Normal Distribution
The normal distribution is a bell-shaped continuous probability distribution function that is parameterized to a mean and standard deviation.
Contents
Description
The distribution is bell-shaped and parameterized to the first and second central 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 like X ~ N(μ, σ2).
The probability density function is given (and commonly notated) as:
Correspondingly, the cumulative probability function is commonly notated as Φ(.) rather than F(.).
Standard Normal Distribution
The standard normal distribution is specified as Z ~ N(0, 1).
Compare to the calculation for Z scores that will be compared to the standard normal distribution: Z = (x - μ)/σ.
The probability density function simplifies to:
.
This is graphed as:
Moments
The expected value is given as E[X] = μ.
Variance is given as Var[X] = σ2.
Usage
Probability Tests
As noted above, Z scores (alt. Z statistics) are test statistics for which likelihood can be calculated using the standard normal distribution.
As an example, for a two-tailed test and a significance level of 5%, the critical Z score value is 1.96.