z-scores & the Standard Normal Distribution
Learn \(z\)-scores and the standard normal distribution for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-07). A \(z\)-score counts how many standard deviations a value sits above or below the mean, putting results from different tests or datasets on one common scale.
You will learn to calculate \(z\)-scores, compare performances fairly, use standard normal tables to find probabilities, and work backwards to an unknown mean or standard deviation — a widely used way to rank results and judge how unusual a value is.
Theory
A z-score measures how many standard deviations a value is from the mean. This Year 12 Mathematics Advanced topic (MAV-12-07) standardises with \(z=\dfrac{x-\mu}{\sigma}\).
\(z=\dfrac{x-\mu}{\sigma}\): \(z>0\) above the mean, \(z<0\) below, \(z=0\) at the mean. Reverse with \(x=\mu+z\sigma\). \(z\)-scores let you compare values from different distributions; the standard normal is \(Z\sim N(0,1)\).
Method
- Subtract the mean, then divide by \(\sigma\).
- Reverse with \(x=\mu+z\sigma\) for a raw score.
- Compare \(z\)-scores across different distributions.
| \(z\) | \(=\) | \(\dfrac{70-50}{10}=2\) |
| \(z\) | \(=\) | \(\dfrac{35-50}{10}=-1.5\) |
| \(x\) | \(=\) | \(60+2(4)=68\) |
Ali (\(1.2>0.8\)).
Common pitfalls
Frequently asked questions
What is a z-score?
It is the number of standard deviations a value is from the mean, found as z equals x minus mu over sigma.
How do you find a raw score from a z-score?
Use x equals mu plus z times sigma.
Why are z-scores useful?
They let you compare values from different distributions on a common scale.
What is the standard normal distribution?
A normal distribution with mean 0 and standard deviation 1.