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<title>Randomness</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Randomness</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Random" redirects here. For a random Wikipedia article, see Special:Random. For information about Wikipedia's random article feature, see Wikipedia:Random. For other uses, see <a href="Random_(disambiguation)" class="mw-disambig" title="Random (disambiguation)">Random (disambiguation)</a>.</div>

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</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert" typeof="mw:File"></span></td></tr><tr><td class="sidebar-content">

<ul><li><a href="Probability" title="Probability">Probability</a>

<ul><li><a href="Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li>

<li><a href="Determinism" title="Determinism">Determinism</a>

<ul><li><a href="Deterministic_system" title="Deterministic system">System</a></li></ul></li>

<li><a href="Indeterminism" title="Indeterminism">Indeterminism</a></li>

</ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Probability_space" title="Probability space">Probability space</a></li>

<li><a href="Sample_space" title="Sample space">Sample space</a></li>

<li><a href="Event_(probability_theory)" title="Event (probability theory)">Event</a>

<ul><li><a href="Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li>

<li><a href="Elementary_event" title="Elementary event">Elementary event</a></li>

<li><a href="Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li>

<li><a href="Outcome_(probability)" title="Outcome (probability)">Outcome</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li>

<li><a href="Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a>

<ul><li><a href="Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a>

<ul><li><a href="Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li>

<li><a href="Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li>

<li><a href="Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li>

<li><a href="Normal_distribution" title="Normal distribution">Normal distribution</a></li>

<li><a href="Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li>

<li><a href="Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li>

<li><a href="Probability_measure" title="Probability measure">Probability measure</a></li>

<li><a href="Random_variable" title="Random variable">Random variable</a>

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li>

<li><a href="Expected_value" title="Expected value">Expected value</a></li>

<li><a href="Variance" title="Variance">Variance</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Realization_(probability)" title="Realization (probability)">Observed value</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a></li>

<li><a href="Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Complementary_event" title="Complementary event">Complementary event</a></li>

<li><a href="Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li>

<li><a href="Marginal_distribution" title="Marginal distribution">Marginal probability</a></li>

<li><a href="Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li>

<li><a href="Conditional_independence" title="Conditional independence">Conditional independence</a></li>

<li><a href="Law_of_total_probability" title="Law of total probability">Law of total probability</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li>

<li><a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a></li>

<li><a href="Boole's_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li>

<li><a href="Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td>

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<p>In common usage, <b>randomness</b> is the apparent or actual lack of definite <a href="Pattern" title="Pattern">pattern</a> or <a href="Predictability" title="Predictability">predictability</a> in information.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A random sequence of events, <a href="Symbol" title="Symbol">symbols</a> or steps often has no <a href="https://en.wiktionary.org/wiki/order" class="extiw external" title="wikt:order">order</a> and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if there is a known <a href="Probability_distribution" title="Probability distribution">probability distribution</a>, the frequency of different outcomes over repeated events (or "trials") is predictable.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> For example, when throwing two <a href="Dice" title="Dice">dice</a>, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, <a href="Probability" title="Probability">probability</a>, and <a href="Information_entropy" class="mw-redirect" title="Information entropy">information entropy</a>.

</p><p>The fields of mathematics, probability, and statistics use formal definitions of randomness, typically assuming that there is some 'objective' probability distribution. In statistics, a <a href="Random_variable" title="Random variable">random variable</a> is an assignment of a numerical value to each possible outcome of an <a href="Event_space" class="mw-redirect" title="Event space">event space</a>. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in <a href="Random_sequence" title="Random sequence">random sequences</a>. A <a href="Random_process" class="mw-redirect" title="Random process">random process</a> is a sequence of random variables whose outcomes do not follow a <a href="Determinism" title="Determinism">deterministic</a> pattern, but follow an evolution described by <a href="Probability_distribution" title="Probability distribution">probability distributions</a>. These and other constructs are extremely useful in <a href="Probability_theory" title="Probability theory">probability theory</a> and the various <a href="Applications_of_randomness" title="Applications of randomness">applications of randomness</a>.

</p><p>Randomness is most often used in <a href="Statistics" title="Statistics">statistics</a> to signify well-defined statistical properties. <a href="Monte_Carlo_method" title="Monte Carlo method">Monte Carlo methods</a>, which rely on random input (such as from <a href="Random_number_generation" title="Random number generation">random number generators</a> or <a href="Pseudorandom_number_generator" title="Pseudorandom number generator">pseudorandom number generators</a>), are important techniques in science, particularly in the field of <a href="Scientific_computing" class="mw-redirect" title="Scientific computing">computational science</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> By analogy, <a href="Quasi-Monte_Carlo_method" title="Quasi-Monte Carlo method">quasi-Monte Carlo methods</a> use <a href="Low-discrepancy_sequence" title="Low-discrepancy sequence">quasi-random number generators</a>.

</p><p>Random selection, when narrowly associated with a <a href="Simple_random_sample" title="Simple random sample">simple random sample</a>, is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. A random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, say research subjects, has the same probability of being chosen, then we can say the selection process is random.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>According to <a href="Ramsey_theory" title="Ramsey theory">Ramsey theory</a>, pure randomness (in the sense of there being no discernible pattern) is impossible, especially for large structures. Mathematician <a href="Theodore_Motzkin" title="Theodore Motzkin">Theodore Motzkin</a> suggested that "while disorder is more probable in general, complete disorder is impossible".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Misunderstanding this can lead to numerous <a href="Conspiracy_theory" title="Conspiracy theory">conspiracy theories</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="Cristian_S._Calude" class="mw-redirect" title="Cristian S. Calude">Cristian S. Calude</a> stated that "given the impossibility of true randomness, the effort is directed towards studying degrees of randomness".<sup id="cite_ref-calude2017_7-0" class="reference"><a href="#cite_note-calude2017-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> It can be proven that there is infinite hierarchy (in terms of quality or strength) of forms of randomness.<sup id="cite_ref-calude2017_7-1" class="reference"><a href="#cite_note-calude2017-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="History_of_randomness" title="History of randomness">History of randomness</a></div>

<p>In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw <a href="Dice" title="Dice">dice</a> to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of <a href="Divination" title="Divination">divination</a> to attempt to circumvent randomness and fate.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Beyond <a href="Religion" title="Religion">religion</a> and <a href="Game_of_chance" title="Game of chance">games of chance</a>, randomness has been attested for <a href="Sortition" title="Sortition">sortition</a> since at least ancient <a href="Athenian_democracy" title="Athenian democracy">Athenian democracy</a> in the form of a <a href="Kleroterion" title="Kleroterion">kleroterion</a>.<sup id="cite_ref-Hansen_10-0" class="reference"><a href="#cite_note-Hansen-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p><p>The formalization of odds and chance was perhaps earliest done by the Chinese of 3,000 years ago. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of <a href="Calculus" title="Calculus">calculus</a> had a positive impact on the formal study of randomness. In the 1888 edition of his book <i>The Logic of Chance</i>, <a href="John_Venn" title="John Venn">John Venn</a> wrote a chapter on <i>The conception of randomness</i> that included his view of the randomness of the digits of <a href="Pi" title="Pi">pi</a> (π), by using them to construct a <a href="Random_walk" title="Random walk">random walk</a> in two dimensions.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p><p>The early part of the 20th century saw a rapid growth in the formal analysis of randomness, as various approaches to the mathematical foundations of probability were introduced. In the mid-to-late-20th century, ideas of <a href="Algorithmic_information_theory" title="Algorithmic information theory">algorithmic information theory</a> introduced new dimensions to the field via the concept of <a href="Algorithmic_randomness" class="mw-redirect" title="Algorithmic randomness">algorithmic randomness</a>.

