Z-Score Calculator

Free z-score and normal distribution calculator: convert a raw value to a z-score and percentile, a z-score to a probability, or a probability back to a z-score. Matches R’s pnorm and qnorm.

Work with the standard normal distribution three ways: turn a raw value into a z-score (and percentile), turn a z-score into a probability, or turn a probability into a z-score. The math matches R’s pnorm() and qnorm().

How to use it

What a z-score is

A z-score tells you how many standard deviations a value sits from the mean: \[z = \frac{x - \mu}{\sigma}\] A z of 0 is exactly average; +1 is one SD above the mean; −2 is two SDs below. Because the standard normal distribution is fixed, a z-score maps directly to a percentile — the share of the population below that value. A z of 1.96, for example, sits at the 97.5th percentile, which is why ±1.96 brackets the middle 95%.

  • Raw value → z-score: standardise a measurement so you can compare across different scales.
  • z-score → probability: find how extreme a value is (one- or two-tailed).
  • Probability → z-score: find the cut-off for a given percentile (e.g. the top 5%).

Do it in R

pnorm(130, mean = 100, sd = 15)   # percentile for a raw value
pnorm(1.96)                        # z-score to cumulative probability
qnorm(0.975)                       # probability back to z-score

FAQ

Frequently asked questions

What’s the difference between a z-score and a percentile?

A z-score is in standard-deviation units; a percentile is the share of the population below the value. They’re two views of the same thing — this calculator converts between them.

When can I use z-scores?

Standardising with z assumes an (approximately) normal distribution. For small samples where you estimate the SD, use the t-distribution instead.

Standardising is step one. When you need the full analysis and a written report, that’s our work.