There are several kinds of mean in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
. Each mean serves to summarize a given group of
data
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted ...
, often to better understand the overall value (
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
and
sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
) of a given
data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...
.
For a
data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...
, the ''
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
'', also known as "arithmetic average", is a measure of
central tendency
In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications ...
of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''
1, ''x''
2, ..., x''
n'' is typically denoted using an
overhead bar
An overline, overscore, or overbar, is a typographical feature of a horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a '' vinculum'', a notation for grouping symbols which is expressed in m ...
,
. If the data set were based on a series of observations obtained by
sampling from a
statistical population
In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypoth ...
, the arithmetic mean is the ''
sample mean
The sample mean (or "empirical mean") and the sample covariance are statistics computed from a Sample (statistics), sample of data on one or more random variables.
The sample mean is the average value (or mean, mean value) of a sample (statistic ...
'' (
) to distinguish it from the mean, or
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
, of the underlying distribution, the ''
population mean
In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypothe ...
'' (denoted
or
).
[Underhill, L.G.; Bradfield d. (1998) ''Introstat'', Juta and Company Ltd.]
p. 181
/ref>
Outside probability and statistics, a wide range of other notions of mean are often used in geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
; examples are given below.
Types of means
Pythagorean means
Arithmetic mean (AM)
The arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
(or simply ''mean'') of a list of numbers, is the sum of all of the numbers divided by the number of numbers. Similarly, the mean of a sample , usually denoted by , is the sum of the sampled values divided by the number of items in the sample.
:
For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:
:
Geometric mean (GM)
The geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
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):
:
For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:
:
Harmonic mean (HM)
The harmonic mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
is an average which is useful for sets of numbers which are defined in relation to some unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
, as in the case of speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
(i.e., distance per unit of time):
:
For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is
:
Relationship between AM, GM, and HM
AM, GM, and HM satisfy these inequalities:
:
Equality holds if all the elements of the given sample are equal.
Statistical location
In descriptive statistics
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and an ...
, the mean may be confused with the median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
, mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
or mid-range
In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set:
:M=\frac.
The mid-range is closely related to the range, a measure of ...
, as any of these may incorrectly be called an "average" (more formally, a measure of central tendency
In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications ...
). 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
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
* Exponential decay, decrease at a rate proportional to value
*Exp ...
and Poisson distributions.
Mean of a probability distribution
The mean of a probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
is the long-run arithmetic average value of a random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
having that distribution. If the random variable is denoted by , then it is also known as the expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of (denoted ). For a discrete probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, the mean is given by , where the sum is taken over all possible values of the random variable and is the probability mass function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
. For a continuous distribution
In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes for an Experiment (probability theory), experiment. ...
, the mean is , where is the probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
. In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
of the random variable with respect to its probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
. The mean need not exist or be finite; for some probability distributions the mean is infinite ( or ), while for others the mean is undefined
Undefined may refer to:
Mathematics
* Undefined (mathematics), with several related meanings
** Indeterminate form, in calculus
Computing
* Undefined behavior, computer code whose behavior is not specified under certain conditions
* Undefined ...
.
Generalized means
Power mean
The generalized mean
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).
De ...
, also known as the power mean or Hölder mean, is an abstraction of the quadratic, arithmetic, geometric, and harmonic means. It is defined for a set of ''n'' positive numbers ''x''i by
By choosing different values for the parameter ''m'', the following types of means are obtained:
''f''-mean
This can be generalized further as the generalized -mean
:
and again a suitable choice of an invertible will give
:
Weighted arithmetic mean
The weighted arithmetic mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
(or weighted average) is used if one wants to combine average values from different sized samples of the same population:
:
Where and are the mean and size of sample respectively. In other applications, they represent a measure for the reliability of the influence upon the mean by the respective values.
Truncated mean
Sometimes, a set of numbers might contain outliers (i.e., data values which are much lower or much higher than the others). Often, outliers are erroneous data caused by artifacts. In this case, one can use a truncated mean
A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, an ...
. It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of the total number of values.
Interquartile mean
The interquartile mean
The interquartile mean (IQM) (or midmean) is a statistical measure of central tendency based on the truncated mean of the interquartile range. The IQM is very similar to the scoring method used in sports that are evaluated by a panel of judges: ''d ...
is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.
:
assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights.
Mean of a function
In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
) set of values. This can happen when calculating the mean value of a function . Intuitively, a mean of a function can be thought of as calculating the area under a section of a curve, and then dividing by the length of that section. This can be done crudely by counting squares on graph paper, or more precisely by integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
. The integration formula is written as:
:
In this case, care must be taken to make sure that the integral converges. But the mean may be finite even if the function itself tends to infinity at some points.