</p><p>Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the 20th century computer scientists began to realize that the <i>deliberate</i> introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases, such <a href="Randomized_algorithms" class="mw-redirect" title="Randomized algorithms">randomized algorithms</a> even outperform the best deterministic methods.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="In_science">In science</h2></div>

<p>Many scientific fields are concerned with randomness:

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<ul><li><a href="Algorithmic_probability" title="Algorithmic probability">Algorithmic probability</a></li>

<li><a href="Chaos_theory" title="Chaos theory">Chaos theory</a></li>

<li><a href="Cryptography" title="Cryptography">Cryptography</a></li>

<li><a href="Game_theory" title="Game theory">Game theory</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Pattern_recognition" title="Pattern recognition">Pattern recognition</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation theory</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a></li>

<li><a href="Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li></ul>

</div>

<div class="mw-heading mw-heading3"><h3 id="In_the_physical_sciences">In the physical sciences</h3></div>

<p>In the 19th century, scientists used the idea of random motions of molecules in the development of <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> to explain phenomena in <a href="Thermodynamics" title="Thermodynamics">thermodynamics</a> and <a href="Gas_laws" title="Gas laws">the properties of gases</a>.

</p><p>According to several standard interpretations of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, microscopic phenomena are objectively random.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> That is, in an experiment that controls all causally relevant parameters, some aspects of the outcome still vary randomly. For example, if a single unstable <a href="Atom" title="Atom">atom</a> is placed in a controlled environment, it cannot be predicted how long it will take for the atom to decay—only the probability of decay in a given time.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Thus, quantum mechanics does not specify the outcome of individual experiments, but only the probabilities. <a href="Hidden_variable_theory" class="mw-redirect" title="Hidden variable theory">Hidden variable theories</a> reject the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are at work behind the scenes, determining the outcome in each case.

</p>

<div class="mw-heading mw-heading3"><h3 id="In_biology">In biology</h3></div>

<p>The <a href="Modern_synthesis_(20th_century)" title="Modern synthesis (20th century)">modern evolutionary synthesis</a> ascribes the observed diversity of life to random genetic <a href="Mutation" title="Mutation">mutations</a> followed by <a href="Natural_selection" title="Natural selection">natural selection</a>. The latter retains some random mutations in the <a href="Gene_pool" title="Gene pool">gene pool</a> due to the systematically improved chance for survival and reproduction that those mutated genes confer on individuals who possess them. The location of the mutation is not entirely random however as e.g. biologically important regions may be more protected from mutations.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>

</p><p>Several authors also claim that evolution (and sometimes development) requires a specific form of randomness, namely the introduction of qualitatively new behaviors. Instead of the choice of one possibility among several pre-given ones, this randomness corresponds to the formation of new possibilities.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>

</p><p>The characteristics of an organism arise to some extent deterministically (e.g., under the influence of genes and the environment), and to some extent randomly. For example, the <i>density</i> of <a href="Freckles" class="mw-redirect" title="Freckles">freckles</a> that appear on a person's skin is controlled by genes and exposure to light; whereas the exact location of <i>individual</i> freckles seems random.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>

</p><p>As far as behavior is concerned, randomness is important if an animal is to behave in a way that is unpredictable to others. For instance, insects in flight tend to move about with random changes in direction, making it difficult for pursuing predators to predict their trajectories.

</p>

<div class="mw-heading mw-heading3"><h3 id="In_mathematics">In mathematics</h3></div>

<p>The mathematical theory of <a href="Probability" title="Probability">probability</a> arose from attempts to formulate mathematical descriptions of chance events, originally in the context of <a href="Gambling" title="Gambling">gambling</a>, but later in connection with physics. <a href="Statistics" title="Statistics">Statistics</a> is used to infer an underlying <a href="Probability_distribution" title="Probability distribution">probability distribution</a> of a collection of empirical observations. For the purposes of <a href="Simulation" title="Simulation">simulation</a>, it is necessary to have a large supply of <a href="Random_sequence" title="Random sequence">random numbers</a>—or means to generate them on demand.

</p><p><a href="Algorithmic_information_theory" title="Algorithmic information theory">Algorithmic information theory</a> studies, among other topics, what constitutes a <a href="Random_sequence" title="Random sequence">random sequence</a>. The central idea is that a string of <a href="Bit" title="Bit">bits</a> is random if and only if it is shorter than any computer program that can produce that string (<a href="Kolmogorov_randomness" class="mw-redirect" title="Kolmogorov randomness">Kolmogorov randomness</a>), which means that random strings are those that cannot be <a href="Data_compression" title="Data compression">compressed</a>. Pioneers of this field include <a href="Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a> and his student <a href="Per_Martin-L%C3%B6f" title="Per Martin-Löf">Per Martin-Löf</a>, <a href="Ray_Solomonoff" title="Ray Solomonoff">Ray Solomonoff</a>, and <a href="Gregory_Chaitin" title="Gregory Chaitin">Gregory Chaitin</a>. For the notion of infinite sequence, mathematicians generally accept <a href="Per_Martin-L%C3%B6f" title="Per Martin-Löf">Per Martin-Löf</a>'s semi-eponymous definition: An infinite sequence is random if and only if it withstands all recursively enumerable null sets.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The other notions of random sequences include, among others, recursive randomness and Schnorr randomness, which are based on recursively computable martingales. It was shown by <a href="Yongge_Wang" title="Yongge Wang">Yongge Wang</a> that these randomness notions are generally different.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>

</p><p>Randomness occurs in numbers such as <a href="Binary_logarithm" title="Binary logarithm">log(2)</a> and <a href="Pi" title="Pi">pi</a>. The decimal digits of pi constitute an infinite sequence and "never repeat in a cyclical fashion." Numbers like pi are also considered likely to be <a href="Normal_number" title="Normal number">normal</a>:

</p>

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</style><blockquote class="templatequote"><p>Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></p></blockquote>

<div class="mw-heading mw-heading3"><h3 id="In_statistics">In statistics</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Statistical_randomness" title="Statistical randomness">Statistical randomness</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Random_error" class="mw-redirect" title="Random error">Random error</a></div>

<p>In statistics, randomness is commonly used to create <a href="Simple_random_samples" class="mw-redirect" title="Simple random samples">simple random samples</a>. This allows surveys of completely random groups of people to provide realistic data that is reflective of the population. Common methods of doing this include drawing names out of a hat or using a random digit chart (a large table of random digits).

</p>

<div class="mw-heading mw-heading3"><h3 id="In_information_science">In information science</h3></div>

<p>In information science, irrelevant or meaningless data is considered noise. Noise consists of numerous transient disturbances, with a statistically randomized time distribution.

</p><p>In <a href="Communication_theory" title="Communication theory">communication theory</a>, randomness in a signal is called "noise", and is opposed to that component of its variation that is causally attributable to the source, the signal.

