Standard Normal Distribution and Standardisation

Junior College 2

TGM Original

Standard normal variable

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

Standardising a 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.

Schematic standardisation: the two bells use separate horizontal scales. The marked values \(\mu-\sigma,\mu,\mu+\sigma\) on the \(X\)-scale correspond to \(-1,0,1\) on the \(Z\)-scale; horizontal drawing positions are not shared numerical coordinates.

Probability conversion

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

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 first sketch uses the normalised standard normal density \(\phi(z)=\dfrac{1}{\sqrt{2\pi}}e^{-z^2/2}\). The paired standardisation sketch is schematic and uses separate \(X\) and \(Z\) scales. Calculate probabilities using the normal cumulative distribution or the correctly normalised density.

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