Chemistry is the scientific discipline involved with elements and compounds composed of atoms and ions: their composition, properties and the changes they undergo during a reaction with other substances. In the scope of its subject, chemistry occupies an intermediate position between physics and biology, it is sometimes called the central science because it provides a foundation for understanding both basic and applied scientific disciplines at a fundamental level. For example, chemistry explains aspects of plant chemistry, the formation of igneous rocks, how atmospheric ozone is formed and how environmental pollutants are degraded, the properties of the soil on the moon, how medications work, how to collect DNA evidence at a crime scene. Chemistry addresses topics such as how atoms and molecules interact via chemical bonds to form new chemical compounds. There are four types of chemical bonds: covalent bonds, in which compounds share one or more electron; the word chemistry comes from alchemy, which referred to an earlier set of practices that encompassed elements of chemistry, philosophy, astronomy and medicine.
It is seen as linked to the quest to turn lead or another common starting material into gold, though in ancient times the study encompassed many of the questions of modern chemistry being defined as the study of the composition of waters, growth, disembodying, drawing the spirits from bodies and bonding the spirits within bodies by the early 4th century Greek-Egyptian alchemist Zosimos. An alchemist was called a'chemist' in popular speech, the suffix "-ry" was added to this to describe the art of the chemist as "chemistry"; the modern word alchemy in turn is derived from the Arabic word al-kīmīā. In origin, the term is borrowed from the Greek χημία or χημεία; this may have Egyptian origins since al-kīmīā is derived from the Greek χημία, in turn derived from the word Kemet, the ancient name of Egypt in the Egyptian language. Alternately, al-kīmīā may derive from χημεία, meaning "cast together"; the current model of atomic structure is the quantum mechanical model. Traditional chemistry starts with the study of elementary particles, molecules, metals and other aggregates of matter.
This matter can be studied in isolation or in combination. The interactions and transformations that are studied in chemistry are the result of interactions between atoms, leading to rearrangements of the chemical bonds which hold atoms together; such behaviors are studied in a chemistry laboratory. The chemistry laboratory stereotypically uses various forms of laboratory glassware; however glassware is not central to chemistry, a great deal of experimental chemistry is done without it. A chemical reaction is a transformation of some substances into one or more different substances; the basis of such a chemical transformation is the rearrangement of electrons in the chemical bonds between atoms. It can be symbolically depicted through a chemical equation, which involves atoms as subjects; the number of atoms on the left and the right in the equation for a chemical transformation is equal. The type of chemical reactions a substance may undergo and the energy changes that may accompany it are constrained by certain basic rules, known as chemical laws.
Energy and entropy considerations are invariably important in all chemical studies. Chemical substances are classified in terms of their structure, phase, as well as their chemical compositions, they can be analyzed using the tools of e.g. spectroscopy and chromatography. Scientists engaged in chemical research are known as chemists. Most chemists specialize in one or more sub-disciplines. Several concepts are essential for the study of chemistry; the particles that make up matter have rest mass as well – not all particles have rest mass, such as the photon. Matter can be a mixture of substances; the atom is the basic unit of chemistry. It consists of a dense core called the atomic nucleus surrounded by a space occupied by an electron cloud; the nucleus is made up of positively charged protons and uncharged neutrons, while the electron cloud consists of negatively charged electrons which orbit the nucleus. In a neutral atom, the negatively charged electrons balance out the positive charge of the protons.
The nucleus is dense. The atom is the smallest entity that can be envisaged to retain the chemical properties of the element, such as electronegativity, ionization potential, preferred oxidation state, coordination number, preferred types of bonds to form. A chemical element is a pure substance, composed of a single type of atom, characterized by its particular number of protons in the nuclei of its atoms, known as the atomic number and represented by the symbol Z; the mass number is the sum of the number of neutrons in a nucleus. Although all the nuclei of all atoms belonging to one element will have the same
The number π is a mathematical constant. Defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics, it is equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is sometimes spelled out as "pi", it is called Archimedes' constant. Being an irrational number, π cannot be expressed as a common fraction. Still, fractions such as 22/7 and other rational numbers are used to approximate π; the digits appear to be randomly distributed. In particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date, no proof of this has been discovered. Π is a transcendental number. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. Ancient civilizations required accurate computed values to approximate π for practical reasons, including the Egyptians and Babylonians.
