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Central tendency

In statistics, a central tendency is a central or typical value for a probability distribution. It may be called a center or location of the distribution. Colloquially, measures of central tendency are called averages; the term central tendency dates from the late 1920s. The most common measures of central tendency are the median and the mode. A middle tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Authors use central tendency to denote "the tendency of quantitative data to cluster around some central value."The central tendency of a distribution is contrasted with its dispersion or variability. Analysis may judge whether data has a weak central tendency based on its dispersion; the following may be applied to one-dimensional data. Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are taking logarithms. Whether a transformation is appropriate and what it should be, depend on the data being analyzed.

Arithmetic mean or mean the sum of all measurements divided by the number of observations in the data set. Median the middle value that separates the higher half from the lower half of the data set; the median and the mode are the only measures of central tendency that can be used for ordinal data, in which values are ranked relative to each other but are not measured absolutely. Mode the most frequent value in the data set; this is the only central tendency measure that can be used with nominal data, which have purely qualitative category assignments. Geometric mean the nth root of the product of the data values; this measure is valid only for data that are measured on a positive scale. Harmonic mean the reciprocal of the arithmetic mean of the reciprocals of the data values; this measure too is valid only for data that are measured on a positive scale. Weighted arithmetic mean. Truncated mean or trimmed mean the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.

Interquartile mean. Midrange the arithmetic mean of minimum values of a data set. Midhinge the arithmetic mean of the third quartiles. Trimean the weighted arithmetic mean of two quartiles. Winsorized mean an arithmetic mean in which extreme values are replaced by values closer to the median. Any of the above may be applied to each dimension of multi-dimensional data, but the results may not be invariant to rotations of the multi-dimensional space. In addition, there are the Geometric median; this is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions. Quadratic mean useful in engineering, but not used in statistics; this is because it is not a good indicator of the center of the distribution when the distribution includes negative values. Simplicial depth the probability that a randomly chosen simplex with vertices from the given distribution will contain the given center Tukey median a point with the property that every halfspace containing it contains many sample points Several measures of central tendency can be characterized as solving a variational problem, in the sense of the calculus of variations, namely minimizing variation from the center.

That is, given a measure of statistical dispersion, one asks for a measure of central tendency that minimizes variation: such that variation from the center is minimal among all choices of center. In a quip, "dispersion precedes location"; this center may not be unique. In the sense of Lp spaces, the correspondence is: The associated functions are called p-norms: 0-"norm", 1-norm, 2-norm, ∞-norm; the function corresponding to the L0 space is not a norm, is thus referred to in quotes: 0-"norm". In equations, for a given data set X, thought of as a vector x =, the dispersion about a point c is the "distance" from x to the constant vector c = in the p-norm: f p = ‖ x − c ‖ p:= 1 / p. For p = 0 and p = ∞ these functions are defined by taking limits as p → 0 and p → ∞ {\d

Fred C. Beyer High School

Fred C. Beyer High School is a high school in Modesto, located in the Stanislaus County. Although when it was opened in 1972 to an enrollment of 975 students freshmen and only a smattering of upperclassmen who transferred from other schools voluntarily, Beyer was designed to serve 2000 students. By the time the first class of freshman had advanced to seniority, enrollment was about 1800. With the addition of numerous portable buildings, the number of enrolled students was as high as 3150, but since Enochs High School opened enrollment has returned to 1800; when built, Beyer was on the outskirts of the city, halfway between Modesto. In anticipation of the 1976 United States Bicentennial, the Patriots were chosen as the school's mascot, the school colors chosen as red and blue. Beyer High was intended to incorporate the latest advances in the science of education, the most prominent of, Daily Demand Scheduling; the DDS or Modular Scheduling as it was called was a dismal failure. It gave all students the ability to chose any classes.

There was little supervision to make sure students met graduation requirements. Hence many did not have the necessary credits at graduation the school districts realized it's and their shortcomings and scraped the program. DDS was terminated in 1981-82 school year by a decision of the Modesto City School Board of Trustees, over the objections of administrators, staff and parents. Under the tutelage of Ron Underwood, Beyer's competitive speech and debate teams and individual competitors won numerous state and national awards in the National Forensics League. Former members of the Central California Conference. Current members of the Modesto Metropolitan Conference. Coach Doug Severe lead the Beyer Patriots to their first trip to the Sac-Joaquin Section Playoffs in 2006. In 2007 the Beyer Patriots made their second trip to the Sac-Joaquin Section Playoffs. Severe played on the 1975 Beyer football team, led by coach Dean Laun, when Beyer won the CCC title while allowing opponents a total of 28 points, including the post-season.

