Z-score
Z-score measures how many standard deviations a value lies from its mean: z = (x − μ) / σ. It standardises values onto a common scale, enabling comparison across different variables and use of normal tables.
See it move
A student scoring 72 where the class mean is 60 and standard deviation is 8 has z = (72 − 60) / 8 = 1.5. Another student scoring 65 on a different exam, mean 50, standard deviation 10, gets z = (65 − 50) / 10 = 1.5 as well. Despite different raw scores and scales, both stand 1.5 standard deviations above their own mean — roughly the 93rd percentile.
The formula
Variables
- Observed data value
- Population mean
- Population standard deviation
- Number of standard deviations x lies from the mean
z = 0 at the mean; z = +1 one standard deviation above; z = −1 one standard deviation below. Standardising allows comparison across variables measured on different scales.
Check yourself
A student sits two end-of-term exams. On the marketing exam she scores 76 (class mean = 68, standard deviation = 4). On the operations exam she scores 84 (class mean = 70, standard deviation = 14). On which exam did she perform better relative to her classmates?