# Dictionary Definition

1 approximating the statistical norm or average or expected value; "the average income in New England is below that of the nation"; "of average height for his age"; "the mean annual rainfall" [syn: average, mean(a)]
2 characterized by malice; "a hateful thing to do"; "in a mean mood" [syn: hateful]
3 having or showing an ignoble lack of honor or morality; "that liberal obedience without which your army would be a base rabble"- Edmund Burke; "taking a mean advantage"; "chok'd with ambition of the meaner sort"- Shakespeare; "something essentially vulgar and meanspirited in politics" [syn: base, meanspirited]
4 excellent; "famous for a mean backhand"
5 marked by poverty befitting a beggar; "a beggarly existence in the slums"; "a mean hut" [syn: beggarly]
6 used of persons or behavior; characterized by or indicative of lack of generosity; "a mean person"; "he left a miserly tip" [syn: mingy, miserly, tight]
7 used of sums of money; so small in amount as to deserve contempt [syn: beggarly] n : an average of n numbers computed by adding some function of the numbers and dividing by some function of n [syn: mean value]

### Verb

1 mean or intend to express or convey; "You never understand what I mean!"; "what do his words intend?" [syn: intend]
2 have as a logical consequence; "The water shortage means that we have to stop taking long showers" [syn: entail, imply]
3 denote or connote; "maison' means house' in French"; "An example sentence would show what this word means" [syn: intend, signify, stand for]
4 have in mind as a purpose; "I mean no harm"; "I only meant to help you"; "She didn't think to harm me"; "We thought to return early that night" [syn: intend, think]
5 have a specified degree of importance; "My ex-husband means nothing to me"; "Happiness means everything"
6 intend to refer to; "I'm thinking of good food when I talk about France"; "Yes, I meant you when I complained about people who gossip!" [syn: think of, have in mind]
7 destine or designate for a certain purpose; "These flowers were meant for you" [also: meant]

# User Contributed Dictionary

## English

• /miːn/
• /mi:n/
• Rhymes: -iːn

### Etymology 1

From mænan, "to mean", "to allude to". Confer Dutch menen, German meinen. Cognate with mind and German Minne, "love".

#### Verb

1. To convey, signify, or indicate.
What does this hieroglyph mean?
The sky is red this morning—does that mean we're in for a storm?
2. To want or intend to convey.
I'm afraid I don't understand what you mean.
Say what you mean and mean what you say.
3. To intend; to plan on doing.
I didn't mean to knock your tooth out.
I mean to go to Baddeck this summer.
I meant to take the car in for a smog check, but it slipped my mind.
4. To have conviction in what one says.
Does she really mean what she said to him last night?
Say what you mean and mean what you say.
5. To have intentions of a some kind.
Don't be angry; she meant well.
Someone's coming up. He means business.
6. To result in; to bring about.
One faltering step means certain death.
##### Translations
convey, signify, indicate
want or intend to convey
intend; plan on doing
have conviction in what one says
have intentions of a some kind

### Etymology 2

mæne. Confer Dutch gemeen, German gemein, Gothic gamains. Cognate with Latin communis.

1. Causing or intending to cause intentional harm; bearing ill will towards another; cruel; malicious.
Watch out for her, she's mean. I said good morning to her, and she punched me in the nose.
2. Miserly; stingy.
He's so mean. I've never seen him spend so much as five pounds on presents for his children.
3. Selfish; acting without consideration of others; unkind.
It was mean to steal the girl's piggy bank, but he just had to get uptown and he had no cash of his own.
4. Powerful; fierce; harsh; damaging.
It must have been a mean typhoon that levelled this town.
5. Accomplished with great skill; deft; hard to compete with.
Your mother can roll a mean cigarette.
He hits a mean backhand.
6. Low in quality; inferior.
##### Translations
causing or intending to cause intentional harm
miserly, stingy
• Spanish: mezquino, tacaño
acting without consideration of others
powerful; fierce; harsh; damaging
• Dutch: gemeen
• Greek: άγριος
• Japanese: 厄介な (yakkai-na)
• Spanish: cruel
low in quality; inferior
accomplished with great skill; deft; hard to compete with
• Spanish: formidable

### Etymology 3

From meien (French moyen), Late Latin medianus, from medius. Cognate with "mid".