</p><p>In terms of the development of random networks, for communication randomness rests on the two simple assumptions of <a href="Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> and <a href="Alfr%C3%A9d_R%C3%A9nyi" title="Alfréd Rényi">Alfréd Rényi</a>, who said that there were a fixed number of nodes and this number remained fixed for the life of the network, and that all nodes were equal and linked randomly to each other.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="In_finance">In finance</h3></div>

<p>The <a href="Random_walk_hypothesis" title="Random walk hypothesis">random walk hypothesis</a> considers that asset prices in an organized <a href="Market_(economics)" title="Market (economics)">market</a> evolve at random, in the sense that the expected value of their change is zero but the actual value may turn out to be positive or negative. More generally, asset prices are influenced by a variety of unpredictable events in the general economic environment.

</p>

<div class="mw-heading mw-heading3"><h3 id="In_politics">In politics</h3></div>

<p>Random selection can be an official method to resolve <a href="Tie_(draw)" title="Tie (draw)">tied</a> elections in some jurisdictions.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Its use in politics originates long ago. Many offices in <a href="Ancient_Athens" class="mw-redirect" title="Ancient Athens">ancient Athens</a> were chosen by lot instead of modern voting.

</p>

<div class="mw-heading mw-heading2"><h2 id="Randomness_and_religion">Randomness and religion</h2></div>

<p>Randomness can be seen as conflicting with the <a href="Deterministic" class="mw-redirect" title="Deterministic">deterministic</a> ideas of some religions, such as those where the universe is created by an omniscient deity who is aware of all past and future events. If the universe is regarded to have a purpose, then randomness can be seen as impossible. This is one of the rationales for religious opposition to <a href="Evolution" title="Evolution">evolution</a>, which states that <a href="Non-random" class="mw-redirect" title="Non-random">non-random</a> selection is applied to the results of random genetic variation.

</p><p><a href="Hindu" class="mw-redirect" title="Hindu">Hindu</a> and <a href="Buddhist" class="mw-redirect" title="Buddhist">Buddhist</a> philosophies state that any event is the result of previous events, as is reflected in the concept of <a href="Karma" title="Karma">karma</a>. As such, this conception is at odds with the idea of randomness, and any reconciliation between both of them would require an explanation.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>

</p><p>In some religious contexts, procedures that are commonly perceived as randomizers are used for divination. <a href="Cleromancy" title="Cleromancy">Cleromancy</a> uses the casting of bones or dice to reveal what is seen as the will of the gods.

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Applications_of_randomness" title="Applications of randomness">Applications of randomness</a></div>

<p>In most of its mathematical, political, social and religious uses, randomness is used for its innate "fairness" and lack of bias.

</p><p><b>Politics</b>: <a href="Athenian_democracy" title="Athenian democracy">Athenian democracy</a> was based on the concept of <a href="Isonomia" title="Isonomia">isonomia</a> (equality of political rights), and used complex allotment machines to ensure that the positions on the ruling committees that ran Athens were fairly allocated. <a href="Sortition" title="Sortition">Allotment</a> is now restricted to selecting jurors in Anglo-Saxon legal systems, and in situations where "fairness" is approximated by <a href="Randomization" title="Randomization">randomization</a>, such as selecting <a href="Juror" class="mw-redirect" title="Juror">jurors</a> and military <a href="Conscription" title="Conscription">draft</a> lotteries.

</p><p><b>Games</b>: Random numbers were first investigated in the context of <a href="Gambling" title="Gambling">gambling</a>, and many randomizing devices, such as <a href="Dice" title="Dice">dice</a>, <a href="Shuffling_playing_cards" class="mw-redirect" title="Shuffling playing cards">shuffling playing cards</a>, and <a href="Roulette" title="Roulette">roulette</a> wheels, were first developed for use in gambling. The ability to produce random numbers fairly is vital to electronic gambling, and, as such, the methods used to create them are usually regulated by government <a href="Gaming_Control_Board" class="mw-redirect" title="Gaming Control Board">Gaming Control Boards</a>. Random drawings are also used to determine <a href="Lottery" title="Lottery">lottery</a> winners. In fact, randomness has been used for games of chance throughout history, and to select out individuals for an unwanted task in a fair way (see <a href="Drawing_straws" class="mw-redirect" title="Drawing straws">drawing straws</a>).

</p><p><b>Sports</b>: Some sports, including <a href="American_football" title="American football">American football</a>, use <a href="Coin_toss" class="mw-redirect" title="Coin toss">coin tosses</a> to randomly select starting conditions for games or <a href="Seed_(sports)" class="mw-redirect" title="Seed (sports)">seed</a> tied teams for <a href="Playoffs" title="Playoffs">postseason play</a>. The <a href="National_Basketball_Association" title="National Basketball Association">National Basketball Association</a> uses a weighted <a href="NBA_draft_lottery" title="NBA draft lottery">lottery</a> to order teams in its draft.

</p><p><b>Mathematics</b>: Random numbers are also employed where their use is mathematically important, such as sampling for <a href="Opinion_poll" title="Opinion poll">opinion polls</a> and for statistical sampling in <a href="Quality_control" title="Quality control">quality control</a> systems. Computational solutions for some types of problems use random numbers extensively, such as in the <a href="Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a> and in <a href="Genetic_algorithm" title="Genetic algorithm">genetic algorithms</a>.

</p><p><b>Medicine</b>: Random allocation of a clinical intervention is used to reduce bias in controlled trials (e.g., <a href="Randomized_controlled_trials" class="mw-redirect" title="Randomized controlled trials">randomized controlled trials</a>).

</p><p><b>Religion</b>: Although not intended to be random, various forms of <a href="Divination" title="Divination">divination</a> such as <a href="Cleromancy" title="Cleromancy">cleromancy</a> see what appears to be a random event as a means for a divine being to communicate their will (see also <a href="Free_will" title="Free will">Free will</a> and <a href="Determinism" title="Determinism">Determinism</a> for more).

</p>

<div class="mw-heading mw-heading2"><h2 id="Generation">Generation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_number_generation" title="Random number generation">Random number generation</a></div>

<p>It is generally accepted that there exist three mechanisms responsible for (apparently) random behavior in systems:

</p>

<ol><li><i>Randomness</i> coming from the environment (for example, <a href="Brownian_motion" title="Brownian motion">Brownian motion</a>, but also <a href="Hardware_random_number_generator" title="Hardware random number generator">hardware random number generators</a>).</li>

<li><i>Randomness</i> coming from the initial conditions. This aspect is studied by <a href="Chaos_theory" title="Chaos theory">chaos theory</a>, and is observed in systems whose behavior is very sensitive to small variations in initial conditions (such as <a href="Pachinko" title="Pachinko">pachinko</a> machines and <a href="Dice" title="Dice">dice</a>).</li>

<li><i>Randomness</i> intrinsically generated by the system. This is also called <a href="Pseudorandomness" title="Pseudorandomness">pseudorandomness</a>, and is the kind used in <a href="Pseudo-random_number_generator" class="mw-redirect" title="Pseudo-random number generator">pseudo-random number generators</a>. There are many algorithms (based on <a href="Arithmetics" class="mw-redirect" title="Arithmetics">arithmetics</a> or <a href="Cellular_automaton" title="Cellular automaton">cellular automaton</a>) for generating pseudorandom numbers. The behavior of the system can be determined by knowing the <a href="Random_seed" title="Random seed">seed state</a> and the algorithm used. These methods are often quicker than getting "true" randomness from the environment.</li></ol>

<p>The many <a href="Applications_of_randomness" title="Applications of randomness">applications of randomness</a> have led to many different methods for generating random data. These methods may vary as to how unpredictable or <a href="Statistical_randomness" title="Statistical randomness">statistically random</a> they are, and how quickly they can generate random numbers.

