Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Web of Science
Web of Science is an online subscription-based scientific citation indexing service produced by the Institute for Scientific Information maintained by Clarivate Analytics, that provides a comprehensive citation search. It gives access to multiple databases that reference cross-disciplinary research, which allows for in-depth exploration of specialized sub-fields within an academic or scientific discipline. A citation index is built on the fact that citations in science serve as linkages between similar research items, lead to matching or related scientific literature, such as journal articles, conference proceedings, etc. In addition, literature which shows the greatest impact in a particular field, or more than one discipline, can be located through a citation index. For example, a paper's influence can be determined by linking to all the papers. In this way, current trends and emerging fields of research can be assessed. Eugene Garfield, the "father of citation indexing of academic literature," who launched the Science Citation Index, which in turn led to the Web of Science, wrote: Citations are the formal, explicit linkages between papers that have particular points in common.
A citation index is built around these linkages. It identifies the sources of the citations. Anyone conducting a literature search can find from one to dozens of additional papers on a subject just by knowing one, cited, and every paper, found provides a list of new citations with which to continue the search. The simplicity of citation indexing is one of its main strengths. Web of Science is described as a unifying research tool which enables the user to acquire and disseminate database information in a timely manner; this is accomplished because of the creation of a common vocabulary, called ontology, for varied search terms and varied data. Moreover, search terms generate related information across categories. Acceptable content for Web of Science is determined by an evaluation and selection process based on the following criteria: impact, timeliness, peer review, geographic representation. Web of Science employs various analysis capabilities. First, citation indexing is employed, enhanced by the capability to search for results across disciplines.
The influence, impact and methodology of an idea can be followed from its first instance, notice, or referral to the present day. This technology points to a deficiency with the keyword-only method of searching. Second, subtle trends and patterns relevant to the literature or research of interest, become apparent. Broad trends indicate significant topics of the day, as well as the history relevant to both the work at hand, particular areas of study. Third, trends can be graphically represented. Expanding the coverage of Web of Science, in November 2009 Thomson Reuters introduced Century of Social Sciences; this service contains files which trace social science research back to the beginning of the 20th century, Web of Science now has indexing coverage from the year 1900 to the present. As of 3 September 2014, the multidisciplinary coverage of the Web of Science encompasses over 50,000 scholarly books, 12,000 journals and 160,000 conference proceedings; the selection is made on the basis of impact evaluations and comprise open-access journals, spanning multiple academic disciplines.
The coverage includes: the sciences, social sciences and humanities, goes across disciplines. However, Web of Science does not index all journals. There is a positive correlation between Impact Factor and CiteScore. However, analysis by Elsevier has identified 216 journals from 70 publishers to be in the top 10 percent of the most-cited journals in their subject category based on the CiteScore while they did not have Impact Factor, it appears that Impact Factor does not provide a comprehensive and an unbiased coverage of high quality journals. Similar results can be observed by comparing Impact Factor with SCImago Journal Rank. Furthermore, as of September 3, 2014 the total file count of the Web of Science was 90 million records, which included over a billion cited references; this citation service on average indexes around 65 million items per year, it is described as the largest accessible citation database. Titles of foreign-language publications are translated into English and so cannot be found by searches in the original language.
The Web of Science Core Collection consists of six online databases: Science Citation Index Expanded covers more than 8,500 notable journals encompassing 150 disciplines. Coverage is from the year 1900 to the present day. Social Sciences Citation Index covers more than 3,000 journals in social science disciplines. Range of coverage is from the year 1900 to the present day. Arts & Humanities Citation Index covers more than 1,700 arts and humanities journals starting from 1975. In addition, 250 major scientific and social sciences journals are covered. Emerging Sources Citation Index covers over 5,000 journals in the sciences, social science, humanities. Book Citation Index covers more than 60,000 editorially selected books starting from 2005. Conference Proceedings Citation Index covers more than 160,000 conference titles in the Sciences starting from 1990 to the present day Since 2008, the Web of Science hosts a number of regional citation indices; the Chinese Science Citation Database, produced in partnership with the Chinese Academy of Sciences, was the first one in a language other than English.
