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45-584: The Dutchess County Agricultural Society, formed in 1841, first organized the fair in 1842 at Washington Hollow in Pleasant Valley, New York . The fair was mainly started as an old-fashioned agricultural event in which farmers would come and show off their livestock, crops, and tools with a popular event being a plow competition. The fair initially alternated between Washington Hollow and Poughkeepsie before moving to Springbrook Park in Rhinebeck in 1919. The move

90-447: A ↦ E ⁡ ( ‖ X − a ‖ ) . {\displaystyle a\mapsto \operatorname {E} (\|X-a\|).\,} The spatial median is unique when the data-set's dimension is two or more. An alternative proof uses the one-sided Chebyshev inequality; it appears in an inequality on location and scale parameters . This formula also follows directly from Cantelli's inequality . For

135-763: A probability density function f ), nor does it require a discrete one . In the former case, the inequalities can be upgraded to equality: a median satisfies P ⁡ ( X ≤ m ) = ∫ − ∞ m f ( x ) d x = 1 2 {\displaystyle \operatorname {P} (X\leq m)=\int _{-\infty }^{m}{f(x)\,dx}={\frac {1}{2}}} and P ⁡ ( X ≥ m ) = ∫ m ∞ f ( x ) d x = 1 2 . {\displaystyle \operatorname {P} (X\geq m)=\int _{m}^{\infty }{f(x)\,dx}={\frac {1}{2}}\,.} Any probability distribution on

180-404: A comment on a subsequent proof by O'Cinneide, Mallows in 1991 presented a compact proof that uses Jensen's inequality twice, as follows. Using |·| for the absolute value , we have The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex. The second inequality comes from the fact that a median minimizes

225-434: A data set x {\displaystyle x} of n {\displaystyle n} elements, ordered from smallest to greatest, Formally, a median of a population is any value such that at least half of the population is less than or equal to the proposed median and at least half is greater than or equal to the proposed median. As seen above, medians may not be unique. If each set contains more than half

270-426: A finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If the data set has an odd number of observations, the middle one is selected (after arranging in ascending order). For example, the following list of seven numbers, has the median of 6 , which is the fourth value. If the data set has an even number of observations, there is no distinct middle value and

315-426: A measure of location when one attaches reduced importance to extreme values, typically because a distribution is skewed , extreme values are not known, or outliers are untrustworthy, i.e., may be measurement or transcription errors. For example, consider the multiset The median is 2 in this case, as is the mode , and it might be seen as a better indication of the center than the arithmetic mean of 4, which

360-419: A measure of variability : the range , the interquartile range , the mean absolute deviation , and the median absolute deviation . For practical purposes, different measures of location and dispersion are often compared on the basis of how well the corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it

405-424: A single pass over the sample. The distributions of both the sample mean and the sample median were determined by Laplace . The distribution of the sample median from a population with a density function f ( x ) {\displaystyle f(x)} is asymptotically normal with mean μ {\displaystyle \mu } and variance where m {\displaystyle m}

450-408: A variable x as med( x ), x͂ , as μ 1/2 , or as M . In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced. The median is a special case of other ways of summarizing the typical values associated with a statistical distribution : it is the 2nd quartile , 5th decile , and 50th percentile . The median can be used as

495-473: A well-defined mean, such as the Cauchy distribution : The mean absolute error of a real variable c with respect to the random variable   X is Provided that the probability distribution of X is such that the above expectation exists, then m is a median of X if and only if m is a minimizer of the mean absolute error with respect to X . In particular, if m is a sample median, then it minimizes

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540-586: Is a town in Dutchess County , within the Hudson Valley of New York , United States. The population was 9,799 at the 2020 census. The town is centrally located in the county, northeast of the city of Poughkeepsie . U.S. Route 44 passes through the town. Settlers began arriving after 1735. The town was part of the Great Nine Partners Patent of 1697. The town was formed in 1821 from part of

585-469: Is a C function, but the reverse does not hold. If f is a C function, then If the medians are not unique, the statement holds for the corresponding suprema. Even though comparison-sorting n items requires Ω ( n log n ) operations, selection algorithms can compute the k th-smallest of n items with only Θ( n ) operations. This includes the median, which is the ⁠ n / 2 ⁠ th order statistic (or for an even number of samples,

630-399: Is an even number of classes. (For odd number classes, one specific class is determined as the median.) A geometric median , on the other hand, is defined in any number of dimensions. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid . There is no widely accepted standard notation for the median, but some authors represent the median of

675-409: Is larger than all but one of the values. However, the widely cited empirical relationship that the mean is shifted "further into the tail" of a distribution than the median is not generally true. At most, one can say that the two statistics cannot be "too far" apart; see § Inequality relating means and medians below. As a median is based on the middle data in a set, it is not necessary to know

720-424: Is less than the mean, as shown in the figure. Jensen's inequality states that for any random variable X with a finite expectation E [ X ] and for any convex function f This inequality generalizes to the median as well. We say a function f : R → R is a C function if, for any t , is a closed interval (allowing the degenerate cases of a single point or an empty set ). Every convex function

