Gaussian (normal) distribution

The normal distribution is the most important continuous model in statistics: it describes measurements subject to many small random errors and its curve is the celebrated “Gaussian bell”.

Property — $X\sim N(\mu,\sigma^2)$

$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$ The curve is the famous “Gaussian bell”: symmetric about $\mu$, with width proportional to $\sigma$.

Three Gaussians. The blue one is the standard normal $N(0,1)$. The red one has mean $\mu=1$ and small $\sigma$ (narrow and tall). The green one has mean $\mu=-1$ and large $\sigma$ (wide and low). The area under each curve is always $1$.

Explore the Gaussian bell in the simulation: move $\mu$ to shift the curve and $\sigma$ to widen or narrow it.

Drag the sliders: $\mu$ shifts the bell, $\sigma$ widens or narrows it (area always $1$).

Links

Topics: Distribuzioni probabilita
Concepts: Densita di probabilita · Deviazione standard · Distribuzione normale · Valore atteso
Skills: Interpretare grafico · Modellizzare
People: Carl Friedrich Gauss