Z-Score Calculator
A value of 130 with a mean of 100 and a standard deviation of 15 has z = 2, which is the 97.7th percentile. The 97.5th percentile has z = 1.96.
How to use
- Choose "Value to z" and enter the value, the mean and the standard deviation of the data.
- Read the z-score and the percentile: the share of a normal population below your value, above it, and in both tails.
- To go the other way choose "Percentile to z", enter a percentile such as 97.5, and add a mean and standard deviation if you also want the matching value.
How it works
z = (x − μ) / σ, where x is the value, μ the mean and σ the standard deviation (σ must be above zero).
Percentile = Φ(z), the cumulative probability of the standard normal distribution: Φ(z) = ½ · erfc(−z / √2). The calculator evaluates erfc with a power series for small |z| and a continued fraction for large |z|, so even far tails keep full precision.
Above the value: 1 − Φ(z) = Φ(−z). Two-tailed p = 2 · min(Φ(z), 1 − Φ(z)). Share within ±|z| = 1 − two-tailed p.
Inverse: given a percentile p, find z with Φ(z) = p. The calculator bisects on Φ and then refines with Newton steps, and the value is x = μ + z · σ.
The standard normal distribution has mean 0 and standard deviation 1. A z-score is the number of standard deviations a value sits from the mean, which lets you compare values measured on different scales.
Worked examples
The percentile is the share of a normal population that scores below the value. It assumes the data follow a normal (bell-shaped) distribution.
| Value | Mean | Standard deviation | z-score | Percentile |
|---|---|---|---|---|
| 130 | 100 | 15 | 2 | 97.72% |
| 85 | 100 | 15 | -1 | 15.87% |
| 72 | 65 | 8 | 0.875 | 80.92% |
| 50 | 50 | 10 | 0 | 50% |
Common critical values
These z values mark the edges of the middle region that holds the stated share of a normal distribution, and the one-sided cut-off for the same level.
| Confidence level | Two-sided z | One-sided z |
|---|---|---|
| 90% | 1.645 | 1.282 |
| 95% | 1.960 | 1.645 |
| 99% | 2.576 | 2.326 |
Reading a z-score
- z = 0 means the value equals the mean. z = 1 means one standard deviation above it, z = -2 means two below.
- About 68% of a normal population lies within z = ±1, 95% within ±1.96 and 99.7% within ±3.
- Two-tailed p is the probability of a value at least this far from the mean in either direction, which is what many significance tests use.
Limits
Percentiles come from the standard normal curve. If your data are skewed or have heavy tails, the z-score is still correct as a distance in standard deviations, but the percentile will not match the real data.
FAQ
What is a z-score?
It is the number of standard deviations a value lies above or below the mean. A z-score of 2 means the value is two standard deviations above the mean.
How do I turn a z-score into a percentile?
Take the cumulative probability of the standard normal curve at that z. For z = 1.96 it is 97.5%, so the value is at about the 97.5th percentile.
Does it matter that my data may not be normal?
The z-score itself is always valid. The percentile and p-values assume a normal distribution, so treat them as approximate if the data are clearly skewed.
What standard deviation should I use for a sample?
Use whichever you were given. For a sample you normally use the sample standard deviation (n − 1 in the denominator); for a whole population, the population one. The z-score formula is the same.
What is the difference between one-tailed and two-tailed?
One-tailed looks at one side only, such as the chance of a value above z. Two-tailed adds both sides and is the probability of being at least |z| away from the mean in either direction.
Why can I not enter 0 or 100 as a percentile?
A normal distribution never reaches exactly 0% or 100%, so the z-score would be minus or plus infinity. Enter a value strictly between 0 and 100.