The bell curve whose averages inherit normality
A normal—or Gaussian—distribution is the continuous bell shaped by mean μ and variance σ². The central limit theorem explains why measurement errors and many sums look nearly normal. Linear mixes of independent normals stay normal, which makes analytic shortcuts possible—and misuse tempting.
Its density is proportional to exp(−(x−μ)²/(2σ²)), scaled by one over σ√(2π). The parameter μ is simultaneously mean, median, and mode; σ is the positive standard deviation. A random variable with this law is called normally distributed or a normal deviate. The standard normal has mean zero and variance one, with density often written φ(z) = e^(−z²/2)/√(2π). Any normal can be standardized by subtracting μ and dividing by σ. Authors also write X ~ N(μ, σ²); some prefer precision τ = 1/σ² as the width parameter. Carl Friedrich Gauss once used a different “standard” scaling, and Stephen Stigler proposed yet another with variance 1/(2π).
Normals matter partly because of the central limit theorem: averages of many independent observations from a law with finite mean and variance converge in distribution toward a normal as the sample size grows. Quantities that behave like sums of many independent influences—classic measurement error—therefore often look nearly Gaussian. Any fixed linear combination of independent normal deviates is again normal, which unlocks closed-form results for uncertainty propagation and least-squares fitting when normality holds.
Caveats keep the romance honest. Many other laws are also bell-shaped—Cauchy, Student’s t, logistic—so “bell curve” is informal nickname, not a synonym. Multivariate and matrix-normal extensions carry the idea to vectors and matrices. Most importantly, normals are frequently assumed where the data are not approximately Gaussian, and then the model is a poor fit. The mathematics is elegant; the modeling judgment still has to earn its keep.
Source: Normal distribution
