What Are the Properties of the Normal Distribution?

A normal distribution is a continuous probability distribution with a symmetric, bell-shaped curve centred at its mean. Its position is determined by the mean, while its spread is determined by the standard deviation.

Core properties

If a random variable is normally distributed, it is written as $X \sim N(\mu,\sigma^2)$, where $\mu$ is the population mean and $\sigma^2$ is the variance.

Changing $\mu$ translates the curve horizontally without changing its shape, whereas changing $\sigma$ changes the spread while preserving total area. Symmetry also gives $P(X<\mu)=P(X>\mu)=0.5$, so the mean divides the distribution into two equal-probability regions.

The normal distribution also follows the 68-95-99.7 rule:

• Approximately $68\%$ of values lie within $\mu \pm \sigma$.
• Approximately $95\%$ lie within $\mu \pm 2\sigma$.
• Approximately $99.7\mathrm{\%}$ lie within .

For example, if heights satisfy $X \sim N(170,6^2)$, approximately $95\%$ of heights lie between $170-12=158$ cm and cm.

IB exam technique

This is SL 4.9 Normal distribution, examined at both SL and HL on GDC-active papers. Use normal CDF or inverse-normal functions directly, state the distribution and parameters, and interpret the answer in context. For the height model, the GDC setup for $P(X<182)$ uses mean $170$ and standard deviation $6$, not variance $36$.

A common misconception is entering $\sigma^2$ into the GDC: the notation displays the variance, but the calculator normally requires the standard deviation $\sigma$. At SL, standardizing to a $z$-score is not required.