Continuous Data
Continuous data can take any value within a range, including decimals and fractions, such as height or temperature.
Definition
Continuous data can be any value within a range, including decimals; it is measured, not counted, and can always be made more precise. It takes any value in an interval (or union of intervals) on the real number line, and because the number of possible values is uncountable, individual values have probability zero, so probabilities are calculated over intervals instead. Formally, a continuous random variable $X$ has a probability density function $f(x)$ such that $P(a \le X \le b) = \int_a^b f(x)\,dx$, with $f(x) \ge 0$ and total integral $=1$; key continuous distributions include Normal, Exponential, Uniform, Beta, Gamma, and Chi-squared, and the CDF $F(x) = P(X \le x)$ is absolutely continuous.
Example
A person's height might be $5.3$ feet, or more precisely $5.28$ feet, or even $5.279$ feet; temperature, weight, and time are all continuous. The time a runner takes to finish a race could be $27.3$ s, $27.31$ s, or $27.314$ s, with sample space the interval $[0, \infty)$, displayed using a histogram and modeled with the normal or exponential distribution. If $X \sim \text{Normal}(\mu=0, \sigma=1)$, then $P(-1 \le X \le 1) = \int_{-1}^{1} \frac{1}{\sqrt{2\pi}} e^{-x^2/2}\,dx$, approximately $0.6827$, the value underlying the empirical rule.
Key Insight
Continuous data lives on a number line with no gaps: you can always zoom in and find a value in between any two measurements, and continuous random variables are described by probability density functions (PDFs), where the probability of a range of values equals the area under the PDF curve over that range. The relationship between discrete and continuous distributions (e.g., Binomial converging to Normal by the CLT, Geometric to Exponential in the limit) shows deep connections across the probability landscape.