Z-score calculator
Find how many standard deviations a value is from the mean and what share of a normal distribution lies below it. Students, teachers and analysts use it for test scores, quality checks and statistics classes.
- Runs in your browser
- Free, no account needed
- Copy or download the result
Result
Example
Calculation: Value to z-score and percentile Value (x): 85 Mean (μ): 70 Standard deviation (σ): 10 Z-score: 1.5 Percentile: 95
Z-score: 1.5 z = (x - μ) / σ = (85 - 70) / 10 Percentile (area to the left of z = 1.5): 93.31928% Area to the right: 6.68072% Area between -1.5 and 1.5: 86.63856% Two-tailed p-value: 0.133614 This assumes the data follow a normal distribution. Normal areas use Hart's double precision approximation (West, 2005), accurate to about 15 decimal places.
How to use the z-score calculator
- Choose a Calculation: Value to z-score and percentile, Z-score to percentile, or Percentile to z-score.
- Fill in Value (x), Mean (μ) and Standard deviation (σ), or the Z-score or Percentile field for the other modes.
- Press Run to see the z-score, areas and p-value.
- Press Copy result or Download to keep the results.
How it works
The z-score is (x - μ) / σ, and the standard deviation must be greater than zero.
Percentiles are areas under the standard normal curve. They come from Hart's double precision algorithm as published by West in 2005, which matched reference values to within about 1e-16 in our tests.
Percentile to z uses Acklam's rational approximation followed by one refinement step. It also gives the ± z for a central interval of that size, such as ±1.96 for 95%.
Percentiles must be strictly between 0 and 100. The results assume your data follow a normal distribution.
Everything runs in your browser. Your input is not uploaded or stored.
Questions and answers
What does a z-score tell you?
It tells you how many standard deviations a value is above or below the mean. A z-score of 1.5 means 1.5 standard deviations above average.
How do I convert a z-score to a percentile?
Find the area under the standard normal curve to the left of z. For z = 1.5 that is about 93.32%, so the value is near the 93rd percentile.
What z-score matches the 95th percentile?
About 1.645. For a two-sided 95% interval, the z value is about ±1.96.