= 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 [[Statistics/Moments|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 [[Statistics/EconometricsNotation|econometrics]]) like X ~ N(μ, σ^2^). 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 {{attachment:stdnorm.svg}}. {{attachment:stdnormgraph.png||width=300px}} 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 [[Statistics/Moments|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 [[Statistics/TestStatistic|significance level]] of 5%, the critical Z score value is 1.96. ---- CategoryRicottone