# Dictionary Definition

median adj

1 relating to or constituting the middle value of
an ordered set of values (or the average of the middle two in an
even-numbered set); "the median value of 17, 20, and 36 is 20";
"the median income for the year was $15,000" [syn: median(a),
average]

2 dividing an animal into right and left halves
[syn: medial]

3 relating to or situated in or extending toward
the middle [syn: medial]
n : the value below which 50% of the cases fall [syn: median
value]

# User Contributed Dictionary

## English

### Etymology

From médian.### Noun

- The measure of central tendency of a set of values computed by ordering the values and taking the value at position ( + 1) / 2 when is odd or the arithmetic mean of the values at positions / 2 and ( / 2) + 1 when is even.
- The area separating two lanes of opposite-direction traffic in the United States. This area is often covered with vegetation, but also may be covered in concrete and possess traffic accident safety devices such as guardrails.
- The middlemost marquee of the body that divides the subject in a symmetrical fashion, such as a median sagittal section', which divides the body symmetrically on a vertical plane, as opposed to a 'parasagittal section', which is a vertical cross section that does not divide on a parallel symmetrical alignment.

#### Translations

statistics: measure of central tendency

the area separating two lanes of
opposite-direction traffic in the United States

anatomy: the middlemost marquee of the body that
divides the subject in a symmetrical fashion

### Adjective

- Having the median as its value.

#### Translations

having the median as its value

### Related terms

## Esperanto

### Adjective

## Finnish

### Noun

median### Anagrams

# Extensive Definition

In probability
theory and statistics, a median is
described as the number separating the higher half of a sample, a
population, or a probability
distribution, from the lower half. The median of a finite list
of numbers can be found by arranging all the observations from
lowest value to highest value and picking the middle one. If there
is an even number of observations, the median is not unique, so one
often takes the mean of the
two middle values. At most half the population have values less
than the median and at most half have values greater than the
median. If both groups contain less than half the population, then
some of the population is exactly equal to the median. For example,
if a < b < c, then the
median of the list is b, and if
a < b < c < d,
then the median of the list is the mean of b and c, i.e. it is
(b + c)/2.

## Notation

The median of some variable x\,\! is denoted either as \tilde\,\! or as \mu_(x).\,\!## Popular explanation

The difference between the median and the mean is
illustrated in this simple example:

Suppose 19 paupers and 1 billionaire are in a
room. Everyone removes all the money from their pockets and puts it
on a table. Each pauper puts $5 on the table; the billionaire puts
$1 billion (i.e. $109) there. The total is then $1,000,000,095. If
that money is divided equally among the 20 people, each gets
$50,000,004.75. That amount is the mean amount of money that the 20
people brought into the room. But the median amount is $5, since
one may divide the group into two groups of 10 people each, and say
that everyone in the first group brought in no more than $5, and
each person in the second group brought in no less than $5. In a
sense, the median is the amount that the typical person brought in.
By contrast, the mean is not at all typical, since nobody in the
room brought in an amount approximating $50,000,004.75.

## Measures of statistical dispersion

When the median is used as a location
parameter in descriptive statistics, there are several choices
for a measure of variability: the range,
the interquartile
range, the mean absolute
deviation, and the median
absolute deviation. Since the median is the same as the second
quartile, its calculation is illustrated in the article on quartiles.

Working with computers, a population of integers
should have an integer median. Thus, for an integer population with
an even number of elements, there are two medians known as lower
median and upper median. For floating point population, the median
lies somewhere between the two middle elements, depending on the
distribution. Median is the middle most value after arranging data
by any order

## Medians of probability distributions

For any probability
distribution on the real line
with
cumulative distribution function F, regardless of whether it is
any kind of continuous probability distribution, in particular an
absolutely
continuous distribution (and therefore has a
probability density function), or a discrete probability
distribution, a median m satisfies the inequalities

- \operatorname(X\leq m) \geq \frac \quad\and\quad \operatorname(X\geq m) \geq \frac\,\!

or

- \int_^m \mathrmF(x) \geq \frac \quad\and\quad \int_m^ \mathrmF(x) \geq \frac\,\!

in which a Riemann-Stieltjes
integral is used. For an absolutely continuous probability
distribution with
probability density function f, we have

- \operatorname(X\leq m) = \operatorname(X\geq m)=\int_^m f(x)\, \mathrmx=0.5.\,\!