1. Having the mean (see noun below) as its value.
mean distance
mean time
mean solar time
mean sun
##### Translations
having the mean as its value

#### Noun

1. The average, the arithmetic mean.
2. Loosely, an intermediate value or range of values; a mid-value; a vague average.
3. Any function of multiple variables that satisfies certain properties and yields a number representative of its arguments.
4. Either of the two numbers in the middle of a proportion, as 2 and 3 in 1:2=3:6.
• 1825, John Farrar, translator, An Elementary Treatise on Arithmetic by Silvestre François Lacroix, third edition, page 102,
...if four numbers be in proportion, the product of the first and last, or of the two extremes, is equal to the product of the second and third, or of the two means.
• 1999, Dawn B. Sova, How to Solve Word Problems in Geometry, McGraw-Hill, ISBN 007134652X, page 85,
Using the means-extremes property of proportions, you know that the product of the extremes equals the product of the means. The ratio t/4 = 5/2 can be rewritten as t:4 = 5:2, in which the extremes are t and 2, and the means are 4 and 5.
• 2007, Carolyn C. Wheater, Homework Helpers: Geometry, Career Press, ISBN 1564147215, page 99,
In \frac=\frac23, the product of the means is 2\cdot27, and the product of the extremes is 18\cdot3. Both products are 54.
##### Translations
arithmetic mean
• Czech: aritmetický průměr
• Spanish: media
the statistical value
intermediate value

# Extensive Definition

In statistics, mean has two related meanings:
It is sometimes stated that the 'mean' means average. This is incorrect if "mean" is taken in the specific sense of "arithmetic mean" as there are different types of averages: the mean, median, and mode. For instance, average house prices almost always use the median value for the average.
For a real-valued random variable X, the mean is the expectation of X. Note that not every probability distribution has a defined mean (or variance); see the Cauchy distribution for an example.
For a data set, the mean is the sum of the observations divided by the number of observations. The mean is often quoted along with the standard deviation: the mean describes the central location of the data, and the standard deviation describes the spread.
An alternative measure of dispersion is the mean deviation, equivalent to the average absolute deviation from the mean. It is less sensitive to outliers, but less mathematically tractable.
As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. These are listed below.

## Examples of means

### Arithmetic mean

The arithmetic mean is the "standard" average, often simply called the "mean".
\bar = \frac\cdot \sum_^n
The mean may often be confused with the median or mode. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is 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. The median or mode are often more intuitive measures of such data.
Nevertheless, many skewed distributions are best described by their mean - such as the Exponential and Poisson distributions.
For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is:
\frac = \frac \approx 36.167.

### Geometric mean

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean). For example rates of growth.
\bar = \left ( \prod_^n \right ) ^
For example, the geometric mean of six values: 34, 27, 45, 55, 22, 34 is:
(34 \cdot 27 \cdot 45 \cdot 55 \cdot 22 \cdot 34)^ = 1,699,493,400^ = 34.545.

### Harmonic mean

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).
\bar = n \cdot \left ( \sum_^n \frac \right ) ^
For example, the harmonic mean of the six values: 34, 27, 45, 55, 22, and 34 is
\frac = \frac \approx 33.0179836.

### Generalized means

#### Power mean

The generalized mean, 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 xi by
\bar(m) = \left ( \frac\cdot\sum_^n \right ) ^
By choosing the appropriate value for the parameter m we get

#### f-mean

This can be generalized further as the generalized f-mean
\bar = f^\left(\right)
and again a suitable choice of an invertible f will give

### Weighted arithmetic mean

The weighted arithmetic mean is used, if one wants to combine average values from samples of the same population with different sample sizes:
\bar = \frac.
The weights w_i represent the bounds of the partial sample. In other applications they represent a measure for the reliability of the influence upon the mean by respective values.

### Truncated mean

Sometimes a set of numbers might contain outliers, i.e. a datum which is 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. 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 total number of values.

### Interquartile mean

The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.
\bar = \sum_^
assuming the values have been ordered.

### Mean of a function

In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by
\bar=\frac\int_a^bf(x)\,dx.
(See also mean value theorem.) In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by
\bar=\frac\int_U f.
This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of f to be
\exp\left(\frac\int_U \log f\right).
More generally, in measure theory and probability theory either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.

### Mean of angles

Most of the usual means fail on circular quantities, like angles, daytimes, fractional parts of real numbers. For those quantities you need a mean of circular quantities.

## Properties

The most general method for defining a mean or average, y, takes any function of a list g(x_1, x_2, ..., x_n), which is symmetric under permutation of the members of the list, and equates it to the same function with the value of the mean replacing each member of the list: g(x_1, x_2, ..., x_n) = g(y, y, ..., y). All means share some properties and additional properties are shared by the most common means. Some of these properties are collected here.

