Z-Score Calculator
Turn a raw score into a z-score and percentile, or go the other way. The bell curve shades the area so you can see what the number means.
Z-score formula
z = (x − μ) ÷ σ ⇔ x = μ + z·σ
Standardising puts every normal distribution on the same ruler, with mean 0 and standard deviation 1. That is what makes one z-table work for heights, test scores and machine tolerances alike.
Comparing two scores
Maya scored 1350 on the SAT (mean 1050, SD 210) and Leo scored 28 on the ACT (mean 19.5, SD 5.9). Maya’s z is 1.43; Leo’s is 1.44. Leo did very slightly better relative to his test-takers, though the gap is too small to mean anything.
When z-scores mislead
Percentiles from z-scores assume a roughly normal shape. For skewed data such as incomes or reaction times, a z of 2 may sit at the 90th percentile rather than the 97.7th. When you have the raw data, the percentile calculator gives the true rank with no assumption.
Frequently asked questions
What is a z-score?
A z-score says how many standard deviations a value lies above or below the mean. z = 2 means two standard deviations above; z = −0.5 means half a standard deviation below. It lets you compare values from different scales, such as an SAT score and an ACT score.
How do I calculate a z-score?
Subtract the mean from the value and divide by the standard deviation: z = (x − μ) ÷ σ. For an SAT score of 1350 with mean 1050 and SD 210, z = 300 ÷ 210 ≈ 1.43.
How do I convert a z-score to a percentile?
Find the cumulative area to the left of z under the standard normal curve (a z-table or this calculator) and multiply by 100. z = 1.43 gives about 0.924, the 92nd percentile. This assumes the data are approximately normal.
What is a good or unusual z-score?
About 95% of normal data have z between −1.96 and 1.96, and 99.7% between −3 and 3. Values beyond ±2 are often called unusual and beyond ±3 very unusual, but the right threshold depends on context.
Can I use a z-score for a sample mean?
Yes, with the standard error in place of the standard deviation: z = (x̄ − μ) ÷ (σ/√n). If σ is unknown and estimated from a small sample, use a t-statistic and the t-test calculator instead.