Medians of particular distributions: The medians
of certain types of distributions can be easily estimated from
their parameters: The median of a normal
distribution with mean μ and variance σ2 is μ. In fact, for a
normal distribution, mean = median = mode. The median of a uniform
distribution in the interval [a, b] is
(a + b) / 2, which is also the mean. The median
of a Cauchy
distribution with location parameter x0 and scale parameter y
is x0, the location parameter. The median of an exponential
distribution with rate
parameter \lambda is the natural log of 2 divided by the rate
parameter: \ln 2 /\lambda. The median of a Weibull
distribution with shape parameter k and scale parameter \lambda
is \lambda (\ln 2)^.

## Medians in descriptive statistics

The median is primarily used for skewed distributions, which it
represents differently than the arithmetic
mean. Consider the multiset . The median is 2 in
this case, as is the mode,
and it might be seen as a better indication of central
tendency than the arithmetic
mean of 3.166….

Calculation of medians is a popular technique in
summary
statistics and
summarizing statistical data, since it is simple to understand
and easy to calculate, while also giving a measure that is more
robust in the presence of outlier values than is the
mean.

## Theoretical properties

### An optimality property

The median is also the central point which
minimizes the average of the absolute deviations; in the example
above this would be (1 + 0 + 0 + 0 + 1 + 7) / 6 = 1.5 using the
median, while it would be 1.944 using the mean. In the language of
probability theory, the value of c that minimizes

- E(\left|X-c\right|)\,

is the median of the probability distribution of
the random
variable X. Note, however, that c is not always unique, and
therefore not well defined in general.

### An inequality relating means and medians

For continuous probability distributions, the
difference between the median and the mean is less than or equal to
one standard
deviation. See
an inequality on location and scale parameters.

## Efficient computation

Even though sorting
n items takes in general O(n log
n) operations, by using a
"divide and conquer" algorithm the median of n items can be
computed with only O(n)
operations (in fact, you can always find the k-th element of a list
of values with this method; this is called the selection
problem).

## Easy explanation (Statistics)

As an example, we will calculate the median of
the following population of numbers: 1, 5, 2, 8, 7.

Start by sorting the numbers: 1, 2, 5, 7,
8.

In this case, 5 is the median, because when the
numbers are sorted, it is the middle number. If there is an even
amount of numbers, the median is the arithmetic
mean of the two middle numbers.

## See also

- Order statistic
- An inequality on location and scale parameters
- The median is the 2nd quartile, 5th decile, and 50th percentile.
- Median voter theory
- The median in general is a biased estimator.
- Median graph
- The centerpoint is a generalization of the median for data in higher dimensions.

## External links

median in Arabic: وسيط حسابي

median in Bulgarian: Медиана (статистика)

median in Catalan: Mediana

median in Czech: Medián

median in Danish: Median

median in German: Median

median in Estonian: Mediaan

median in Spanish: Mediana (estadística)

median in Esperanto: Mediano (statistiko)

median in Basque: Mediana

median in Persian: میانه (آمار)

median in French: Médiane (centre)

median in Galician: Mediana

median in Croatian: Mediana

median in Icelandic: Miðgildi

median in Italian: Mediana (statistica)

median in Hebrew: חציון

median in Lithuanian: Mediana

median in Hungarian: Medián

median in Dutch: Mediaan (statistiek)

median in Japanese: 中央値

median in Norwegian: Median

median in Polish: Mediana

median in Portuguese: Mediana
(estatística)

median in Russian: Медиана (статистика)

median in Sicilian: Mediana

median in Simple English: Median

median in Slovak: Medián

median in Slovenian: Mediana

median in Serbian: Медијана (статистика)

median in Sundanese: Median

median in Finnish: Mediaani

median in Swedish: Median

median in Tamil: இடைநிலையளவு

median in Thai: มัธยฐาน

median in Vietnamese: Số trung vị

median in Tajik: Медиана

median in Turkish: Medyan

median in Chinese: 中位數

# Synonyms, Antonyms and Related Words

amidships, average, balance, banal, center, centermost, central, common, core, diameter, diaphragm, equator, equatorial, equidistant, generality, golden mean,
halfway, happy medium,
heart, intercurrent, interior, interjacent, intermediary, intermediate, intervenient, intervening, juste-milieu,
kernel, mean, medial, mediocre, mediocrity, mediterranean, medium, mesial, mesne, mezzo, mid, middle, middle course, middle
ground, middle point, middle position, middle state,
middle-of-the-road, middlemost, middling, midland, midmost, midpoint, midriff, midships, midst, midway, moderate, norm, normal, nuclear, nucleus, ordinary, par, routine, rule, run, standard, thick, thick of things, usual, via media, waist, waistline, zone