### Weighted mean

A weighted mean M is a function which maps tuples of positive numbers to a positive number (\mathbb_^n\to\mathbb_).
• "Fixed point": M(1,1,\dots,1) = 1
• Homogenity: \forall\lambda\ \forall x\ M(\lambda\cdot x_1, \dots, \lambda\cdot x_n) = \lambda \cdot M(x_1, \dots, x_n)
(using vector notation: \forall\lambda\ \forall x\ M(\lambda\cdot x) = \lambda \cdot M x )
• Monotony: \forall x\ \forall y\ (\forall i\ x_i \le y_i) \Rightarrow M x \le M y
It follows
Sketch of a proof: Because \forall x\ \forall y\ \left(||x-y||_\infty\le\varepsilon\cdot\min x \Rightarrow \forall i\ |x_i-y_i|\le\varepsilon\cdot x_i\right) and M((1+\varepsilon)\cdot x) = (1+\varepsilon)\cdot M x it follows \forall x\ \forall \varepsilon>0\ \forall y\ ||x-y||_\infty\le\varepsilon\cdot\min x \Rightarrow |Mx-My|\le\varepsilon.
• There are means, which are not differentiable. For instance, the maximum number of a tuple is considered a mean (as an extreme case of the power mean, or as a special case of a median), but is not differentiable.
• All means listed above, with the exception of most of the Generalized f-means, satisfy the presented properties.
• If f is bijective, then the generalized f-mean satisfies the fixed point property.
• If f is strictly monotonic, then the generalized f-mean satisfy also the monotony property.
• In general a generalized f-mean will miss homogenity.
The above properties imply techniques to construct more complex means:
If C, M_1, \dots, M_m are weighted means, p is a positive real number, then A, B with
\forall x\ A x = C(M_1 x, \dots, M_m x)
\forall x\ B x = \sqrt[p]
are also a weighted mean.

### Unweighted mean

Intuitively spoken, an unweighted mean is a weighted mean with equal weights. Since our definition of weighted mean above does not expose particular weights, equal weights must be asserted by a different way. A different view on homogeneous weighting is, that the inputs can be swapped without altering the result.
Thus we define M being an unweighted mean if it is a weighted mean and for each permutation \pi of inputs, the result is the same. Let P be the set of permutations of n-tuples.
Symmetry: \forall x\ \forall \pi\in P \ M x = M(\pi x)
Analogously to the weighted means, if C is a weighted mean and M_1, \dots, M_m are unweighted means, p is a positive real number, then A, B with
\forall x\ A x = C(M_1 x, \dots, M_m x)
\forall x\ B x = \sqrt[p]
are also unweighted means.

### Convert unweighted mean to weighted mean

An unweighted mean can be turned into a weighted mean by repeating elements. This connection can also be used to state that a mean is the weighted version of an unweighted mean. Say you have the unweighted mean M and weight the numbers by natural numbers a_1,\dots,a_n. (If the numbers are rational, then multiply them with the least common denominator.) Then the corresponding weighted mean A is obtained by
A(x_1,\dots,x_n) = M(\underbrace_,x_2,\dots,x_,\underbrace_).

### Means of tuples of different sizes

If a mean M is defined for tuples of several sizes, then one also expects that the mean of a tuple is bounded by the means of partitions. More precisely
• Given an arbitrary tuple x, which is partitioned into y_1, \dots, y_k, then it holds M x \in \mathrm(M y_1, \dots, M y_k). (See Convex hull)

## Population and sample means

The mean of a normally distributed population has an expected value of μ, known as the population mean. The sample mean makes a good estimator of the population mean, as its expected value is the same as the population mean. The sample mean of a population is a random variable, not a constant, and consequently it will have its own distribution. For a random sample of n observations from a normally distributed population, the sample mean distribution is
\bar \thicksim N\left\.
Often, since the population variance is an unknown parameter, it is estimated by the mean sum of squares, which changes the distribution of the sample mean from a normal distribution to a Student's t distribution with n − 1 degrees of freedom.

## Mathematics education

In many state and government curriculum standards, students are traditionally expected to learn either the meaning or formula for computing the mean by the fourth grade. However, in many standards-based mathematics curricula, students are encouraged to invent their own methods, and may not be taught the traditional method. Reform based texts such as TERC in fact discourage teaching the traditional "add the numbers and divide by the number of items" method in favor of spending more time on the concept of median, which does not require division. However, mean can be computed with a simple four-function calculator, while median requires a computer. The same teacher guide devotes several pages on how to find the median of a set, which is judged to be simpler than finding the mean.

mean in Danish: Gennemsnit
mean in German: Mittelwert
mean in Spanish: Promedio
mean in Esperanto: Averaĝo
mean in Persian: میانگین
mean in French: Moyenne
mean in Korean: 평균
mean in Italian: Media (statistica)
mean in Hebrew: ממוצע
mean in Dutch: Gemiddelde
mean in Japanese: 平均
mean in Lao: ຄ່າສະເຫຼ່ຍ
mean in Norwegian: Gjennomsnitt
mean in Polish: Średnia
mean in Portuguese: Média
mean in Russian: Среднее значение
mean in Slovenian: Srednja vrednost
mean in Sundanese: Mean
mean in Thai: มัชฌิม
mean in Finnish: Keskiluku
mean in Chinese: 平均数