text:
John Singer Sargent

John Singer Sargent (January 12, 1856 – April 14, 1925) was an American expatriate artist, considered the "leading portrait painter of his generation" for his evocations of Edwardian-era luxury. He created roughly 900 oil paintings and more than 2,000 watercolors, as well as countless sketches and charcoal drawings. His oeuvre documents worldwide travel, from Venice to the Tyrol, Corfu, Spain, the Middle East, Montana, Maine, and Florida.

Born in Florence to American parents, he was trained in Paris before moving to London, living most of his life in Europe. He enjoyed international acclaim as a portrait painter. An early submission to the Paris Salon in the 1880s, his Portrait of Madame X, was intended to consolidate his position as a society painter in Paris but instead resulted in scandal. During the year following the scandal, Sargent departed for England, where he continued a successful career as a portrait artist.

From the beginning, Sargent's work is characterized by remarkable technical facility, particularly in his ability to draw with a brush, which in later years inspired admiration as well as criticism for its supposed superficiality. His commissioned works were consistent with the grand manner of portraiture, while his informal studies and landscape paintings displayed a familiarity with Impressionism. In later life, Sargent expressed ambivalence about the restrictions of formal portrait work and devoted much of his energy to mural painting and working en plein air. Art historians generally ignored society artists such as Sargent until the late 20th century.

The exhibition in the 1980s of Sargent's previously hidden male nudes served to spark a reevaluation of his life and work, and its psychological complexity. In addition to the beauty, sensation and innovation of his oeuvre, his same-sex interests, unconventional friendships with women and engagement with race, gender nonconformity and emerging globalism are now viewed as socially and aesthetically progressive and radical.

Early life
Sargent was a descendant of Epes Sargent, a colonial military leader and jurist. Before John Singer Sargent's birth, his father, FitzWilliam (b. 1820 in Gloucester, Massachusetts), was an eye surgeon at the Wills Eye Hospital in Philadelphia from 1844 to 1854. After John's older sister died at the age of two, his mother, Mary Newbold Sargent (née Singer, 1826–1906), suffered a breakdown, and the couple decided to go abroad to recover. They remained nomadic expatriates for the rest of their lives. Although based in Paris, Sargent's parents moved regularly, spending seasons at the sea and at mountain resorts in France, Germany, Italy, and Switzerland.

While Mary was pregnant, they stopped in Florence, Tuscany, because of a cholera epidemic. Sargent was born there in 1856. A year later, his sister Mary was born. After her birth, FitzWilliam reluctantly resigned his post in Philadelphia and accepted his wife's request to remain abroad. They lived modestly on a small inheritance and savings, leading a quiet life with their children. They generally avoided society and other Americans, except for friends in the art world. Four more children were born abroad, of whom only two lived past childhood.

Although his father was a patient teacher of basic subjects, young Sargent was a rambunctious child, more interested in outdoor activities than his studies. As his father wrote home: "He is quite a close observer of animated nature." His mother was convinced that traveling around Europe, and visiting museums and churches, would give young Sargent a satisfactory education. Several attempts to have him formally schooled failed, owing mostly to their itinerant life. His mother was a capable amateur artist and his father was a skilled medical illustrator. Early on, she gave him sketchbooks and encouraged drawing excursions. Sargent worked on his drawings, and he enthusiastically copied images from The Illustrated London News of ships and made detailed sketches of landscapes. FitzWilliam had hoped that his son's interest in ships and the sea might lead him toward a naval career.

At thirteen, his mother reported that John "sketches quite nicely, & has a remarkably quick and correct eye. If we could afford to give him really good lessons, he would soon be quite a little artist". At the age of thirteen, he received some watercolor lessons from Carl Welsch, a German landscape painter. Although his education was far from complete, Sargent grew up to be a highly literate and cosmopolitan young man, accomplished in art, music, and literature. He was fluent in English, French, Italian, and German. At seventeen, Sargent was described as "willful, curious, determined and strong" (after his mother) yet shy, generous, and modest (after his father). He was well-acquainted with many of the great masters from first-hand observation, as he wrote in 1874: "I have learned in Venice to admire Tintoretto immensely and to consider him perhaps second only to Michelangelo and Titian."

Training

An attempt to study at the Academy of Florence failed, as the school was reorganizing at the time. After returning to Paris from Florence, Sargent began his art studies with the young French portraitist Carolus-Duran. Following a meteoric rise, the artist was noted for his bold technique and modern teaching methods; his influence would be pivotal to Sargent during the period from 1874 to 1878.

In 1874, Sargent passed on his first attempt the rigorous exam required to gain admission to the École des Beaux-Arts, the premier art school in France. He took drawing classes, which included anatomy and perspective, and gained a silver prize. He also spent much time in self-study, drawing in museums and painting in a studio he shared with James Carroll Beckwith. He became both a valuable friend and Sargent's primary connection with the American artists abroad. Sargent also took some lessons from Léon Bonnat.

Carolus-Duran's atelier was progressive, dispensing with the traditional academic approach, which required careful drawing and underpainting, in favor of the alla prima method of working directly on the canvas with a loaded brush, derived from Diego Velázquez. It was an approach that relied on the proper placement of tones of paint. Sargent would later create a painting in this style that prompted comments such as: "The student has surpassed the teacher."

This approach also permitted spontaneous flourishes of color not bound to an underdrawing. It was markedly different from the traditional atelier of Jean-Léon Gérôme, where Americans Thomas Eakins and Julian Alden Weir had studied. Sargent was the star student in short order. Weir met Sargent in 1874 and noted that Sargent was "one of the most talented fellows I have ever come across; his drawings are like the old masters, and his color is equally fine". Sargent's excellent command of French and his superior talent made him both popular and admired. Through his friendship with Paul César Helleu, Sargent would meet giants of the art world, including Degas, Rodin, Monet, and Whistler.

Sargent's early enthusiasm was for landscapes, not portraiture, as evidenced by his voluminous sketches full of mountains, seascapes, and buildings. Carolus-Duran's expertise in portraiture finally influenced Sargent in that direction. Commissions for history paintings were still considered more prestigious, but were much harder to get. Portrait painting, on the other hand, was the best way of promoting an art career, getting exhibited in the Salon, and gaining commissions to earn a livelihood. Sargent's first major portrait was of his friend Fanny Watts in 1877, and was also his first Salon admission. Its particularly well-executed pose drew attention. His second salon entry was the Oyster Gatherers of Cançale, an impressionistic painting of which he made two copies, one of which he sent back to the United States, and both received warm reviews.

Early career

In 1879 at the age of 23, Sargent painted a portrait of his teacher Carolus-Duran; the virtuoso effort met with public approval and announced the direction his mature work would take. Its showing at the Paris Salon was both a tribute to his teacher and an advertisement for portrait commissions. Of Sargent's early work, Henry James wrote that the artist offered "the slightly 'uncanny' spectacle of a talent which on the very threshold of its career has nothing more to learn".

After leaving Carolus-Duran's atelier, Sargent visited Spain. There he studied the paintings of Velázquez with a passion, absorbing the master's technique, and in his travels gathered ideas for future works. He was entranced with Spanish music and dance. The trip also re-awakened his own talent for music (which was nearly equal to his artistic talent), and which found visual expression in his early masterpiece El Jaleo (1882). Music would continue to play a major part in his social life as well, as he was a skillful accompanist of both amateur and professional musicians. Sargent became a strong advocate for modern composers, especially Gabriel Fauré.

Trips to Italy provided sketches and ideas for several Venetian street-scene genre paintings, which effectively captured gestures and postures he would find useful in later portraiture.

Upon his return to Paris, Sargent quickly received several portrait commissions. His career was launched. He immediately demonstrated the concentration and stamina that enabled him to paint with workman-like steadiness for the next twenty-five years. He filled in the gaps between commissions with many non-commissioned portraits of friends and colleagues. His fine manners, perfect French, and great skill made him a standout among the newer portraitists, and his fame quickly spread. He confidently set high prices and turned down unsatisfactory sitters. He mentored his friend Emil Fuchs, who was learning to paint portraits in oils.

Nineteenth-century portraits
In the early 1880s, Sargent regularly exhibited portraits at the Salon, and these were mostly full-length portrayals of women, such as Madame Edouard Pailleron (1880) (done en plein-air) and Madame Ramón Subercaseaux (1881). He continued to receive positive critical notice.

