
Mean
There are several kinds of mean in various branches of mathematics (especially statistics).
For a data set, the arithmetic mean, also called the mathematical expectation or average, is the central value of a discrete set of numbers: specifically, 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 by
x
¯
{displaystyle {bar {x}}}
, pronounced “x bar”. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (denoted
x
¯
{displaystyle {bar {x}}}
) to distinguish it from the mean of the underlying distribution, the population mean (denoted
μ
{displaystyle mu }
or
μ
x
{displaystyle mu _{x}}
).In probability and statistics, the population mean, or expected value, are a measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p(x), and then adding all these products together, giving
μ
=
∑
x
p
(
x
)
{displaystyle mu =sum xp(x)}
. An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite.
For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.Outside probability and statistics, a wide range of other notions of “mean” are often used in geometry and analysis; examples are given below.

Mean (verb)
To intend.

Mean (verb)
To intend, to plan (to do); to have as one’s intention. from 8th c.
“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.”

Mean (verb)
To have intentions of a given kind. from 14th c.
“Don’t be angry; she meant well.”

Mean (verb)
To convey meaning.

Mean (verb)
To intend (something) for a given purpose or fate; to predestine. from 16th c.
“Actually this desk was meant for the subeditor.”
“Man was not meant to question such things.”

Mean (verb)
To convey (a given sense); to signify, or indicate (an object or idea). from 8th c.
“I’m afraid I don’t understand what you mean.”
“The sky is red this morning—does that mean we’re in for a storm?”

Mean (verb)
To have conviction in (something said or expressed); to be sincere in (what one says). from 18th c.
“Does she really mean what she said to him last night?”
“Say what you mean and mean what you say.”

Mean (verb)
To result in; to bring about. from 19th c.
“One faltering step means certain death.”

Mean (verb)
To be important (to). from 19th c.
“My home life means a lot to me.”

Mean (verb)
To lament.

Mean (adjective)
Common; general.

Mean (adjective)
Of a common or low origin, grade, or quality; common; humble.
“a man of mean parentage”
“/ a mean abode”

Mean (adjective)
Low in quality or degree; inferior; poor; shabby.
“a mean appearance”
“/ mean dress”

Mean (adjective)
Without dignity of mind; destitute of honour; lowminded; spiritless; base.
“a mean motive”

Mean (adjective)
Of little value or account; worthy of little or no regard; contemptible; despicable.

Mean (adjective)
Ungenerous; stingy, tightfisted; North American English: cheap; formal: niggardly, penurious, miserly.
“He’s so mean. I’ve never seen him spend so much as five pounds on presents for his children.”

Mean (adjective)
Disobliging; pettily offensive or unaccommodating; small.

Mean (adjective)
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.”

Mean (adjective)
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.”

Mean (adjective)
Powerful; fierce; harsh; damaging.
“It must have been a mean typhoon that levelled this town.”

Mean (adjective)
Accomplished with great skill; deft; hard to compete with.
“Your mother can roll a mean cigarette.”
“He hits a mean backhand.”

Mean (adjective)
Difficult, tricky.
“This problem is mean!”

Mean (adjective)
Having the mean (see noun below) as its value.

Mean (adjective)
Middling; intermediate; moderately good, tolerable.

Mean (noun)
A method or course of action used to achieve some result. from 14th c.

Mean (noun)
An intermediate step or intermediate steps.

Mean (noun)
Something which is intermediate or in the middle; an intermediate value or range of values; a medium. from 14th c.

Mean (noun)
The middle part of threepart polyphonic music; now specifically, the alto part in polyphonic music; an alto instrument. from 15th c.

Mean (noun)
The average of a set of values, calculated by summing them together and dividing by the number of terms; the arithmetic mean. from 15th c.

Mean (noun)
Any function of multiple variables that satisfies certain properties and yields a number representative of its arguments; or, the number so yielded; a measure of central tendency.

Mean (noun)
Either of the two numbers in the middle of a conventionally presented proportion, as 2 and 3 in 1:2=3:6.

Mean (verb)
intend to convey or refer to (a particular thing); signify
“I don’t know what you mean”
“I meant you, not Jones”
“he was asked to clarify what his remarks meant”

Mean (verb)
(of a word) have (something) as its signification in the same language or its equivalent in another language
“its name means ‘painted rock’ in Cherokee”

Mean (verb)
genuinely intend to express (something)
“when she said that she meant it”

Mean (verb)
be of a specified degree of importance to (someone)
“animals have always meant more to him than people”

Mean (verb)
intend (something) to occur or be the case
“it was meant to be a secret”
“they mean no harm”

Mean (verb)
be supposed to do something
“we were meant to go over yesterday”

Mean (verb)
design or destine for a particular purpose
“the jacket was meant for a much larger person”

Mean (verb)
have something as a motive or explanation in saying or doing
“what do you mean by leaving me out here in the cold?”

Mean (verb)
be generally considered to be
“this one’s meant to be priceless”

Mean (verb)
have as a consequence or result
“the proposals are likely to mean another hundred closures”
“heavy rain meant that the pitch was waterlogged”

Mean (verb)
necessarily or usually entail or involve
“coal stoves mean a lot of smoke”

Mean (adjective)
unwilling to give or share things, especially money; not generous
“they’re not mean with the garlic”
“she felt mean not giving a tip”

Mean (adjective)
unkind, spiteful, or unfair
“I was mean to them over the festive season”

Mean (adjective)
vicious or aggressive in behaviour
“the dogs were considered mean, vicious, and a threat”

Mean (adjective)
(especially of a place) poor in quality and appearance; shabby
“her home was mean and small”

Mean (adjective)
(of a person’s mental capacity or understanding) inferior
“it was obvious to even the meanest intelligence”

Mean (adjective)
of low birth or social class
“a muffler like that worn by the meanest of people”

Mean (adjective)
very skilful or effective; excellent
“she dances a mean tango”
“he’s a mean cook”

Mean (adjective)
(of a quantity) calculated as a mean; average
“participants in the study had a mean age of 35 years”

Mean (adjective)
equally far from two extremes
“hope is the mean virtue between despair and presumption”

Mean (noun)
the value obtained by dividing the sum of several quantities by their number; an average
“acid output was calculated by taking the mean of all three samples”

Mean (noun)
a condition, quality, or course of action equally removed from two opposite extremes
“the measure expresses a mean between saving and splashing out”