There being no Sac-Joaquin Section playoffs in the 1970s, Beyer and Stagg High School played a Sac-Joaquin post-season game called the Turkey Bowl. In 2011, Beyer's varsity football team went undefeated in league, 8-2 overall, they beat Downey in decision to who won the MMC title, won that in their division, continuing to proceed into the playoffs. Beyer's baseball team won conference championships in the 1970s. Won frequent conference championships in the 1980s including a C. I. F. Sac-Joaquin Section AAA Tournament 2nd-place finish under P. Cornwell in 1985. Beyer basketball saw a number of conference championships under coach Davis. Beyer boys' cross-country team has taken three Division I Sac-Joaquin Section championships. Beyer girls' cross-country team went undefeated in the Central California Conference for six years. Beyer's golf team competed in section playoffs and went to state finals under coach Streeter. Coached by Doug Severe, Beyer High has had several wrestlers place at the California Interscholastic Federation State Wrestling Championships.

Eric Bell, MLB player Timothy Olyphant – class of 1986, Primetime Emmy-nominated actor Jeremy Renner – class of 1989, Oscar-nominated actor Joel N. Ward Official website

West Harptree

West Harptree is a small village and civil parish in the Chew Valley, Somerset within the unitary district of Bath and North East Somerset. The parish has a population of 439; the village is 8 miles south of Bristol and 10 miles from Bath. It is just south of Chew Valley Lake on the A368; the village has several shops including a post office. With its close neighbour East Harptree the villages are collectively known as the Harptrees. According to Stephen Robinson it is listed in the 1086 Domesday Book as Herpetreu meaning'The military road by the wood' from the Old English herepoep and treow. Between 1154 and 1172 an estate at West Harptree was granted by William FitzJohn to the Knights Templar; the shape of some of the existing fields with cross-slope and down-slope field banks and cultivated ridges forming an interleaving irregular mosaic suggest they are of medieval origin. The parish was part of the hundred of Chewton; the parish council has responsibility for local issues, including setting an annual precept to cover the council's operating costs and producing annual accounts for public scrutiny.

The parish council evaluates local planning applications and works with the local police, district council officers, neighbourhood watch groups on matters of crime and traffic. The parish council's role includes initiating projects for the maintenance and repair of parish facilities, such as the village hall or community centre, playing fields and playgrounds, as well as consulting with the district council on the maintenance and improvement of highways, footpaths, public transport, street cleaning. Conservation matters and environmental issues are of interest to the council. Along with East Harptree and Hinton Blewett, West Harptree is part of the Mendip Ward, represented by one councillor on the unitary authority of Bath and North East Somerset, created in 1996, as established by the Local Government Act 1992, it provides a single tier of local government with responsibility for all local government functions within its area including local planning and building control, local roads, council housing, environmental health and fairs, refuse collection, cemeteries, leisure services and tourism.

They are responsible for education, social services, main roads, public transport, Trading Standards, waste disposal and strategic planning, although fire and ambulance services are provided jointly with other authorities through the Avon Fire and Rescue Service and Somerset Constabulary and the Great Western Ambulance Service. Bath and North East Somerset's area covers part of the ceremonial county of Somerset but it is administered independently of the non-metropolitan county, its administrative headquarters is in Bath. Between 1 April 1974 and 1 April 1996, it was the Wansdyke district and the City of Bath of the county of Avon. Before 1974 that the parish was part of the Clutton Rural District; the parish is represented in the House of Commons of the Parliament of the United Kingdom as part of North East Somerset. It elects one Member of Parliament by the first past the post system of election. According to the 2001 Census the Mendip Ward, had 1,465 residents, living in 548 households, with an average age of 39.0 years.

