Statistics

Standard Normal Variable and z-scores

Standardise a normal variable so probabilities can be compared on the same scale.

Formula

A z-score tells us how many standard deviations a value is from the mean.

\[ Z=\frac{X-\mu}{\sigma} \]
01−12Normal distribution
The standard normal distribution has mean 0 and standard deviation 1.

Worked example

Heights are normally distributed with mean \(170\) cm and standard deviation \(6\) cm. Find the z-score for \(182\) cm.

\[z=\frac{182-170}{6}=2.\]

A height of 182 cm is 2 standard deviations above the mean.

Exam tip: z-scores are useful when comparing values from different normal distributions.