Around 250 BC the Greek mathematician Archimedes created an algorithm for calculating it. In the 5th century AD Chinese mathematics approximated π to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques; the first exact formula for π, based on infinite series, was not available until a millennium when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics. In the 20th and 21st centuries and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits after the decimal point. All scientific applications require no more than a few hundred digits of π, many fewer, so the primary motivation for these computations is the quest to find more efficient algorithms for calculating lengthy numeric series, as well as the desire to break records; the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.
Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry those concerning circles and spheres. In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry, it appears therefore in areas of mathematics and the sciences having little to do with the geometry of circles, such as number theory and statistics, as well as in all areas of physics. The ubiquity of π makes it one of the most known mathematical constants both inside and outside the scientific community. Several books devoted to π have been published, record-setting calculations of the digits of π result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits; the symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter π, sometimes spelled out as pi, derived from the first letter of the Greek word perimetros, meaning circumference.
In English, π is pronounced as "pie". In mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation; the choice of the symbol π is discussed in the section Adoption of the symbol π. Π is defined as the ratio of a circle's circumference C to its diameter d: π = C d The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will have twice the circumference, preserving the ratio C/d; this definition of π implicitly makes use of flat geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits, a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x2 + y2 = 1, as the integral: π = ∫ − 1 1 d x 1 − x 2.
An integral such as this was adopted as the definition of π by Karl Weierstrass, who defined it directly as an integral in 1841. Definitions of π such as these that rely on a notion of circumference, hence implicitly on concepts of the integral calculus, are no longer common in the literature. Remmert explains that this is because in many modern treatments of calculus, differential calculus precedes integral calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, due to Richard Baltzer, popularized by Edmund Landau, is the following: π is twice the smallest positive number at which the cosine function equals 0; the cosine can be defined independently of geometry as a power series, or as the solution of a differen
The kilogram or kilogramme is the base unit of mass in the International System of Units. Until 20 May 2019, it remains defined by a platinum alloy cylinder, the International Prototype Kilogram, manufactured in 1889, stored in Saint-Cloud, a suburb of Paris. After 20 May, it will be defined in terms of fundamental physical constants; the kilogram was defined as the mass of a litre of water. That was an inconvenient quantity to replicate, so in 1799 a platinum artefact was fashioned to define the kilogram; that artefact, the IPK, have been the standard of the unit of mass for the metric system since. In spite of best efforts to maintain it, the IPK has diverged from its replicas by 50 micrograms since their manufacture late in the 19th century; this led to efforts to develop measurement technology precise enough to allow replacing the kilogram artifact with a definition based directly on physical phenomena, now scheduled to take place in 2019. The new definition is based on invariant constants of nature, in particular the Planck constant, which will change to being defined rather than measured, thereby fixing the value of the kilogram in terms of the second and the metre, eliminating the need for the IPK.
The new definition was approved by the General Conference on Weights and Measures on 16 November 2018. The Planck constant relates a light particle's energy, hence mass, to its frequency; the new definition only became possible when instruments were devised to measure the Planck constant with sufficient accuracy based on the IPK definition of the kilogram. The gram, 1/1000 of a kilogram, was provisionally defined in 1795 as the mass of one cubic centimetre of water at the melting point of ice; the final kilogram, manufactured as a prototype in 1799 and from which the International Prototype Kilogram was derived in 1875, had a mass equal to the mass of 1 dm3 of water under atmospheric pressure and at the temperature of its maximum density, 4 °C. The kilogram is the only named SI unit with an SI prefix as part of its name; until the 2019 redefinition of SI base units, it was the last SI unit, still directly defined by an artefact rather than a fundamental physical property that could be independently reproduced in different laboratories.