Mean of angles and cyclical quantities
Angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
s, times of day, and other cyclical quantities require modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
to add and otherwise combine numbers. In all these situations, there will not be a unique mean. For example, the times an hour before and after midnight are equidistant to both midnight and noon. It is also possible that no mean exists. Consider a color wheel
A color wheel or color circle is an abstract illustrative organization of color hues around a circle, which shows the relationships between primary colors, secondary colors, tertiary colors etc.
Some sources use the terms ''color wheel'' & ' ...
—there is no mean to the set of all colors. In these situations, you must decide which mean is most useful. You can do this by adjusting the values before averaging, or by using a specialized approach for the mean of circular quantities.
Fréchet mean
The Fréchet mean In mathematics and statistics, the Fréchet mean is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. Karcher mean is the renaming o ...
gives a manner for determining the "center" of a mass distribution on a surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
or, more generally, Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
. Unlike many other means, the Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars.
It is sometimes also known as the ''Karcher mean'' (named after Hermann Karcher).
Swanson's rule
This is an approximation to the mean for a moderately skewed distribution.[Hurst A, Brown GC, Swanson RI (2000) Swanson's 30-40-30 Rule. American Association of Petroleum Geologists Bulletin 84(12) 1883-1891] It is used in hydrocarbon exploration and is defined as:
:
where ''P''10, ''P''50 and ''P''90 10th, 50th and 90th percentiles of the distribution.
Other means
* Arithmetic-geometric mean
*Arithmetic-harmonic mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
* Cesàro mean
*Chisini mean
In mathematics, a function ''f'' of ''n'' variables
:''x''1, ..., ''x'n''
leads to a Chisini mean ''M'' if for every vector <''x''1, ..., ''x'n''>, there exists a unique ''M'' such that
:''f''(''M'',''M'', ..., ''M'') = ''f''(''x' ...
*Contraharmonic mean In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, L_p, where ''p'' = 2.
Definition
The contraharmonic mean of a set of positive numbers is def ...
* Elementary symmetric mean
* Geometric-harmonic mean
*Grand mean The grand mean or pooled mean is the average of the means of several subsamples, as long as the subsamples have the same number of data points. For example, consider several lots, each containing several items. The items from each lot are sampling ( ...
*Heinz mean
In mathematics, the Heinz mean (named after E. Heinz) of two non-negative real numbers ''A'' and ''B'', was defined by Bhatia as:
:\operatorname_x(A, B) = \frac,
with 0 ≤ ''x'' ≤ .
For different values of ''x'', th ...
*Heronian mean In mathematics, the Heronian mean ''H'' of two non-negative real numbers ''A'' and ''B'' is given by the formula:
:H = \frac \left(A + \sqrt +B \right).
It is named after Hero of Alexandria.
Properties
*Just like all means, the Heronian mean is ...
*Identric mean
The identric mean of two positive real numbers ''x'', ''y'' is defined as:
:
\begin
I(x,y)
&=
\frac\cdot
\lim_
\sqrt xi-\eta\\ pt&=
\lim_
\exp\left(\frac-1\right)
\\ pt&=
\begin
x & \textx=y \\ pt\frac \sqrt -y& \text
\end
\end
It can be de ...
*Lehmer mean In mathematics, the Lehmer mean of a tuple x of positive real numbers, named after Derrick Henry Lehmer, is defined as:
:L_p(\mathbf) = \frac.
The weighted Lehmer mean with respect to a tuple w of positive weights is defined as:
:L_(\mathbf) = \fra ...
*Logarithmic mean
In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient.
This calculation is applicable in engineering problems involving heat and mass trans ...
*Moving average
In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
* Neuman–Sándor mean
*Quasi-arithmetic mean
In mathematics and statistics, the quasi-arithmetic mean or generalised ''f''-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f. It is a ...
*Root mean square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
(quadratic mean)
* Rényi's entropy (a generalized f-mean
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
)
*Spherical mean
In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point.
Definition
Consider an open set ''U'' in the Euclidean space R''n'' and a continuou ...
*Stolarsky mean In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.
Definition
For two positive real numbers ''x'', ''y'' the Stolarsky Mean is defined as:
:
\begin
S_p(x,y)
& = ...
*Weighted geometric mean
In statistics, the weighted geometric mean is a generalization of the geometric mean using the weighted arithmetic mean.
Given a sample x=(x_1,x_2\dots,x_n) and weights w=(w_1, w_2,\dots,w_n), it is calculated as:
: \bar = \left(\prod_^n x_i^\ri ...
*Weighted harmonic mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
See also
*Central tendency
In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications ...
**Median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
**Mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
*Descriptive statistics
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and an ...
*Kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
*Law of averages
The law of averages is the commonly held belief that a particular outcome or event will, over certain periods of time, occur at a frequency that is similar to its probability. Depending on context or application it can be considered a valid common ...
*Mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
*Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mas ...
*Summary statistics
In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in
* a measure of ...
*Taylor's law
Taylor's power law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship. It is named after the ecologist who first propos ...
Notes
References
{{Authority control
Moment (mathematics)