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<title>Statistical classification</title>

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

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<p>When <a href="Classification" title="Classification">classification</a> is performed by a computer, statistical methods are normally used to develop the algorithm.

</p><p>Often, the individual observations are analyzed into a set of quantifiable properties, known variously as <a href="Explanatory_variables" class="mw-redirect" title="Explanatory variables">explanatory variables</a> or <i>features</i>. These properties may variously be <a href="Categorical_data" class="mw-redirect" title="Categorical data">categorical</a> (e.g. "A", "B", "AB" or "O", for <a href="Blood_type" title="Blood type">blood type</a>), <a href="Ordinal_data" title="Ordinal data">ordinal</a> (e.g. "large", "medium" or "small"), <a href="Integer" title="Integer">integer-valued</a> (e.g. the number of occurrences of a particular word in an <a href="Email" title="Email">email</a>) or <a href="Real_number" title="Real number">real-valued</a> (e.g. a measurement of <a href="Blood_pressure" title="Blood pressure">blood pressure</a>). Other classifiers work by comparing observations to previous observations by means of a <a href="Similarity_function" class="mw-redirect" title="Similarity function">similarity</a> or <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">distance</a> function.

</p><p>An <a href="Algorithm" title="Algorithm">algorithm</a> that implements classification, especially in a concrete implementation, is known as a <b>classifier</b>. The term "classifier" sometimes also refers to the mathematical <a href="Function_(mathematics)" title="Function (mathematics)">function</a>, implemented by a classification algorithm, that maps input data to a category.

</p><p>Terminology across fields is quite varied. In <a href="Statistics" title="Statistics">statistics</a>, where classification is often done with <a href="Logistic_regression" title="Logistic regression">logistic regression</a> or a similar procedure, the properties of observations are termed <a href="Explanatory_variable" class="mw-redirect" title="Explanatory variable">explanatory variables</a> (or <a href="Independent_variable" class="mw-redirect" title="Independent variable">independent variables</a>, regressors, etc.), and the categories to be predicted are known as outcomes, which are considered to be possible values of the <a href="Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variable</a>. In <a href="Machine_learning" title="Machine learning">machine learning</a>, the observations are often known as <i>instances</i>, the explanatory variables are termed <i>features</i> (grouped into a <a href="Feature_vector" class="mw-redirect" title="Feature vector">feature vector</a>), and the possible categories to be predicted are <i>classes</i>. Other fields may use different terminology: e.g. in <a href="Community_ecology" class="mw-redirect" title="Community ecology">community ecology</a>, the term "classification" normally refers to <a href="Cluster_analysis" title="Cluster analysis">cluster analysis</a>.

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<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_problems">Relation to other problems</h2></div>

<p><a href="Classification" title="Classification">Classification</a> and clustering are examples of the more general problem of <a href="Pattern_recognition" title="Pattern recognition">pattern recognition</a>, which is the assignment of some sort of output value to a given input value. Other examples are <a href="Regression_analysis" title="Regression analysis">regression</a>, which assigns a real-valued output to each input; <a href="Sequence_labeling" title="Sequence labeling">sequence labeling</a>, which assigns a class to each member of a sequence of values (for example, <a href="Part_of_speech_tagging" class="mw-redirect" title="Part of speech tagging">part of speech tagging</a>, which assigns a <a href="Part_of_speech" title="Part of speech">part of speech</a> to each word in an input sentence); <a href="Parsing" title="Parsing">parsing</a>, which assigns a <a href="Parse_tree" title="Parse tree">parse tree</a> to an input sentence, describing the <a href="Syntactic_structure" class="mw-redirect" title="Syntactic structure">syntactic structure</a> of the sentence; etc.

</p><p>A common subclass of classification is <a href="Probabilistic_classification" title="Probabilistic classification">probabilistic classification</a>. Algorithms of this nature use <a href="Statistical_inference" title="Statistical inference">statistical inference</a> to find the best class for a given instance. Unlike other algorithms, which simply output a "best" class, probabilistic algorithms output a <a href="Probability" title="Probability">probability</a> of the instance being a member of each of the possible classes. The best class is normally then selected as the one with the highest probability. However, such an algorithm has numerous advantages over non-probabilistic classifiers:

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<ul><li>It can output a confidence value associated with its choice (in general, a classifier that can do this is known as a <i>confidence-weighted classifier</i>).</li>