Three other base units and 17 derived units in the SI system are defined in relation to the kilogram, thus its stability is important. The definitions of only eight other named SI units do not depend on the kilogram: those of temperature and frequency, angle; the IPK is used or handled. Copies of the IPK kept by national metrology laboratories around the world were compared with the IPK in 1889, 1948, 1989 to provide traceability of measurements of mass anywhere in the world back to the IPK; the International Prototype Kilogram was commissioned by the General Conference on Weights and Measures under the authority of the Metre Convention, in the custody of the International Bureau of Weights and Measures who hold it on behalf of the CGPM. After the International Prototype Kilogram had been found to vary in mass over time relative to its reproductions, the International Committee for Weights and Measures recommended in 2005 that the kilogram be redefined in terms of a fundamental constant of nature.
At its 2011 meeting, the CGPM agreed in principle that the kilogram should be redefined in terms of the Planck constant, h. The decision was deferred until 2014. CIPM has proposed revised definitions of the SI base units, for consideration at the 26th CGPM; the formal vote, which took place on 16 November 2018, approved the change, with the new definitions coming into force on 20 May 2019. The accepted redefinition defines the Planck constant as 6.62607015×10−34 kg⋅m2⋅s−1, thereby defining the kilogram in terms of the second and the metre. Since the second and metre are defined in terms of physical constants, the kilogram is defined in terms of physical constants only; the avoirdupois pound, used in both the imperial and US customary systems, is now defined in terms of the kilogram. Other traditional units of weight and mass around the world are now defined in terms of the kilogram, making the kilogram the primary standard for all units of mass on Earth; the word kilogramme or kilogram is derived from the French kilogramme, which itself was a learned coinage, prefixing the Greek stem of χίλιοι khilioi "a thousand" to gramma, a Late Latin term for "a small weight", itself from Greek γράμμα.
The word kilogramme was written into French law in 1795, in the Decree of 18 Germinal, which revised the older system of units introduced by the French National Convention in 1793, where the gravet had been defined as weight of a cubic centimetre of water, equal to 1/1000 of a grave. In the decree of 1795, the term gramme thus replaced gravet, kilogramme replaced grave; the French spelling was adopted in Great Britain when the word was used for the first time in English in 1795, with the spelling kilogram being adopted in the United States. In the United Kingdom both spellings are used, with "kilogram" having become by far the more common. UK law regulating the units to be used when trading by weight or measure does not prevent the use of either spelling. In the 19th century the French word kilo, a shortening of kilogramme, was imported into the English language where it has been used to mean both kilogram and kilometre. While kilo is acceptable in many generalist texts
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. It is an irrational number, a solution to the quadratic equation x 2 − x − 1 = 0, with a value of: φ = 1 + 5 2 = 1.6180339887 …. The golden ratio is called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number. Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio; the golden ratio has been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.
The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ. One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a = a a + b a = 1 + b a = 1 + 1 φ. Therefore, 1 + 1 φ = φ. Multiplying by φ gives φ + 1 = φ 2 which can be rearranged to φ 2 − φ − 1 = 0. Using the quadratic formula, two solutions are obtained: 1 + 5 2 = 1.618 033 988 7 … and 1 − 5 2 = − 0.618 033 988 7 … Because φ is the ratio between positive quantities, φ is positive: φ = 1 + 5 2 = 1.61803 39887 … The golden ratio has been claimed to have held a special fascination for at least 2,400 years, although without reliable evidence.
According to Mario Livio: Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, musicians, architects and mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction, surprising Pythagoreans. Euclid's Elements provides several propositions and their proofs employing the golden ratio and contains the first known definition: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.
The golden ratio was studied peripherally over the next millennium. Abu Kamil employed it in his geometric calculati
The Avogadro constant, named after scientist Amedeo Avogadro, is the number of constituent particles molecules, atoms or ions that are contained in the amount of substance given by one mole. It is the proportionality factor that relates the molar mass of a substance to the mass of a sample, is designated with the symbol NA or L, has the value 6.022140857×1023 mol−1 in the International System of Units. Previous definitions of chemical quantity involved the Avogadro number, a historical term related to the Avogadro constant, but defined differently: the Avogadro number was defined by Jean Baptiste Perrin as the number of atoms in one gram-molecule of atomic hydrogen, meaning one gram of hydrogen; this number is known as Loschmidt constant in German literature. The constant was redefined as the number of atoms in 12 grams of the isotope carbon-12, still generalized to relate amounts of a substance to their molecular weight. For instance, the number of nucleons in one mole of any sample of ordinary matter is, to a first approximation, 6×1023 times its molecular weight.
