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Statistical data type
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<h2 id="simple-data-types">Simple data types</h2> <p>The following table classifies the various simple data types, associated distributions, permissible operations, etc. Regardless of the logical possible values, all of these data types are generally coded using <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, because the theory of <a href="/facts/Random_variable/TwTBXnLT">random variables</a> often explicitly assumes that they hold real numbers. </p> <table><tbody><tr><th>Data Type</th><th>Possible values</th><th>Example usage</th><th>Level ofmeasurement</th><th>Common<p>Distributions</p></th><th>Scale ofrelativedifferences</th><th>Permissible statistics</th><th>Common model</th></tr><tr><th><a href="/facts/Binary_variable/PSFa5S2Q">binary</a></th><td>0, 1 (arbitrary labels)</td><td>binary outcome ("yes/no", "true/false", "success/failure", etc.)</td><td rowspan="2"><a href="/facts/Nominal_scale/8GAxPxFv">nominal scale</a></td><td><a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli</a></td><td rowspan="2"><a href="/facts/Comparability/PvLeJXol">incomparable</a></td><td rowspan="2"><a href="/facts/Mode_(statistics)/vq5tGp0G">mode</a>, <a href="/facts/Chi-squared_test/EbjRpLJP">chi-squared</a></td><td><a href="/facts/Logistic_regression/mGj6mc8y">logistic</a>, <a href="/facts/Probit_regression/k7hCvtYj">probit</a></td></tr><tr><th><a href="/facts/Categorical_variable/3UQNkLqs">categorical</a></th><td>"name1", "name2", "name3", ... "nameK" (arbitrary labels)</td><td>categorical outcome with names or places like "Rome", "Amsterdam", "Madrid", "London", "Washington" (specific <a href="/facts/Blood_type/zdxFTV6F">blood type</a>, <a href="/facts/Political_party/IAKw7oEK">political party</a>, word, etc.)</td><td><a href="/facts/Categorical_distribution/xpc6RWM2">categorical</a></td><td><a href="/facts/Multinomial_logit/aeF5lqDE">multinomial logit</a>, <a href="/facts/Multinomial_probit/HxwRDglc">multinomial probit</a></td></tr><tr><th><a href="/facts/Ordinal_variable/kgLhAaCn">ordinal</a></th><td>ordering categories or <a href="/facts/Integer/PUwwrawx">integer</a> or <a href="/facts/Real_number/R02gw5Pb">real number</a> (arbitrary scale)</td><td>Ordering adverbs like "Small", "Medium", "Large", relative score, significant only for creating a ranking</td><td><a href="/facts/Ordinal_scale/kgLhAaCn">ordinal scale</a></td><td><a href="/facts/Categorical_distribution/xpc6RWM2">categorical</a></td><td>relativecomparison</td><td></td><td><a href="/facts/Ordinal_regression/2oWy4EYh">ordinal regression</a> (<a href="/facts/Ordered_logit/tVm2X3Qd">ordered logit</a>, <a href="/facts/Ordered_probit/mwTHy4U2">ordered probit</a>)</td></tr><tr><th><a href="/facts/Binomial_variable/UMoFMjDj">binomial</a></th><td>0, 1, ..., N</td><td>number of successes (e.g. yes votes) out of <i>N</i> possible</td><td><a href="/facts/Interval_scale/8GAxPxFv">interval scale</a></td><td><a href="/facts/Binomial_distribution/UMoFMjDj">binomial</a>, <a href="/facts/Beta-binomial_distribution/N8n5PpHv">beta-binomial</a></td><td>additive</td><td><a href="/facts/Mean/swcEd4Pg">mean</a>, <a href="/facts/Median/YwXGWTyr">median</a>, <a href="/facts/Mode_(statistics)/vq5tGp0G">mode</a>, <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a>, <a href="/facts/Correlation/egkluAEm">correlation</a></td><td><a href="/facts/Binomial_regression/PKoTBPja">binomial regression</a> (<a href="/facts/Logistic_regression/mGj6mc8y">logistic</a>, <a href="/facts/Probit_regression/k7hCvtYj">probit</a>)</td></tr><tr><th><a href="/facts/Count_variable/vPP2VEyk">count</a></th><td>nonnegative <a href="/facts/Integer/PUwwrawx">integers</a> (0, 1, ...)</td><td>number of items (<a href="/facts/Telephone_call/4wUtUXpY">telephone calls</a>, people, <a href="/facts/Molecules/N9ljSfFq">molecules</a>, births, deaths, etc.) in given interval/area/volume</td><td><a href="/facts/Ratio_scale/VFXU56H9">ratio scale</a></td><td><a href="/facts/Poisson_distribution/CvCzXkHr">Poisson</a>, <a href="/facts/Negative_binomial_distribution/znzj365Y">negative binomial</a></td><td>multiplicative</td><td>All statistics permitted for interval scales plus the following: <a href="/facts/Geometric_mean/6Nf6uk1A">geometric mean</a>, <a href="/facts/Harmonic_mean/ejNdKrvn">harmonic mean</a>, <a href="/facts/Coefficient_of_variation/V8ddKkz4">coefficient of variation</a></td><td><a href="/facts/Poisson_regression/XWn6AW7A">Poisson</a>, negative binomial regression</td></tr><tr><th><a href="/facts/Real-valued/mGYQu2xt">real-valued</a>additive</th><td><a href="/facts/Real_number/R02gw5Pb">real