Continuous Data
Statistics & ProbabilityContinuous 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.
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.
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.
Definition
Continuous data takes any value in an interval (or union of intervals) on the real number line. Because the number of possible values is uncountable, individual values have probability zero; probabilities are calculated over intervals. Histograms are the standard display for continuous data.
Example
The time a runner takes to finish a race could be $27.3$ s, $27.31$ s, or $27.314$ s. The sample space is the interval $[0, \infty)$. The data is displayed using a histogram, and models include the normal or exponential distribution.
Key Insight
Continuous random variables are described by probability density functions (PDFs). The probability of a range of values equals the area under the PDF curve over that range.
Definition
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: Normal, Exponential, Uniform, Beta, Gamma, Chi-squared. The CDF $F(x) = P(X \le x)$ is absolutely continuous.
Example
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 $0.6827$. This value underlies the empirical rule.
Key Insight
The relationship between discrete and continuous distributions (e.g., Binomial converging to Normal by CLT, Geometric to Exponential in the limit) shows deep connections across the probability landscape.