12 grams of 12C, with the mass number 12, has a similar number of carbon atoms, 6.022×1023. The Avogadro number is a dimensionless quantity, has the same numerical value of the Avogadro constant when given in base units. In contrast, the Avogadro constant has the dimension of reciprocal amount of substance; the Avogadro constant can be expressed as 0.6023... mL⋅mol−1⋅Å−3, which can be used to convert from volume per molecule in cubic ångströms to molar volume in millilitres per mole. Pending revisions in the base set of SI units necessitated redefinitions of the concepts of chemical quantity; the Avogadro number, its definition, was deprecated in favor of the Avogadro constant and its definition. Based on measurements made through the middle of 2017 which calculated a value for the Avogadro constant of NA = 6.022140758×1023 mol−1, the redefinition of SI units is planned to take effect on 20 May 2019. The value of the constant will be fixed to 6.02214076×1023 mol−1. The Avogadro constant is named after the early 19th-century Italian scientist Amedeo Avogadro, who, in 1811, first proposed that the volume of a gas is proportional to the number of atoms or molecules regardless of the nature of the gas.
The French physicist Jean Perrin in 1909 proposed naming the constant in honor of Avogadro. Perrin won the 1926 Nobel Prize in Physics for his work in determining the Avogadro constant by several different methods; the value of the Avogadro constant was first indicated by Johann Josef Loschmidt, who in 1865 estimated the average diameter of the molecules in the air by a method, equivalent to calculating the number of particles in a given volume of gas. This latter value, the number density n0 of particles in an ideal gas, is now called the Loschmidt constant in his honor, is related to the Avogadro constant, NA, by n 0 = p 0 N A R T 0, where p0 is the pressure, R is the gas constant, T0 is the absolute temperature; the connection with Loschmidt is the origin of the symbol L sometimes used for the Avogadro constant, German-language literature may refer to both constants by the same name, distinguished only by the units of measurement. Accurate determinations of the Avogadro constant require the measurement of a single quantity on both the atomic and macroscopic scales using the same unit of measurement.
This became possible for the first time when American physicist Robert Millikan measured the charge on an electron in 1910. The electric charge per mole of electrons is a constant called the Faraday constant and had been known since 1834 when Michael Faraday published his works on electrolysis. By dividing the charge on a mole of electrons by the charge on a single electron the value of the Avogadro number is obtained. Since 1910, newer calculations have more determined the values for the Faraday constant and the elementary charge. Perrin proposed the name Avogadro's number to refer to the number of molecules in one gram-molecule of oxygen, this term is still used in introductory works; the change in name to Avogadro constant came with the introduction of the mole as a base unit in the International System of Units in 1971, which regarded amount of substance as an independent dimension of measurement. With this recognition, the Avogadro constant was no longer a pure number, but had a unit of measurement, the reciprocal mole.
While it is rare to use units of amount of substance other than the mole, the Avogadro constant can be expressed by pound-mole and ounce-mole. The current definition of the mole links it to the kilogram; the revised definition breaks that link by making a mole a specific number of entities of the substance in question. Previous definition: The mole is the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12. When the mole is used, the elementary entities must be specified and may be atoms, ions, other particles, or specified groups of such particles. 2019 definition: The mole, symbol mol, is the SI unit of amount of substance. One mole contains 6.02214076×1023 elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit mol−1 and is called the Avogadro number; the am
The metre or meter is the base unit of length in the International System of Units. The SI unit symbol is m; the metre is defined as the length of the path travelled by light in vacuum in 1/299 792 458 of a second. The metre was defined in 1793 as one ten-millionth of the distance from the equator to the North Pole – as a result the Earth's circumference is 40,000 km today. In 1799, it was redefined in terms of a prototype metre bar. In 1960, the metre was redefined in terms of a certain number of wavelengths of a certain emission line of krypton-86. In 1983, the current definition was adopted; the imperial inch is defined as 0.0254 metres. One metre is about 3 3⁄8 inches longer than a yard, i.e. about 39 3⁄8 inches. Metre is the standard spelling of the metric unit for length in nearly all English-speaking nations except the United States and the Philippines, which use meter. Other Germanic languages, such as German and the Scandinavian languages spell the word meter. Measuring devices are spelled "-meter" in all variants of English.
