The Normal Distribution Topic Review summary

Key learning points

• $X\\sim N(\\mu,\\sigma^2)$ ⇒ $Z=\\frac{X-\\mu}{\\sigma}\\sim N(0,1)$.
• Symmetric bell curve; mean = median = mode.
• Use tables or a calculator for $\\Phi(z)=P(Z\\le z)$.
• Inverse Normal for percentiles given a probability.
• Continuity corrections appear when approximating discrete distributions.

Worked example 1 — Standardise

$X\\sim N(10,4)$, $P(X<12)$.

Working

$\\sigma=2$.

Worked example 2 — Symmetry

$\\Phi(-z)$.

Working

Standard Normal symmetry.

Worked example 3 — X\\\\sim N(\\\\mu,\\\\sigma^2) ⇒ Z=\\\X-\\

$X\\\\sim N(\\\\mu,\\\\sigma^2)$ ⇒ $Z=\\\\frac{X-\\\\mu}{\\\\sigma}\\\\sim N(0,1)$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Symmetric bell curve; mean = median = mode

Apply this idea from the Topic Review: Symmetric bell curve; mean = median = mode.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.