Sargent's best portraits reveal the individuality and personality of the sitters; his most ardent admirers think he is matched in this only by Velázquez, who was one of Sargent's great influences. The Spanish master's spell is apparent in Sargent's The Daughters of Edward Darley Boit, 1882, a haunting interior that echoes Velázquez's Las Meninas. As in many of his early portraits, Sargent confidently tries different approaches with each new challenge, here employing both unusual composition and lighting to striking effect. One of his most widely exhibited and best loved works of the 1880s was Lady with the Rose (1882), a portrait of Charlotte Burckhardt, a close friend and possible romantic attachment.

Portrait of Madame X

His most controversial work, Portrait of Madame X (Madame Pierre Gautreau) (1884) is now considered one of his best works, and was the artist's personal favorite; he stated in 1915: "I suppose it is the best thing I have done." When unveiled in Paris at the 1884 Salon, it aroused such a negative reaction that it likely prompted Sargent's move to London. Sargent's self-confidence had led him to attempt a risque experiment in portraiture—but this time it unexpectedly backfired.

The painting was not commissioned by her, and he pursued her for the opportunity, quite unlike most of his portrait work where clients sought him out. Sargent wrote to a common acquaintance:

"I have a great desire to paint her portrait and have reason to think she would allow it and is waiting for someone to propose this homage to her beauty. ...you might tell her that I am a man of prodigious talent."

It took well over a year to complete the painting. The first version of the portrait of Madame Gautreau, with the famously plunging neckline, white-powdered skin, and arrogantly cocked head, featured an intentionally suggestive off-the-shoulder dress strap, on her right side only, which made the overall effect more daring and sensual. Sargent repainted the strap to its expected over-the-shoulder position to try to dampen the furor, but the damage had been done. French commissions dried up and he told his friend Edmund Gosse in 1885 that he contemplated giving up painting for music or business.

Writing of the reaction of visitors, Judith Gautier observed:

"Is it a woman? a chimera, the figure of a unicorn rearing as on a heraldic coat of arms or perhaps the work of some oriental decorative artist to whom the human form is forbidden and who, wishing to be reminded of woman, has drawn the delicious arabesque? No, it is none of these things, but rather the precise image of a modern woman scrupulously drawn by a painter who is a master of his art.'"

Prior to the Madame X scandal of 1884, Sargent had painted exotic beauties such as Rosina Ferrara of Capri, and the Spanish expatriate model Carmela Bertagna, but the earlier pictures had not been intended for broad public reception. Sargent kept the painting prominently displayed in his London studio until he sold it to the Metropolitan Museum of Art in 1916 after moving to the United States, and a few months after Gautreau's death.

Move to England

Before arriving in England, Sargent began sending paintings for exhibition at the Royal Academy. These included the portraits of Dr. Pozzi at Home (1881), a flamboyant essay in red and his first full-length male portrait, and the more traditional Mrs. Henry White (1883). The ensuing portrait commissions encouraged Sargent to complete his move to London in 1886, where he settled in the artistic community of Chelsea.

Notwithstanding the Madame X scandal, he had considered moving to London as early as 1882; he had been urged to do so repeatedly by his new friend, the novelist Henry James. In retrospect his transfer to London may be seen to have been inevitable.

English critics were not warm at first, faulting Sargent for his "clever" "Frenchified" handling of paint. One reviewer seeing his portrait of Mrs. Henry White described his technique as "hard" and "almost metallic" with "no taste in expression, air, or modeling". With help from Mrs. White, however, Sargent soon gained the admiration of English patrons and critics. Henry James also gave the artist "a push to the best of my ability". Sargent spent much time painting outdoors in the English countryside when not in his studio. On a visit to Monet at Giverny in 1885, Sargent painted one of his most Impressionistic portraits, of Monet at work painting outdoors with his new bride nearby. Sargent is usually not thought of as an Impressionist painter, but he sometimes used impressionistic techniques to great effect. His Claude Monet Painting at the Edge of a Wood is rendered in his own version of the Impressionist style. In the 1880s, he attended the Impressionist exhibitions and he began to paint outdoors in the plein-air manner after that visit to Monet. Sargent purchased four Monet works for his personal collection during that time.

Sargent was similarly inspired to do a portrait of his artist friend Paul César Helleu, also painting outdoors with his wife by his side. A photograph very similar to the painting suggests that Sargent occasionally used photography as an aid to composition. Through Helleu, Sargent met and painted the famed French sculptor Auguste Rodin in 1884, a rather somber portrait reminiscent of works by Thomas Eakins. Although the British critics classified Sargent in the Impressionist camp, the French Impressionists thought otherwise. As Monet later stated: "He is not an Impressionist in the sense that we use the word, he is too much under the influence of Carolus-Duran."

Sargent's first major success at the Royal Academy of Arts came in 1887, with the enthusiastic response to Carnation, Lily, Lily, Rose, a large piece, painted on site, of two young girls lighting lanterns in an English garden in Broadway in the Cotswolds. The painting was immediately purchased by the Tate Gallery.

His first trip to New York and Boston as a professional artist in 1887–88 produced over 20 important commissions, including portraits of Isabella Stewart Gardner, the famed Boston art patron. His portrait of Mrs. Adrian Iselin, wife of a New York businessman, revealed her character in one of his most insightful works. In Boston, Sargent was honored with his first solo exhibition, which presented 22 of his paintings. Here he became friends with painter Dennis Miller Bunker, who traveled to England in the summer of 1888 to paint with him en plein air, and is the subject of Sargent's 1888 painting Dennis Miller Bunker Painting at Calcot.

Back in London, Sargent was quickly busy again. His working methods were by then well-established, following many of the steps employed by other master portrait painters before him. After securing a commission through negotiations which he carried out, Sargent would visit the client's home to see where the painting was to hang. He would often review a client's wardrobe to pick suitable attire. Some portraits were done in the client's home, but more often in his studio, which was well-stocked with furniture and background materials he chose for proper effect. He usually required eight to ten sittings from his clients, although he would try to capture the face in one sitting. He usually kept up pleasant conversation and sometimes he would take a break and play the piano for his sitter. Sargent seldom used pencil or oil sketches, and instead laid down oil paint directly. Finally, he would select an appropriate frame.

Sargent had no assistants; he handled all the tasks, such as preparing his canvases, varnishing the painting, arranging for photography, shipping, and documentation. He commanded about $5,000 per portrait, or about $130,000 in current dollars. Some American clients traveled to London at their own expense to have Sargent paint their portrait. The range of pigments regularly used by Sargent was: "Mars yellow (a synthetic iron oxide) and cadmium yellow; viridian and emerald green, sometimes mixed; vermillion and Mars red, both alone and mixed; madder; synthetic ultramarine or cobalt blue; and ivory black, sienna and Mars brown".

Around 1890, Sargent painted two daring non-commissioned portraits as show pieces—one of actress Ellen Terry as Lady Macbeth and one of the popular Spanish dancer La Carmencita. Sargent was elected an associate of the Royal Academy, and was made a full member three years later.

In the 1890s, he averaged fourteen portrait commissions per year, none more beautiful than the genteel Lady Agnew of Lochnaw, 1892. His portrait of Mrs. Hugh Hammersley (Mrs. Hugh Hammersley, 1892) was equally well received for its lively depiction of one of London's most notable hostesses. As a portrait painter in the grand manner, Sargent had unmatched success; he portrayed subjects who were at once ennobled and often possessed of nervous energy. Sargent was referred to as "the Van Dyck of our times".

Although Sargent was an American expatriate, he returned to the United States many times, often to answer the demand for commissioned portraits. Sargent exhibited nine of his portraits in the Palace of Fine Arts at the 1893 World's Columbian Exposition in Chicago.

Sargent painted a series of three portraits of Robert Louis Stevenson. The second, Portrait of Robert Louis Stevenson and his Wife (1885), was one of his best known. He also completed portraits of two U.S. presidents: Theodore Roosevelt and Woodrow Wilson. In 1896, the Trustees of the Metropolitan Museum of Art commissioned Sargent to produce a portrait of Henry Gurdon Marquand. Marquand served as the second president of the museum, and was instrumental in its founding. In 1888, Sargent released his portrait of Alice Vanderbilt Shepard, great-granddaughter of Cornelius Vanderbilt. Many of his most important works are in museums in the United States. In 1897, a friend sponsored a famous portrait in oil of Mr. and Mrs. I. N. Phelps Stokes, by Sargent, as a wedding gift. Jean Zimmerman documents the creation of the Stokes portrait in her dual biography of the couple, Love, Fiercely.

