A distribution describes the pattern of values that a variable takes — which values occur and how frequently. For a discrete variable, the distribution is specified by its probability mass function: P(X = x) for each possible value x. For a continuous variable, it is specified by a probability density function f(x), where the probability of X falling in an interval [a, b] is the integral of f from a to b.
Named distributions formalize common patterns. The normal (Gaussian) distribution is the bell curve, characterized by mean and standard deviation. The binomial distribution counts successes in independent trials. The Poisson distribution models rare events. The exponential distribution models waiting times. Each is a mathematical model that data may approximately follow.
A distribution is fully characterized by its cumulative distribution function F(x) = P(X ≤ x). Summary statistics — mean, median, mode, variance — capture specific features of the distribution but not the entire shape. The shape itself — symmetry, skewness, kurtosis, number of modes — conveys information that no single summary can.