<li>Correspondingly, it can <i>abstain</i> when its confidence of choosing any particular output is too low.</li>

<li>Because of the probabilities which are generated, probabilistic classifiers can be more effectively incorporated into larger machine-learning tasks, in a way that partially or completely avoids the problem of <i>error propagation</i>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Frequentist_procedures">Frequentist procedures</h2></div>

<p>Early work on statistical classification was undertaken by <a href="Ronald_Fisher" title="Ronald Fisher">Fisher</a>,<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-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> in the context of two-group problems, leading to <a href="Fisher's_linear_discriminant" class="mw-redirect" title="Fisher's linear discriminant">Fisher's linear discriminant</a> function as the rule for assigning a group to a new observation.<sup id="cite_ref-G1977_3-0" class="reference"><a href="#cite_note-G1977-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This early work assumed that data-values within each of the two groups had a <a href="Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a>. The extension of this same context to more than two groups has also been considered with a restriction imposed that the classification rule should be <a href="Linear" class="mw-redirect" title="Linear">linear</a>.<sup id="cite_ref-G1977_3-1" class="reference"><a href="#cite_note-G1977-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Later work for the multivariate normal distribution allowed the classifier to be <a href="Nonlinear" class="mw-redirect" title="Nonlinear">nonlinear</a>:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> several classification rules can be derived based on different adjustments of the <a href="Mahalanobis_distance" title="Mahalanobis distance">Mahalanobis distance</a>, with a new observation being assigned to the group whose centre has the lowest adjusted distance from the observation.

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<div class="mw-heading mw-heading2"><h2 id="Bayesian_procedures">Bayesian procedures</h2></div>

<p>Unlike frequentist procedures, Bayesian classification procedures provide a natural way of taking into account any available information about the relative sizes of the different groups within the overall population.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Bayesian procedures tend to be computationally expensive and, in the days before <a href="Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> computations were developed, approximations for Bayesian clustering rules were devised.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>Some Bayesian procedures involve the calculation of <a href="Group-membership_probabilities" class="mw-redirect" title="Group-membership probabilities">group-membership probabilities</a>: these provide a more informative outcome than a simple attribution of a single group-label to each new observation.

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<div class="mw-heading mw-heading2"><h2 id="Binary_and_multiclass_classification">Binary and multiclass classification</h2></div>

<p>Classification can be thought of as two separate problems – <a href="Binary_classification" title="Binary classification">binary classification</a> and <a href="Multiclass_classification" title="Multiclass classification">multiclass classification</a>. In binary classification, a better understood task, only two classes are involved, whereas multiclass classification involves assigning an object to one of several classes.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Since many classification methods have been developed specifically for binary classification, multiclass classification often requires the combined use of multiple binary classifiers.

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<div class="mw-heading mw-heading2"><h2 id="Feature_vectors">Feature vectors</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Feature_vector" class="mw-redirect" title="Feature vector">Feature vector</a></div>

<p>Most algorithms describe an individual instance whose category is to be predicted using a <a href="Feature_vector" class="mw-redirect" title="Feature vector">feature vector</a> of individual, measurable properties of the instance. Each property is termed a <a href="Feature_(pattern_recognition)" class="mw-redirect" title="Feature (pattern recognition)">feature</a>, also known in statistics as an <a href="Explanatory_variable" class="mw-redirect" title="Explanatory variable">explanatory variable</a> (or <a href="Independent_variable" class="mw-redirect" title="Independent variable">independent variable</a>, although features may or may not be <a href="Statistically_independent" class="mw-redirect" title="Statistically independent">statistically independent</a>). Features may variously be <a href="Binary_data" title="Binary data">binary</a> (e.g. "on" or "off"); <a href="Categorical_data" class="mw-redirect" title="Categorical data">categorical</a> (e.g. "A", "B", "AB" or "O", for <a href="Blood_type" title="Blood type">blood type</a>); <a href="Ordinal_data" title="Ordinal data">ordinal</a> (e.g. "large", "medium" or "small"); <a href="Integer" title="Integer">integer-valued</a> (e.g. the number of occurrences of a particular word in an email); or <a href="Real_number" title="Real number">real-valued</a> (e.g. a measurement of blood pressure). If the instance is an image, the feature values might correspond to the pixels of an image; if the instance is a piece of text, the feature values might be occurrence frequencies of different words. Some algorithms work only in terms of discrete data and require that real-valued or integer-valued data be <i>discretized</i> into groups (e.g. less than 5, between 5 and 10, or greater than 10).

