continousvariable

Continuous variable, sometimes misspelled as continousvariable, is a concept in statistics and probability referring to a quantitative variable that can take an uncountable set of values within an interval or range. It contrasts with discrete variables, which take only distinct, separate values.

In probability theory, a continuous random variable X has a probability distribution described by a probability density function f(x) such that for any a < b, P(a < X ≤ b) = ∫ from a to b of f(x) dx. The probability of X taking any specific value is zero. The cumulative distribution function F(x) = P(X ≤ x) is nondecreasing, right-continuous, and satisfies lim x→−∞ F(x) = 0 and lim x→∞ F(x) = 1.

Common continuous distributions include the normal, uniform, exponential, and lognormal. A density f(x) uniquely determines the distribution, and moments such as the mean and variance are given by integrals: E[X] = ∫ x f(x) dx, Var(X) = ∫ (x − E[X])^2 f(x) dx.

In measurement and data analysis, real-valued data are often modeled as continuous variables, even when measurement precision is finite; the continuous-variable model is an idealization that facilitates calculus-based methods such as regression, hypothesis testing, and Bayesian inference. Distinguishing continuous from discrete variables influences statistical methods: probabilities for continuous variables are assigned to intervals via densities, whereas discrete variables use probability mass functions for individual outcomes.