765-418: Is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the efficiency of candidate estimators shows that the sample mean is more statistically efficient when—and only when— data is uncontaminated by data from heavy-tailed distributions or from mixtures of distributions. Even then, the median has a 64% efficiency compared to

810-416: Is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of the center. Median income , for example, may be a better way to describe the center of the income distribution because increases in the largest incomes alone have no effect on the median. For this reason, the median is of central importance in robust statistics . The median of

855-520: Is the median of f ( x ) {\displaystyle f(x)} and n {\displaystyle n} is the sample size: A modern proof follows below. Laplace's result is now understood as a special case of the asymptotic distribution of arbitrary quantiles . For normal samples, the density is f ( m ) = 1 / 2 π σ 2 {\displaystyle f(m)=1/{\sqrt {2\pi \sigma ^{2}}}} , thus for large samples

900-517: The Dutchess County Fairgrounds . The fair closed early in 2011 due to Hurricane Irene . Recent additions to the fairgrounds include the restored Pleasant Valley Train Station. This article related to an American festival is a stub . You can help Misplaced Pages by expanding it . Pleasant Valley (town), New York Jennifer Chapman ( R ) Dan Degan ( WOR ) Pleasant Valley

945-410: The absolute deviation function a ↦ E ⁡ ( | X − a | ) {\displaystyle a\mapsto \operatorname {E} (|X-a|)} . Mallows's proof can be generalized to obtain a multivariate version of the inequality simply by replacing the absolute value with a norm : where m is a spatial median , that is, a minimizer of the function

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990-422: The arithmetic mean of the two middle order statistics). Selection algorithms still have the downside of requiring Ω( n ) memory, that is, they need to have the full sample (or a linear-sized portion of it) in memory. Because this, as well as the linear time requirement, can be prohibitive, several estimation procedures for the median have been developed. A simple one is the median of three rule, which estimates

1035-626: The drawing in the definition of expected value for arbitrary real-valued random variables ). An equivalent phrasing uses a random variable X distributed according to F : P ⁡ ( X ≤ m ) ≥ 1 2  and  P ⁡ ( X ≥ m ) ≥ 1 2 . {\displaystyle \operatorname {P} (X\leq m)\geq {\frac {1}{2}}{\text{ and }}\operatorname {P} (X\geq m)\geq {\frac {1}{2}}\,.} Note that this definition does not require X to have an absolutely continuous distribution (which has

1080-459: The Board meets on the third Thursday of each month. Median The median of a set of numbers 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 feature of the median in describing data compared to the mean (often simply described as the "average")

1125-453: The arithmetic mean of the absolute deviations. Note, however, that in cases where the sample contains an even number of elements, this minimizer is not unique. More generally, a median is defined as a minimum of as discussed below in the section on multivariate medians (specifically, the spatial median ). This optimization-based definition of the median is useful in statistical data-analysis, for example, in k -medians clustering . If

1170-407: The average family size was 3.09. In the town, the population was spread out, with 26.1% under the age of 18, 6.4% from 18 to 24, 30.6% from 25 to 44, 26.2% from 45 to 64, and 10.6% who were 65 years of age or older. The median age was 38 years. For every 100 females, there were 97.6 males. For every 100 females age 18 and over, there were 95.1 males. The median income for a household in the town

1215-576: The case of unimodal distributions, one can achieve a sharper bound on the distance between the median and the mean: A similar relation holds between the median and the mode: A typical heuristic is that positively skewed distributions have mean > median. This is true for all members of the Pearson distribution family . However this is not always true. For example, the Weibull distribution family has members with positive mean, but mean < median. Violations of

1260-412: The distribution has finite variance, then the distance between the median X ~ {\displaystyle {\tilde {X}}} and the mean X ¯ {\displaystyle {\bar {X}}} is bounded by one standard deviation . This bound was proved by Book and Sher in 1979 for discrete samples, and more generally by Page and Murty in 1982. In

1305-470: The formula for the case of discrete variables is given below in § Empirical local density . The sample can be summarized as "below median", "at median", and "above median", which corresponds to a trinomial distribution with probabilities F ( v ) {\displaystyle F(v)} , f ( v ) {\displaystyle f(v)} and 1 − F ( v ) {\displaystyle 1-F(v)} . For

1350-411: The library offers services such as printing, faxing, photocopying, and tech help. Digital services offered by the library include Overdrive, Inc., Hoopla, and NovelNY, among others. The library's programming includes book discussion groups, children's storytimes, in-depth tech help, and yoga. On November 6, 2018, an electrical fire started in the basement of the library and spread through the walls to

1395-459: The median as the median of a three-element subsample; this is commonly used as a subroutine in the quicksort sorting algorithm, which uses an estimate of its input's median. A more robust estimator is Tukey 's ninther , which is the median of three rule applied with limited recursion: if A is the sample laid out as an array , and then The remedian is an estimator for the median that requires linear time but sub-linear memory, operating in

