standardnormal

Standardnormal, commonly referred to as the standard normal distribution, is the normal distribution with mean 0 and standard deviation 1. It is the canonical form of a normal variable and is denoted by Z ~ N(0,1). The probability density function is f(z) = (1/√(2π)) exp(-z^2/2) and the cumulative distribution function is Φ(z) = ∫_{-∞}^z f(t) dt. The distribution is symmetric, unimodal, and its graph is the familiar bell curve. The standard normal plays a central role in statistics as a reference distribution for standardization and hypothesis testing.

Any normal variable X ~ N(μ, σ^2) can be transformed to Z by z = (X - μ)/σ. Then Z ~ N(0,1). This standardization enables direct comparison between values from different normal distributions and allows the calculation of p-values, confidence intervals, and critical values using either standard normal tables or software.

Key properties of the standard normal include a mean of 0 and a variance of 1, with E[Z] = 0 and Var(Z) = 1. The distribution is symmetric around zero, with skewness 0 and excess kurtosis 0, reflecting its bell-shaped form independent of scale.

Applications of the standard normal include computing z-scores to assess extremes, conducting z-tests and construction of confidence intervals for means when population variance is known, and obtaining percentiles or p-values via Φ or its inverse. Since the cumulative distribution lacks a closed elementary form, numerical methods and tables are used for practical calculations.