Normal Distribution
The normal distribution is a bell-shaped continuous probability distribution function that is parameterized to a mean and standard deviation.
Description
The distribution is bell-shaped and parameterized to the first and second moments. This has useful consequences for estimating the probability that a given value is in the distribution. For example:
- 68.27% of the cumulative distribution is within 1 standard deviation of the mean
- 95.45% within 2
- 99.73% within 3
A variable distributed this way is notated (especially in econometrics) like X ~ N(μ, σ2).
When the mean is 0 and the variance is 1, the p.d.f. is specifically referred to as the standard normal distribution. This is defined as 
.
More generally, the p.d.f. is given by f(x) = (1/σ) * φ(x-μ/σ).
The c.d.f. for the standard normal distribution is notated as Φ(.), while the c.d.f. for the generic normal distribution is sometimes notated as F(.).
Moments
The first and second moments are intrinsic to the distribution definition.
Usage
Probability Tests
The standard normal distribution is referenced for Z scores (alternatively called Z statistics). As an example, for a two-tailed test and a significance level of 5%, the critical Z score value is 1.96.