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60-426: The district includes the following municipalities: Media Borough , Edgmont Township , Middletown Township , and Upper Providence Township . In the school year 2022-2023, the district provided basic educational services to 4143 pupils with 300 teachers, 395 other staff, and 19.5 administrators. The district received more than $ 17 million in state funding in school year 2022–2023. According to 2017-21 ACS -ED data,

120-447: 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

180-498: A Quaker , was an early Media resident, settling just outside the small village then known as "Providence", along Providence Great Road. The village then included a tailor shop, blacksmith shop, wheelwright shop, barn and other buildings. Minshall bought 625 acres (253 ha) from William Penn and arrived in 1682. The Providence Friends Meetinghouse was established at his house in February 1688. The original Friends Meetinghouse

240-763: 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

300-554: A vaudeville house in 1927. The first talkie film, The Jazz Singer , was shown there. It remained a popular cinema through the 1970s and 1980s. In 1994, the theater underwent a $ 1 million (~$ 1.87 million in 2023) restoration by Walter Strine Sr. and re-opened as the Media Theatre for the Performing Arts. Shows produced there have included The Full Monty , Carousel and Miss Saigon . On March 8, 1971,

360-428: 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 a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If

420-404: 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

480-408: A median is based on the middle data in a set, it is not necessary to know 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

540-430: 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. the drawing in the definition of expected value for arbitrary real-valued random variables ). An equivalent phrasing uses

600-428: 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 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

660-452: 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 a data set x {\displaystyle x} of n {\displaystyle n} elements, ordered from smallest to greatest, Formally,

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720-501: 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

780-552: A rectangular grid around the location of the new courthouse, lots were sold at public auctions, and the construction of houses began. Sources agree that Minshall Painter, a descendant of Thomas Minshall, suggested the name "Media", but do not agree on the reason. The name most likely comes from the borough's median location in the direct center of Delaware County. In 1940, the Pennsylvania guide described Media by noting that "[t]he majority of its houses, almost all built since

840-410: A robust approximation to the mean , the median is a popular summary statistic in descriptive statistics . In this context, there are several choices for 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

900-431: 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") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides

960-424: 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}

1020-477: 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

1080-525: Is 7a. As of the 2020 census, the racial makeup of the borough was 82.9% White , 3.5% African American , 0.4% Native American , 5.7% Asian , 2.6% from Hawaiian and Pacific Islander, 3.9% from two or more races, and 2.5% from Hispanic or Latino of any race. At the time of the 2010 Census, the racial makeup of the borough was 83.4% White , 10.6% African American , 0.1% Native American , 3.5% Asian , 0.5% from other races , and 1.9% from two or more races. Hispanic or Latino of any race were 2.5% of

1140-469: 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,

1200-465: 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 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

1260-400: 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 is an even number of classes. (For odd number classes, one specific class is determined as the median.) A geometric median , on

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1320-424: 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

1380-539: Is part of the Delaware Valley , also known as the Philadelphia metropolitan area. Media was incorporated in 1850 at the same time that it was named the county seat. As of the 2020 U.S. census , the population was 5,991. The history of the area goes back to William Penn , but the area remained predominantly rural until the 20th century. Land in the area was sold and settled soon after William Penn

1440-1133: Is run by a mayor and an elected council. Mayor Bob McMahon was first elected in 1992. Mark Paikoff is the president of the Media Borough Council, and Elizabeth Romaine is the vice-president. As of April 2024, the other Council members are Kevin Boyer, Lisa Gelman, Jen Malkoun, Tray Herman, and Joi Washington. Media lies within the Rose Tree Media School District , created by a merger with the Rose Tree Union School District and Media Borough School District in 1966. Public school students living within borough boundaries attend Media Elementary School, located in Downtown Media, for grades K-5. Springton Lake Middle School serves students in grades 6–8, and Penncrest High School serves students in grades 9–12. The Media-Upper Providence Friends School