number</a></td><td>temperature in degree Celsius or degree Fahrenheit, relative distance, <a href="/facts/Location_parameter/UQTyS3JU">location parameter</a>, etc. (or approximately, anything not varying over a large scale)</td><td><a href="/facts/Interval_scale/8GAxPxFv">interval scale</a></td><td><a href="/facts/Normal_distribution/UapjjPyQ">normal</a>, etc. (usually symmetric about the <a href="/facts/Mean/swcEd4Pg">mean</a>)</td><td>additive</td><td><a href="/facts/Mean/swcEd4Pg">mean</a>, <a href="/facts/Median/YwXGWTyr">median</a>, <a href="/facts/Mode_(statistics)/vq5tGp0G">mode</a>, <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a>, <a href="/facts/Correlation/egkluAEm">correlation</a></td><td>standard <a href="/facts/Linear_regression/5n998IhK">linear regression</a></td></tr><tr><th><a href="/facts/Real-valued/mGYQu2xt">real-valued</a>multiplicative</th><td>positive <a href="/facts/Real_number/R02gw5Pb">real number</a></td><td>temperature in <a href="/facts/Kelvin/E3trzC4C">kelvin</a>, price, income, size, <a href="/facts/Scale_parameter/6lxcyKUf">scale parameter</a>, etc. (especially when varying over a large scale)</td><td><a href="/facts/Ratio_scale/VFXU56H9">ratio scale</a></td><td><a href="/facts/Log-normal_distribution/V2RXmmlE">log-normal</a>, <a href="/facts/Gamma_distribution/lczcdJmw">gamma</a>, <a href="/facts/Exponential_distribution/yY3EdV0u">exponential</a>, etc. (usually a <a href="/facts/Skewed/rdS8luFa">skewed</a> distribution)</td><td>multiplicative</td><td>All statistics permitted for interval scales plus the following: <a href="/facts/Geometric_mean/6Nf6uk1A">geometric mean</a>, <a href="/facts/Harmonic_mean/ejNdKrvn">harmonic mean</a>, <a href="/facts/Coefficient_of_variation/V8ddKkz4">coefficient of variation</a></td><td><a href="/facts/Generalized_linear_model/Sf1htm5W">generalized linear model</a> with <a href="/facts/Logarithm/QeXaV97q">logarithmic</a> link</td></tr></tbody></table> <h2 id="multivariate-data-types">Multivariate data types</h2> <p>Data that cannot be described using a single number are often shoehorned into <a href="/facts/Random_vector/qMfooyVf">random vectors</a> of real-valued <a href="/facts/Random_variable/TwTBXnLT">random variables</a>, although there is an increasing tendency to treat them on their own. Some examples: </p> <ul><li><a href="/facts/Random_vector/qMfooyVf">Random vectors</a>. The individual elements may or may not be <a href="/facts/Correlated/egkluAEm">correlated</a>. Examples of distributions used to describe correlated random vectors are the <a href="/facts/Multivariate_normal_distribution/2Xfegqz2">multivariate normal distribution</a> and <a href="/facts/Multivariate_t-distribution/WspgrIac">multivariate t-distribution</a>. In general, there may be arbitrary correlations between any elements and any others; however, this often becomes unmanageable above a certain size, requiring further restrictions on the correlated elements.</li> <li><a href="/facts/Random_matrix/Yqo1lwFW">Random matrices</a>. Random matrices can be laid out linearly and treated as random vectors; however, this may not be an efficient way of representing the correlations between different elements. Some probability distributions are specifically designed for random matrices, e.g. the <a href="/facts/Matrix_normal_distribution/TkP6C4eu">matrix normal distribution</a> and <a href="/facts/Wishart_distribution/MEcqTump">Wishart distribution</a>.</li> <li><a href="/facts/Random_sequence/z2uHfvK4">Random sequences</a>. These are sometimes considered to be the same as random vectors, but in other cases the term is applied specifically to cases where each random variable is only correlated with nearby variables (as in a <a href="/facts/Markov_model/Mlndun4Q">Markov model</a>). This is a particular case of a <a href="/facts/Bayes_network/Ry31vq6T">Bayes network</a> and often used for very long sequences, e.g. gene sequences or lengthy text documents. A number of models are specifically designed for such sequences, e.g. <a href="/facts/Hidden_Markov_model/ur1zTAhP">hidden Markov models</a>.</li> <li><a href="/facts/Random_process/Ng1n1dnB">Random processes</a>. These are similar to random sequences, but where the length of the sequence is indefinite or infinite and the elements in the sequence are processed one-by-one. This is often used for data that can be described as a <a href="/facts/Time_series/fSXPR817">time series</a>, e.g. the price of a stock on successive days. Random processes are also used to model values that vary continuously (e.g. the temperature at successive moments in time), rather than at discrete intervals.