In 1898, Asher Wertheimer, a wealthy Jewish art dealer living in London, commissioned from Sargent a series of a dozen portraits of his family, the artist's largest commission from a single patron. The Wertheimer portraits reveal a pleasant familiarity between the artist and his subjects. Even though Wertheimer bequeathed most of the paintings to the National Gallery, nowadays they are on display at the Tate Britain.

Twentieth-century portraits
By 1900, Sargent was at the height of his fame. Cartoonist Max Beerbohm completed one of his seventeen caricatures of Sargent, making well known to the public the artist's paunchy physique. Although only in his forties, Sargent began to travel more and to devote relatively less time to portrait painting. His An Interior in Venice (1900), a portrait of four members of the Curtis family in their elegant palatial home, Palazzo Barbaro, was a resounding success. But, Whistler did not approve of the looseness of Sargent's brushwork, which he summed up as "smudge everywhere". One of Sargent's last major portraits in his bravura style was that of Lord Ribblesdale, in 1902, finely attired in an elegant hunting uniform. Between 1900 and 1907, Sargent continued his high productivity, which included, in addition to dozens of oil portraits, hundreds of portrait drawings at about $400 each. In 1901, he purchased the next door property to his home in Tite Street, to create a larger studio.

In 1907, at the age of fifty-one, Sargent officially closed his studio. Relieved, he stated: "Painting a portrait would be quite amusing if one were not forced to talk while working.... What a nuisance having to entertain the sitter and to look happy when one feels wretched." In that same year, Sargent painted his modest and serious self-portrait, his last, for the celebrated self-portrait collection of the Uffizi Gallery in Florence, Italy.

Sargent made several summer visits to the Swiss Alps with his sisters Emily Sargent, an accomplished painter in her own right, and Violet Sargent (Mrs Ormond) and also Violet's daughters Rose-Marie and Reine, who were the subject of a number of paintings between 1906 and 1913 like The Black Brook (1908) or Nonchaloir (Repose) (1911).

By the time Sargent finished his portrait of John D. Rockefeller in 1917, most critics began to consign him to the masters of the past, "a brilliant ambassador between his patrons and posterity". Modernists treated him more harshly, considering him completely out of touch with the reality of American life and with emerging artistic trends including Cubism and Futurism. Sargent quietly accepted the criticism, but refused to alter his negative opinions of modern art. He retorted: "Ingres, Raphael and El Greco, these are now my admirations, these are what I like."

Sometime between 1917 and 1920, Sargent painted the portrait of Thomas E. McKeller, a young African-American elevator operator and WWI veteran. The canvas was kept in the painter's studio until his death and only began to be displayed permanently to the public in 1986 when it was acquired by the Museum of Fine Arts in Boston. McKeller also posed as a model for the mythological murals that Sargent painted at the stairway and the rotunda of the MFA Boston and for the World War I memorial murals at Harvard's Widener Library.

In 1925, shortly before he died, Sargent painted his last oil portrait, a canvas of aristocrat Grace Curzon, Marchioness Curzon of Kedleston. The painting was purchased in 1936 by the Currier Museum of Art, in Manchester, New Hampshire, where it has been on display since then.

Watercolors

During Sargent's long career, he painted more than 2,000 watercolors, roving from the English countryside to Venice to the Tyrol, Corfu, the Middle East, Montana, Maine, and Florida. Each destination offered pictorial stimulation and treasure. Even at his leisure, in escaping the pressures of the portrait studio, he painted with restless intensity, often painting from morning until night.

His hundreds of watercolors of Venice are especially notable, many done from the perspective of a gondola. His colors were sometimes extremely vivid and as one reviewer noted: "Everything is given with the intensity of a dream." In the Middle East and North Africa Sargent painted Bedouins, goatherds, and fishermen. In the last decade of his life, he produced many watercolors in Maine, Florida, and in the American West, of fauna, flora, and native peoples.

With his watercolors, Sargent was able to indulge his earliest artistic inclinations for nature, architecture, exotic peoples, and noble mountain landscapes. And it is in some of his late works where one senses Sargent painting most purely for himself. His watercolors were executed with a joyful fluidness. He also painted extensively family, friends, gardens, and fountains. In watercolors, he playfully portrayed his friends and family dressed in Orientalist costume, relaxing in brightly lit landscapes that allowed for a more vivid palette and experimental handling than did his commissions (The Chess Game, 1906). His first major solo exhibit of watercolor works was at the Carfax Gallery in London in 1905. In 1909, he exhibited eighty-six watercolors in New York City, eighty-three of which were bought by the Brooklyn Museum. Evan Charteris wrote in 1927:

"To live with Sargent's water-colours is to live with sunshine captured and held, with the luster of a bright and legible world, 'the refluent shade' and 'the Ambient ardours of the noon'."

Although not generally accorded the critical respect given Winslow Homer, perhaps America's greatest watercolorist, scholarship has revealed that Sargent was fluent in the entire range of opaque and transparent watercolor technique, including the methods used by Homer.

Other work

As a concession to the insatiable demand of wealthy patrons for portraits, Sargent dashed off hundreds of rapid charcoal portrait sketches, which he called "Mugs". Forty-six of these, spanning the years 1890–1916, were exhibited at the Royal Society of Portrait Painters in 1916.

All of Sargent's murals are to be found in the Boston/Cambridge area in Massachusetts. They are in the Boston Public Library, the Museum of Fine Arts, and Harvard's Widener Library. Sargent's largest scale works are the mural decorations Triumph of Religion that grace the Boston Public Library, depicting the history of religion and the gods of polytheism. They were attached to the walls of the library by means of marouflage. He worked on the cycle for almost thirty years but never completed the final mural. Sargent drew on his extensive travels and museum visits to create a dense art historical mélange. The murals were most recently restored in 2003–2004 by a team from the Straus Center for Conservation and Technical Studies, Harvard Art Museums.

Sargent worked on the murals from 1895 through 1919; they were intended to show religion's (and society's) progress from pagan superstition up through the ascension of Christianity, concluding with a painting depicting Jesus delivering the Sermon on the Mount. But Sargent's paintings of "The Church" and "The Synagogue", installed in late 1919, inspired a debate about whether the artist had represented Judaism in a stereotypical, or even an anti-Semitic, manner. Drawing upon iconography that was used in medieval paintings, Sargent portrayed Judaism and the synagogue as a blind, ugly hag, and Christianity and the church as a lovely, radiant young woman. He also failed to understand how these representations might be problematic for the Jews of Boston; he was both surprised and hurt when the paintings were criticized. The paintings were objectionable to Boston Jews since they seemed to show Judaism defeated, and Christianity triumphant. The Boston newspapers also followed the controversy, noting that while many found the paintings offensive, not everyone agreed. In the end, Sargent abandoned his plan to finish the murals, and the controversy eventually died down.

Upon his return to England in 1918 after a visit to the United States, Sargent was commissioned as a war artist by the British Ministry of Information. In his large painting Gassed and in many watercolors, he depicted scenes from the Great War. Sargent had been affected by the death of his niece Rose-Marie in the shelling of the St Gervais church, Paris, on Good Friday 1918.

Relationships and personal life
Sargent was a life-long bachelor with a wide circle of friends, including Oscar Wilde (with whom he was neighbors for several years), gay author Violet Paget and his likely lover Albert de Belleroche. Biographers once portrayed him as staid and reticent. However, recent scholarship has theorised he was a private, complex and passionate man whose homosexual identity was integral to shaping his art. This view is based on statements by his friends and associates, the general alluring remoteness of his portraits, the way his works challenge 19th-century notions of gender difference, his previously ignored male nudes, and some male portraits, including those of Thomas E. McKeller, Bartholomy Maganosco, Olimpio Fusco, and that of aristocratic artist Albert de Belleroche, which hung in his Chelsea dining room. Sargent had a long friendship with Belleroche, whom he met in 1882 and traveled with frequently. A surviving drawing suggests Sargent might have used him as a model for Madame X, following a coincidence of dates for Sargent drawing each of them separately around the same time, and the delicate pose suggestive more of Sargent's sketches of the male form than his often stiff commissions.