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1440-421: The median is usually defined to be the arithmetic mean of the two middle values. For example, this data set of 8 numbers has a median value of 4.5 , that is ( 4 + 5 ) / 2 {\displaystyle (4+5)/2} . (In more technical terms, this interprets the median as the fully trimmed mid-range ). In general, with this convention, the median can be defined as follows: For

1485-557: The minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean. For any real -valued probability distribution with cumulative distribution function   F , a median is defined as any real number  m that satisfies the inequalities lim x → m − F ( x ) ≤ 1 2 ≤ F ( m ) {\displaystyle \lim _{x\to m-}F(x)\leq {\frac {1}{2}}\leq F(m)} (cf.

1530-567: The northern part of the town. As of the census of 2000, there were 9,066 people, 3,467 households, and 2,485 families residing in the town. The population density was 275.4 inhabitants per square mile (106.3/km ). There were 3,614 housing units at an average density of 109.8 per square mile (42.4/km ). The racial makeup of the town was 95.46% white , 1.93% black or African American , .12% Native American , .64% Asian , .02% Pacific Islander , .83% from other races , and 1% from two or more races. Hispanic or Latino of any race were 2.65% of

1575-426: The population, then some of the population is exactly equal to the unique median. The median is well-defined for any ordered (one-dimensional) data and is independent of any distance metric . The median can thus be applied to school classes which are ranked but not numerical (e.g. working out a median grade when student test scores are graded from F to A), although the result might be halfway between classes if there

1620-408: The population. There were 3,467 households, out of which 35.7% had children under the age of 18 living with them, 58.9% were married couples living together, 8.7% had a female householder with no husband present, and 28.3% were non-families. 23.2% of all households were made up of individuals, and 7.9% had someone living alone who was 65 years of age or older. The average household size was 2.61 and

1665-430: The real number set R {\displaystyle \mathbb {R} } has at least one median, but in pathological cases there may be more than one median: if F is constant 1/2 on an interval (so that f = 0 there), then any value of that interval is a median. The medians of certain types of distributions can be easily calculated from their parameters; furthermore, they exist even for some distributions lacking

1710-400: The rule are particularly common for discrete distributions. For example, any Poisson distribution has positive skew, but its mean < median whenever μ mod 1 > ln ⁡ 2 {\displaystyle \mu {\bmod {1}}>\ln 2} . See for a proof sketch. When the distribution has a monotonically decreasing probability density, then the median

1755-547: The town of Clinton . According to the United States Census Bureau , the town has a total area of 33.1 square miles (85.8 km ), of which 32.6 square miles (84.4 km ) is land and 0.58 square miles (1.5 km ), or 1.7%, is water. A section of U.S. Route 44 goes through the town in a southwestern-northeastern direction. It intersects with the Taconic State Parkway , a north–south highway, in

1800-479: The upper floor. It was estimated that the library lost about 52,000 items in the collection, primarily due to smoke damage, though in the year after the fire, the library acquired 30,000 items mostly through donations. The historic building, built in the 1870s, required about $ 3 million dollars to rebuild and expand, and finally reopened in early February 2022. The Pleasant Valley Free Library Board of Trustees currently has 12 members, though it can have up to 13, and

1845-492: The value of extreme results in order to calculate it. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated. Because the median is simple to understand and easy to calculate, while also a robust approximation to the mean , the median is a popular summary statistic in descriptive statistics . In this context, there are several choices for

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1890-420: The variance of the median equals ( π / 2 ) ⋅ ( σ 2 / n ) . {\displaystyle ({\pi }/{2})\cdot (\sigma ^{2}/n).} (See also section #Efficiency below.) We take the sample size to be an odd number N = 2 n + 1 {\displaystyle N=2n+1} and assume our variable continuous;

1935-436: Was $ 54,578, and the median income for a family was $ 62,264. Males had a median income of $ 47,647 versus $ 31,496 for females. The per capita income for the town was $ 25,942. About 3.9% of families and 5.6% of the population were below the poverty line , including 6.6% of those under age 18 and 6.2% of those age 65 or over. The Pleasant Valley Free Library was founded in 1903, and originally operated under State Charter and

1980-405: Was governed by five trustees. The library has moved locations twice, the first time after it burned in 1913, and the second in 1974, where it remains currently. The library offers adult, young adult, and children's materials in the form of books, audiobooks, music CDs, and DVDs. There are eight public computers available for use, as well as two public computers in the young adult area. Additionally,

2025-583: Was made following financial issues in the early 1900s and the subsequent cancellations of the 1916 and 1918 fairs. The Society faced heavy debt in the Depression era after the cancellations of the 1927 and 1928 fairs due to flooding. The fair was also cancelled in 1942 due to World War II. A financial upturn in the late 1940s allowed the Society to repay its debt and acquire the Springbrook Park land renaming it to

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