How many standard deviations from the mean?
A z-score strips the units off a value and answers one question: how far from the average does this sit, measured in standard deviations? A test score of 85 means little until you know the class averaged 70 with a standard deviation of 10 — then it is a z of +1.5, better than about 93.3% of the class. This calculator converts a value to its z-score, percentile and tail probabilities (and back from a z-score), using a high-accuracy approximation of the normal CDF (error under 10−7).
Φ is the standard normal cumulative distribution. One-tailed probability is the area beyond z in one direction; the two-tailed figure doubles the smaller tail and is the one most hypothesis tests report.
Worked example
X = 85, μ = 70, σ = 10: z = (85 − 70) ÷ 10 = 1.5. Φ(1.5) ≈ 0.9332, so the percentile is 93.32% — about 6.68% of values sit above it (one tail), and 13.36% sit at least that far from the mean in either direction (two tails).
How to use this calculator
- Value mode: enter the value, the mean and the standard deviation of its distribution.
- Or switch to z mode and enter a z-score directly to get its percentile.
- Read the percentile as “percentage of the distribution below this point.”
- For a test or experiment, the one- and two-tailed probabilities are the p-value ingredients — the Standard Deviation Calculator next door computes σ from raw data.
Reading z-scores
z = 0 is exactly average. About 68% of a normal distribution lies within z = ±1, 95% within ±2 and 99.7% within ±3 — so a |z| past 2 is unusual and past 3 is rare, which is why z-scores run everything from growth charts to quality control to the p-values in research papers.