Continuous or discrete variable: Difference between revisions

Revision as of 18:39, 8 December 2023

Revision as of 19:59, 8 December 2023

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{{Short description|Types of quantitative variables in mathematics}}

{{distinguish|Discrete-time and continuous-time variables}}

{{more references|date=November 2015}}

{{Probability fundamentals}}

In [[mathematics]] and [[statistics]], a quantitative [[variable (mathematics)|variable]] may be '''continuous''' or '''discrete''' if they are typically obtained by ''measuring'' or ''[[counting]]'', respectively.<ref>{{cite journal |last1=Ali |first1=Zulfiqar |last2=Bhaskar |first2=S. Bala |title=Basic statistical tools in research and data analysis |journal=Indian Journal of Anaesthesia |date=September 2016 |volume=60 |issue=9 |page=662–669 |doi=10.4103/0019-5049.190623|doi-access=free}}</ref> If it can take on two particular [[real number|real]] values such that it can also take on all real values between them (including values that are arbitrarily or infinitesimally close together), the variable is continuous in that [[Interval (mathematics)|interval]].<ref>{{cite journal |last1=Kaliyadan |first1=Feroze |last2=Kulkarni |first2=Vinay |title=Types of Variables, Descriptive Statistics, and Sample Size |journal=Indian Dermatology Online Journal |date=January 2019 |volume=10 |issue=1 |pages=82–86 |doi=10.4103/idoj.IDOJ_468_18 |pmid=30775310 |pmc=6362742 |doi-access=free }}</ref> If it can take on a value such that there is a non-[[infinitesimal]] gap on each side of it containing no values that the variable can take on, then it is discrete around that value.<ref>K.D. Joshi, ''Foundations of Discrete Mathematics'', 1989, New Age International Limited, [https://books.google.com/books?id=RM1D3mFw2u0C&dq=continuous+discrete+variable+math&pg=PA7], page 7.</ref> In some contexts a variable can be discrete in some ranges of the [[number line]] and continuous in others.

[[File:Continuous and discrete variables.png|thumb|upright=1.3|right|Variables can be divided into two main categories: qualitative (categorical) and quantitative (numerical). Continuous and discrete variables are subcategories of quantitative variables. Note that this schematic is not exhaustive in terms of the types of variables.]]

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Methods of [[calculus]] are often used in problems in which the variables are continuous, for example in continuous [[optimization]] problems.<ref>{{Cite book |last1=Griva |first1=Igor |url=https://www.worldcat.org/oclc/236082842 |title=Linear and nonlinear optimization |last2=Nash |first2=Stephen |last3=Sofer |first3=Ariela|author3-link= Ariela Sofer |publisher=Society for Industrial and Applied Mathematics |year=2009 |isbn=978-0-89871-661-0 |edition=2nd |___location=Philadelphia |pages=7 |language=en |oclc=236082842}}</ref>

In [[statistics|statistical theory]], the [[probability distribution]]s of continuous variables can be expressed in terms of [[probability density function]]s. <ref name="Springer Texts in Statistics">{{Citecite journal |last1=Dekking |first1=Frederik Michel |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |title=A Modern Introduction to Probability and Statistics |url=https://doi.org/10.1007/1-84628-168-7 |journal=Springer Texts in Statistics |language=en |doi=10.1007/1-84628-168-7 |isbn=978-1-85233-896-1 |issn=1431-875X}}</ref>

In [[continuous time|continuous-time]] [[dynamical system|dynamics]], the variable ''time'' is treated as continuous, and the equation describing the evolution of some variable over time is a [[differential equation]].<ref>{{cite journal |last1=Poyton |first1=A. A. |last2=Varziri |first2=Mohammad Saeed |last3=McAuley |first3=Kimberley B. |last4=MclellanPat James |first4=Pat James |last5=Ramsay |first5=James O. |title=Parameter estimation in continuous-time dynamic models using principal differential analysis |journal=Computers & Chemical Engineering |date=February 15, 2006 |volume=30 |issue=4 |page=698-708 |doi=10.1016/j.compchemeng.2005.11.008}}</ref> The [[instantaneous rate of change]] is a well-defined concept that takes the ratio of the change in the dependent variable to the independent variable at a specific instant.

==Discrete variable==

In contrast, a variable is a '''discrete variable''' if and only if there exists a one-to-one correspondence between this variable and <math>\mathbb{N}</math>, the set of [[natural numbers]].<ref>{{cite book |last1=Odifreddi |first1=Piergiorgio |title=Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers |date=February 18, 1992 |publisher=North Holland Publishing Company |isbn=978-0444894830 |page=18}}</ref> In other words; a discrete variable over a particular interval of real values is one for which, for any value in the range that the variable is permitted to take on, there is a positive minimum distance to the nearest other permissible value. The value of a discrete variable can be obtained by counting, and the number of permitted values is either finite or [[countably infinite]]. Common examples are variables that must be [[Integer|integers]], non-negative integers, positive integers, or only the integers 0 and 1.<ref>{{cite book |last1=van Douwen |first1=Eric |title=Handbook of Set-Theoretic Topology |date=1984 |publisher=Elsevier |___location=North Holland |isbn=978-0-444-86580-9 |pages=113-167}}</ref>

Methods of calculus do not readily lend themselves to problems involving discrete variables. Especially in multivariable calculus, many models rely on the assumption of continuity.<ref>{{cite book |last1=Clogg |first1=Clifford C. |last2=Shockey |first2=James W. |title=Handbook of Multivariate Experimental Psychology |date=1988 |publisher=Springer Publishing Company |___location=Boston, Massachusetts |isbn=978-1-4613-0893-5 |page=337-365}}</ref> Examples of problems involving discrete variables include [[integer programming]].