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<div class="mw-heading mw-heading2"><h2 id="Linear_classifiers">Linear classifiers</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Linear_classifier" title="Linear classifier">Linear classifier</a></div>

<p>A large number of <a href="Algorithm" title="Algorithm">algorithms</a> for classification can be phrased in terms of a <a href="Linear_function" title="Linear function">linear function</a> that assigns a score to each possible category <i>k</i> by <a href="Linear_combination" title="Linear combination">combining</a> the feature vector of an instance with a vector of weights, using a <a href="Dot_product" title="Dot product">dot product</a>. The predicted category is the one with the highest score. This type of score function is known as a <a href="Linear_predictor_function" title="Linear predictor function">linear predictor function</a> and has the following general form:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {score} (\mathbf {X} _{i},k)={\boldsymbol {\beta }}_{k}\cdot \mathbf {X} _{i},}</annotation>

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where <b>X</b><sub><i>i</i></sub> is the feature vector for instance <i>i</i>, <b>β</b><sub><i>k</i></sub> is the vector of weights corresponding to category <i>k</i>, and score(<b>X</b><sub><i>i</i></sub>, <i>k</i>) is the score associated with assigning instance <i>i</i> to category <i>k</i>. In <a href="Discrete_choice" title="Discrete choice">discrete choice</a> theory, where instances represent people and categories represent choices, the score is considered the <a href="Utility" title="Utility">utility</a> associated with person <i>i</i> choosing category <i>k</i>.

</p><p>Algorithms with this basic setup are known as <a href="Linear_classifier" title="Linear classifier">linear classifiers</a>. What distinguishes them is the procedure for determining (training) the optimal weights/coefficients and the way that the score is interpreted.

</p><p>Examples of such algorithms include

</p>

<ul><li><a href="Logistic_regression" title="Logistic regression">Logistic regression</a> – Statistical model for a binary dependent variable

<ul><li><a href="Multinomial_logistic_regression" title="Multinomial logistic regression">Multinomial logistic regression</a> – Regression for more than two discrete outcomes</li></ul></li>

<li><a href="Probit_regression" class="mw-redirect" title="Probit regression">Probit regression</a> – Statistical regression where the dependent variable can take only two values<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li>The <a href="Perceptron" title="Perceptron">perceptron</a> algorithm</li>

<li><a href="Support_vector_machine" title="Support vector machine">Support vector machine</a> – Set of methods for supervised statistical learning</li>

<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Linear discriminant analysis</a> – Method used in statistics, pattern recognition, and other fields</li></ul>

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

<p>Since no single form of classification is appropriate for all data sets, a large toolkit of classification algorithms has been developed. The most commonly used include:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

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<ul><li><a href="Artificial_neural_networks" class="mw-redirect" title="Artificial neural networks">Artificial neural networks</a> – Computational model used in machine learning, based on connected, hierarchical functions<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting (machine learning)</a> – Ensemble learning method</li>

<li><a href="Random_forest" title="Random forest">Random forest</a> – Tree-based ensemble machine learning method</li>

<li><a href="Genetic_programming" title="Genetic programming">Genetic programming</a> – Evolving computer programs with techniques analogous to natural genetic processes

<ul><li><a href="Gene_expression_programming" title="Gene expression programming">Gene expression programming</a> – Evolutionary algorithm</li>

<li><a href="Multi_expression_programming" title="Multi expression programming">Multi expression programming</a></li>

<li><a href="Linear_genetic_programming" title="Linear genetic programming">Linear genetic programming</a></li></ul></li>

<li><a href="Kernel_estimation" class="mw-redirect" title="Kernel estimation">Kernel estimation</a> – Window function<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span>

<ul><li><a href="K-nearest_neighbor_algorithm" class="mw-redirect" title="K-nearest neighbor algorithm">k-nearest neighbor</a> – Non-parametric classification method<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul></li>

<li><a href="Learning_vector_quantization" title="Learning vector quantization">Learning vector quantization</a></li>

<li><a href="Linear_classifier" title="Linear classifier">Linear classifier</a> – Statistical classification in machine learning