It has been suggested that Sargent's reputation in the 1890s as "the painter of the Jews" may have been due to his empathy with and complicit enjoyment of their mutual social foreignness. One such Jewish client, Betty Wertheimer, wrote that when in Venice, Sargent "was only interested in the Venetian gondoliers". The painter Jacques-Émile Blanche, who was one of his early sitters, said after Sargent's death that his sex life "was notorious in Paris, and in Venice, positively scandalous. He was a frenzied bugger."

He had many relationships with women. It has been suggested that those with his sitters Rosina Ferrara, Virginie Gautreau, and Judith Gautier may have tipped into infatuation. As a young man, Sargent also for a time courted Louise Burkhardt, the model for Lady with the Rose.

Sargent's friends and supporters included Henry James and Isabella Stewart Gardner, of both of whom he painted portraits, and Gardner also commissioned and purchased works from Sargent and sought his advice on other acquisitions. Edward VII and Paul César Helleu were also friends and supporters of Sargent. His associations also included Prince Edmond de Polignac and Count Robert de Montesquiou. Other artists Sargent associated with were Dennis Miller Bunker, James Carroll Beckwith, Edwin Austin Abbey, and John Elliott (who also worked on the Boston Public Library murals), Francis David Millet, Joaquín Sorolla, and Claude Monet, whom Sargent portrayed with Monet's wife "by the edge of a wood". Between 1905 and 1914, Sargent's frequent traveling companions were the married artist couple Wilfrid de Glehn and Jane Emmet de Glehn. The trio would often spend summers in France, Spain, or Italy, and all three would depict one another in their paintings during their travels.

Critical assessment

In a time when the art world focused, in turn, on Impressionism, Fauvism, and Cubism, Sargent practiced his own form of Realism, which made brilliant references to Velázquez, Van Dyck, and Gainsborough. His seemingly effortless facility for paraphrasing the masters in a contemporary fashion led to a stream of commissioned portraits of remarkable virtuosity (Arsène Vigeant, 1885, Musées de Metz; Mr. and Mrs. Isaac Newton Phelps-Stokes, 1897, Metropolitan Museum of Art, New York) and earned Sargent the moniker, "the Van Dyck of our times". In 1916, he was awarded an honorary degree by Harvard University.

Still, during his life his work engendered negative responses from some of his colleagues: Camille Pissarro wrote "he is not an enthusiast but rather an adroit performer", and Walter Sickert published a satirical turn under the heading "Sargentolatry". By the time of his death he was dismissed as an anachronism, a relic of the Gilded Age and out of step with the artistic sentiments of post-World War I Europe. Elizabeth Prettejohn suggests that the decline of Sargent's reputation was due partly to the rise of anti-Semitism, and the resultant intolerance of 'celebrations of Jewish prosperity.' It has been suggested that the exotic qualities inherent in his work appealed to the sympathies of the Jewish clients whom he painted from the 1890s on.

Nowhere is this more apparent than in his portrait Almina, Daughter of Asher Wertheimer (1908), in which the subject is seen wearing a Persian costume, a pearl encrusted turban, and strumming an Indian tambura, accoutrements all meant to convey sensuality and mystery. If Sargent used this portrait to explore issues of sexuality and identity, it seems to have met with the satisfaction of the subject's father, Asher Wertheimer, a wealthy Jewish art dealer.

Foremost of Sargent's detractors was the influential English art critic Roger Fry, of the Bloomsbury Group, who at the 1926 Sargent retrospective in London dismissed Sargent's work as lacking aesthetic quality: "Wonderful indeed, but most wonderful that this wonderful performance should ever have been confused with that of an artist." And, in the 1930s, Lewis Mumford led a chorus of the severest critics: "Sargent remained to the end an illustrator ... the most adroit appearance of workmanship, the most dashing eye for effect, cannot conceal the essential emptiness of Sargent's mind, or the contemptuous and cynical superficiality of a certain part of his execution."

Part of Sargent's devaluation is also attributed to his expatriate life, which made him seem less American at a time when "authentic" socially conscious American art, as exemplified by the Stieglitz circle and by the Ashcan School, was on the ascent.

After such a long period of critical disfavor, Sargent's reputation has increased steadily since the 1950s. In the 1960s, a revival of Victorian art and new scholarship directed at Sargent strengthened his reputation. Sargent has been the subject of large-scale exhibitions in major museums, including a retrospective exhibition at the Whitney Museum of American Art in 1986 and a major 1999 traveling show that exhibited at the Museum of Fine Arts, Boston, the National Gallery of Art Washington, and the National Gallery, London. In 2022, the National Gallery of Art in Washington, DC, and in 2023 the Legion of Honor in San Francisco, California, hosted an exhibition of Sargent's paintings from Spain.

In 1986, Andy Warhol commented to Sargent scholar Trevor Fairbrother that Sargent "made everybody look glamorous. Taller. Thinner. But they all have mood, every one of them has a different mood." In a Time magazine article from the 1980s, critic Robert Hughes praised Sargent as "the unrivaled recorder of male power and female beauty in a day that, like ours, paid excessive court to both".

Later life

In 1922, Sargent co-founded New York City's Grand Central Art Galleries together with Edmund Greacen, Walter Leighton Clark, and others. Sargent actively participated in the Grand Central Art Galleries and their academy, the Grand Central School of Art, until his death in 1925. The Galleries held a major retrospective exhibit of Sargent's work in 1924. He then returned to England, where he died at his Chelsea home on April 14, 1925, of heart disease. Sargent is interred in Brookwood Cemetery near Woking, Surrey.

Memorial exhibitions of Sargent's work were held in Boston in 1925, at the Metropolitan Museum of Art in New York and at the Royal Academy and Tate Gallery in London in 1926. The Grand Central Art Galleries also organized a posthumous exhibition in 1928 of previously unseen sketches and drawings from throughout his career.

Sales
Portrait of Robert Louis Stevenson and his Wife was sold in 2004 for US$8.8 million and is located at Crystal Bridges Museum of American Art at Bentonville, Arkansas.

In December 2004, Group with Parasols (A Siesta) (1905) sold for US$23.5 million, nearly double the Sotheby's estimate of $12 million. The previous highest price for a Sargent painting was US$11 million.

In popular culture
In 2018, Comedy Central star Jade Esteban Estrada wrote, directed, and starred in Madame X: A Burlesque Fantasy, a stage production (premiered in San Antonio, Texas) based on the life of Sargent and his famous painting, Portrait of Madame X.

The works of Sargent feature prominently in Maggie Stiefvater's 2021 novel Mister Impossible.

In 2024, Exhibition on Screen produced a documentary John Singer Sargent: Fashion & Swagger, filmed at the Museum of Fine Arts, Boston and the Tate Britain, London. It was based on the Sargent and fashion exhibits at those two museums in 2023 and 2024.

title: John Singer Sargent


text:
Student's t-distribution

In probability and statistics, Student's $t$ distribution (or simply the $t$ distribution) $$\ t_\nu\ $$ is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.

However, $$\ t_\nu\ $$ has heavier tails and the amount of probability mass in the tails is controlled by the parameter $$\ \nu ~ ~.$$ For $$\ \nu = 1\ $$ the Student's $t$ distribution $$t_\nu$$ becomes the standard Cauchy distribution, which has very "fat" tails; whereas for $$\ \nu \rightarrow \infty\ $$ it becomes the standard normal distribution $$\ \mathcal{N}(0,1) \,$$ which has very "thin" tails.

The Student's $t$ distribution plays a role in a number of widely used statistical analyses, including Student's $t$ test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.

In the form of the location-scale $t$ distribution $$lst(\mu, \tau^2, \nu)$$ it generalizes the normal distribution and also arises in the Bayesian analysis of data from a normal family as a compound distribution when marginalizing over the variance parameter.

History and etymology

In statistics, the $t$ distribution was first derived as a posterior distribution in 1876 by Helmert  and Lüroth. As such, Student's t-distribution is an example of Stigler's Law of Eponymy. The $t$ distribution also appeared in a more general form as Pearson type IV distribution in Karl Pearson's 1895 paper.

In the English-language literature, the distribution takes its name from William Sealy Gosset's 1908 paper in Biometrika under the pseudonym "Student". One version of the origin of the pseudonym is that Gosset's employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. Another version is that Guinness did not want their competitors to know that they were using the $t$ test to determine the quality of raw material.

Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the problems of small samples – for example, the chemical properties of barley where sample sizes might be as few as 3. Gosset's paper refers to the distribution as the "frequency distribution of standard deviations of samples drawn from a normal population". It became well known through the work of Ronald Fisher, who called the distribution "Student's distribution" and represented the test value with the letter $t$.

Probability density function
Student's $t$ distribution has the probability density function (PDF) given by

 * $$f(t) = \frac{\ \Gamma\left(\frac{\ \nu+1\ }{ 2 }\right)\ }{\ \sqrt{\pi\ \nu\ }\ \Gamma\left(\frac{\nu}{2}\right)} \left(\ 1 + \frac{~ t^2\ }{ \nu }\ \right)^{-(\nu+1)/2}\ ,$$

where $$\nu$$ is the number of degrees of freedom and $$\Gamma$$ is the gamma function. This may also be written as

 * $$f(t) = \frac{ 1 }{\ \sqrt{\nu\ }\ {\mathrm B}\left( \frac{\ 1\ }{ 2 },\ \frac{\ \nu\ }{ 2 }\right)\ } \ \left(\ 1 + \frac{\ t^2\ }{ \nu }\ \right)^{-(\nu+1)/2}\ ,$$

where $$\ {\mathrm B}\ $$is the Beta function. In particular for integer valued degrees of freedom $$\ \nu\ $$ we have:

For $$\ \nu > 1\ $$ and even,

 * $$\ \frac{\ \Gamma\left( \frac{\ \nu+1\ }{ 2 }\right)\ }{\ \sqrt{\pi\ \nu\ }\ \Gamma\left( \frac{\ \nu\ }{ 2 }\right)\ } = \frac{\ (\nu - 1)\cdot(\nu -3)\cdots 5 \cdot 3\ } { 2 \sqrt{\nu\ }\ (\nu -2)\cdot(\nu -4) \cdots 4 \cdot 2\ } ~.$$

For $$\ \nu > 1\ $$ and odd,

 * $$\ \frac{\ \Gamma\left( \frac{\ \nu+1\ }{ 2 } \right)\ }{\ \sqrt{\pi\ \nu\ }\ \Gamma\left( \frac{\ \nu\ }{ 2 } \right)} = \frac{(\nu -1)\cdot(\nu -3)\cdots 4 \cdot 2\ }{\ \pi \sqrt{\nu\ }\ (\nu -2)\cdot(\nu -4)\cdots 5 \cdot 3\ } ~.$$

The probability density function is symmetric, and its overall shape resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the $t$ distribution approaches the normal distribution with mean 0 and variance 1. For this reason $${\ \nu\ }$$ is also known as the normality parameter.

The following images show the density of the $t$ distribution for increasing values of $$\ \nu ~.$$ The normal distribution is shown as a blue line for comparison. Note that the $t$ distribution (red line) becomes closer to the normal distribution as $$\ \nu\ $$ increases.

Cumulative distribution function
The cumulative distribution function (CDF) can be written in terms of $t$, the regularized incomplete beta function. For $∞$ ,

 * $$F(t) = \int_{-\infty}^t\ f(u)\ \operatorname{d}u ~=~ 1 - \frac{1}{2} I_{x(t)}\!\left( \frac{\ \nu\ }{ 2 },\ \frac{\ 1\ }{ 2 } \right)\ ,$$

where

 * $$x(t) = \frac{ \nu }{\ t^2+\nu\ } ~.$$

Other values would be obtained by symmetry. An alternative formula, valid for $$\ t^2 < \nu\ ,$$ is

 * $$\int_{-\infty}^t f(u)\ \operatorname{d}u ~=~ \frac{1}{2} + t\ \frac{\ \Gamma\!\left( \frac{\ \nu+1\ }{ 2 } \right)\ }{\ \sqrt{\pi\ \nu\ }\ \Gamma\!\left( \frac{ \nu }{\ 2\ }\right)\ } \ {}_{2}F_1\!\left(\ \frac{1}{2}, \frac{\ \nu+1\ }{2}\ ; \frac{ 3 }{\ 2\ }\ ;\ -\frac{~ t^2\ }{ \nu }\ \right)\ ,$$

where $$\ {}_{2}F_1(\ ,\ ;\ ;\ )\ $$ is a particular instance of the hypergeometric function.

For information on its inverse cumulative distribution function, see.

Special cases
Certain values of $$\ \nu\ $$ give a simple form for Student's t-distribution.

Moments
For $$\nu > 1\ ,$$ the raw moments of the $t$ distribution are

 * $$\operatorname{\mathbb E}\left\{\ T^k\ \right\} = \begin{cases}

\quad 0 & k \text{ odd }, \quad 0 < k < \nu\, \\ {} \\ \frac{1}{\ \sqrt{\pi\ }\ \Gamma\left(\frac{\ \nu\ }{ 2 }\right)}\ \left[\ \Gamma\!\left(\frac{\ k + 1\ }{ 2 }\right)\ \Gamma\!\left(\frac{\ \nu - k\ }{ 2 }\right)\ \nu^{\frac{\ k\ }{ 2 }}\ \right] & k \text{ even }, \quad 0 < k < \nu ~.\\ \end{cases}$$

Moments of order $$\ \nu\ $$ or higher do not exist.

The term for $$\ 0 < k < \nu\ ,$$ $t$ even, may be simplified using the properties of the gamma function to

 * $$\operatorname{\mathbb E}\left\{\ T^k\ \right\} = \nu^{ \frac{\ k\ }{ 2 } }\ \prod_{j=1}^{k/2}\ \frac{~ 2j - 1 ~}{ \nu - 2j } \qquad k \text{ even}, \quad 0 < k < \nu ~.$$

For a $t$ distribution with $$\ \nu\ $$ degrees of freedom, the expected value is $$\ 0\ $$ if $$\ \nu > 1\ ,$$ and its variance is $$\ \frac{ \nu }{\ \nu-2\ }\ $$ if $$\ \nu > 2 ~.$$ The skewness is 0 if $$\ \nu > 3\ $$ and the excess kurtosis is $$\ \frac{ 6 }{\ \nu - 4\ }\ $$ if $$\ \nu > 4 ~.$$

Location-scale transformation
Student's $t$ distribution generalizes to the three parameter location-scale $t$ distribution $$\ \mathcal{lst}(\mu,\ \tau^2,\ \nu)\ $$ by introducing a location parameter $$\ \mu\ $$ and a scale parameter $$\ \tau ~.$$ With
 * $$\ T \sim t_\nu\ $$

and location-scale family transformation
 * $$\ X = \mu + \tau\ T\ $$

we get
 * $$\ X \sim \mathcal{lst}(\mu,\ \tau^2,\ \nu) ~.$$

The resulting distribution is also called the non-standardized Student's $I$ distribution.

Density and first two moments
The location-scale $t$ distribution has a density defined by:

 * $$p(x\mid \nu,\mu,\tau) = \frac{\ \Gamma\left(\frac{\ \nu + 1\ }{ 2 } \right)\ }{\ \Gamma\left( \frac{\ \nu\ }{ 2 }\right)\ \sqrt{\pi\ \nu\ }\ \tau\ }\ \left( 1 + \frac{\ 1\ }{ \nu }\ \left(\ \frac{\ x-\mu\ }{ \tau }\ \right)^2\ \right)^{-(\nu+1)/2}\ $$

Equivalently, the density can be written in terms of $$\tau^2$$:

 * $$\ p(x\ \mid\ \nu,\ \mu,\ \tau^2) = \frac{\ \Gamma( \frac{\nu + 1}{2})\ }{\ \Gamma\left(\frac{\ \nu\ }{ 2 }\right)\ \sqrt{\pi\ \nu\ \tau^2}\ }\ \left(\ 1 + \frac{\ 1\ }{ \nu }\ \frac{\ (x - \mu)^2\ }{\ \tau^2\ }\ \right)^{-(\nu+1)/2}\ $$

Other properties of this version of the distribution are:

 * $$\begin{align}

\operatorname{\mathbb E}\{\ X\ \} &= \mu & \text{ for } \nu > 1\ ,\\ \operatorname{var}\{\ X\ \} &= \tau^2\frac{\nu}{\nu-2} & \text{ for } \nu > 2\ ,\\ \operatorname{mode}\{\ X\ \} &= \mu ~. \end{align} $$

