Standardise a normal variable to a z-score and calculate normal probabilities, including finding a value from a given probability

What this dot point is asking

SCSA Unit 4 uses standardisation to compute probabilities for any normal distribution. This dot point asks you to convert to a $z$-score, find probabilities (often with the calculator), and solve inverse problems where a value is found from a given probability. It is examined in both sections.

Standardising

A $z$-score measures how many standard deviations a value lies from the mean.

Standardising lets one standard normal handle every normal distribution, and is the basis for both table-based and calculator-based probability work.

Using symmetry

The standard normal is symmetric about $0$, so $P(Z<-a)=P(Z>a)$ and $P(Z>a)=1-P(Z<a)$. These identities convert any tail or interval probability into a form the calculator or the empirical rule handles, and they are essential in the calculator-free section.

Inverse problems

An inverse problem gives a probability and asks for the corresponding value of $X$. Find the $z$-score matching that cumulative probability, then unstandardise with $x=\mu+z\sigma$.

Choosing the direction

The most common slip is solving the wrong inequality. Sketch the bell, shade the region the probability describes, and check whether the required $z$ is positive (above the mean) or negative (below) before unstandardising.