<ul><li><a href="Fisher's_linear_discriminant" class="mw-redirect" title="Fisher's linear discriminant">Fisher's linear discriminant</a> – Method used in statistics, pattern recognition, and other fields<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic regression</a> – Statistical model for a binary dependent variable</li>

<li><a href="Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes classifier</a> – Probabilistic classification algorithm</li>

<li><a href="Perceptron" title="Perceptron">Perceptron</a> – Algorithm for supervised learning of binary classifiers</li></ul></li>

<li><a href="Quadratic_classifier" title="Quadratic classifier">Quadratic classifier</a></li>

<li><a href="Support_vector_machine" title="Support vector machine">Support vector machine</a> – Set of methods for supervised statistical learning

<ul><li><a href="Least_squares_support_vector_machine" class="mw-redirect" title="Least squares support vector machine">Least squares support vector machine</a></li></ul></li></ul>

<p>Choices between different possible algorithms are frequently made on the basis of quantitative <a href="Classification#Evaluation_of_accuracy" title="Classification">evaluation of accuracy</a>.

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<div class="mw-heading mw-heading2"><h2 id="Application_domains">Application domains</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Cluster_analysis#Applications" title="Cluster analysis">Cluster analysis § Applications</a></div>

<p>Classification has many applications. In some of these, it is employed as a <a href="Data_mining" title="Data mining">data mining</a> procedure, while in others more detailed statistical modeling is undertaken.

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<ul><li><a href="Biological_classification" class="mw-redirect" title="Biological classification">Biological classification</a> – The science of identifying, describing, defining and naming groups of biological organisms</li>

<li><a href="Biometric" class="mw-redirect" title="Biometric">Biometric</a> – Metrics related to human characteristics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span> identification</li>

<li><a href="Computer_vision" title="Computer vision">Computer vision</a> – Computerized information extraction from images

<ul><li>Medical image analysis and <a href="Medical_imaging" title="Medical imaging">medical imaging</a> – Technique and process of creating visual representations of the interior of a body</li>

<li><a href="Optical_character_recognition" title="Optical character recognition">Optical character recognition</a> – Computer recognition of visual text</li>

<li><a href="Video_tracking" title="Video tracking">Video tracking</a> – Locating a moving object by analyzing frames of a video</li></ul></li>

<li><a href="Credit_scoring" class="mw-redirect" title="Credit scoring">Credit scoring</a> – Numerical expression representing a person's creditworthiness<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Document_classification" title="Document classification">Document classification</a> – Process of categorizing documents</li>

<li><a href="Drug_discovery" title="Drug discovery">Drug discovery</a> and <a href="Drug_development" title="Drug development">development</a> – Process of bringing a new pharmaceutical drug to the market

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

<li><a href="Quantitative_structure-activity_relationship" class="mw-redirect" title="Quantitative structure-activity relationship">Quantitative structure-activity relationship</a> – Predictive chemical model<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul></li>

<li><a href="Geostatistics" title="Geostatistics">Geostatistics</a> – Branch of statistics focusing on spatial data sets</li>

<li><a href="Handwriting_recognition" title="Handwriting recognition">Handwriting recognition</a> – Ability of a computer to receive and interpret intelligible handwritten input</li>

<li>Internet <a href="Search_engines" class="mw-redirect" title="Search engines">search engines</a></li>

<li>Micro-array classification</li>

<li><a href="Pattern_recognition" title="Pattern recognition">Pattern recognition</a> – Automated recognition of patterns and regularities in data</li>

<li><a href="Recommender_system" title="Recommender system">Recommender system</a> – System to predict users' preferences</li>

<li><a href="Speech_recognition" title="Speech recognition">Speech recognition</a> – Automatic conversion of spoken language into text</li>

<li><a href="Statistical_natural_language_processing" class="mw-redirect" title="Statistical natural language processing">Statistical natural language processing</a> – Processing of natural language by a computer<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul>

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<ul><li><a href="Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a> – Intelligence of machines</li>

<li><a href="Binary_classification" title="Binary classification">Binary classification</a> – Dividing things between two categories</li>

<li><a href="Multiclass_classification" title="Multiclass classification">Multiclass classification</a> – Problem in machine learning and statistical classification</li>

<li><a href="Class_membership_probabilities" class="mw-redirect" title="Class membership probabilities">Class membership probabilities</a> – Machine learning problem<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Classification_rule" title="Classification rule">Classification rule</a></li>