</li> <li><a href="/facts/Bayes_network/Ry31vq6T">Bayes networks</a>. These correspond to aggregates of random variables described using <a href="/facts/Graphical_model/XxfmKhmM">graphical models</a>, where individual random variables are linked in a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> structure with <a href="/facts/Conditional_distribution/0eGm3P9W">conditional distributions</a> relating variables to nearby variables. <ul><li><a href="/facts/Multilevel_model/1DQbaCEx">Multilevel models</a> are subclasses of Bayes networks that can be thought of as having multiple levels of <a href="/facts/Linear_regression/5n998IhK">linear regression</a>.</li> <li><a href="/facts/Random_tree/5T6dH57r">Random trees</a>. These are a subclass of Bayes network, where the variables are linked in a <a href="/facts/Tree_structure/R2vT5UIu">tree structure</a>. An example is the problem of <a href="/facts/Parsing/nRKB01S3">parsing</a> a sentence, when statistical parsing techniques are used, such as <a href="/facts/Probabilistic_context-free_grammar/84Ps7P9j">probabilistic context-free grammars</a> (PCFG's).</li></ul></li> <li><a href="/facts/Random_field/8N7Id48z">Random fields</a>. These represent the extension of <a href="/facts/Random_process/Ng1n1dnB">random processes</a> to multiple dimensions, and are common in <a href="/facts/Physics/a7aH2y08">physics</a>, where they are used in <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a> to describe properties such as <a href="/facts/Force/533Q1uzq">force</a> or <a href="/facts/Electric_field/tt4MooBs">electric field</a> that can vary continuously over three dimensions (or four dimensions, when time is included).</li></ul> <p>These concepts originate in various scientific fields and frequently overlap in usage. As a result, it is very often the case that multiple concepts could potentially be applied to the same problem. </p> <h2 id="comparison-to-programming-data-types">Comparison to programming data types</h2> <p>Most data types in statistics have comparable types in computer programming, and vice versa, as shown in the following table: </p> <table><tbody><tr><th>Statistics</th><th>Programming</th></tr><tr><td><a href="/facts/Real-valued/mGYQu2xt">real-valued</a> (<a href="/facts/Interval_scale/8GAxPxFv">interval scale</a>)</td><td rowspan="2"><a href="/facts/Floating-point/eIckahxe">floating-point</a></td></tr><tr><td><a href="/facts/Real-valued/mGYQu2xt">real-valued</a> (<a href="/facts/Ratio_scale/VFXU56H9">ratio scale</a>)</td></tr><tr><td><a href="/facts/Count_data/vPP2VEyk">count data</a> (usually non-negative)</td><td><a href="/facts/Integer_(computer_science)/zWabo5M9">integer</a></td></tr><tr><td><a href="/facts/Binary_data/PSFa5S2Q">binary data</a></td><td><a href="/facts/Boolean_data_type/0GWYLDoC">Boolean</a></td></tr><tr><td><a href="/facts/Categorical_data/3UQNkLqs">categorical data</a></td><td><a href="/facts/Enumerated_type/oQITgIYb">enumerated type</a></td></tr><tr><td><a href="/facts/Random_vector/qMfooyVf">random vector</a></td><td><a href="/facts/List_(abstract_data_type)/9tzhAH6M">list</a> or <a href="/facts/Array_data_type/gHJJ3XPw">array</a></td></tr><tr><td><a href="/facts/Random_matrix/Yqo1lwFW">random matrix</a></td><td>two-dimensional <a href="/facts/Array_data_type/gHJJ3XPw">array</a></td></tr><tr><td><a href="/facts/Random_tree/5T6dH57r">random tree</a></td><td><a href="/facts/Tree_(data_structure)/RcXz9noF">tree</a></td></tr></tbody></table> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Mosteller, F.; Tukey, J.W. (1977). Data analysis and regression. Addison-Wesley. ISBN 978-0-201-04854-4. <a href="978-0-201-04854-4" target="_blank">978-0-201-04854-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Nelder, J.A. (1990). "The knowledge needed to computerise the analysis and interpretation of statistical information". Expert systems and artificial intelligence: the need for information about data. London: Library Association. OCLC 27042489. <a href="/wiki/OCLC_(identifier)" target="_blank">/wiki/OCLC_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Chrisman, Nicholas R. (1998). "Rethinking Levels of Measurement for Cartography". Cartography and Geographic Information Science. 25 (4): 231–242. Bibcode:1998CGISy..25..231C. doi:10.1559/152304098782383043. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>van den Berg, G. (1991). Choosing an analysis method. Leiden: DSWO Press. ISBN 978-90-6695-062-7. <a href="978-90-6695-062-7" target="_blank">978-90-6695-062-7</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Hand, D.J. (2004). Measurement theory and practice: The world through quantification. Wiley. p. 82. ISBN 978-0-470-68567-9. <a href="978-0-470-68567-9" target="_blank">978-0-470-68567-9</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>