Z scores explained
A z-score measures how many standard deviations a value lies from the mean. This guide shows how to calculate one, how to read it against the normal curve, and why standardizing makes different data sets comparable.
What a z-score is
A z-score is found with z equals the value minus the mean, divided by the standard deviation. A z-score of 0 is exactly average, a positive z-score is above the mean, and a negative one is below. The size tells you how far, measured in standard deviations.
How to calculate it
- Find the mean and standard deviation of the data.
- Subtract the mean from your value.
- Divide by the standard deviation.
- The result is the z-score.
A score of 86 in a test with mean 70 and standard deviation 8 gives z = (86 − 70) / 8 = 2, two standard deviations above the mean.
Reading z-scores with the normal curve
On a normal distribution, a z-score maps directly to a percentile. A z of 0 is the 50th percentile, a z of 1 is about the 84th, and a z of 2 is about the 97.5th. Standard normal tables, or a calculator, turn any z-score into the probability of scoring below that value.
Why standardize
Converting to z-scores removes the original units, so values from different tests or measurements become directly comparable. A z of 1.5 means the same relative standing whether the data is heights, exam marks, or reaction times. This is why z-scores underpin grading curves, quality control, and significance testing.
- A z-score is the number of standard deviations from the mean.
- z = (value − mean) / standard deviation.
- Positive z is above the mean; negative z is below.
- On a normal curve, a z-score maps to a percentile.
- Standardizing makes different data sets comparable.
Related references
See the Z Score Formula and the Normal Distribution Guide.
FAQ
How do I calculate a z-score?
Subtract the mean from the value and divide by the standard deviation: z = (x − mean) / SD.
What does a z-score of 2 mean?
The value is two standard deviations above the mean, higher than about 97.5 percent of a normal distribution.
Why use z-scores?
They standardize values so data measured on different scales can be compared directly.