Special cases

 * If $$\ X\ $$ follows a location-scale $k$ distribution $$\ X \sim \mathcal{lst}\left(\mu,\ \tau^2,\ \nu\right)\ $$ then for $$\ \nu \rightarrow \infty\ $$ $$\ X\ $$ is normally distributed $$X \sim \mathrm{N}\left(\mu, \tau^2\right)$$ with mean $$\mu$$ and variance $$\ \tau^2 ~.$$
 * The location-scale $t$ distribution $$\ \mathcal{lst}\left(\mu,\ \tau^2,\ \nu=1 \right)\ $$ with degree of freedom $$\nu=1$$ is equivalent to the Cauchy distribution $$\mathrm{Cau}\left(\mu, \tau\right) ~.$$
 * The location-scale $t$ distribution $$\ \mathcal{lst}\left(\mu=0,\ \tau^2=1,\ \nu\right)\ $$ with $$\mu=0$$ and $$\ \tau^2=1\ $$ reduces to the Student's $t$ distribution $$\ t_\nu ~.$$

As the distribution of a test statistic
Student's t-distribution with $$\nu$$ degrees of freedom can be defined as the distribution of the random variable T with

 * $$ T=\frac{Z}{\sqrt{V/\nu}} = Z \sqrt{\frac{\nu}{V}},$$

where
 * Z is a standard normal with expected value 0 and variance 1;
 * V has a chi-squared distribution ( χ 2-distribution) with $$\nu$$ degrees of freedom;
 * Z and V are independent;

A different distribution is defined as that of the random variable defined, for a given constant μ, by
 * $$(Z+\mu)\sqrt{\frac{\nu}{V}}.$$

This random variable has a noncentral t-distribution with noncentrality parameter μ. This distribution is important in studies of the power of Student's t-test.

Derivation
Suppose X1, ..., Xn are independent realizations of the normally-distributed, random variable X, which has an expected value μ and variance σ2. Let

 * $$\overline{X}_n = \frac{1}{n}(X_1+\cdots+X_n)$$

be the sample mean, and

 * $$S_n^2 = \frac{1}{n-1} \sum_{i=1}^n \left(X_i - \overline{X}_n\right)^2$$

be an unbiased estimate of the variance from the sample. It can be shown that the random variable

 * $$V = (n-1)\frac{S_n^2}{\sigma^2} $$

has a chi-squared distribution with $$\nu = n - 1$$ degrees of freedom (by Cochran's theorem). It is readily shown that the quantity

 * $$Z = \left(\overline{X}_n - \mu\right) \frac{\sqrt{n}}{\sigma}$$

is normally distributed with mean 0 and variance 1, since the sample mean $$\overline{X}_n$$ is normally distributed with mean μ and variance σ2/n. Moreover, it is possible to show that these two random variables (the normally distributed one Z and the chi-squared-distributed one V) are independent. Consequently the pivotal quantity

 * $T \equiv \frac{Z}{\sqrt{V/\nu}} = \left(\overline{X}_n - \mu\right) \frac{\sqrt{n}}{S_n},$

which differs from Z in that the exact standard deviation σ is replaced by the random variable Sn, has a Student's t-distribution as defined above. Notice that the unknown population variance σ2 does not appear in T, since it was in both the numerator and the denominator, so it canceled. Gosset intuitively obtained the probability density function stated above, with $$\nu$$ equal to n − 1, and Fisher proved it in 1925.

The distribution of the test statistic T depends on $$\nu$$, but not μ or σ; the lack of dependence on μ and σ is what makes the t-distribution important in both theory and practice.

Sampling distribution of t-statistic
The $t$ distribution arises as the sampling distribution of the $t$ statistic. Below the one-sample $t$ statistic is discussed, for the corresponding two-sample $t$ statistic see Student's t-test.

Unbiased variance estimate
Let $$\ x_1, \ldots, x_n \sim {\mathcal N}(\mu, \sigma^2)\ $$ be independent and identically distributed samples from a normal distribution with mean $$\mu$$ and variance $$\ \sigma^2 ~.$$ The sample mean and unbiased sample variance are given by:

\begin{align} \bar{x} &= \frac{\ x_1+\cdots+x_n\ }{ n }\, \\[5pt] s^2    &= \frac{ 1 }{\ n-1\ }\ \sum_{i=1}^n (x_i - \bar{x})^2 ~. \end{align} $$

The resulting (one sample) $t$ statistic is given by

 * $$ t = \frac{\bar{x} - \mu}{\ \sqrt{s^2/n\ }\ } \sim t_{n-1} ~.$$

and is distributed according to a Student's $t$ distribution with $$\ n - 1\ $$ degrees of freedom.

Thus for inference purposes the $t$ statistic is a useful "pivotal quantity" in the case when the mean and variance $$(\mu, \sigma^2)$$ are unknown population parameters, in the sense that the $t$ statistic has then a probability distribution that depends on neither $$\mu$$ nor $$\ \sigma^2 ~.$$

ML variance estimate
Instead of the unbiased estimate $$\ s^2\ $$ we may also use the maximum likelihood estimate
 * $$\ s^2_\mathsf{ML} = \frac{\ 1\ }{ n }\ \sum_{i=1}^n (x_i - \bar{x})^2\ $$

yielding the statistic
 * $$\ t_\mathsf{ML} = \frac{\bar{x} - \mu}{\sqrt{s^2_\mathsf{ML}/n\ }} = \sqrt{\frac{n}{n-1}\ }\ t ~.$$

This is distributed according to the location-scale $t$ distribution:
 * $$ t_\mathsf{ML} \sim \mathcal{lst}(0,\ \tau^2=n/(n-1),\ n-1) ~.$$

Compound distribution of normal with inverse gamma distribution
The location-scale $t$ distribution results from compounding a Gaussian distribution (normal distribution) with mean $$\ \mu\ $$ and unknown variance, with an inverse gamma distribution placed over the variance with parameters $$\ a = \frac{\ \nu\ }{ 2 }\ $$ and $$b = \frac{\ \nu\ \tau^2\ }{ 2 } ~.$$ In other words, the random variable X is assumed to have a Gaussian distribution with an unknown variance distributed as inverse gamma, and then the variance is marginalized out (integrated out).

Equivalently, this distribution results from compounding a Gaussian distribution with a scaled-inverse-chi-squared distribution with parameters $$\nu$$ and $$\ \tau^2 ~.$$ The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. $$\ \nu = 2\ a, \; {\tau}^2 = \frac{\ b\ }{ a } ~.$$

The reason for the usefulness of this characterization is that in Bayesian statistics the inverse gamma distribution is the conjugate prior distribution of the variance of a Gaussian distribution. As a result, the location-scale $t$ distribution arises naturally in many Bayesian inference problems.

Maximum entropy distribution
Student's $t$ distribution is the maximum entropy probability distribution for a random variate X for which $$\ \operatorname{\mathbb E}\left\{\ \ln(\nu+X^2)\ \right\}\ $$ is fixed.

Monte Carlo sampling
There are various approaches to constructing random samples from the Student's $t$ distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a quantile function to uniform samples; e.g., in the multi-dimensional applications basis of copula-dependency. In the case of stand-alone sampling, an extension of the Box–Muller method and its polar form is easily deployed. It has the merit that it applies equally well to all real positive degrees of freedom, $t$, while many other candidate methods fail if $t$ is close to zero.

Integral of Student's probability density function and $t$-value
The function $t > 0$ is the integral of Student's probability density function, $A(t | ν)$ between $t$ and $t$, for $f(t)$. It thus gives the probability that a value of t less than that calculated from observed data would occur by chance. Therefore, the function $t ≥ 0$ can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value of $t$ and the probability of its occurrence if the two sets of data were drawn from the same population. This is used in a variety of situations, particularly in $t$ tests. For the statistic $ν$, with $ν$ degrees of freedom, $A(t | ν)$ is the probability that $t$ would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that $A(t | ν)$ ). It can be easily calculated from the cumulative distribution function $t ≥ 0$ of the $t$ distribution:

 * $$ A( t \mid \nu) = F_\nu(t) - F_\nu(-t) = 1 - I_{ \frac{\nu}{\nu +t^2} }\!\left(\frac{\nu}{2},\frac{1}{2}\right),$$

where $Fν(t)$ is the regularized incomplete beta function.