1500-520: 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

1560-676: Is the only private school in the borough. Mother of Providence Regional Catholic School in Wallingford is the area Catholic school of the Roman Catholic Archdiocese of Philadelphia . It formed in 2012 from a merger of Nativity BVM School, which was Media's only pariochal school and St. John Chrysostom in Wallingford. Nativity BVM school opened in 1912, with its last building occupied in 1949. The Nativity BVM school administration chose not to file an appeal against

1620-401: Is uncontaminated by data from heavy-tailed distributions or from mixtures of distributions. Even then, the median has a 64% efficiency compared to 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 ,

1680-618: The American Civil War , sit far back on shaded lawns and seem somewhat gloomy. The borough has a large and prosperous business section and a few small industrial plants; many townspeople work in Philadelphia or Chester ." The John J. Tyler Arboretum occupies part of Thomas Minshall's original 625 acres (253 ha). This farm and a nearby Village of Lima was used by the Underground Railroad . The land

1740-589: The Citizens' Commission to Investigate the FBI raided an FBI "resident agency" in Media. They later released thousands of documents to major newspapers around the country. These documents revealed FBI tactics, like the recruitment of Boy Scouts as informants, and confirmed for the first time the existence of COINTELPRO , an FBI program to "expose, disrupt, misdirect, discredit, or otherwise neutralize" dissident groups in

1800-581: The Pennsylvania Department of Transportation (PennDOT) and 14.89 miles (23.96 km) were maintained by the borough. Route 252 , Providence Road, is the only numbered highway presently traversing the borough. It follows a north-south alignment along the eastern border of Media. U.S. 1 formerly ran through the borough until the "Media bypass" was completed in 1960. The bypass has an unusual "volleyball" or three-level diamond interchange with Interstate 476. The former Route 1 through

1860-553: The United States Census Bureau , the borough has a total area of 0.77 square miles (2.0 km ), of which 0.004 square miles (0.01 km ), or 0.42%, is water. Media is situated on high ground (250 to 350 feet (76 to 107 m) above sea level) draining west to Ridley Creek , a south-flowing tributary of the Delaware River . Media has a humid subtropical climate ( Cfa ) and the hardiness zone

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1920-410: 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

1980-422: 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

2040-498: 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 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

2100-579: The 2012 order to merge. Some parents had lobbied for the continued operation of the school. The archdiocese had originally planned to make Nativity BVM the regional campus, but changed when St. John Chrystosom had appealed the decision. After the closure, Media Elementary School occupied the campus while renovations of the permanent Media Elementary occurred. The following are in townships around Media, and have Media mailing addresses: As of 2018, there were 18.29 miles (29.43 km) of public roads in Media, of which 3.40 miles (5.47 km) were maintained by

2160-567: The Commonwealth of Pennsylvania by Special Act of Assembly incorporated the Borough of Media, and made the sale of malt and spirituous liquors unlawful within its borders. At the same time, the county seat of Delaware County was moved to Media from Chester. The borough was formed from four farms purchased by the county, totaling only 480 acres (190 ha). The borders of the borough have not changed since that time. Streets were plotted in

2220-481: The Idlewild Hotel (1871) on Lincoln Street at Gayley Terrace, Chestnut Grove House or "The Colonial" (1860) on Orange Street, and Brooke Hall on Orange Street and Washington Avenue (now Baltimore Avenue). The Chestnut Grove was used for a year by nearby Swarthmore College due to a fire on its campus. The West Chester and Philadelphia Railroad was built through Media on October 19, 1854. Electrified service

2280-604: The United States. In June 2006, Media became the first town in the United States to follow over 300 towns in Europe in attaining fair trade certification . To meet the criteria for certification, Media passed a council resolution in support of fair trade , served fair-trade coffee and tea in local government meetings and offices, ensured that a range of fair-trade products were available in local restaurants and businesses, raised popular support and provided media coverage for

2340-453: 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

2400-586: 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

2460-422: The center of Media is known by its older name, Baltimore Avenue, changing to " Baltimore Pike " outside the borough limits. Philadelphia International Airport (PHL), the 21st-busiest airport in the nation in 2022, is 11 miles' driving distance (about a 15-minute drive) from downtown Media, following Baltimore Pike east, then Interstate 476 south and Interstate 95 northeast. Median The median of

