Misplaced Pages

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49-679: The TWHBEA was founded in 1935 in Lewisburg, Tennessee for the purpose of registering Tennessee Walking Horses and maintaining their bloodlines . The first president was Burt Hunter. The association was originally named the Tennessee Walking Horse Breeders' Association of America, but was renamed in 1974 to cover the popular show industry. The TWHBEA "closed the studbook" in 1947. This meant that after that date, new horses could only be registered if both their dam and stud stallion were each already registered. As of 2017,

98-447: 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

147-427: 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 a better representation of the center. Median income , for example, may be

196-763: 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

245-400: 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 the data set has an odd number of observations,

294-404: 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

343-466: A gaited breed, is considered to have been developed in Middle Tennessee, and the area is still a center of breeding and exhibitions for this horse. Marshall County is significant partly because of three men who have served as Governor of Tennessee were living here when they were elected to office: Henry Horton , Jim Nance McCord , and Buford Ellington . Today, Lewisburg/Marshall County is

392-408: 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

441-430: 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

490-428: 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

539-589: A mob that week. They are likely the same men. Until 1925, Lewisburg served the area principally as a trading and shipping center for its livestock and farm produce. As the county seat, it was a center of the justice system and active as a retail center for the area farmers. The world headquarters of the Tennessee Walking Horse Breeders' and Exhibitors' Association has been based in Lewisburg since 1939. The Tennessee Walking Horse ,

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588-501: 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

637-410: 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

686-424: 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}

735-556: A slow economy and soring allegations, 2015 saw a 7.9% increase in foal registration. The TWHBEA has a variety of programs through which horses can earn points and awards, including distance, competitive trail riding , versatility, and specific programs for mares or geldings . Individuals may work to become certified as riding instructors through the TWHBEA. The TWHBEA publishes the Voice magazine six times per year. The TWHBEA houses

784-477: 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

833-422: 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,

882-469: 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,

931-465: 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

980-400: 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

1029-424: 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

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1078-555: Is located was long occupied by various cultures of indigenous peoples. Historic Native Americans were here when French, Spanish and English explorers entered the area. Revolutionary War veterans were given land grants in this area by the State of North Carolina for services rendered during the war followed in the 1780s. North Carolina still claimed this territory under its colonial charter, but later gave up that claim and Tennessee became an independent state. Marshall County, named in honor of

1127-520: 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

1176-401: 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 ,

1225-487: The United States Census Bureau , the city has a total area of 11.7 square miles (30 km ), almost all of which is land (0.09% is water). As of the 2020 United States census , there were 12,288 people, 4,394 households, and 2,758 families residing in the city. As of the census of 2000, there were 10,413 people, 4,242 households, and 2,740 families residing in the city. The population density

1274-410: 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

1323-422: 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

1372-498: 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

1421-444: The TWHBEA has registered more than 520,000 horses , and has approximately 19,000 members. Only purebred Tennessee Walking Horses are eligible for TWHBEA registration. Beginning January 1, 2007, all foals must be DNA tested to verify their parentage before they are issued registration papers. An individual does not have to be a member of the TWHBEA to register a horse with the group. Although registration had been down since 2004, due to

1470-608: The Tennessee Walking Horse Hall of Fame, which honors notable trainers, breeders and others who made major contributions to the breed. Lewisburg, Tennessee Lewisburg is a city in and the county seat of Marshall County, Tennessee , United States. The population was 12,288 in 2020. Lewisburg is located in Middle Tennessee , fifty miles south of Nashville and fifty-two miles north of Huntsville , Alabama . Residents have access to

1519-453: 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

Tennessee Walking Horse Breeders' and Exhibitors' Association - Misplaced Pages Continue

1568-406: The average family size was 2.94. In the city, the population was spread out, with 24.2% under the age of 18, 10.2% from 18 to 24, 27.3% from 25 to 44, 22.1% from 45 to 64, and 16.3% who were 65 years of age or older. The median age was 37 years. For every 100 females, there were 90.0 males. For every 100 females age 18 and over, there were 86.4 males. The median income for a household in the city

1617-586: 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

1666-436: 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

1715-412: 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

1764-470: 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

1813-399: The home of several nationally known industries. Due to restructuring and movement of industry offshore, it has suffered thousands of job losses in recent years, but it maintains a viable work force. It is served by Marshall County Schools , including Marshall County High School . Median The median of a set of numbers is the value separating the higher half from the lower half of

1862-541: The larger cities via nearby I-65 . Lewisburg was named for the explorer Meriwether Lewis . By 1838, the town was supporting a newspaper and a bank. The downtown area is similar to many other small southern towns, with a courthouse on a square, surrounded by retail and commercial businesses. Shopping centers are located on the east and west ends of town. Lewisburg is located at 35°26′57″N 86°47′35″W  /  35.44917°N 86.79306°W  / 35.44917; -86.79306 (35.449034, -86.793112). According to

1911-459: 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

1960-420: 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 a median value of 4.5 , that

2009-474: 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

Tennessee Walking Horse Breeders' and Exhibitors' Association - Misplaced Pages Continue

2058-409: The population. There were 4,242 households, out of which 29.2% had children under the age of 18 living with them, 43.4% were married couples living together, 16.6% had a female householder with no husband present, and 35.4% were non-families. 31.0% of all households were made up of individuals, and 13.7% had someone living alone who was 65 years of age or older. The average household size was 2.38 and

2107-430: 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

2156-400: 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

2205-420: 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;

2254-641: The young nation's first Supreme Court Chief Justice and noted American jurist, John Marshall, was established by an act of the Tennessee General Assembly in 1836. The act which created the county specified that the county seat be named Lewisburg to commemorate the deeds of frontier explorer Meriweather Lewis . He had been leader of the Lewis and Clark Expedition to explore the Louisiana Territory soon after its purchase. Lewisburg

2303-422: Was $ 31,033, and the median income for a family was $ 38,246. Males had a median income of $ 30,619 versus $ 21,765 for females. The per capita income (the average income a person in the city is earning.) was $ 16,401. About 12.7% of families and 16.1% of the population were below the poverty line , including 20.1% of those under age 18 and 14.2% of those age 65 or over. The area in which Lewisburg and Marshall County

2352-410: Was 891.5 inhabitants per square mile (344.2/km ). There were 4,584 housing units at an average density of 392.4 units per square mile (151.5 units/km ). The racial makeup of the city was 79.96% White , 15.44% African American , 0.19% Native American , 0.47% Asian , 0.03% Pacific Islander , 2.85% from other races , and 1.06% from two or more races. Hispanic or Latino of any race were 5.15% of

2401-705: Was incorporated in 1837, on a site of 50 acres (200,000 m ) donated for civic purpose. There was white violence against freedmen during and after Reconstruction, extending into the early 20th century. Some lynchings of African Americans took place at the Marshall County Courthouse in Lewisburg during the period of highest violence around the turn of the 20th century. Two unidentified black men were lynched without trial in Lewisburg on August 5, 1903. Another account said that John Milligan (Millikin) and John L. Hunter had been killed in Needmore by

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