Of these 79% of residents describing their health as'good', 22% of 16- to 74-year-olds had no qualifications. In the Index of Multiple Deprivation 2004, it was ranked at 25,387 out of 32,482 wards in England, where 1 was the most deprived LSOA and 32,482 the least deprived. Gournay Court is a Grade II* Country house. Circa 1600 The entrance Gates and railings are grade II as are the Gatepiers to the west The Church of St Mary dates from the 12th century, although the tower is a much addition, is a Grade II* listed building "Area 4 – Mendip Slopes". BANES Environmental Services. Retrieved 5 October 2010. Map of West Harptree circa 1900 West Harptree Memorial Hall

Boris Vyacheslavich

Boris Vyacheslavich was Prince of Chernigov for eight days in 1077. He was the son of Prince of Smolensk. Following his father's death in 1057, the child Boris was debarred from his inheritance, he died fighting against his uncles—Vsevolod Yaroslavich, Prince of Chernigov and Izyaslav Yaroslavich, Grand Prince of Kiev—on 3 October 1078. Boris was the son of Vyacheslav Yaroslavich, Prince of Smolensk, a younger son of Yaroslav the Wise, Grand Prince of Kiev. According to the historian Martin Dimnik, Boris was a child when his father died in 1057. Boris became an izgoi—a member of the Rurik dynasty debarred from ruling—after his father's death, because his uncle, Igor Yaroslavich succeeded his father in Smolensk. Boris's close relationship with his cousins, Oleg Svyatoslavich and Roman Svyatoslavich implies that their father, Sviatoslav Iaroslavich, Prince of Chernigov appeased Boris "in some manner, undoubtedly, by giving him a town", according to Martin. Upon the death of Sviatoslav Yaroslavich in 1077, his brothers Vsevolod Yaroslavich and Izyaslav Yaroslavich started a bitter rivalry over the Kievan throne.

Vsevolod left Chernigov and headed towards Izyaslav, who had set out on a military campaign against Kiev. Boris seized control of Chernigov, he only managed to remain in power for eight days and had to flee to Tmutarakan upon hearing the news of Vsevolod's return. In Tmutarakan, Boris was accepted by Prince Roman Svyatoslavich; the two were soon joined by Roman's brother, banished by his uncles from the Principality of Vladimir. Boris and Oleg allied themselves with the Cumans and attacked Vsevolod on the Sozh River, defeating his army in a bloody battle and capturing Chernigov on 25 August 1078. Soon and Izyaslav were able to muster a new army with the help of their sons and headed for Chernigov. Boris and Oleg had left the city by the time Vsevolod and Izyaslav approached it, but the citizens of Chernigov closed the gates and prepared for the siege; the attackers burned the outer parts of the city and wanted to proceed further, but received the news on Oleg and Boris coming to Chernigov's rescue.

Oleg tried to convince his cousin Boris not to seek direct confrontation with the four princes, but Boris decided to take them on. Boris died in a fierce battle "at a place near a village on the meadow of Nezhata" on 3 October, according to the Russian Primary Chronicle. Profile at

St. Mary's Church, Walthamstow

St. Mary’s Church, Walthamstow, is in Walthamstow Village, a conservation area in Walthamstow, London, it is still a working church. It retains over one hundred and fifty brasses and monuments, the oldest dating from 1436, though all that now remains of the original Norman church is some pillar bases and the chisel marks on them; the first church building on the site was constructed in the 12th century, measured about 15 metres by 8 metres with a sanctuary or chancel of unknown size at the east end. It was built of flint rubble and stood where the present nave now stands, it is believed that the north aisle of the church was rebuilt in the 13th century and the south aisle in the 14th century, both extending east only as far as the current chancel arch. In the 15th century a tower was added at the west end and the chancel was extended eastward. In 1535, Sir George Monoux, Alderman of London and Master of the Worshipful Company of Drapers, one of the guilds of the City of London, repaired the north aisle and built a chapel on the east end of it running along the north wall of the chancel.

At the same time, Monoux demolished the top 40 feet of the tower and rebuilt it in Tudor brick, adding a spiral stairway in a turret at the south-east corner of the tower. In the same year, money from Robert Thorne was used to rebuild the south aisle and to add a chapel on its eastern end. Internal galleries to provide extra seating for the congregation were added in the 18th century, the first being constructed at the west end of the church in 1710, with further galleries being built in the north and south aisles during the course of the century. In 1819, the aisle galleries were linked with the western one, the walls of the nave at the west end were increased in height. At the same time, the old windows of the nave were bricked up and the present Gothic revival ones cut. In 1843, the rest of the walls of the church were increased in height to match that of the west end, the pillars of the nave were heightened and remodelled to their present shape, leaving only the column bases as the final remnant of the 12th-century church above ground level.