</p><p>Before the advent of computational <a href="Random_number_generator" class="mw-redirect" title="Random number generator">random number generators</a>, generating large amounts of sufficiently random numbers (which is important in statistics) required a lot of work. Results would sometimes be collected and distributed as <a href="Random_number_table" title="Random number table">random number tables</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Measures_and_tests">Measures and tests</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Randomness_tests" class="mw-redirect" title="Randomness tests">Randomness tests</a></div>

<p>There are many practical measures of randomness for a binary sequence. These include measures based on frequency, <a href="Discrete_transform" title="Discrete transform">discrete transforms</a>, <a href="Complexity" title="Complexity">complexity</a>, or a mixture of these, such as the tests by Kak, Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Quantum_nonlocality" title="Quantum nonlocality">Quantum nonlocality</a> has been used to certify the presence of genuine or strong form of randomness in a given string of numbers.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Misconceptions_and_logical_fallacies">Misconceptions and logical fallacies</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Gambler's_fallacy" title="Gambler's fallacy">Gambler's fallacy</a></div>

<p>Popular perceptions of randomness are frequently mistaken, and are often based on fallacious reasoning or intuitions.

</p>

<div class="mw-heading mw-heading3"><h3 id="Fallacy:_a_number_is_"due"">Fallacy: a number is "due"</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Coupon_collector's_problem" title="Coupon collector's problem">Coupon collector's problem</a></div>

<p>This argument is, "In a random selection of numbers, since all numbers eventually appear, those that have not come up yet are 'due', and thus more likely to come up soon." This logic is only correct if applied to a system where numbers that come up are removed from the system, such as when <a href="Playing_card" title="Playing card">playing cards</a> are drawn and not returned to the deck. In this case, once a jack is removed from the deck, the next draw is less likely to be a jack and more likely to be some other card. However, if the jack is returned to the deck, and the deck is thoroughly reshuffled, a jack is as likely to be drawn as any other card. The same applies in any other process where objects are selected independently, and none are removed after each event, such as the roll of a die, a coin toss, or most <a href="Lottery" title="Lottery">lottery</a> number selection schemes. Truly random processes such as these do not have memory, which makes it impossible for past outcomes to affect future outcomes. In fact, there is no finite number of trials that can guarantee a success.

</p>

<div class="mw-heading mw-heading3"><h3 id="Fallacy:_a_number_is_"cursed"_or_"blessed"">Fallacy: a number is "cursed" or "blessed"</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Benford's_law" title="Benford's law">Benford's law</a></div>

<p>In a random sequence of numbers, a number may be said to be cursed because it has come up less often in the past, and so it is thought that it will occur less often in the future. A number may be assumed to be blessed because it has occurred more often than others in the past, and so it is thought likely to come up more often in the future. This logic is valid only if the randomisation might be biased, for example if a die is suspected to be loaded then its failure to roll enough sixes would be evidence of that loading. If the die is known to be fair, then previous rolls can give no indication of future events.

</p><p>In nature, events rarely occur with a frequency that is known <i><a href="A_priori" class="mw-redirect" title="A priori">a priori</a></i>, so observing outcomes to determine which events are more probable makes sense. However, it is fallacious to apply this logic to systems designed and known to make all outcomes equally likely, such as shuffled cards, dice, and roulette wheels.

</p>

<div class="mw-heading mw-heading3"><h3 id="Fallacy:_odds_are_never_dynamic">Fallacy: odds are never dynamic</h3></div>

<p>In the beginning of a scenario, one might calculate the probability of a certain event. However, as soon as one gains more information about the scenario, one may need to re-calculate the probability accordingly.

</p>

<p>For example, when being told that a woman has two children, one might be interested in knowing if either of them is a girl, and if yes, the probability that the other child is also a girl. Considering the two events independently, one might expect that the probability that the other child is female is ½ (50%), but by building a <a href="Probability_space" title="Probability space">probability space</a> illustrating all possible outcomes, one would notice that the probability is actually only ⅓ (33%).

</p><p>To be sure, the probability space does illustrate four ways of having these two children: boy-boy, girl-boy, boy-girl, and girl-girl. But once it is known that at least one of the children is female, this rules out the boy-boy scenario, leaving only three ways of having the two children: boy-girl, girl-boy, girl-girl. From this, it can be seen only ⅓ of these scenarios would have the other child also be a girl<sup id="cite_ref-NYOdds_29-0" class="reference"><a href="#cite_note-NYOdds-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> (see <a href="Boy_or_girl_paradox" title="Boy or girl paradox">Boy or girl paradox</a> for more).

</p><p>In general, by using a probability space, one is less likely to miss out on possible scenarios, or to neglect the importance of new information. This technique can be used to provide insights in other situations such as the <a href="Monty_Hall_problem" title="Monty Hall problem">Monty Hall problem</a>, a game show scenario in which a car is hidden behind one of three doors, and two goats are hidden as <a href="Booby_prize" title="Booby prize">booby prizes</a> behind the others. Once the contestant has chosen a door, the host opens one of the remaining doors to reveal a goat, eliminating that door as an option. With only two doors left (one with the car, the other with another goat), the player must decide to either keep their decision, or to switch and select the other door. Intuitively, one might think the player is choosing between two doors with equal probability, and that the opportunity to choose another door makes no difference. However, an analysis of the probability spaces would reveal that the contestant has received new information, and that changing to the other door would increase their chances of winning.<sup id="cite_ref-NYOdds_29-1" class="reference"><a href="#cite_note-NYOdds-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>

</p>

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<ul><li><a href="Chaitin's_constant" title="Chaitin's constant">Chaitin's constant</a></li>

<li><a href="Chance_(disambiguation)" class="mw-redirect mw-disambig" title="Chance (disambiguation)">Chance (disambiguation)</a></li>

<li><a href="Frequentist_probability" title="Frequentist probability">Frequentist probability</a></li>

<li><a href="Indeterminism" title="Indeterminism">Indeterminism</a></li>

<li><a href="Nonlinear_system" title="Nonlinear system">Nonlinear system</a></li>

<li><a href="Probability_interpretations" title="Probability interpretations">Probability interpretations</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Pseudorandomness" title="Pseudorandomness">Pseudorandomness</a></li>

<li><a href="Random.org" title="Random.org">Random.org</a>—generates random numbers using atmospheric noise</li>

<li><a href="Sortition" title="Sortition">Sortition</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Strictly speaking, the frequency of an outcome will converge <a href="Almost_surely" title="Almost surely">almost surely</a> to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has <a href="Probability" title="Probability">probability</a> zero. Consistent non-convergence is thus evidence of the lack of a fixed probability distribution, as in many evolutionary processes.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The <i><a href="Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a></i> defines "random" as "Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice; haphazard."</span>

</li>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Ted.com, (May 2016). <a rel="nofollow" class="external text" href="https://www.ted.com/talks/patrickjmt_the_origin_of_countless_conspiracy_theories/transcript">The origin of countless conspiracy theories</a></span>

