Standard Normal Distribution

The standard normal distribution is the particular normal distribution that has a mean of 0 and a standard deviation of 1. Its density is $\varphi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$, the familiar bell curve centred on zero. Any normally distributed value can be converted to this scale by subtracting its mean and dividing by its standard deviation, which gives its z-score. Probabilities can then be read from a single table or function, written $N(z)$ or $\Phi(z)$. About 68 percent of the distribution lies between -1 and 1, about 95 percent between -1.96 and 1.96, and about 99.7 percent between -3 and 3.

In finance the function $N(\cdot)$ appears in the Black-Scholes-Merton model, where it converts the model's inputs into probabilities and hedge ratios. It supplies the multipliers used in parametric value at risk, such as 1.645 for a 95 percent level, and the reference distribution for the z-test. Its tails are thinner than those of actual market returns.