Percentile included
A z of 1.5 is abstract. Saying it beats about 93% of the distribution is not.
Tools / Z Score Calculator
Z Score Calculator
Enter a value, the mean, and the standard deviation to get the z-score, its percentile rank, and the tail probabilities that go with it.
z = (x − μ) ÷ σ
Enter values to see the result
- 1
Enter the value
The raw observation you want to place, such as one student's score or one order's total.
- 2
Enter mean and standard deviation
From the population if you have it, or from your sample. The standard deviation must be above zero.
- 3
Read z and percentile
The z-score is the distance in standard deviations. The percentile is the share of a normal distribution below that point.
Why convert to a z-score
A z-score puts any measurement on the same scale, so a test result and a response time become directly comparable.
Tail probabilities too
Left, right, and two-tailed probabilities are shown, which is what you need when the z-score is a test statistic.
Signs handled properly
Negative z-scores below the mean are reported as such, with the matching percentile below 50%.
Where z-scores help
Grading and test scores
Compare a result across exams that used different scales and difficulty.
Outlier screening
Values beyond about ±3 are unusual in normal data and worth checking before analysis.
Quality control
Flag measurements that drift too far from the process mean.
The formula
z = (x − μ) ÷ σ
x is your value, μ the mean, and σ the standard deviation. The percentile shown is Φ(z), the area under the standard normal curve to the left of z.
Common z-scores and percentiles
z = −2
About the 2nd percentile
z = −1
About the 16th percentile
z = 0
Exactly the 50th percentile
z = 1.96
About the 97.5th percentile
Related calculators
Frequently asked questions
How do you calculate a z-score?
Subtract the mean from your value and divide by the standard deviation:
z = (x − μ) ÷ σ
A score of 85 in a distribution with mean 70 and standard deviation 10 gives z = 1.5, so it sits one and a half standard deviations above average.
What does a negative z-score mean?
The value is below the mean. A z of −1.2 sits 1.2 standard deviations below average, at roughly the 12th percentile.
Negative is not bad in itself. For response times or error rates, below average is the good direction.
What is a good z-score?
It depends on what you are measuring. For a test result, higher is better. For a defect rate, lower is better.
As a rough guide to how unusual a value is: about 68% of normal data falls between −1 and 1, about 95% between −2 and 2, and about 99.7% between −3 and 3.
How do I turn a z-score into a percentile?
The percentile is the area under the standard normal curve to the left of z. This calculator does it for you, so a z of 1.0 shows as roughly the 84th percentile.
The conversion assumes the underlying data is approximately normal.
What is the difference between a z-score and a p-value?
A z-score is a position. A p-value is the probability of seeing a result at least that extreme under the null hypothesis.
Use the p value calculator to convert a z-score into a one or two-tailed p-value.
Do I need the population standard deviation?
Ideally yes. With a small sample and an estimated standard deviation, a t-distribution is more appropriate than a normal one, so use the t test calculator for comparing means.
For descriptive placement of a single value, the sample standard deviation is usually fine.
Are my numbers stored?
No. Everything runs locally in your browser.
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