Z-Score Calculator
Convert any value to a z-score and find its probability in a normal distribution. Essential for statistics and quality control.
The data point to standardize.
Z-score
1
Below
84.1%
Above
15.9%
Z = (x − μ) ÷ σ. Uses the standard normal distribution — assumes your data is roughly bell-shaped.
How it works
z = (x − μ) ÷ σ. Probability from the standard normal CDF (erf approximation).
Example: x=70, μ=60, σ=10 → z=1.0, 84.1% below.
Last updated: August 2026
How this calculator is verified
Checked by Rahim Virani on
- z = (x - mean) / standard deviation verified on a known sample
- Sample vs population standard deviation selected explicitly, not inferred
- Probability tails reported both directions so left/right is unambiguous
The full verification method is on our how we verify page. Found an error? Tell us and we will re-check it.
When to Use This Calculator
In a group with an average height of 165 cm and a standard deviation of 6 cm, someone 177 cm tall is exactly 2 standard deviations above the mean, so the z-score is 2. The tool converts this to a normal-distribution probability: about 97.7% of values fall below that height. Enter the value, mean and standard deviation, and it returns the z-score plus the percentages below, above and between. Z-scores make scores from different tests comparable, because each one is expressed in standard-deviation units.
How to Use This Calculator
- Step 1: Enter the value you want to evaluate, such as a test score or a height measurement.
- Step 2: Enter the mean and the standard deviation of the group that value belongs to.
- Step 3: Read the z-score, which tells you how many standard deviations the value sits from the mean.
- Step 4: Read the probabilities shown: the percentage of values below, above, and between the mean and your value.
Worked Example
In a group with an average height of 165 cm and a standard deviation of 6 cm, someone 177 cm tall is exactly 2 standard deviations above the mean, so the z-score is 2. The tool converts this to a normal-distribution probability: about 97.7% of values fall below that height. Enter the value, mean and standard deviation, and it returns the z-score plus the percentages below, above and between. Z-scores make scores from different tests comparable, because each one is expressed in standard-deviation units.
Tips and Common Mistakes
- •Tip 1: Use z-scores to compare results from different tests, since converting each score into standard-deviation units puts them on one scale.
- •Tip 2: For quality control, a process output beyond roughly 3 standard deviations from the mean signals that the process needs attention.
- •Tip 3: Check that your standard deviation is the population figure if that is how your reference data reports it, or results will shift.
- ✗Mistake 1: Entering variance instead of standard deviation, which inflates every z-score and probability in the output.
- ✗Mistake 2: Assuming the data is normally distributed when it may be skewed, which makes the probability estimates less reliable.
Frequently Asked Questions
What is a z-score?
A z-score tells how many standard deviations a value is from the mean. A z of 2 means the value is two standard deviations above average.
How is the probability from a z-score found?
The calculator integrates the standard normal curve (via the error function) to find the area below the z-score, which equals the percentile.
When should I not use z-scores?
Z-score probabilities assume a normal (bell-shaped) distribution. For skewed or heavy-tailed data the probabilities can mislead.
What is a z-score table?
It lists the area under the normal curve for each z-value so you can look up probabilities by hand. This calculator computes the same numbers directly.
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