Definition: A z-score tells how many standard deviations a value lies above or below the mean, computed as the value minus the mean, divided by the standard deviation.
What is a z-score?
It converts a raw value into a common scale: z = (x – mean) / standard deviation. A positive z means above average, a negative z below average, and z = 0 means exactly average. It lets you compare values from different distributions.
How do you use it?
Worked example: a student scores 128 on a test with mean 100 and standard deviation 15. Then z = (128 – 100)/15 = 28/15, about 1.87.
Another student scores 41 on a quiz with mean 35 and standard deviation 2.5, so z = 6/2.5 = 2.4. The second performance is farther above its class average, even though the raw numbers look smaller.
You can also work backward: a z of -1.5 on that test gives x = 100 – 1.5(15) = 77.5.
What mistake is common?
Dividing by the mean or by the variance instead of the standard deviation. Mathbot also reminds you that z-scores describe position, not probability; to get a probability you need a normal model and a table or calculator.