Z score formula
A z-score tells you how many standard deviations a value lies from the mean. It puts any value on a common scale, making it easy to compare data from different distributions and to look up probabilities.
- z = the z-score, the number of standard deviations from the mean
- x = the value being measured
- μ = the mean of the data
- σ = the standard deviation
What the z-score means
A z-score of 0 is exactly average. A positive z-score sits above the mean, a negative one below, and the size says how far in standard deviations. A z-score of 2 means the value is two standard deviations above the mean, which on a normal curve is higher than about 97.5 percent of values.
Why standardize
Converting raw values to z-scores removes the original units, so scores from different tests or measurements become directly comparable. It also lets you use a single standard normal table to find the probability of any value, regardless of the original mean and spread.
- Exam scores have mean μ = 70 and standard deviation σ = 8.
- A student scores x = 86.
- z = (86 − 70) / 8 = 16 / 8 = 2.
- The score is 2 standard deviations above the mean.
Related guide
Read the Normal Distribution Guide to see how z-scores map to probabilities.
FAQ
What is a z-score?
The number of standard deviations a value is from the mean, found with z = (x − μ) / σ.
What does a negative z-score mean?
The value is below the mean. A z-score of −1 is one standard deviation below average.