For statistical hypothesis testing this function is used to construct the p-value.

Related distributions

 * The noncentral $ν$ distribution generalizes the $ν$ distribution to include a noncentrality parameter. Unlike the nonstandardized $p$ distributions, the noncentral distributions are not symmetric (the median is not the same as the mode).

 * The discrete Student's $-t$ distribution is defined by its probability mass function at r being proportional to: $$ \prod_{j=1}^k \frac{1}{(r+j+a)^2+b^2} \quad \quad r=\ldots, -1, 0, 1, \ldots ~.$$ Here a, b, and k are parameters. This distribution arises from the construction of a system of discrete distributions similar to that of the Pearson distributions for continuous distributions.

 * One can generate Student $Ix(a, b)$ samples by taking the ratio of variables from the normal distribution and the square-root of the $A(t | ν)$ distribution. If we use instead of the normal distribution, e.g., the Irwin–Hall distribution, we obtain over-all a symmetric 4 parameter distribution, which includes the normal, the uniform, the triangular, the Student $t$ and the Cauchy distribution. This is also more flexible than some other symmetric generalizations of the normal distribution.

 * $t$ distribution is an instance of ratio distributions.

In frequentist statistical inference
Student's $t$ distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors. If (as in nearly all practical statistical work) the population standard deviation of these errors is unknown and has to be estimated from the data, the $t$ distribution is often used to account for the extra uncertainty that results from this estimation. In most such problems, if the standard deviation of the errors were known, a normal distribution would be used instead of the $ν$ distribution.

Confidence intervals and hypothesis tests are two statistical procedures in which the quantiles of the sampling distribution of a particular statistic (e.g. the standard score) are required. In any situation where this statistic is a linear function of the data, divided by the usual estimate of the standard deviation, the resulting quantity can be rescaled and centered to follow Student's $t$ distribution. Statistical analyses involving means, weighted means, and regression coefficients all lead to statistics having this form.

Quite often, textbook problems will treat the population standard deviation as if it were known and thereby avoid the need to use the Student's $t$ distribution. These problems are generally of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of the variance as if it were certain, and (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining.

Hypothesis testing
A number of statistics can be shown to have $t$ distributions for samples of moderate size under null hypotheses that are of interest, so that the $t$ distribution forms the basis for significance tests. For example, the distribution of Spearman's rank correlation coefficient $t$, in the null case (zero correlation) is well approximated by the $t$ distribution for sample sizes above about 20.

Confidence intervals
Suppose the number A is so chosen that

 * $$\ \operatorname{\mathbb P}\left\{\ -A < T < A\ \right\} = 0.9\ ,$$

when $t$ has a $t$ distribution with $χ²$   degrees of freedom. By symmetry, this is the same as saying that $t$ satisfies

 * $$\ \operatorname{\mathbb P}\left\{\ T < A\ \right\} = 0.95\ ,$$

so A is the "95th percentile" of this probability distribution, or $$\ A = t_{(0.05,n-1)} ~.$$ Then

 * $$\ \operatorname{\mathbb P}\left\{\ -A < \frac{\ \overline{X}_n - \mu\ }{ S_n/\sqrt{n\ } } < A\ \right\} = 0.9\ ,$$

and this is equivalent to

 * $$\ \operatorname{\mathbb P}\left\{\ \overline{X}_n - A \frac{ S_n }{\ \sqrt{n\ }\ } < \mu < \overline{X}_n + A\ \frac{ S_n }{\ \sqrt{n\ }\ }\ \right\} = 0.9.$$

Therefore, the interval whose endpoints are

 * $$\ \overline{X}_n\ \pm A\ \frac{ S_n }{\ \sqrt{n\ }\ }\ $$

is a 90% confidence interval for μ. Therefore, if we find the mean of a set of observations that we can reasonably expect to have a normal distribution, we can use the $t$ distribution to examine whether the confidence limits on that mean include some theoretically predicted value – such as the value predicted on a null hypothesis.

It is this result that is used in the Student's $t$ tests: since the difference between the means of samples from two normal distributions is itself distributed normally, the $t$ distribution can be used to examine whether that difference can reasonably be supposed to be zero.

If the data are normally distributed, the one-sided $n − 1$ upper confidence limit (UCL) of the mean, can be calculated using the following equation:

 * $$\mathsf{UCL}_{1-\alpha} = \overline{X}_n + t_{\alpha,n-1}\ \frac{ S_n }{\ \sqrt{n\ }\ } ~.$$

The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words, $$\overline{X}_n$$ being the mean of the set of observations, the probability that the mean of the distribution is inferior to UCL$t$ is equal to the confidence level $(1 − α)$.

Prediction intervals
The $t$ distribution can be used to construct a prediction interval for an unobserved sample from a normal distribution with unknown mean and variance.

In Bayesian statistics
The Student's $t$ distribution, especially in its three-parameter (location-scale) version, arises frequently in Bayesian statistics as a result of its connection with the normal distribution. Whenever the variance of a normally distributed random variable is unknown and a conjugate prior placed over it that follows an inverse gamma distribution, the resulting marginal distribution of the variable will follow a Student's $ρ$ distribution. Equivalent constructions with the same results involve a conjugate scaled-inverse-chi-squared distribution over the variance, or a conjugate gamma distribution over the precision. If an improper prior proportional to $t$ is placed over the variance, the $T$ distribution also arises. This is the case regardless of whether the mean of the normally distributed variable is known, is unknown distributed according to a conjugate normally distributed prior, or is unknown distributed according to an improper constant prior.

Related situations that also produce a $t$ distribution are:
 * The marginal posterior distribution of the unknown mean of a normally distributed variable, with unknown prior mean and variance following the above model.
 * The prior predictive distribution and posterior predictive distribution of a new normally distributed data point when a series of independent identically distributed normally distributed data points have been observed, with prior mean and variance as in the above model.

Robust parametric modeling
The $A$ distribution is often used as an alternative to the normal distribution as a model for data, which often has heavier tails than the normal distribution allows for; see e.g. Lange et al. The classical approach was to identify outliers (e.g., using Grubbs's test) and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially in high dimensions), and the $t$ distribution is a natural choice of model for such data and provides a parametric approach to robust statistics.

A Bayesian account can be found in Gelman et al. The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. The likelihood can have multiple local maxima and, as such, it is often necessary to fix the degrees of freedom at a fairly low value and estimate the other parameters taking this as given. Some authors report that values between 3 and 9 are often good choices. Venables and Ripley suggest that a value of 5 is often a good choice.

Student's $t$ process
For practical regression and prediction needs, Student's $t$ processes were introduced, that are generalisations of the Student $$1 − α$$ distributions for functions. A Student's $t$ process is constructed from the Student $t$ distributions like a Gaussian process is constructed from the Gaussian distributions. For a Gaussian process, all sets of values have a multidimensional Gaussian distribution. Analogously, $$X(t)$$ is a Student $t$ process on an interval $$I=[a,b]$$ if the correspondent values of the process $$\ X(t_1),\ \ldots\, X(t_n)\ $$ ($$t_i \in I$$) have a joint multivariate Student $1⁄σ²$ distribution. These processes are used for regression, prediction, Bayesian optimization and related problems. For multivariate regression and multi-output prediction, the multivariate Student $t$ processes are introduced and used.

Table of selected values
The following table lists values for $t$ distributions with $t$ degrees of freedom for a range of one-sided or two-sided critical regions. The first column is $t$, the percentages along the top are confidence levels $$\ \alpha\ ,$$ and the numbers in the body of the table are the $$t_{\alpha,n-1}$$ factors described in the section on confidence intervals.

The last row with infinite $t$ gives critical points for a normal distribution since a $t$ distribution with infinitely many degrees of freedom is a normal distribution. (See Related distributions above).

 * Calculating the confidence interval :

Let's say we have a sample with size 11, sample mean 10, and sample variance 2. For 90% confidence with 10 degrees of freedom, the one-sided $t$ value from the table is 1.372. Then with confidence interval calculated from

 * $$\ \overline{X}_n \pm t_{\alpha,\nu}\ \frac{S_n}{\ \sqrt{n\ }\ }\ ,$$

we determine that with 90% confidence we have a true mean lying below

 * $$\ 10 + 1.372\ \frac{ \sqrt{2\ } }{\ \sqrt{11\ }\ } = 10.585 ~.$$

In other words, 90% of the times that an upper threshold is calculated by this method from particular samples, this upper threshold exceeds the true mean.

