Mean

A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers.

The arithmetic mean, also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted using an overhead bar, <math>\bar{x}</math>. If the numbers are from observing a sample of a larger group, the arithmetic mean is termed the sample mean (<math>\bar{x}</math>) to distinguish it from the group mean (or expected value) of the underlying distribution, denoted <math>\mu</math> or <math>\mu_x</math>.

Outside probability and statistics, a wide range of other notions of mean are often used in geometry and mathematical analysis; examples are given below.

Types of means

Pythagorean means

In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.

Arithmetic mean (AM)

The arithmetic mean (or simply mean or average) of a list of numbers, is the sum of all of the numbers divided by their count. Similarly, the mean of a sample <math>x_1,x_2,\ldots,x_n</math>, usually denoted by <math>\bar{x}</math>, is the sum of the sampled values divided by the number of items in the sample.

:<math> \bar{x} = \frac{1}{n}\sum_{i=1}^n{x_i} = \frac{x_1+x_2+\cdots +x_n}{n} </math>

For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:

:<math>\frac{4+36+45+50+75}{5} = \frac{210}{5} = 42.</math>

Geometric mean (GM)

The geometric mean is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean):

:<math>\bar{x} = \left( \prod_{i=1}^n{x_i} \right )^\frac{1}{n} = \left(x_1 x_2 \cdots x_n \right)^\frac{1}{n}</math>

For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:

:<math>(4 \times 36 \times 45 \times 50 \times 75)^\frac{1}{5} = \sqrt[5]{24\;300\;000} = 30.</math>

Harmonic mean (HM)

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, as in the case of speed (i.e., distance per unit of time):

:<math> \bar{x} = n \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}</math>

For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is

:<math>\frac{5}{\tfrac{1}{4}+\tfrac{1}{36}+\tfrac{1}{45} + \tfrac{1}{50} + \tfrac{1}{75 = \frac{5}{\;\tfrac{1}{3}\;} = 15.</math>

If we have five pumps that can empty a tank of a certain size in respectively 4, 36, 45, 50, and 75 minutes, then the harmonic mean of <math>15</math>

tells us that these five different pumps working together will pump at the same rate as five pumps that can each empty the tank in <math>15</math> minutes.

Relationship between AM, GM, and HM

AM, GM, and HM of nonnegative real numbers satisfy these inequalities:

:<math> \mathrm{AM} \ge \mathrm{GM} \ge \mathrm{HM} \, </math>

Equality holds if all the elements of the given sample are equal.

Statistical location

thumb|Comparison of the [[arithmetic mean, median, and mode of two skewed (log-normal) distributions]]

thumb|upright|Geometric visualization of the mode, median and mean of an arbitrary probability density function

In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may colloquially be called an "average" (more formally, a measure of central tendency). The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions.

Mean of a probability distribution

The mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. If the random variable is denoted by <math>X</math>, then the mean is also known as the expected value of <math>X</math> (denoted <math>E(X)</math>). For a discrete probability distribution, the mean is given by <math>\textstyle \sum xP(x)</math>, where the sum is taken over all possible values of the random variable and <math>P(x)</math> is the probability mass function. For a continuous distribution, the mean is <math>\textstyle \int_{-\infty}^{\infty} xf(x)\,dx</math>, where <math>f(x)</math> is the probability density function. In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure. The mean need not exist or be finite; for some probability distributions the mean is infinite ( or ), while for others the mean is undefined.

Generalized means

Power mean

The generalized mean, also known as the power mean or Hölder mean, abstracts several other means. It is defined for positive numbers <math>x_1, \dots, x_n</math> by

:<math>M_p(x_1, \dots, x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}.</math>

This, as a function of <math>p</math>, is well defined on <math>\mathbb{R}\setminus \{0\}</math>, but can be extended continuously to <math>\mathbb{R} \cup \{-\infty, +\infty\}</math>.

By choosing different values for <math>p</math>, other well known means are retrieved.

{| class="wikitable"

! Name

! Exponent

! Value

| Minimum

| <math>p = -\infty</math>

| <math>\min \{x_1, \dots, x_n\}</math>

| Harmonic mean

| <math>p = -1</math>

| <math>\frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n</math>

| Geometric mean

| <math>p = 0</math>

| <math>\sqrt[n]{x_1\dots x_n}</math>

| Arithmetic mean

| <math>p = 1</math>

| <math>\frac{x_1 + \dots + x_n}{n}</math>

| Root mean square

| <math>p = 2</math>

| <math>\sqrt{\frac{x_1^2 + \dots + x_n^2}{n</math>

| Cubic mean

| <math>p = 3</math>

| <math>\sqrt[3]{\frac{x_1^3 + \dots + x_n^3}{n</math>

| Maximum

| <math>p = +\infty</math>

| <math>\max\{x_1, \dots, x_n\}</math>

Quasi-arithmetic mean

A similar approach to the power mean is the <math>f</math>-mean, also known as the quasi-arithmetic mean.

For an injective function <math>f \colon I \rightarrow \mathbb{R}</math> on an interval <math>I \subset \mathbb{R}</math> and real numbers <math>x_1, \dots, x_n \in I</math> we define their <math>f</math>-mean as

: <math> M_f(x_1, \dots, x_n) = f^{-1}\left({\frac{1}{n} \sum_{i=1}^n{f\left(x_i\right)\right). </math>

By choosing different functions <math>f</math>, other well known means are retrieved.

{| class="wikitable"

! Mean

! <math>I</math>

! Function

| Arithmetic mean

| <math>\mathbb{R}</math>

| <math>x \mapsto x</math>

| Geometric mean

| <math>]0, +\infty[</math>

| <math>x \mapsto \ln(x)</math>

| Harmonic mean

| <math>\mathbb{R} \setminus \{0\}</math>

| <math>x \mapsto x^{-1}</math>

| Power mean

| <math>\mathbb{R} \setminus \{0\}</math>

| <math>x \mapsto x^m</math>

Weighted arithmetic mean

The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from different sized samples of the same population, and is define by

Swanson's rule

This is an approximation to the mean for a moderately skewed distribution. It is used in hydrocarbon exploration and is defined as:

: <math> m = 0.3P_{10} + 0.4P_{50} + 0.3P_{90} </math>

where <math display="inline">P_{10}</math>, <math display="inline">P_{50}</math> and <math display="inline">P_{90}</math> are the 10th, 50th and 90th percentiles of the distribution, respectively.

Other means

Arithmetic-geometric mean
Arithmetic-harmonic mean
Cesàro mean
Chisini mean
Contraharmonic mean
Elementary symmetric mean
Geometric-harmonic mean
Grand mean
Heinz mean
Heronian mean
Identric mean
Lehmer mean
Logarithmic mean
Moving average
Neuman–Sándor mean
Quasi-arithmetic mean
Root mean square (quadratic mean)
Rényi's entropy (a generalized f-mean)
Spherical mean
Stolarsky mean
Weighted geometric mean
Weighted harmonic mean