<li><a href="Compound_term_processing" class="mw-redirect" title="Compound term processing">Compound term processing</a></li>

<li><a href="Confusion_matrix" title="Confusion matrix">Confusion matrix</a> – Table layout for visualizing performance; also called an error matrix</li>

<li><a href="Data_mining" title="Data mining">Data mining</a> – Process of extracting and discovering patterns in large data sets</li>

<li><a href="Data_warehouse" title="Data warehouse">Data warehouse</a> – Centralized storage of knowledge</li>

<li><a href="Fuzzy_logic" title="Fuzzy logic">Fuzzy logic</a> – System for reasoning about vagueness</li>

<li><a href="Information_retrieval" title="Information retrieval">Information retrieval</a> – Obtaining information resources relevant to an information need</li>

<li><a href="List_of_datasets_for_machine_learning_research" class="mw-redirect" title="List of datasets for machine learning research">List of datasets for machine learning research</a></li>

<li><a href="Machine_learning" title="Machine learning">Machine learning</a> – Study of algorithms that improve automatically through experience</li>

<li><a href="Recommender_system" title="Recommender system">Recommender system</a> – System to predict users' preferences</li></ul>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></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:Statistical_classification" class="extiw external" title="commons:Category:Statistical classification">Statistical classification</a></span>.</div></div>

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</style><cite id="CITEREFFisher1936" class="citation journal cs1">Fisher, R. A. (1936). "The Use of Multiple Measurements in Taxonomic Problems". <i><a href="Annals_of_Eugenics" class="mw-redirect" title="Annals of Eugenics">Annals of Eugenics</a></i>. <b>7</b> (2): <span class="nowrap">179–</span>188. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1469-1809.1936.tb02137.x">10.1111/j.1469-1809.1936.tb02137.x</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2440%2F15227">2440/15227</a></span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFFisher1938" class="citation journal cs1">Fisher, R. A. (1938). "The Statistical Utilization of Multiple Measurements". <i><a href="Annals_of_Eugenics" class="mw-redirect" title="Annals of Eugenics">Annals of Eugenics</a></i>. <b>8</b> (4): <span class="nowrap">376–</span>386. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1469-1809.1938.tb02189.x">10.1111/j.1469-1809.1938.tb02189.x</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2440%2F15232">2440/15232</a></span>.</cite></span>

</li>

<li id="cite_note-G1977-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-G1977_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-G1977_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Gnanadesikan, R. (1977) <i>Methods for Statistical Data Analysis of Multivariate Observations</i>, Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-30845-5</bdi> (p. 83–86)</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="C._R._Rao" title="C. R. Rao">Rao, C.R.</a> (1952) <i>Advanced Statistical Methods in Multivariate Analysis</i>, Wiley. (Section 9c)</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="T._W._Anderson" class="mw-redirect" title="T. W. Anderson">Anderson, T.W.</a> (1958) <i>An Introduction to Multivariate Statistical Analysis</i>, Wiley.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFBinder1978" class="citation journal cs1">Binder, D. A. (1978). "Bayesian cluster analysis". <i><a href="Biometrika" title="Biometrika">Biometrika</a></i>. <b>65</b>: <span class="nowrap">31–</span>38. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F65.1.31">10.1093/biomet/65.1.31</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFBinder1981" class="citation journal cs1">Binder, David A. (1981). "Approximations to Bayesian clustering rules". <i><a href="Biometrika" title="Biometrika">Biometrika</a></i>. <b>68</b>: <span class="nowrap">275–</span>285. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F68.1.275">10.1093/biomet/68.1.275</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="Sariel_Har-Peled" title="Sariel Har-Peled">Har-Peled, S.</a>, Roth, D., Zimak, D. (2003) "Constraint Classification for Multiclass Classification and Ranking." In: Becker, B., <a href="Sebastian_Thrun" title="Sebastian Thrun">Thrun, S.</a>, Obermayer, K. (Eds) <i>Advances in Neural Information Processing Systems 15: Proceedings of the 2002 Conference</i>, MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-02550-7</bdi></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://builtin.com/data-science/tour-top-10-algorithms-machine-learning-newbies">"A Tour of The Top 10 Algorithms for Machine Learning Newbies"</a>. <i>Built In</i>. 2018-01-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-06-10</span></span>.</cite></span>

</li>

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

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</style><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 uncollapsed 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="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><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<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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