z-scores & the Standard Normal Distribution
z-scores standardise a value with \(z=\dfrac{x-\mu}{\sigma}\), letting you compare data and read probabilities from the standard normal distribution.
Part of the NSW Year 12 Mathematics Advanced course, in the Statistical analysis area of study (Random variables focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
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.