Z-Score Calculator
This z-score calculator tells you how far a value sits from the average, measured in standard deviations. Type a value, the mean, and the standard deviation, and you'll get the z-score along with its percentile and probability, drawn on a live normal curve. Need to go the other way? It'll also find the original value from a z-score, or work the z-score straight from a data set you paste in.
- Value to z-score
- Percentile & probability
- Reverse: z to value
- From a data set
- Steps shown
Last updated June 16, 2026 z = (x − μ) ÷ σ Reviewed by the Calcowa math team
Enter numbers to see the z-score.
Decimals and negatives are fine. A z-score needs a standard deviation above zero.
z = (85 − 70) ÷ 10 = 1.5
What is a z-score?
A z-score, also called a standard score, tells you how many standard deviations a value sits above or below the mean. It rescales any value onto one common ruler, so a z of 1 always means one standard deviation above average, whether you're looking at test scores, heights, or blood pressure. That's what makes it so useful: a 90 on one test and an 1800 on another become directly comparable once you turn them into z-scores.
How do you calculate a z-score?
Calculating a z-score takes one subtraction and one division. Here's the method the z-score calculator runs, using a value of 85 with a mean of 70 and a standard deviation of 10:
- 1
Subtract the meanTake the value and subtract the mean: 85 minus 70 is 15. This gap is how far the value is from average in raw units.
- 2
Divide by the standard deviationDivide that gap by the standard deviation: 15 divided by 10 is 1.5. Now the distance is in standard deviations, not raw units.
- 3
Read the z-scoreThe z-score is 1.5, so the value sits 1.5 standard deviations above the mean. A positive z is above average, a negative z is below.
- 4
Look up the percentileMatch the z-score to the normal curve and a z of 1.5 lands at about the 93rd percentile, meaning roughly 93% of values fall below it.
From z-score to percentile and probability
A z-score on its own is a distance, but paired with the standard normal curve it becomes a probability. The percentile is the share of values below your z-score, which is why a z of 0 is the 50th percentile, a z of 1 is about the 84th, and a z of 1.96 is about the 97.5th. The calculator reads these straight off the curve, so you don't have to chase a number across a z-score table. It also reports the probability above your value and a two-tailed p-value, the figure you'd quote when you're testing how surprising a result is.
Finding the value from a z-score
Sometimes a problem gives you the z-score and asks for the raw value behind it. Switch to the reverse mode and the tool rearranges the formula to x = μ + zσ. So a z of 1.5 with a mean of 70 and a standard deviation of 10 gives 70 + 1.5 × 10, which is 85, right back where we started. This is how you turn a target percentile into a cutoff score, like the mark you'd need to land in the top 10%.
Calculating a z-score from a data set
If you've only got a list of numbers, you don't need the mean and standard deviation in advance. Paste the data set into the From-data mode, pick the value you want to score, and the calculator finds the mean and the standard deviation for you, then applies the z-score formula. By default it uses the sample standard deviation, since a list of readings is usually a sample; tick the population box if your numbers cover the entire group. For the spread step on its own, our standard deviation calculator shows that working in full.
Frequently asked questions
What does a z-score of 0 mean?
A z-score of 0 means the value equals the mean exactly, so it sits at the 50th percentile, right in the middle. There's no distance from the average to report.
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) ÷ σ. The answer tells you how many standard deviations the value sits above or below the average. A positive z is above the mean, a negative z is below it, and a z of 0 lands right on the mean. This z-score calculator runs that formula and shows each step.
There's no single good value, it depends on what you're measuring. A z-score between −2 and 2 covers about 95% of a normal distribution, so anything in that band is fairly typical. Past ±2 a value starts to look unusual, and past ±3 it's rare. For a test you'd want a high positive z; for an error rate you'd want a low one.
A negative z-score just means the value is below the mean. A z of −1 sits one standard deviation below average, and a z of −2 sits two below. It isn't bad on its own, it only tells you the direction and distance from the mean. The size of the number, not its sign, is what tells you how far out the value is.
The percentile is the share of a normal distribution that falls below your z-score. The calculator works it out from the standard normal curve, so a z of 0 is the 50th percentile, a z of 1 is about the 84th, and a z of 1.96 is about the 97.5th. That's the same number a z-score table gives you, without the hunting.
For a single value it's z = (x − μ) ÷ σ, where x is the value, μ is the mean, and σ is the standard deviation. When you're working from a sample of data instead of a known population, you use the sample mean and the sample standard deviation in their place. The tool accepts either, including a raw data set you paste in.
Yes. Switch to the reverse mode and the calculator rearranges the formula to x = μ + zσ. Give it the z-score, the mean, and the standard deviation, and it returns the original value. That's handy when a problem hands you a z and a percentile and asks what raw score they point to.
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