It was followed in 2013 by the SciELO Citation Index, covering Brazil, Portugal, the Cari
Germany the Federal Republic of Germany, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, the Alps to the south. It borders Denmark to the north and the Czech Republic to the east and Switzerland to the south, France to the southwest, Luxembourg and the Netherlands to the west. Germany includes 16 constituent states, covers an area of 357,386 square kilometres, has a temperate seasonal climate. With 83 million inhabitants, it is the second most populous state of Europe after Russia, the most populous state lying in Europe, as well as the most populous member state of the European Union. Germany is a decentralized country, its capital and largest metropolis is Berlin, while Frankfurt serves as its financial capital and has the country's busiest airport. Germany's largest urban area is the Ruhr, with its main centres of Essen; the country's other major cities are Hamburg, Cologne, Stuttgart, Düsseldorf, Dresden, Bremen and Nuremberg. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity.
A region named Germania was documented before 100 AD. During the Migration Period, the Germanic tribes expanded southward. Beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation. After the collapse of the Holy Roman Empire, the German Confederation was formed in 1815; the German revolutions of 1848–49 resulted in the Frankfurt Parliament establishing major democratic rights. In 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the revolution of 1918–19, the Empire was replaced by the parliamentary Weimar Republic; the Nazi seizure of power in 1933 led to the establishment of a dictatorship, the annexation of Austria, World War II, the Holocaust. After the end of World War II in Europe and a period of Allied occupation, Austria was re-established as an independent country and two new German states were founded: West Germany, formed from the American and French occupation zones, East Germany, formed from the Soviet occupation zone.
Following the Revolutions of 1989 that ended communist rule in Central and Eastern Europe, the country was reunified on 3 October 1990. Today, the sovereign state of Germany is a federal parliamentary republic led by a chancellor, it is a great power with a strong economy. As a global leader in several industrial and technological sectors, it is both the world's third-largest exporter and importer of goods; as a developed country with a high standard of living, it upholds a social security and universal health care system, environmental protection, a tuition-free university education. The Federal Republic of Germany was a founding member of the European Economic Community in 1957 and the European Union in 1993, it is part of the Schengen Area and became a co-founder of the Eurozone in 1999. Germany is a member of the United Nations, NATO, the G7, the G20, the OECD. Known for its rich cultural history, Germany has been continuously the home of influential and successful artists, musicians, film people, entrepreneurs, scientists and inventors.
Germany has a large number of World Heritage sites and is among the top tourism destinations in the world. The English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine; the German term Deutschland diutisciu land is derived from deutsch, descended from Old High German diutisc "popular" used to distinguish the language of the common people from Latin and its Romance descendants. This in turn descends from Proto-Germanic *þiudiskaz "popular", derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- "people", from which the word Teutons originates; the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a coal mine in Schöningen between 1994 and 1998 where eight 380,000-year-old wooden javelins of 1.82 to 2.25 m length were unearthed. The Neander Valley was the location where the first non-modern human fossil was discovered.
The Neanderthal 1 fossils are known to be 40,000 years old. Evidence of modern humans dated, has been found in caves in the Swabian Jura near Ulm; the finds included 42,000-year-old bird bone and mammoth ivory flutes which are the oldest musical instruments found, the 40,000-year-old Ice Age Lion Man, the oldest uncontested figurative art discovered, the 35,000-year-old Venus of Hohle Fels, the oldest uncontested human figurative art discovered. The Nebra sky disk is a bronze artefact created during the European Bronze Age attributed to a site near Nebra, Saxony-Anhalt, it is part of UNESCO's Memory of the World Programme. The Germanic tribes are thought to date from the Pre-Roman Iron Age. From southern Scandinavia and north Germany, they expanded south and west from the 1st century BC, coming into contact with the Celtic tribes of Gaul as well
Theoretical computer science
Theoretical computer science is a subset of general computer science and mathematics that focuses on more mathematical topics of computing and includes the theory of computation. It is difficult to circumscribe the theoretical areas precisely; the ACM's Special Interest Group on Algorithms and Computation Theory provides the following description: TCS covers a wide variety of topics including algorithms, data structures, computational complexity and distributed computation, probabilistic computation, quantum computation, automata theory, information theory, program semantics and verification, machine learning, computational biology, computational economics, computational geometry, computational number theory and algebra. Work in this field is distinguished by its emphasis on mathematical technique and rigor. While logical inference and mathematical proof had existed in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved.
