How to use the Z-Score Calculator
- Enter the raw value you want to place in context.
- Enter the mean of the distribution it came from.
- Enter the standard deviation — sample or population, as long as it matches the mean you used.
- Read the z-score: positive is above the mean, negative below.
- Use the percentile to say what share of the distribution falls below the value, and the p-value to judge how unusual it is.
How the calculation works
Standardising subtracts the mean and divides by the standard deviation, mapping any normal distribution onto the standard normal with mean 0 and standard deviation 1. This is what makes an IQ of 130 and a height of 190 cm directly comparable: both become pure counts of standard deviations, stripped of their original units.
The percentile comes from the standard normal cumulative distribution function, which has no closed form and is evaluated here with the Abramowitz and Stegun 7.1.26 rational approximation, accurate to better than 7.5 × 10⁻⁸. That is far tighter than any real sampling uncertainty, so the reported percentile is limited by your inputs, not by the arithmetic.
Two things routinely go wrong. First, the percentile is only meaningful if the underlying distribution really is approximately normal — a z of 2 on a heavily skewed variable does not correspond to the 97.7th percentile. Second, when the standard deviation is estimated from a small sample, the t distribution has fatter tails than the normal and the p-value here will be optimistic; below roughly n = 30 use a t-based test instead.
z = (x − μ) / σ; percentile = Φ(z); two-tailed p = 2 × (1 − Φ(|z|))Source: Abramowitz & Stegun, Handbook of Mathematical Functions (1964), formula 7.1.26; NIST/SEMATECH e-Handbook, section 1.3.6.6.1.
Worked example
An IQ test is scaled to mean 100 and standard deviation 15. How unusual is a score of 130?
- z = (130 − 100) ÷ 15 = 2.0.
- Φ(2.0) = 0.97725, so the score is at the 97.7th percentile.
- Area above = 1 − 0.97725 = 2.275%.
- Two-tailed p = 2 × 0.02275 = 0.0455.
A z of 2.0 puts the score above 97.7% of the population — roughly 1 person in 44 scores at least this high.
Frequently asked questions
What counts as a high z-score?+
Beyond ±2 covers about 4.6% of a normal distribution and is conventionally 'unusual'; beyond ±3 covers 0.27% and is genuinely rare.
Can a z-score be negative?+
Yes, and it simply means the value is below the mean. The magnitude, not the sign, tells you how unusual it is.
Does the data have to be normally distributed?+
The z-score itself is defined for any distribution, but the percentile and p-value assume normality. For skewed data they can be badly wrong.
When should I use t instead of z?+
When the standard deviation is estimated from a small sample, roughly under 30 observations. The t distribution accounts for that extra uncertainty.
Last reviewed September 1, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.