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<li id="cite_note-calude2017-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-calude2017_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-calude2017_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="Cristian_S._Calude" class="mw-redirect" title="Cristian S. Calude">Cristian S. Calude</a>, (2017). <a rel="nofollow" class="external text" href="https://www.cs.auckland.ac.nz/~cristian/crispapers/QR_TP.pdf">"Quantum Randomness: From Practice to Theory and Back"</a> in "The Incomputable Journeys Beyond the Turing Barrier" Editors: <a href="S._Barry_Cooper" title="S. Barry Cooper">S. Barry Cooper</a>, Mariya I. Soskova, 169–181, doi:10.1007/978-3-319-43669-2_11.</span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><i>Handbook to life in ancient Rome</i> by Lesley Adkins 1998 <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-512332-8</bdi> page 279</span>

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<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">"Each nucleus decays spontaneously, at random, in accordance with the blind workings of chance." <i>Q for Quantum</i>, <a href="John_Gribbin" title="John Gribbin">John Gribbin</a></span>

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<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite id="CITEREFBelfieldDingJamiesonVisscher2018" class="citation journal cs1">Belfield, Eric J.; Ding, Zhong Jie; Jamieson, Fiona J.C.; Visscher, Anne M.; Zheng, Shao Jian; Mithani, Aziz; Harberd, Nicholas P. (January 2018). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5749183">"DNA mismatch repair preferentially protects genes from mutation"</a>. <i>Genome Research</i>. <b>28</b> (1): <span class="nowrap">66–</span>74. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1101%2Fgr.219303.116">10.1101/gr.219303.116</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5749183">5749183</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/29233924">29233924</a>.</cite></span>

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<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite id="CITEREFLongoMontévil2013" class="citation journal cs1">Longo, Giuseppe; Montévil, Maël (1 October 2013). <a rel="nofollow" class="external text" href="https://www.academia.edu/11720575">"Extended criticality, phase spaces and enablement in biology"</a>. <i>Chaos, Solitons & Fractals</i>. Emergent Critical Brain Dynamics. <b>55</b>: <span class="nowrap">64–</span>79. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013CSF....55...64L">2013CSF....55...64L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.chaos.2013.03.008">10.1016/j.chaos.2013.03.008</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:55589891">55589891</a>.</cite></span>

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<li id="cite_note-NYOdds-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-NYOdds_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-NYOdds_29-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFJohnson2008" class="citation news cs1">Johnson, George (8 June 2008). <a rel="nofollow" class="external text" href="https://www.nytimes.com/2008/06/08/books/review/Johnson-G-t.html?_r=1">"Playing the Odds"</a>. <i>The New York Times</i>.</cite></span>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><i>Randomness</i> by <a href="Deborah_J._Bennett" title="Deborah J. Bennett">Deborah J. Bennett</a>. Harvard University Press, 1998. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-674-10745-4</bdi>.</li>

<li><i>Random Measures, 4th ed.</i> by <a href="Olav_Kallenberg" title="Olav Kallenberg">Olav Kallenberg</a>. Academic Press, New York, London; Akademie-Verlag, Berlin, 1986. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0854102">0854102</a>.</li>

<li><i>The Art of Computer Programming. Vol. 2: Seminumerical Algorithms, 3rd ed.</i> by <a href="Donald_Knuth" title="Donald Knuth">Donald E. Knuth</a>. Reading, MA: Addison-Wesley, 1997. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-89684-2</bdi>.</li>

<li><i><a href="Fooled_by_Randomness" title="Fooled by Randomness">Fooled by Randomness</a>, 2nd ed.</i> by <a href="Nassim_Nicholas_Taleb" title="Nassim Nicholas Taleb">Nassim Nicholas Taleb</a>. Thomson Texere, 2004. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58799-190-X</bdi>.</li>

<li><i>Exploring Randomness</i> by <a href="Gregory_Chaitin" title="Gregory Chaitin">Gregory Chaitin</a>. Springer-Verlag London, 2001. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-85233-417-7</bdi>.</li>

<li><i>Random</i> by Kenneth Chan includes a "Random Scale" for grading the level of randomness.</li>

<li><i>The Drunkard’s Walk: How Randomness Rules our Lives</i> by <a href="Leonard_Mlodinow" title="Leonard Mlodinow">Leonard Mlodinow</a>. Pantheon Books, New York, 2008. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-375-42404-5</bdi>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Random" class="extiw external" title="v:Random">Random</a></b></i></div></div>

</div>

<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/randomness" class="extiw external" title="wiktionary:randomness">randomness</a></b></i> in Wiktionary, the free dictionary.</div></div>

</div>

<div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Randomness" class="extiw external" title="q:Special:Search/Randomness">Randomness</a></b></i>.</div></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Randomness" class="extiw external" title="commons:Category:Randomness">Randomness</a></span>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://www.quantumlab.de">QuantumLab</a> Quantum random number generator with single photons as interactive experiment.</li>

<li><a rel="nofollow" class="external text" href="http://www.fourmilab.ch/hotbits/">HotBits</a> generates random numbers from radioactive decay.</li>

<li><a rel="nofollow" class="external text" href="http://random.irb.hr">QRBG</a> Quantum Random Bit Generator</li>

<li><a rel="nofollow" class="external text" href="http://qrng.physik.hu-berlin.de/">QRNG</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101212025631/http://qrng.physik.hu-berlin.de/">Archived</a> 12 December 2010 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Fast Quantum Random Bit Generator</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070406063940/http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer.html">Chaitin: Randomness and Mathematical Proof</a></li>

<li><a rel="nofollow" class="external text" href="http://www.fourmilab.ch/random/">A Pseudorandom Number Sequence Test Program (Public Domain)</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20050404045817/http://etext.lib.virginia.edu/cgi-local/DHI/dhi.cgi?id=dv1-46"><i>Dictionary of the History of Ideas</i>:</a> Chance</li>

<li><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/chance-randomness/">Chance versus Randomness</a>, from the <a href="Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Chaos_theory160" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Chaos_theory160" style="font-size:114%;margin:0 4em"><a href="Chaos_theory" title="Chaos theory">Chaos theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Core84" scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Core</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Attractor" title="Attractor">Attractor</a></li>

<li><a href="Bifurcation_theory" title="Bifurcation theory">Bifurcation</a></li>

<li><a href="Fractal" title="Fractal">Fractal</a></li>

<li><a href="Limit_set" title="Limit set">Limit set</a></li>

<li><a href="Lyapunov_exponent" title="Lyapunov exponent">Lyapunov exponent</a></li>

<li><a href="Orbit_(dynamics)" title="Orbit (dynamics)">Orbit</a></li>

<li><a href="Periodic_point" title="Periodic point">Periodic point</a></li>

<li><a href="Phase_space" title="Phase space">Phase space</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anosov_diffeomorphism" title="Anosov diffeomorphism">Anosov diffeomorphism</a></li>

<li><a href="Arnold_tongue" title="Arnold tongue">Arnold tongue</a></li>

<li><a href="Axiom_A" title="Axiom A">axiom A dynamical system</a></li>

<li><a href="Bifurcation_diagram" title="Bifurcation diagram">Bifurcation diagram</a></li>

<li><a href="Box-counting_dimension" class="mw-redirect" title="Box-counting dimension">Box-counting dimension</a></li>