These developments have led to the modern study of logic and computability, indeed the field of theoretical computer science as a whole. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and parallel distributed processing were established. In 1971, Stephen Cook and, working independently, Leonid Levin, proved that there exist relevant problems that are NP-complete – a landmark result in computational complexity theory. With the development of quantum mechanics in the beginning of the 20th century came the concept that mathematical operations could be performed on an entire particle wavefunction. In other words, one could compute functions on multiple states simultaneously; this led to the concept of a quantum computer in the latter half of the 20th century that took off in the 1990s when Peter Shor showed that such methods could be used to factor large numbers in polynomial time, which, if implemented, would render most modern public key cryptography systems uselessly insecure.
Modern theoretical computer science research is based on these basic developments, but includes many other mathematical and interdisciplinary problems that have been posed, as shown below: An algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, automated reasoning. An algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states producing "output" and terminating at a final ending state; the transition from one state to the next is not deterministic. A data structure is a particular way of organizing data in a computer so that it can be used efficiently. Different kinds of data structures are suited to different kinds of applications, some are specialized to specific tasks. For example, databases use B-tree indexes for small percentages of data retrieval and compilers and databases use dynamic hash tables as look up tables.
Data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services. Efficient data structures are key to designing efficient algorithms; some formal design methods and programming languages emphasize data structures, rather than algorithms, as the key organizing factor in software design. Storing and retrieving can be carried out on data stored in both main memory and in secondary memory. Computational complexity theory is a branch of the theory of computation that focuses on classifying computational problems according to their inherent difficulty, relating those classes to each other. A computational problem is understood to be a task, in principle amenable to being solved by a computer, equivalent to stating that the problem may be solved by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used.
The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Other complexity measures are used, such as the amount of communication, the number of gates in a circuit and the number of processors. One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do. Distributed computing studies distributed systems. A distributed system is a software system in which components located on networked computers communicate and coordinate their actions by passing messages; the components interact with each other. Three significant characteristics of distributed systems are: concurrency of components, lack of a global clock, independent failure of components. Examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications, and now a perfect example could be the blockcain.
A computer program that runs in a distributed system is called a distributed program, distributed programming is the process of writing such programs. There are m
Russia the Russian Federation, is a transcontinental country in Eastern Europe and North Asia. At 17,125,200 square kilometres, Russia is by far or by a considerable margin the largest country in the world by area, covering more than one-eighth of the Earth's inhabited land area, the ninth most populous, with about 146.77 million people as of 2019, including Crimea. About 77 % of the population live in the European part of the country. Russia's capital, Moscow, is one of the largest cities in the world and the second largest city in Europe. Extending across the entirety of Northern Asia and much of Eastern Europe, Russia spans eleven time zones and incorporates a wide range of environments and landforms. From northwest to southeast, Russia shares land borders with Norway, Estonia, Latvia and Poland, Ukraine, Azerbaijan, China and North Korea, it shares maritime borders with Japan by the Sea of Okhotsk and the U. S. state of Alaska across the Bering Strait. However, Russia recognises two more countries that border it, Abkhazia and South Ossetia, both of which are internationally recognized as parts of Georgia.
