Normal distribution

A random variable $X$ with the normal distribution $N(\mu, \sigma^2)$ has a bell-shaped density: the mean $\mu$ is at the peak and the standard deviation $\sigma$ sets the width. The probability that $X$ falls between two values is the area under the curve between them. Here the curve stands over two rulers, $x$ for values and $z$ for standard units; you can move $\mu$ and $\sigma$, cut the line at $a$ and $b$, and choose which ruler stays fixed on the screen.

Why is every normal curve the same curve? Fix the $z$ ruler and change $\mu$ or $\sigma$: the curve on the screen stays still. Instead the $x$ ruler slides and stretches underneath it, and the numbers on the density scale change. The cuts are values of $x$, so they ride along with that ruler, and the shaded area changes only when their places on the $z$ ruler change. Every normal distribution is the standard normal $N(0, 1)$ read on another ruler, so $P(a \le X \le b) = \Phi(z_b) - \Phi(z_a)$, where $\Phi(z)$ is the area to the left of $z$ under the standard curve.

Where is $\sigma$ on the curve? At the bend points. On the ordinary scale the curve bends downward between $\mu - \sigma$ and $\mu + \sigma$ and upward outside them. At these two inflection points the tangent crosses the curve. Because the shape never changes, the area within $k$ standard deviations of the mean does not depend on $\mu$ or $\sigma$: about 68.27% within one, 95.45% within two and 99.73% within three.

Why is a narrower curve taller? The whole area is always 1. Build it shows the two steps from the standard curve: moved to $\mu$ and stretched sideways by $\sigma$, the curve encloses an area of $\sigma$; divided in height by $\sigma$, it encloses 1 again. So the peak is $1/(\sigma\sqrt{2\pi})$, and halving $\sigma$ doubles it. The height is a density, probability per unit of $x$; only areas are probabilities.

How thin are the tails? On the ordinary scale the curve seems to meet the axis by about $3\sigma$, yet it never reaches zero. On the log scale $\log f$ is a quadratic in $x$, so the curve becomes a downward parabola and the tails fall ever faster: beyond $3\sigma$ on either side lies about 0.270% of the probability, beyond $4\sigma$ about 1 in 15,800, beyond $6\sigma$ about 1 in 507 million. On a log scale the area under the curve is no longer the probability, so the region is marked only on the $x$ ruler.