<li><a href="Correlation_dimension" title="Correlation dimension">Correlation dimension</a></li>

<li><a href="Conservative_system" title="Conservative system">Conservative system</a></li>

<li><a href="Ergodicity" title="Ergodicity">Ergodicity</a></li>

<li><a href="False_nearest_neighbors" class="mw-redirect" title="False nearest neighbors">False nearest neighbors</a></li>

<li><a href="Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a></li>

<li><a href="Invariant_measure" title="Invariant measure">Invariant measure</a></li>

<li><a href="Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li>

<li><a href="Measure-preserving_dynamical_system" title="Measure-preserving dynamical system">Measure-preserving dynamical system</a></li>

<li><a href="Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li>

<li><a href="Poincar%C3%A9_section" class="mw-redirect" title="Poincaré section">Poincaré section</a></li>

<li><a href="Recurrence_plot" title="Recurrence plot">Recurrence plot</a></li>

<li><a href="SRB_measure" class="mw-redirect" title="SRB measure">SRB measure</a></li>

<li><a href="Stable_manifold" title="Stable manifold">Stable manifold</a></li>

<li><a href="Topological_conjugacy" title="Topological conjugacy">Topological conjugacy</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Theorems</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ergodic_theory#Ergodic_theorems" title="Ergodic theory">Ergodic theorem</a></li>

<li><a href="Liouville's_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville's theorem</a></li>

<li><a href="Krylov%E2%80%93Bogolyubov_theorem" title="Krylov–Bogolyubov theorem">Krylov–Bogolyubov theorem</a></li>

<li><a href="Poincar%C3%A9%E2%80%93Bendixson_theorem" title="Poincaré–Bendixson theorem">Poincaré–Bendixson theorem</a></li>

<li><a href="Poincar%C3%A9_recurrence_theorem" title="Poincaré recurrence theorem">Poincaré recurrence theorem</a></li>

<li><a href="Stable_manifold_theorem" title="Stable manifold theorem">Stable manifold theorem</a></li>

<li><a href="Takens's_theorem" title="Takens's theorem">Takens's theorem</a></li></ul>

</div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span>

<p><br>

</p>

<span typeof="mw:File"></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theoretical<br>branches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bifurcation_theory" title="Bifurcation theory">Bifurcation theory</a></li>

<li><a href="Control_of_chaos" title="Control of chaos">Control of chaos</a></li>

<li><a href="Dynamical_system" title="Dynamical system">Dynamical system</a></li>

<li><a href="Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li>

<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>

<li><a href="Stability_theory" title="Stability theory">Stability theory</a></li>

<li><a href="Synchronization_of_chaos" title="Synchronization of chaos">Synchronization of chaos</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Chaotic<br>maps (<a href="List_of_chaotic_maps" title="List of chaotic maps">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Discrete</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arnold's_cat_map" title="Arnold's cat map">Arnold's cat map</a></li>

<li><a href="Baker's_map" title="Baker's map">Baker's map</a></li>

<li><a href="Complex_quadratic_polynomial" title="Complex quadratic polynomial">Complex quadratic map</a></li>

<li><a href="Coupled_map_lattice" title="Coupled map lattice">Coupled map lattice</a></li>

<li><a href="Duffing_map" title="Duffing map">Duffing map</a></li>

<li><a href="Dyadic_transformation" title="Dyadic transformation">Dyadic transformation</a></li>

<li><a href="Dynamical_billiards" title="Dynamical billiards">Dynamical billiards</a>

<ul><li><a href="Outer_billiards" title="Outer billiards">outer</a></li></ul></li>

<li><a href="Exponential_map_(discrete_dynamical_systems)" title="Exponential map (discrete dynamical systems)">Exponential map</a></li>

<li><a href="Gauss_iterated_map" title="Gauss iterated map">Gauss map</a></li>

<li><a href="Gingerbreadman_map" title="Gingerbreadman map">Gingerbreadman map</a></li>

<li><a href="H%C3%A9non_map" title="Hénon map">Hénon map</a></li>

<li><a href="Horseshoe_map" title="Horseshoe map">Horseshoe map</a></li>

<li><a href="Ikeda_map" title="Ikeda map">Ikeda map</a></li>

<li><a href="Interval_exchange_transformation" title="Interval exchange transformation">Interval exchange map</a></li>

<li><a href="Irrational_rotation" title="Irrational rotation">Irrational rotation</a></li>

<li><a href="Kaplan%E2%80%93Yorke_map" title="Kaplan–Yorke map">Kaplan–Yorke map</a></li>

<li><a href="Langton's_ant" title="Langton's ant">Langton's ant</a></li>

<li><a href="Logistic_map" title="Logistic map">Logistic map</a></li>

<li><a href="Standard_map" title="Standard map">Standard map</a></li>

<li><a href="Tinkerbell_map" title="Tinkerbell map">Tinkerbell map</a></li>

<li><a href="Zaslavskii_map" title="Zaslavskii map">Zaslavskii map</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Continuous</div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Double_scroll_attractor" class="mw-redirect" title="Double scroll attractor">Double scroll attractor</a></li>

<li><a href="Duffing_equation" title="Duffing equation">Duffing equation</a></li>

<li><a href="Lorenz_system" title="Lorenz system">Lorenz system</a></li>

<li><a href="Lotka%E2%80%93Volterra_equations" title="Lotka–Volterra equations">Lotka–Volterra equations</a></li>

<li><a href="Mackey%E2%80%93Glass_equations" title="Mackey–Glass equations">Mackey–Glass equations</a></li>

<li><a href="Rabinovich%E2%80%93Fabrikant_equations" title="Rabinovich–Fabrikant equations">Rabinovich–Fabrikant equations</a></li>

<li><a href="R%C3%B6ssler_attractor" title="Rössler attractor">Rössler attractor</a></li>

<li><a href="Three-body_problem" title="Three-body problem">Three-body problem</a></li>

<li><a href="Van_der_Pol_oscillator" title="Van der Pol oscillator">Van der Pol oscillator</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical<br>systems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chua's_circuit" title="Chua's circuit">Chua's circuit</a></li>

<li><a href="Rayleigh%E2%80%93B%C3%A9nard_convection" title="Rayleigh–Bénard convection">Convection</a></li>

<li><a href="Double_pendulum" title="Double pendulum">Double pendulum</a></li>

<li><a href="Elastic_pendulum" title="Elastic pendulum">Elastic pendulum</a></li>

<li><a href="Fermi%E2%80%93Pasta%E2%80%93Ulam%E2%80%93Tsingou_problem" title="Fermi–Pasta–Ulam–Tsingou problem">FPUT problem</a></li>

<li><a href="H%C3%A9non%E2%80%93Heiles_system" title="Hénon–Heiles system">Hénon–Heiles system</a></li>

<li><a href="Kicked_rotator" title="Kicked rotator">Kicked rotator</a></li>

<li><a href="Multiscroll_attractor" title="Multiscroll attractor">Multiscroll attractor</a></li>

<li><a href="Population_dynamics" title="Population dynamics">Population dynamics</a></li>

<li><a href="Swinging_Atwood's_machine" title="Swinging Atwood's machine">Swinging Atwood's machine</a></li>

