Normal (Gaussian) distribution

The normal distribution $N(\mu,\sigma)$ is the most important distribution in statistics — the “bell curve”. Standardization converts a value $x$ into a z-score:

$z=\frac{x-\mu}{\sigma}$

which tells you how many standard deviations $x$ lies from the mean. The cumulative distribution $\Phi(z)$ gives the probability $P(X<x)$.

The 68–95–99.7 rule (three-sigma rule) states what fraction of the data falls within intervals around the mean:

$\mu\pm\sigma\approx68\%\quad \mu\pm2\sigma\approx95\%\quad \mu\pm3\sigma\approx99.7\%$