How it works
A z-score tells you how many standard deviations a value sits above or below the mean. A z-score of 0 is exactly average, +1 is one standard deviation above, and −2 is two standard deviations below.
If the data are roughly normally distributed, the z-score converts directly to a percentile using the standard normal curve. For example, z = 1.5 is about the 93rd percentile: 93% of values fall below it.
Z-scores let you compare results measured on different scales, such as a test score and a reaction time, by putting them in the same standard units.
Formula
z = (x − μ) ÷ σ · Percentile = Φ(z) × 100
Questions people ask
What is a good z-score?
It depends on whether higher is better. Values between −1 and 1 cover about 68% of a normal distribution; beyond ±2 covers only about 5%, and beyond ±3 about 0.3%.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. A z-score of −1 corresponds to roughly the 16th percentile.
Should I use the sample or population standard deviation?
Use the population value if you know it. If you only have a sample, the sample standard deviation is the usual stand-in, and the result is technically a t-like score.