Normal distribution & z-scores

The normal distribution is a symmetric bell-shaped model for many real measurements.

Normal model basics

A normal variable \(X\sim N(\mu,\sigma^2)\) has mean \(\mu\) and standard deviation \(\sigma\).

• About 68% of values lie within \(\mu\pm\sigma\).
• About 95% lie within \(\mu\pm2\sigma\).
• About 99.7% lie within \(\mu\pm3\sigma\).

Standardizing with z-score

\[ z=\frac{x-\mu}{\sigma}. \]

This converts values to the standard normal \(Z\sim N(0,1)\).

Finding probabilities

1. Convert boundary value(s) to z-score(s).
2. Use normal table/calculator for area.
3. Apply complement or subtraction if needed.

Finding values from probabilities

If \(P(X\le x)=p\), find corresponding \(z_p\), then:

\[ x=\mu+z_p\sigma. \]

Exam strategy

• Draw a quick bell curve and shade required region.
• Label mean, SD marks, and inequality direction.
• Use continuity correction when question requires discrete-to-normal approximation.
• Round only at final step.

Checkpoints