A z-score standardises a normal value to the standard normal distribution, enabling exact probabilities to be found by technology or tables.

What this dot point is asking

The empirical rule only handles whole-number multiples of $\sigma$. For any value, you standardise it to a z-score and read the probability from the standard normal distribution $N(0,1)$.

The z-score

Standardising maps $X\sim N(\mu,\sigma^2)$ onto $Z\sim N(0,1)$, which has mean $0$ and standard deviation $1$. Equal z-scores represent equally extreme positions in different distributions, so z-scores let you compare values from different normal distributions.

Finding probabilities

In Stage 2 you use a calculator's normal CDF (or the standard normal table) to find the area. The standardisation tells you which area to look up.

The inverse problem

Sometimes you know the probability and want the value - the inverse normal. Find the z-score for that area, then unstandardise with $x=\mu+z\sigma$.

Common errors

Why it matters

Z-scores are the exact-probability engine of Topic 5 and a guaranteed exam skill, both forward (value to probability) and inverse (probability to value). Standardisation is also the mechanism behind Topic 6's confidence intervals, where critical z-values such as $1.96$ come straight from inverse-normal reasoning.