The standard normal variable satisfies Z N(0,1), with E(Z)=0, Var(Z)=1 and SD(Z)=1. Standardising a normal variable If X N(mu,sigma2), then the increasing linear transformation Z= X-musigma gives Z N(0,1). It preserves corresponding cumulative areas. P(X<x)=P(Z<(x-mu)/(sigma)) In particular, mu-sigma, mu and mu+sigma correspond respectively to -1, 0 and 1. Standardisation is useful when a probability condition is used to determine an unknown mean or variance.

The standard normal variable satisfies \(Z\sim N(0,1)\), with \(E(Z)=0\), \(\operatorname{Var}(Z)=1\) and \(\operatorname{SD}(Z)=1\).

Standard normal variable

If \(X\sim N(\mu,\sigma^2)\), then the increasing linear transformation \(Z=\dfrac{X-\mu}{\sigma}\) gives \(Z\sim N(0,1)\). It preserves corresponding cumulative areas.

\[P(X<x)=P\left(Z<\frac{x-\mu}{\sigma}\right)\]

Probability conversion

In particular, \(\mu-\sigma\), \(\mu\) and \(\mu+\sigma\) correspond respectively to \(-1\), \(0\) and \(1\). Standardisation is useful when a probability condition is used to determine an unknown mean or variance.

Standard Normal Distribution and Standardisation

Standard Normal Distribution and Standardisation