Z-Score Calculator
Calculate a z-score from a value, mean, and standard deviation, and convert it to a percentile and tail probabilities on the normal distribution.
Como usar esta calculadora
- 1Enter the value you want to standardise.
- 2Enter the mean and standard deviation of the distribution it comes from.
- 3Read the z-score, its percentile, and how unusual the value is.
Como funciona
Z-score (standard score)
z = (x − μ) ÷ σ x = the value, μ = mean, σ = standard deviation z is the number of standard deviations from the mean percentile = share of a normal distribution below z
A z-score, or standard score, expresses a value not in its own units but in standard deviations from the mean. Subtracting the mean centres the value at zero; dividing by the standard deviation rescales it so that one unit is one standard deviation. This puts any value from any normal distribution on the same universal scale, which is what lets you compare a test score, a height, and a temperature, and read off exactly how unusual each one is.
Exemplo resolvido
An IQ of 130, on a scale with mean 100 and standard deviation 15, has a z-score of (130 − 100) ÷ 15 = 2.0. That places it two standard deviations above the mean, at about the 97.7th percentile — higher than roughly 98% of the population.
Z-Score Calculator: o guia completo
Putting everything on one scale
The power of the z-score is that it strips away units. A value of '130' means nothing on its own — 130 what, from what distribution? Converting to a z-score of 2.0 says something universal: this value is two standard deviations above average, wherever it came from. That common scale is what makes different measurements comparable.
This is why standardisation underpins so much of statistics. It lets you ask whether a student did better on a maths test (mean 70, sd 10, scored 85) or an English test (mean 60, sd 5, scored 70) by comparing their z-scores — 1.5 versus 2.0 — rather than the raw marks, which are on incomparable scales. The higher z-score wins regardless of the original numbers.
The empirical rule
For data that follows a normal distribution, z-scores map neatly onto known proportions. About 68% of values fall within one standard deviation of the mean (z between −1 and 1), about 95% within two, and about 99.7% within three. This 68–95–99.7 rule is the quickest way to judge how unusual a value is.
It also frames rarity in concrete terms. A z-score beyond 2 puts a value in the outer 5% of the distribution; beyond 3, in the outer 0.3%. Scientific results are often held to a threshold of statistical significance expressed in these terms, and particle physics famously demands 'five sigma' — a z-score of 5, a one-in-3.5-million chance of arising by luck — before claiming a discovery.
Percentiles, and the normal-distribution caveat
The percentile answers the intuitive question: what fraction of the population is at or below this value? A z-score of 0 is the 50th percentile (the mean, in a symmetric distribution); a z-score of 2 is roughly the 98th. This is exactly how standardised test scores, growth charts, and IQ scales report where an individual sits relative to everyone else.
The one caveat is that percentiles and tail probabilities assume the data is normally distributed — the familiar bell curve. Many natural measurements are close enough for this to work well: heights, measurement errors, and test scores by design. But skewed data, like income or reaction times, has a long tail, and there a z-score still measures distance from the mean correctly while the percentile derived from the normal curve becomes only approximate. Know your distribution before trusting the percentile.
Perguntas frequentes
What is a z-score?
A z-score, or standard score, is the number of standard deviations a value lies from the mean: z = (x − μ) ÷ σ. A z-score of 0 is exactly average, +2 is two standard deviations above, and −1.5 is one and a half below. It puts any value on a universal scale.
How do I convert a z-score to a percentile?
Look up the cumulative probability of the z-score on the standard normal distribution, which this calculator does. A z of 2.0 corresponds to about the 97.7th percentile, meaning roughly 97.7% of a normal population scores at or below that value.
What is a good or unusual z-score?
About 68% of values fall within ±1, 95% within ±2, and 99.7% within ±3 (the empirical rule). So a z-score beyond 2 is somewhat unusual, beyond 3 is rare, and beyond 5 is the 'five sigma' threshold physicists require to announce a discovery.
Does the data have to be normally distributed?
The z-score itself just measures distance from the mean and works for any data. But converting it to a percentile or probability assumes a normal distribution. For strongly skewed data those percentiles become approximate, so check the shape of your distribution first.