Normal Distribution PDF/CDF

Normal PDF: the Gaussian density

The normal (Gaussian) probability density function is the formula behind the familiar bell-shaped curve: f(x) = (1 / (σ√(2π)))·e^(−(x − μ)² / (2σ²)). Its peak lands on the mean μ, and how wide it spreads depends on the standard deviation σ. One thing trips people up: the PDF returns density rather than probability. To get an actual probability you need the area under the curve (the CDF), which has no elementary closed form and gets computed through the error function erf. Standardizing with Z = (X − μ)/σ turns any normal into the standard normal N(0, 1). There's also the handy 68-95-99.7 rule of thumb: roughly 68% of the mass sits within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. As an example, at μ = 0, σ = 1, x = 1.96, f(x) ≈ 0.0584.

Applications

You see it in biometric measurements like height, weight and IQ, in measurement error across physics and engineering, and in finance (Black-Scholes assumes log-normal returns). It shows up in statistical process control, where Six Sigma targets 3.4 defects per million, and in regression analysis, since OLS assumes normal residuals. The Central Limit Theorem is the deeper reason behind all of this: add up enough independent variables and the result drifts toward a normal distribution.

FAQ

Why can f(x) exceed 1? Because density isn't the same thing as probability. A small σ makes the curve tall and narrow, so f(x) can climb above 1 with no problem. What's constrained to equal 1 is the integral, the total area.

PDF vs CDF — what is the difference? The PDF tells you the density at a single point, while the CDF gives the accumulated probability P(X ≤ x). Put another way, the CDF is just the integral of the PDF.

When is the normal a bad fit? Whenever the data is skewed, like income or time-to-failure, or when the tails are heavy, as financial returns tend to be during crises. In those cases a log-normal, Student's t, or an extreme value distribution usually fits better.

What does the Central Limit Theorem say? Take the average of many independent samples from any distribution with finite variance, and that average converges to a normal. That convergence is the reason the Gaussian keeps turning up in practice.