<li><a href="Tilt-A-Whirl" title="Tilt-A-Whirl">Tilt-A-Whirl</a></li>

<li><a href="Weather_forecasting" title="Weather forecasting">Weather</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Chaos<br>theorists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Michael_Berry_(physicist)" title="Michael Berry (physicist)">Michael Berry</a></li>

<li><a href="Rufus_Bowen" title="Rufus Bowen">Rufus Bowen</a></li>

<li><a href="Mary_Cartwright" title="Mary Cartwright">Mary Cartwright</a></li>

<li><a href="Chen_Guanrong" title="Chen Guanrong">Chen Guanrong</a></li>

<li><a href="Mitchell_Feigenbaum" title="Mitchell Feigenbaum">Mitchell Feigenbaum</a></li>

<li><a href="Peter_Grassberger" title="Peter Grassberger">Peter Grassberger</a></li>

<li><a href="Celso_Grebogi" title="Celso Grebogi">Celso Grebogi</a></li>

<li><a href="Martin_Gutzwiller" title="Martin Gutzwiller">Martin Gutzwiller</a></li>

<li><a href="Brosl_Hasslacher" title="Brosl Hasslacher">Brosl Hasslacher</a></li>

<li><a href="Michel_H%C3%A9non" title="Michel Hénon">Michel Hénon</a></li>

<li><a href="Svetlana_Jitomirskaya" title="Svetlana Jitomirskaya">Svetlana Jitomirskaya</a></li>

<li><a href="Bryna_Kra" title="Bryna Kra">Bryna Kra</a></li>

<li><a href="Edward_Norton_Lorenz" title="Edward Norton Lorenz">Edward Norton Lorenz</a></li>

<li><a href="Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a></li>

<li><a href="Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoît Mandelbrot</a></li>

<li><a href="Edward_Ott" title="Edward Ott">Edward Ott</a></li>

<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></li>

<li><a href="Itamar_Procaccia" title="Itamar Procaccia">Itamar Procaccia</a></li>

<li><a href="Otto_R%C3%B6ssler" title="Otto Rössler">Otto Rössler</a></li>

<li><a href="David_Ruelle" title="David Ruelle">David Ruelle</a></li>

<li><a href="Caroline_Series" title="Caroline Series">Caroline Series</a></li>

<li><a href="Yakov_Sinai" title="Yakov Sinai">Yakov Sinai</a></li>

<li><a href="Oleksandr_Mykolayovych_Sharkovsky" class="mw-redirect" title="Oleksandr Mykolayovych Sharkovsky">Oleksandr Mykolayovych Sharkovsky</a></li>

<li><a href="Nina_Snaith" title="Nina Snaith">Nina Snaith</a></li>

<li><a href="Floris_Takens" title="Floris Takens">Floris Takens</a></li>

<li><a href="Audrey_Terras" title="Audrey Terras">Audrey Terras</a></li>

<li><a href="Mary_Tsingou" title="Mary Tsingou">Mary Tsingou</a></li>

<li><a href="Marcelo_Viana" title="Marcelo Viana">Marcelo Viana</a></li>

<li><a href="Amie_Wilkinson" title="Amie Wilkinson">Amie Wilkinson</a></li>

<li><a href="James_A._Yorke" title="James A. Yorke">James A. Yorke</a></li>

<li><a href="Lai-Sang_Young" title="Lai-Sang Young">Lai-Sang Young</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br>articles</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Butterfly_effect" title="Butterfly effect">Butterfly effect</a></li>

<li><a href="Complexity" title="Complexity">Complexity</a></li>

<li><a href="Edge_of_chaos" title="Edge of chaos">Edge of chaos</a></li>

<li><a href="Predictability" title="Predictability">Predictability</a></li>

<li><a href="Santa_Fe_Institute" title="Santa Fe Institute">Santa Fe Institute</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Statistics654" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistics654" style="font-size:114%;margin:0 4em"><a href="Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Outline_of_statistics" title="Outline of statistics">Outline</a></li>

<li><a href="List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics654" style="font-size:114%;margin:0 4em"><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li>

<li><a href="Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li>

<li><a href="Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li>

<li><a href="Cubic_mean" title="Cubic mean">Cubic</a></li>

<li><a href="Generalized_mean" title="Generalized mean">Generalized/power</a></li>

<li><a href="Geometric_mean" title="Geometric mean">Geometric</a></li>

<li><a href="Harmonic_mean" title="Harmonic mean">Harmonic</a></li>

<li><a href="Heronian_mean" title="Heronian mean">Heronian</a></li>

<li><a href="Heinz_mean" title="Heinz mean">Heinz</a></li>

<li><a href="Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li>

<li><a href="Median" title="Median">Median</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li>

<li><a href="Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>

<li><a href="Interquartile_range" title="Interquartile range">Interquartile range</a></li>

<li><a href="Percentile" title="Percentile">Percentile</a></li>

<li><a href="Range_(statistics)" title="Range (statistics)">Range</a></li>

<li><a href="Standard_deviation" title="Standard deviation">Standard deviation</a></li>

<li><a href="Variance#Sample_variance" title="Variance">Variance</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Moment_(mathematics)" title="Moment (mathematics)">Moments</a>

<ul><li><a href="Kurtosis" title="Kurtosis">Kurtosis</a></li>

<li><a href="L-moment" title="L-moment">L-moments</a></li>

<li><a href="Skewness" title="Skewness">Skewness</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li>

<li><a href="Grouped_data" title="Grouped data">Grouped data</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>

<li><a href="Rank_correlation" title="Rank correlation">Rank correlation</a>

<ul><li><a href="Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li>

<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bar_chart" title="Bar chart">Bar chart</a></li>

<li><a href="Biplot" title="Biplot">Biplot</a></li>

<li><a href="Control_chart" title="Control chart">Control chart</a></li>

<li><a href="Correlogram" title="Correlogram">Correlogram</a></li>

<li><a href="Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>

<li><a href="Forest_plot" title="Forest plot">Forest plot</a></li>

<li><a href="Histogram" title="Histogram">Histogram</a></li>

<li><a href="Pie_chart" title="Pie chart">Pie chart</a></li>

<li><a href="Radar_chart" title="Radar chart">Radar chart</a></li>

<li><a href="Run_chart" title="Run chart">Run chart</a></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li>

<li><a href="Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>

<li><a href="Violin_plot" title="Violin plot">Violin plot</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection654" style="font-size:114%;margin:0 4em"><a href="Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Effect_size" title="Effect size">Effect size</a></li>

<li><a href="Missing_data" title="Missing data">Missing data</a></li>

<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li>

<li><a href="Statistical_population" title="Statistical population">Population</a></li>

<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication</a></li>

<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>

<ul><li><a href="Cluster_sampling" title="Cluster sampling">Cluster</a></li>

<li><a href="Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li>

<li><a href="Opinion_poll" title="Opinion poll">Opinion poll</a></li>

<li><a href="Questionnaire" title="Questionnaire">Questionnaire</a></li>

<li><a href="Standard_error" title="Standard error">Standard error</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>

<li><a href="Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li>

<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>

<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>

<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>

<li><a href="Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>

<li><a href="Scientific_control" title="Scientific control">Scientific control</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li>

<li><a href="Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li>

<li><a href="Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohort_study" title="Cohort study">Cohort study</a></li>

<li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>

<li><a href="Natural_experiment" title="Natural experiment">Natural experiment</a></li>