A rose window was inserted in the east wall of the chancel. The new parish of St Peter's was split off from the parish in 1845. In 1876, the galleries were'thrown back' from the nave pillars, an older plaster ceiling was removed and a roof of stained wood installed; the private box pews which had accrued in the nave since the Reformation were removed and replaced by open benches which were, in turn, superseded by the current pews in the 1920s, together with oak panelling to the rear and sides of the church and in the galleries as a First World War memorial. Choir stalls of carved oak were given to the church by Sir William Mallinson in 1939 in memory of his father. In 1936 the east wall was found to be suffering from structural failure and, during the course of repairs, the opportunity was taken to rebuild that part of the church and extend the chancel 3 metres east and to build vestries on either side. In 1939, a large window in perpendicular gothic style was added to the east end of the church.

St Mary's suffered serious damage during Second World War. On 4 October 1940 the south aisle roof was destroyed by incendiary bomb and the gallery on that side was subsequently demolished to provide timber for repairing the church. On 8 October 1944, a bomb damaged the north side of the church tower, it was reported that a bomb disposal officer who investigated the scene in the dark could "feel the shape of a bomb and the jagged edge of its fins." The churchyard was closed for two Sundays pending investigations. It transpired. During the spring of 1942 all the railings surrounding the churchyard and most of those round the monuments were removed to provide scrap iron help the war effort. All that remained were south gates. Post-war restoration of the church included the rebuilding of the organ; the railings outside were replaced in 1955. In the early 1960s the east end of the south aisle was remodelled to create a side chapel. Between 1995 and 2001, extensive refurbishment of the church took place, which included removing three rows of pews from the west end of the nave and creating a larger entrance area adjacent to the west door.

When the floor was removed some of the crypt underneath the church were rediscovered. During the same work, when the ceiling from the entrance lobby underneath the bell tower was removed, it revealed several large oak beams used during the reconstruction of the tower in the 16th century, it was decided that these beams would be left permanently exposed. At the same period a disabled toilet and refreshment area were installed and the sanctuary was reordered. In 2001, the floor of the chancel was lowered and extended into the nave to provide an open space for worship. Two of the choir stalls were retained in the sanctuary and the rest relocated to the west end of the gallery. Communion can now be served on three sides; the churchyard contains war graves of three service personnel of World War II and nine of World War II. St. Mary's is an active church today, with a large multi-cultural congregation and involved in supporting its local community; the church is now a music and recording venue. Peter McCarthy runs the "Music in the

Phoebe Blair-White

Rosetta Phoebe "Binky" Blair-White was an Irish tennis player. Phoebe Blair-White was born Rosetta Phoebe Newell in Omagh, County Tyrone, in 1895 or on 10 September 1894, her parents were R. J. Newell, DL and JP, of Hillside and Anna Frances Scott, she began playing tennis when the family moved to Monkstown, County Dublin playing everyday against a wall. On 31 December 1918 she married a cricketer, they had three daughters, Rachel Majory, Juliet Francis, Rosemary. Blair-White was first noted as a tennis player in 1919 when she won the Monkstown lawn tennis club's ladies’ championships, she went on to win this event again in 1920 and 1921. In 1923, she won the prestigious ladies’ singles championships at the Boat Club tournament in Belfast, she was ranked alongside Hilda Wallis as the best female tennis players in Ireland. She was part of the Irish Olympic tennis team, she was defeated in both her first ladies singles and ladies doubles with Wallis. With her mixed double partner, William G. Ireland, she was defeated in the first round of the mixed doubles competition.

They lost in straight sets. Following the Olympics, she continued to succeed at domestic games, she won the 1925 Boat Club tournament in Belfast. In 1928 she won the all-comers ladies’ singles title at the Fitzwilliam Club in Dublin and the all-comers ladies’ doubles event with Rosie Fleming, she won the all-comers singles event again in 1931. She was knocked out of the tournament early, she represented Ireland numerous times against England. Blair-White continued her interest in tennis for the rest of her life playing the game in her seventies with her grandchildren. In life, she moved to Surrey, England, she died in Surrey on 6 March 1991, is buried in Lifford, County Donegal beside her husband