The East Slavs emerged as a recognizable group in Europe between the 3rd and 8th centuries AD. Founded and ruled by a Varangian warrior elite and their descendants, the medieval state of Rus arose in the 9th century. In 988 it adopted Orthodox Christianity from the Byzantine Empire, beginning the synthesis of Byzantine and Slavic cultures that defined Russian culture for the next millennium. Rus' disintegrated into a number of smaller states; the Grand Duchy of Moscow reunified the surrounding Russian principalities and achieved independence from the Golden Horde. By the 18th century, the nation had expanded through conquest and exploration to become the Russian Empire, the third largest empire in history, stretching from Poland on the west to Alaska on the east. Following the Russian Revolution, the Russian Soviet Federative Socialist Republic became the largest and leading constituent of the Union of Soviet Socialist Republics, the world's first constitutionally socialist state; the Soviet Union played a decisive role in the Allied victory in World War II, emerged as a recognized superpower and rival to the United States during the Cold War.
The Soviet era saw some of the most significant technological achievements of the 20th century, including the world's first human-made satellite and the launching of the first humans in space. By the end of 1990, the Soviet Union had the world's second largest economy, largest standing military in the world and the largest stockpile of weapons of mass destruction. Following the dissolution of the Soviet Union in 1991, twelve independent republics emerged from the USSR: Russia, Belarus, Uzbekistan, Azerbaijan, Kyrgyzstan, Tajikistan and the Baltic states regained independence: Estonia, Lithuania, it is governed as a federal semi-presidential republic. Russia's economy ranks as the twelfth largest by nominal GDP and sixth largest by purchasing power parity in 2018. Russia's extensive mineral and energy resources are the largest such reserves in the world, making it one of the leading producers of oil and natural gas globally; the country is one of the five recognized nuclear weapons states and possesses the largest stockpile of weapons of mass destruction.
Russia is a great power as well as a regional power and has been characterised as a potential superpower. It is a permanent member of the United Nations Security Council and an active global partner of ASEAN, as well as a member of the Shanghai Cooperation Organisation, the G20, the Council of Europe, the Asia-Pacific Economic Cooperation, the Organization for Security and Co-operation in Europe, the World Trade Organization, as well as being the leading member of the Commonwealth of Independent States, the Collective Security Treaty Organization and one of the five members of the Eurasian Economic Union, along with Armenia, Belarus and Kyrgyzstan; the name Russia is derived from Rus', a medieval state populated by the East Slavs. However, this proper name became more prominent in the history, the country was called by its inhabitants "Русская Земля", which can be translated as "Russian Land" or "Land of Rus'". In order to distinguish this state from other states derived from it, it is denoted as Kievan Rus' by modern historiography.
The name Rus itself comes from the early medieval Rus' people, Swedish merchants and warriors who relocated from across the Baltic Sea and founded a state centered on Novgorod that became Kievan Rus. An old Latin version of the name Rus' was Ruthenia applied to the western and southern regions of Rus' that were adjacent to Catholic Europe; the current name of the country, Россия, comes from the Byzantine Greek designation of the Rus', Ρωσσία Rossía—spelled Ρωσία in Modern Greek. The standard way to refer to citizens of Russia is rossiyane in Russian. There are two Russian words which are commonly
Outline of academic disciplines
An academic discipline or field of study is a branch of knowledge and researched as part of higher education. A scholar's discipline is defined by the university faculties and learned societies to which she or he belongs and the academic journals in which she or he publishes research. Disciplines vary between well-established ones that exist in all universities and have well-defined rosters of journals and conferences and nascent ones supported by only a few universities and publications. A discipline may have branches, these are called sub-disciplines. There is no consensus on how some academic disciplines should be classified, for example whether anthropology and linguistics are disciplines of the social sciences or of the humanities; the following outline is provided as topical guide to academic disciplines. Biblical studies Religious studies Biblical Hebrew, Biblical Greek, Aramaic Buddhist theology Christian theology Anglican theology Baptist theology Catholic theology Eastern Orthodox theology Protestant theology Hindu theology Jewish theology Muslim theology Biological anthropology Linguistic anthropology Cultural anthropology Social anthropology Archaeology Accounting Business