And with 90% confidence we have a true mean lying above

 * $$\ 10 - 1.372\ \frac{ \sqrt{2\ } }{\ \sqrt{11\ }\ } = 9.414 ~.$$

In other words, 90% of the times that a lower threshold is calculated by this method from particular samples, this lower threshold lies below the true mean.

So that at 80% confidence (calculated from 100% − 2 × (1 − 90%) = 80%), we have a true mean lying within the interval

 * $$\left(\ 10 - 1.372\ \frac{ \sqrt{2\ } }{\ \sqrt{11\ }\ },\ 10 + 1.372\ \frac{ \sqrt{2\ } }{\ \sqrt{11\ }\ }\ \right) = (\ 9.414,\ 10.585\ ) ~.$$

Saying that 80% of the times that upper and lower thresholds are calculated by this method from a given sample, the true mean is both below the upper threshold and above the lower threshold is not the same as saying that there is an 80% probability that the true mean lies between a particular pair of upper and lower thresholds that have been calculated by this method; see confidence interval and prosecutor's fallacy.

Nowadays, statistical software, such as the R programming language, and functions available in many spreadsheet programs compute values of the $t$ distribution and its inverse without tables.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Student's t-distribution


text:
Squeeze play

Squeeze play may refer to:

In games and sports

 * Squeeze play (baseball)
 * Squeeze play (bridge)
 * Squeeze play (poker)

Arts and media

 * Squeeze Play (album), a 1956 album by John Serry, Sr
 * Squeeze Play!, a 1979 comedy film
 * "Squeeze Play" (song), by Snoop Dogg
 * Squeeze Play (The Price Is Right), a segment game from The Price Is Right
 * Squeeze Play (novel), a semi-autobiographical baseball novel by sportswriter Jane Leavy.
 * Squeezeplay, a Transformers character in the Headmaster sub-group

Other uses

 * Operation Squeeze Play, a 2005 military operation in the Iraq War

title: Squeeze play


text:
Talk:Disulfide bond

Cytoplasmic disulfide bonds in eukaryotes
The Hatahet et al. paper says that disulfide bonds are not stable in the cytoplasm of E. coli because it is a reducing environment. However, neither that paper nor Sevier et al. say the same about eukaryotes. Certainly the proteins are stable in the endoplasmic reticulum where they are created. There are probably other organelles where they are stable as well. If "usually proteins secreted to the extracellular medium" is true for eukaryotes, an additional source is needed to support this. 140.233.203.193 (talk) 18:17, 2 March 2012 (UTC)

Cys-cys bonds
Quoting from the text:
 * When two cysteine amino acids bond to each other, they do so through a disulfide bond.

Ouch. No cys-cys dipeptides? David M

Disulphide bonds in bacteria
I think that the section on disuplhide bonds in bacteria should be made clearer. I am a microbiologist and I'm really unsure to what you are refering when you say as a reversible switch that turns a protein on or off when bacterial cells are exposed to oxidation reactions.

Also bacteria do have the ability to form disulphide bonds but only in the periplasm where it is an oxidizing enviroment.

Nick

Formation rates
Is there any info on rates of formation? Or even some references. This is a complicated question, however it may be good to address it rather than ignore it. For instance: rate of thiol to thiol disulfide formation vs rate of CYS to CYS disulfide formation, etc.

Spamming Wikipedia
Biologicalworld.com has spammed wikipedia like no tomorrow. He is a site of only a few pages and a LOT of adsense. Not much information is given except for "protocols" which are not referenced, and cannot be trusted from a site of that quality.

check: Links from Wikipedia

The following have been cleaned up:

and many more Sciencetalks (talk) —Preceding comment was added at 03:03, 4 January 2008 (UTC)
 * en.wikipedia.org/wiki/Plasmid
 * en.wikipedia.org/wiki/Gel_electrophoresis
 * en.wikipedia.org/wiki/Green_fluorescent_protein
 * en.wikipedia.org/wiki/Homology_(biology)
 * en.wikipedia.org/wiki/HeLa
 * en.wikipedia.org/wiki/Protease
 * en.wikipedia.org/wiki/Restriction_enzyme
 * en.wikipedia.org/wiki/Petri_dish
 * en.wikipedia.org/wiki/Structural_domain
 * en.wikipedia.org/wiki/Trypsin
 * en.wikipedia.org/wiki/Oligonucleotide
 * en.wikipedia.org/wiki/Transmission_electron_microscope
 * en.wikipedia.org/wiki/Agar_plate
 * en.wikipedia.org/wiki/Calcium_phosphate
 * en.wikipedia.org/wiki/Disulfide_bond
 * en.wikipedia.org/wiki/Denaturation_(biochemistry)
 * en.wikipedia.org/wiki/DNA_ligase
 * en.wikipedia.org/wiki/Wild_type
 * en.wikipedia.org/wiki/Tissue_culture
 * en.wikipedia.org/wiki/Transmission_electron_microscopy
 * en.wikipedia.org/wiki/Reporter_gene
 * en.wikipedia.org/wiki/Northern_blot
 * en.wikipedia.org/wiki/Protein_engineering
 * en.wikipedia.org/wiki/Sticky_end/blunt_end
 * en.wikipedia.org/wiki/Taq_polymerase
 * en.wikipedia.org/wiki/Protein_domain
 * en.wikipedia.org/wiki/Coomassie
 * en.wikipedia.org/wiki/Native_state
 * en.wikipedia.org/wiki/Chinese_Hamster_Ovary_cell
 * en.wikipedia.org/wiki/Peptidase
 * en.wikipedia.org/wiki/Visking_tubing
 * en.wikipedia.org/wiki/Streptavidin
 * en.wikipedia.org/wiki/Microtiter_plate
 * en.wikipedia.org/wiki/Subcloning
 * en.wikipedia.org/wiki/Ion_exchange_chromatography
 * en.wikipedia.org/wiki/Thermal_cycler
 * en.wikipedia.org/wiki/Bovine_serum_albumin
 * en.wikipedia.org/wiki/Phosphate_buffered_saline
 * en.wikipedia.org/wiki/Glutathione_S-transferase
 * en.wikipedia.org/wiki/HEPES
 * en.wikipedia.org/wiki/Ortholog
 * en.wikipedia.org/wiki/Proteases
 * en.wikipedia.org/wiki/Salting_out
 * en.wikipedia.org/wiki/Fetal_bovine_serum
 * en.wikipedia.org/wiki/Proteolytic_enzyme
 * en.wikipedia.org/wiki/DNA_end
 * en.wikipedia.org/wiki/Supernatant
 * en.wikipedia.org/wiki/ABTS
 * en.wikipedia.org/wiki/Conserved_sequence
 * en.wikipedia.org/wiki/Peptide_mass_fingerprinting
 * en.wikipedia.org/wiki/Dithiothreitol
 * en.wikipedia.org/wiki/Uranyl_acetate

Corrections needed
There are quite a few mistakes in this article. I have fixed some, but many remain. For example, it can't be stated that in-vivo disulfide rearrangement reactions (or other reactions involving disulfides) are intra-molecular. Disulfide formation/destruction/isomerization is generally a catalyzed process in living cells, and catalysis usually involves formation of inter-molecular mixed disulfide bond with the catalyst. I hope to find some time to add a section on how disulfide bonds are formed and destroyed in different compartments of prokaryotic and eukaryotic cells. The importance of disulfide bonds in living organisms is understated. For example, the production of deoxyribonucleotides (building blocks of the genetic code) involves an enzyme called ribonucleotide reductase, with a disulfide bond at it's catalytic heart. The numbering of the figures should be removed, as now some figures are numbered while others are not. Thioredoxins usually do not have vicinal Cys. The catalytic motif in Trx is CXXC. It usually participates in the reduction of disulfide bonds in cytoplasm with an indirect role of oxidation of extracytoplasmic proteins through an electron relay such as DsbD in E. coli. Naming cytoplasm (redox potential ~-300 mV) as reducing environment and e.g. endoplasmic reticulum (-160 mV) as oxidizing is misleading, i.e. standard redox potential of oxygen (O2) is +620 mV. Fermmyt (talk) 23:35, 4 October 2013 (UTC)

title: Talk:Disulfide bond