<li><a href="Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference654" style="font-size:114%;margin:0 4em"><a href="Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a></li>

<li><a href="Sampling_distribution" title="Sampling distribution">Sampling distribution</a>

<ul><li><a href="Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>

<li><a href="Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>

<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>

<li><a href="Statistical_model" title="Statistical model">Statistical model</a>

<ul><li><a href="Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>

<li><a href="Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>

<li><a href="Statistical_parameter" title="Statistical parameter">Parameter</a>

<ul><li><a href="Location_parameter" title="Location parameter">location</a></li>

<li><a href="Scale_parameter" title="Scale parameter">scale</a></li>

<li><a href="Shape_parameter" title="Shape parameter">shape</a></li></ul></li>

<li><a href="Parametric_statistics" title="Parametric statistics">Parametric family</a>

<ul><li><a href="Likelihood_function" title="Likelihood function">Likelihood</a> <a href="Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size: 85%;">(monotone)</span></a></li>

<li><a href="Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>

<li><a href="Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>

<li><a href="Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>

<li><a href="Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>

<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Optimal_decision" title="Optimal decision">Optimal decision</a>

<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></li>

<li><a href="Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>

<li><a href="Statistical_distance" title="Statistical distance">Statistical distance</a>

<ul><li><a href="Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>

<li><a href="Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>

<li><a href="Robust_statistics" title="Robust statistics">Robustness</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Estimating_equations" title="Estimating equations">Estimating equations</a>

<ul><li><a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>

<li><a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>

<li><a href="M-estimator" title="M-estimator">M-estimator</a></li>

<li><a href="Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>

<li><a href="Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>

<ul><li><a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>

<ul><li><a href="Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>

<li><a href="Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>

<li><a href="Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Confidence_interval" title="Confidence interval">Confidence interval</a></li>

<li><a href="Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>

<li><a href="Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>

<li><a href="Prediction_interval" title="Prediction interval">Prediction interval</a></li>

<li><a href="Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>

<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li>

<li><a href="Power_(statistics)" title="Power (statistics)">Power</a>

<ul><li><a href="Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>

<li><a href="Permutation_test" title="Permutation test">Permutation test</a>

<ul><li><a href="Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>

<li><a href="Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>

<li><a href="Score_test" title="Score test">Score/Lagrange multiplier</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Z-test" title="Z-test"><i>Z</i>-test <span style="font-size: 85%;">(normal)</span></a></li>

<li><a href="Student's_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>

<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>

<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>

<li><a href="Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>

<li><a href="Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>

<li><a href="Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size: 85%;">(Shapiro–Wilk)</span></a></li>

<li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>

<li><a href="Model_selection" title="Model selection">Model selection</a>

<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>

<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size: 85%;">(Wilcoxon)</span></a>

<ul><li><a href="Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>

<li><a href="Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size: 85%;">(Mann–Whitney)</span></a></li>

<li><a href="Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="Analysis_of_variance" title="Analysis of variance">anova</a>

<ul><li><a href="Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size: 85%;">(Kruskal–Wallis)</span></a></li>

<li><a href="Friedman_test" title="Friedman test">2-way <span style="font-size: 85%;">(Friedman)</span></a></li>

<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size: 85%;">(Jonckheere–Terpstra)</span></a></li></ul></li>

<li><a href="Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>

<ul><li><a href="Prior_probability" title="Prior probability">prior</a></li>

<li><a href="Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>

<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>

<li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a></li>

<li><a href="Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>

<ul><li><a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis654" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>

<li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Confounding" title="Confounding">Confounding variable</a></li>

<li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>

<li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>

<li><a href="Mixed_model" title="Mixed model">Mixed effects models</a></li>

<li><a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>

<li><a href="Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li>

<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>

<li><a href="General_linear_model" title="General linear model">General linear model</a></li>

<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>

<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>

<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>

<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>

<li><a href="Robust_regression" title="Robust regression">Robust</a></li>

<li><a href="Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Exponential_family" title="Exponential family">Exponential families</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic <span style="font-size: 85%;">(Bernoulli)</span></a> / <a href="Binomial_regression" title="Binomial regression">Binomial</a> / <a href="Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>

<li><a href="Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>

<li><a href="Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_multivariate_/_time-series_/_survival_analysis654" style="font-size:114%;margin:0 4em"><a href="Categorical_variable" title="Categorical variable">Categorical</a> / <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> / <a href="Time_series" title="Time series">time-series</a> / <a href="Survival_analysis" title="Survival analysis">survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a></li>

<li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Graphical_model" title="Graphical model">Graphical model</a></li>

<li><a href="Poisson_regression" title="Poisson regression">Log-linear model</a></li>

<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a></li>

<li><a href="Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="General_linear_model" title="General linear model">Regression</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>

<li><a href="Principal_component_analysis" title="Principal component analysis">Principal components</a></li>

<li><a href="Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>

<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>

<li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>

<li><a href="Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>

<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>

<li><a href="Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>

<ul><li><a href="Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>

<ul><li><a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>

<li><a href="Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationarity</a></li>

<li><a href="Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>

<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>

<li><a href="Cointegration" title="Cointegration">Cointegration</a></li>

<li><a href="Structural_break" title="Structural break">Structural break</a></li>

<li><a href="Granger_causality" title="Granger causality">Granger causality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>

<li><a href="Johansen_test" title="Johansen test">Johansen</a></li>

<li><a href="Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size: 85%;">(Ljung–Box)</span></a></li>

<li><a href="Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>

<li><a href="Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>

<ul><li><a href="Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>

<li><a href="Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li>

<li><a href="Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size: 85%;">(Box–Jenkins)</span></a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>

<li><a href="Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>

<li><a href="Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>

<li><a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>

<li><a href="Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>

<li><a href="First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications654" style="font-size:114%;margin:0 4em"><a href="List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>

<li><a href="Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="Clinical_study_design" title="Clinical study design">studies</a></li>

<li><a href="Epidemiology" title="Epidemiology">Epidemiology</a></li>

<li><a href="Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chemometrics" title="Chemometrics">Chemometrics</a></li>

<li><a href="Methods_engineering" title="Methods engineering">Methods engineering</a></li>

<li><a href="Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li>

<li><a href="Statistical_process_control" title="Statistical process control">Process</a> / <a href="Quality_control" title="Quality control">quality control</a></li>

<li><a href="Reliability_engineering" title="Reliability engineering">Reliability</a></li>

<li><a href="System_identification" title="System identification">System identification</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_science" title="Actuarial science">Actuarial science</a></li>

<li><a href="Census" title="Census">Census</a></li>

<li><a href="Crime_statistics" title="Crime statistics">Crime statistics</a></li>

<li><a href="Demographic_statistics" title="Demographic statistics">Demography</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Jurimetrics" title="Jurimetrics">Jurimetrics</a></li>

<li><a href="National_accounts" title="National accounts">National accounts</a></li>

<li><a href="Official_statistics" title="Official statistics">Official statistics</a></li>

<li><a href="Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li>

<li><a href="Psychometrics" title="Psychometrics">Psychometrics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cartography" title="Cartography">Cartography</a></li>

<li><a href="Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>

<li><a href="Geographic_information_system" title="Geographic information system">Geographic information system</a></li>

<li><a href="Geostatistics" title="Geostatistics">Geostatistics</a></li>

<li><a href="Kriging" title="Kriging">Kriging</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

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<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw external" title="commons:Category:Statistics">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>

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