={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}}$

(See summation for an explanation of the summation operator).

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

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

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. If the data set is a statistical sample (a subset of the population), we call the statistic resulting from this calculation a sample mean.

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.

The arithmetic mean has several properties that make it useful, 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 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.
• 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, in the sense of minimizing the sum of squared deviations from the typical value: the sum of ${\displaystyle (x_{i}-{\bar {x}})^{2}}$. (It follows that the sample mean is also the best single predictor in the sense of having 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 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 example, consider the data sample ${\displaystyle {1,2,3,4}}$. The average is ${\displaystyle 2.5}$, as is the median. However, when we consider a sample that cannot be arranged so as 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 6.2 and the median is 4. In general, the average value can vary significantly from most values in the sample, and can be larger or smaller than most of them.

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.[3]

Particular care must be taken when using cyclic data, such as phases or angles. Naïvely 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°, each of which 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 difference as a modular distance (i.e., the distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°).

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

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 ¯).[5]

In some texts, such as pdfs, the x̄ symbol may be replaced by a cent (¢) symbol (Unicode &#162) when copied to text processor such as Microsoft Word.

• Fréchet mean
• Generalized mean
• Geometric mean
• Harmonic mean
• Mode
• Sample mean and covariance
• 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
• Weisstein, Eric W. "Arithmetic Mean". MathWorld.
• Calculate the arithmetic mean of a series of numbers on fxSolver