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2520-436: 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 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

2580-438: 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 the median is usually defined to be the arithmetic mean of the two middle values. For example, this data set of 8 numbers has

2640-412: 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

2700-399: The district serves a resident population of 37,141. The median household income is $ 103,810, versus a state median income of $ 67,587, and national median income of $ 69,021. Current Board Members: Media, Pennsylvania Media is a borough in and the county seat of Delaware County, Pennsylvania , United States. It is located about 13 miles (21 km) west of Philadelphia . It

2760-517: The fair-trade campaign, and convened a fair-trade steering committee to ensure continued commitment. Three locally recognized historic districts were designated by the borough in 1975. These districts are: There are several parks located within the borough of Media and shared with surrounding communities. Media is located in central Delaware County at 39°55′8″N 75°23′17″W  /  39.91889°N 75.38806°W  / 39.91889; -75.38806 (39.918761, -75.388127). According to

2820-428: The following addresses are included in the 19063 ZIP Code: Elwyn , Garden City , Glen Riddle , and Rose Valley . Other areas at least partially included in the 19063 zip code are Upper Providence Township ; Nether Providence Township , the neighborhoods of South Media , Bowling Green, Pine Ridge and Ridgewood; and most of Middletown Township , including, Bortondale, Riddlewood , and Lima . The borough of Media

2880-470: 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

2940-459: 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

3000-474: 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 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

3060-607: The population. The term "Media" is often used to include not only the borough of Media, but other municipalities but that share the ZIP Code . The borough of Media covers only 0.8 square miles (2.1 square kilometres) and less than 6,000 residents, but the Media ZIP Code 19063 covers 23.08 sq mi (59.8 km ) and a population of 35,704. According to the United States Postal Service ,

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3120-430: 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

3180-400: 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

3240-470: The time. The current borough, formed in 1850, sits between the two townships. In 1683, the Court of Chester County approved the construction of "Providence Great Road", now Pennsylvania Route 252 . The road, which runs north from Chester to within a few blocks of today's downtown, is shown on a 1687 map along with the names of local landowners. It forms the eastern border of the borough. Thomas Minshall,

3300-420: 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;

3360-418: Was built out of logs in 1699 or 1700, and the current building was completed in 1814. A house on Minshall's property, built around 1750, still stands and was given to the citizens of the borough in 1975. In 1789, Chester County, Pennsylvania was divided, with the eastern portion becoming Delaware County, Pennsylvania . The area in the center of the new county remained rural through 1850. On March 11, 1850,

3420-492: Was donated to a public trust in 1944 by an eighth-generation descendant. The arboretum was started as a private collection by brothers Jacob and Minshall Painter. In 1825, they began systematically planting over 1,000 varieties of trees and shrubs. Over twenty of their original trees survive, including a giant sequoia . Minshall Painter was also a leader of the Delaware County Institute of Science , which

3480-464: Was formed on September 21, 1833, with just four other members: George Miller, John Miller, George Smith, M.D., and John Cassin . The institute was incorporated in 1836. About 1850, Painter gave the institute the land where its building currently stands at 11 Veterans Square, and the building was constructed in 1867. In the second half of the 19th century, Media was a summer resort for well-to-do Philadelphians. The borough's large vacation hotels included

3540-495: Was named proprietor of the colonial-era Province of Pennsylvania in 1681 by King Charles II of England . Peter and William Taylor bought the land where Media is now located, directly from Penn. At the time, the land was located in Chester County . Providence Township was organized in 1684, and later divided into Upper Providence and Nether Providence townships by 1690, even though they only had 40 taxable properties at

3600-559: Was opened on December 2, 1928. Up to 50 trains passed through each day. The railroad became part of the Philadelphia, Wilmington and Baltimore Railroad and eventually the Penn Central . SEPTA took over operations in 1983. Woodrow Wilson spoke at the Media Station in 1912 during his first election campaign . Trolley transportation lines spread to and through Media in the 1890s and early 1900s. The Media Theatre opened as

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