The suffix "-meter" has the same Greek origin as the unit of length. The etymological roots of metre can be traced to the Greek verb μετρέω and noun μέτρον, which were used for physical measurement, for poetic metre and by extension for moderation or avoiding extremism; this range of uses is found in Latin, French and other languages. The motto ΜΕΤΡΩ ΧΡΩ in the seal of the International Bureau of Weights and Measures, a saying of the Greek statesman and philosopher Pittacus of Mytilene and may be translated as "Use measure!", thus calls for both measurement and moderation. In 1668 the English cleric and philosopher John Wilkins proposed in an essay a decimal-based unit of length, the universal measure or standard based on a pendulum with a two-second period; the use of the seconds pendulum to define length had been suggested to the Royal Society in 1660 by Christopher Wren. Christiaan Huygens had observed that length to be 39.26 English inches. No official action was taken regarding these suggestions.
In 1670 Gabriel Mouton, Bishop of Lyon suggested a universal length standard with decimal multiples and divisions, to be based on a one-minute angle of the Earth's meridian arc or on a pendulum with a two-second period. In 1675, the Italian scientist Tito Livio Burattini, in his work Misura Universale, used the phrase metro cattolico, derived from the Greek μέτρον καθολικόν, to denote the standard unit of length derived from a pendulum; as a result of the French Revolution, the French Academy of Sciences charged a commission with determining a single scale for all measures. On 7 October 1790 that commission advised the adoption of a decimal system, on 19 March 1791 advised the adoption of the term mètre, a basic unit of length, which they defined as equal to one ten-millionth of the distance between the North Pole and the Equator. In 1793, the French National Convention adopted the proposal. In 1791, the French Academy of Sciences selected the meridional definition over the pendular definition because the force of gravity varies over the surface of the Earth, which affects the period of a pendulum.
To establish a universally accepted foundation for the definition of the metre, more accurate measurements of this meridian were needed. The French Academy of Sciences commissioned an expedition led by Jean Baptiste Joseph Delambre and Pierre Méchain, lasting from 1792 to 1799, which attempted to measure the distance between a belfry in Dunkerque and Montjuïc castle in Barcelona to estimate the length of the meridian arc through Dunkerque; this portion of the meridian, assumed to be the same length as the Paris meridian, was to serve as the basis for the length of the half meridian connecting the North Pole with the Equator. The problem with this approach is that the exact shape of the Earth is not a simple mathematical shape, such as a sphere or oblate spheroid, at the level of precision required for defining a standard of length; the irregular and particular shape of the Earth smoothed to sea level is represented by a mathematical model called a geoid, which means "Earth-shaped". Despite these issues, in 1793 France adopted this definition of the metre as its official unit of length based on provisional results from this expedition.
However, it was determined that the first prototype metre bar was short by about 200 micrometres because of miscalculation of the flattening of the Earth, making the prototype about 0.02% shorter than the original proposed definition of the metre. Regardless, this length became the French standard and was progressively adopted by other countries in Europe; the expedition was fictionalised in Le mètre du Monde. Ken Alder wrote factually about the expedition in The Measure of All Things: the seven year odyssey and hidden error that transformed the world. In 1867 at the second general conference of the International Association of Geodesy held in Berlin, the question of an international standard unit of length was discussed in order to combine the measurements made in different countries to determine the size and shape of the Earth; the conference recommended the adoption of the metre and the creation of an internatio
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