management Finance Marketing Operations management Edaphology Environmental chemistry Environmental science Gemology Geochemistry Geodesy Physical geography Atmospheric science / Meteorology Biogeography / Phytogeography Climatology / Paleoclimatology / Palaeogeography Coastal geography / Oceanography Edaphology / Pedology or Soil science Geobiology Geology Geostatistics Glaciology Hydrology / Limnology / Hydrogeology Landscape ecology Quaternary science Geophysics Paleontology Paleobiology Paleoecology Astrobiology Astronomy Observational astronomy Gamma ray astronomy Infrared astronomy Microwave astronomy Optical astronomy Radio astronomy UV astronomy X-ray astronomy Astrophysics Gravitational astronomy Black holes Interstellar medium Numerical simulations Astrophysical plasma Galaxy formation and evolution High-energy astrophysics Hydrodynamics Magnetohydrodynamics Star formation Physical cosmology Stellar astrophysics Helioseismology Stellar evolution Stellar nucleosynthesis Planetary science Also a branch of electrical engineering Pure mathematics Applied mathematics Astrostatistics Biostatistics Academia Academic genealogy Curriculum Multidisciplinary approach Interdisciplinarity Transdisciplinarity Professions Classification of Instructional Programs Joint Academic Coding System List of fields of doctoral studies in the United States List of academic fields Abbott, Andrew.
Chaos of Disciplines. University of Chicago Press. ISBN 978-0-226-00101-2. Oleson, Alexandra; the Organization of knowledge in modern America, 1860-1920. ISBN 0-8018-2108-8. US Department of Education Institute of Education Sciences. Classification of Instructional Programs. National Center for Education Statistics. Classification of Instructional Programs: Developed by the U. S. Department of Education's National Center for Education Statistics to provide a taxonomic scheme that will support the accurate tracking and reporting of fields of study and program completions activity. Complete JACS from Higher Education Statistics Agency in the United Kingdom Australian and New Zealand Standard Research Classification Chapter 3 and Appendix 1: Fields of research classification. Fields of Knowledge, a zoomable map allowing the academic disciplines and sub-disciplines in this article be visualised. Sandoz, R. Interactive Historical Atlas of the Disciplines, University of Geneva
Statistics is a branch of mathematics dealing with data collection, analysis and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics; when census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements.
In contrast, an observational study does not involve experimental manipulation. Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, inferential statistics, which draw conclusions from data that are subject to random variation. Descriptive statistics are most concerned with two sets of properties of a distribution: central tendency seeks to characterize the distribution's central or typical value, while dispersion characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets.
Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors and Type II errors. Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are subject to error. Many of these errors are classified as random or systematic, but other types of errors can be important; the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more from calculus and probability theory. In more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis.
Some definitions are: Merriam-Webster dictionary defines statistics as "a branch of mathematics dealing with the collection, analysis and presentation of masses of numerical data." Statistician Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed in relation to each other."Statistics is a mathematical body of science that pertains to the collection, interpretation or explanation, presentation of data, or as a branch of mathematics. Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty and decision making in the face of uncertainty. Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, measure-theoretic probability theory.
In applying statistics to a problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Ideally, statisticians compile data about the entire population; this may be organized by governmental statistical institutes. Descriptive statistics can be used to summarize the population data. Numerical descriptors include mean and standard deviation for continuous data types, while frequency and percentage are more useful in terms of describing categorical data; when a census is not feasible, a chosen subset of the population called. Once a sample, representative of the population is determined, data is collected for the sample members in an observational or experimental setting. Again, descriptive statistics can be used to summarize the sample data. However, the drawing of the sample has been subject to an element of randomness, hence the established numerical descriptors from the sample are due to uncertainty.
To still draw meaningful conclusions about the entire population, in