Normal Probability Density

This is the bell curve itself. The exponent measures how far x sits from the mean in standard deviations and squares it, so height falls away fast; the constant 1/(σ√(2π)) is exactly what is needed to make the total area equal 1. At the peak, x = μ, the density is 0.3989/σ — for the standard normal with σ = 1 that is 0.39894, the number at the top of every z-table's density column. One standard deviation out it has already dropped to 0.2420. Abraham de Moivre derived the curve in 1733 as an approximation to binomial coin-flip probabilities; Gauss and Laplace later made it the law of measurement error, which is why it carries Gauss's name.

The trap is calling f(x) a probability. It is a density — a height, not an area — and for a continuous variable the probability of any exact value is zero. Probabilities come from areas under the curve between two points, which is what the z-table's cumulative column gives you. A density can even exceed 1: set σ = 0.1 and the peak height is 3.99, perfectly legal because the curve is narrow enough that the area still comes to 1. Use this formula for plotting the curve, for likelihood calculations, and for finding which x produces a given height — never for reading off a probability directly.