Z-Score Calculator
Standardize a value: get its z-score, its percentile and the probability of a value below or above it in a normal distribution.
What a z-score is
The z-score tells you how many standard deviations a value lies from the mean: z = (x − μ) / σ. A z-score of 0 is exactly average, +1 is one standard deviation above, −2 two below.
Worked example
A test has a mean of 70 and a standard deviation of 10. A score of 85 has z = (85 − 70) / 10 = 1.5. In a normal distribution 93.32% of values are lower, so the score is at the 93rd percentile.
The 68–95–99.7 rule
- About 68% of values lie within ±1 standard deviation (z between −1 and 1)
- About 95% within ±2 (more exactly ±1.96)
- About 99.7% within ±3
A z-score beyond ±3 is rare – often a sign of an outlier or a measurement error.
Comparing different scales
z-scores make different tests comparable. 85 points on a test with mean 70 and σ 10 (z = 1.5) is a better result than 120 on an IQ test with mean 100 and σ 15 (z = 1.33).
The percentiles assume the data are normally distributed. For skewed data, such as incomes, the z-score still measures distance from the mean but the percentile will be off.
Frequently asked questions
What does a z-score of 2 mean?
The value is 2 standard deviations above the mean; about 97.7% of values are lower.
Can a z-score be negative?
Yes – a negative z-score means the value is below the mean.
What z-score is the 95th percentile?
1.645.