A-Level — mathsStandard2 min read

The Normal Distribution

Work with the normal distribution at A-Level. Standardise, find probabilities, and use inverse normal.

Tutor AI Team2026-02-25T01:58:08.630Z

The normal distribution $X \sim N(\mu, \sigma^2)$ is the most important continuous distribution. Its bell-shaped curve models many natural phenomena.

Core Concepts

$Z = \frac{X - \mu}{\sigma}$

Converts $X \sim N(\mu, \sigma^2)$ to $Z \sim N(0, 1)$ (standard normal).

$P(X < a) = P\left(Z < \frac{a - \mu}{\sigma}\right)$ . Look up in tables or use calculator.

Given a probability, find the value of $x$ .

If $P(X < a) = 0.95$ , standardise: $\frac{a - \mu}{\sigma} = 1.645$ → $a = \mu + 1.645\sigma$ .

Problem

$X \sim N(50, 16)$ (so $\sigma = 4$ ). Find $P(X > 56)$ .

$Z = \frac{56-50}{4} = 1.5$ . $P(Z > 1.5) = 1 - 0.9332 = 0.0668$ .

Solution

Problem

$X \sim N(100, 225)$ . Find $a$ such that $P(X < a) = 0.9$ .

$\sigma = 15$ . $Z = 1.2816$ . $a = 100 + 1.2816(15) = 119.2$ .

Solution

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