How many standard deviations a value sits from the mean, and the percentile that puts it at.
This updates immediately as you edit any of the three fields above.
A z-score is how many standard deviations a value sits from the mean: z = (x − μ) ÷ σ. For the default scenario, a score of 85 with a class mean of 75 and an SD of 8: z = (85 − 75) ÷ 8 = 1.25, which lands at roughly the 89th percentile.
These come from the same cumulative-normal calculation running the panel above, at a handful of values worth memorizing:
The 1.645 row is why 95% one-tailed thresholds and 90% two-tailed confidence levels use the same number; see the confidence interval calculator for that connection worked through, or the normal distribution calculator for both-tail probabilities in one view.
Z-scores are also how test results from different scales get compared honestly. A score of 85 on one exam and 640 on another are meaningless side by side until you convert both to standard deviations from their own mean; whichever z-score is larger represents the stronger relative performance, regardless of how different the original point scales were.
Same underlying math, different framing. This page solves for a single z-score and its percentile from a value, mean, and SD. The normal distribution calculator reports both P(X less than x) and P(X greater than x) as separate two-tailed figures. Use whichever output shape your worksheet or write-up actually asks for.
Yes. A negative z-score just means the value sits below the mean rather than above it; the size of the number still tells you how many standard deviations away it is, the sign only tells you which direction.
Very. A z of 3 lands at roughly the 99.87th percentile, meaning fewer than 2 in 1,000 values would be expected to sit that far above the mean in a genuinely normal distribution. Most fields treat |z| above 2 as worth a second look and above 3 as a likely outlier.
The calculator only blocks an SD of exactly zero; it does not check for a negative one. A negative SD is not meaningful (standard deviation can't be less than zero), and entering one will flip the sign of your z-score silently rather than showing an error. Always enter SD as a positive number.