Part of a series onstatistics
Probability theory
Probability Axioms Determinism System Indeterminism Randomness
Probability space Sample space Event Collectively exhaustive events Elementary event Mutual exclusivity Outcome Singleton Experiment Bernoulli trial Probability distribution Bernoulli distribution Binomial distribution Exponential distribution Normal distribution Pareto distribution Poisson distribution Probability measure Random variable Bernoulli process Continuous or discrete Expected value Variance Markov chain Observed value Random walk Stochastic process
Complementary event Joint probability Marginal probability Conditional probability
Independence Conditional independence Law of total probability Law of large numbers Bayes' theorem Boole's inequality
Venn diagram Tree diagram
v t e

Inmathematics andstatistics, a quantitativevariable may becontinuous ordiscrete. If it can take on tworeal values and all the values between them, the variable is continuous in thatinterval.^[1] 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.^[2] In some contexts, a variable can be discrete in some ranges of thenumber line and continuous in others. In statistics, continuous and discrete variables are distinctstatistical data types which are described with differentprobability distributions.

Acontinuous variable is a variable such that there are possible values between any two values.

For example, a variable over a non-empty range of thereal numbers is continuous if it can take on any value in that range.^[3]

Methods ofcalculus are often used in problems in which the variables are continuous, for example in continuousoptimization problems.^[4]

Instatistical theory, theprobability distributions of continuous variables can be expressed in terms ofprobability density functions.^[5]

Incontinuous-timedynamics, the variabletime is treated as continuous, and the equation describing the evolution of some variable over time is adifferential equation.^[6] Theinstantaneous 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.

In contrast, a variable is adiscrete variable if and only if there exists a one-to-one correspondence between this variable and a subset of${\displaystyle \mathbb{N}}$, the set ofnatural numbers.^[7] 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 number of permitted values is either finite orcountably infinite. Common examples are variables that must beintegers, non-negative integers, positive integers, or only the integers 0 and 1.^[8]

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.^[9] Examples of problems involving discrete variables includeinteger programming.

In statistics, the probability distributions of discrete variables can be expressed in terms ofprobability mass functions.^[5]

Indiscrete time dynamics, the variabletime is treated as discrete, and the equation of evolution of some variable over time is called adifference equation.^[10] For certain discrete-time dynamical systems, the system response can be modelled by solving the difference equation for an analytical solution.

Ineconometrics and more generally inregression analysis, sometimes some of the variables beingempirically related to each other are 0-1 variables, being permitted to take on only those two values.^[11] The purpose of the discrete values of 0 and 1 is to use the dummy variable as a ‘switch’ that can ‘turn on’ and ‘turn off’ by assigning the two values to different parameters in an equation. A variable of this type is called adummy variable. If thedependent variable is a dummy variable, thenlogistic regression orprobit regression is commonly employed. In the case of regression analysis, a dummy variable can be used to represent subgroups of the sample in a study (e.g. the value 0 corresponding to a constituent of the control group).^[12]

A mixed multivariate model can contain both discrete and continuous variables. For instance, a simple mixed multivariate model could have a discrete variable${\displaystyle x}$, which only takes on values 0 or 1, and a continuous variable${\displaystyle y}$.^[13] An example of a mixed model could be a research study on the risk of psychological disorders based on one binary measure of psychiatric symptoms and one continuous measure of cognitive performance.^[14] Mixed models may also involve a single variable that is discrete over some range of the number line and continuous at another range.

In probability theory and statistics, the probability distribution of a mixed random variable consists of both discrete and continuous components. A mixed random variable does not have acumulative distribution function that is discrete or everywhere-continuous. An example of a mixed type random variable is the probability of wait time in a queue. The likelihood of a customer experiencing a zero wait time is discrete, while non-zero wait times are evaluated on a continuous time scale.^[15] In physics (particularly quantum mechanics, where this sort of distribution often arises),Dirac delta functions are often used to treat continuous and discrete components in a unified manner. For example, the previous example might be described by a probability density${\displaystyle p(t)=\alpha \delta (t)+g(t)}$, such that${\displaystyle P(t>0)={\int }_0^{\mathrm{\infty }}g(t)=1-\alpha }$, and${\displaystyle P(t=0)=\alpha }$.

• Mathematical terminology
• Statistical data types

• Articles with short description
• Short description is different from Wikidata