The Normal Distribution summary

Key learning points

• $Z=\frac{X-\mu}{\sigma}$ when $X\sim N(\mu,\sigma^2)$.
• Empirical rule: roughly 68%/95%/99.7% within 1/2/3 s.d.
• Sketch and shade the required region.
• Inverse normal finds $x$ from a probability.

Worked example 1 — Standardise

$X\sim N(50,16)$. Set up $P(X<46)$.

Working

$\sigma=4$; then use $\Phi(-1)$.

Worked example 2 — when

Use this method from the The Normal Distribution sheet: $Z=\frac{X-\mu}{\sigma}$ when $X\sim N(\mu,\sigma^2)$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — Empirical rule: roughly 68%/95%/99.7% within 1/2/3

Use this method from the The Normal Distribution sheet: Empirical rule: roughly 68%/95%/99.7% within 1/2/3 s.d.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Sketch and shade the required region

Use this method from the The Normal Distribution sheet: Sketch and shade the required region.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Inverse normal finds from a probability

Use this method from the The Normal Distribution sheet: Inverse normal finds $x$ from a probability.

Working

Each key learning point matches a question type on the printable worksheet.

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.