Z-Score (Standard Score) Calculator
A z-score (or standard score) expresses how many standard deviations a value sits from its mean, letting you compare measurements drawn from different distributions on a common scale. Enter the value, the mean and the standard deviation to get Z, the normal-distribution percentile and one- and two-sided p-values.
Default is IQ 130 (mean 100, SD 15) → Z = 2, ~97.7th percentile — about the top 2–3 in every 100.
Enter finite numbers; standard deviation must be greater than 0.
Z-score
2.00
Positive = above mean, negative = below. |Z| ≤ 1 is common, ±2 is notable, ±3 is rare.
Percentile (normal distribution)
97.7%
Share of the population below x. E.g. 95 means "higher than 95 % of values".
One-sided p (≤ x)
One-sided p (≥ x)
Two-sided p (|Z| ≥ |z|)
Probability of drawing a value at least this extreme, assuming a normal distribution.
Formula
Z = (x − μ) ÷ σ; percentile = Φ(Z) × 100%
- · x is the observed value, μ the population (or sample) mean, σ the population standard deviation.
- · Φ is the standard-normal cumulative distribution function. We evaluate it from erf using Abramowitz & Stegun 7.1.26 (max error ≈ 1.5×10⁻⁷).
- · Two-sided p = 2·(1 − Φ(|Z|)). Standard rule of thumb: |Z| ≥ 1.96 ↔ p ≈ 0.05.
- · Percentiles only reflect reality when the underlying distribution is approximately normal; for skewed data prefer non-parametric methods.
- · If you only have a sample SD with a small n, use the t-distribution instead — this tool assumes σ is known or n is large.
Frequently asked
What is the difference between a z-score and a percentile?
A z-score is a continuous scale that can be positive or negative — it tells you how many σ a value sits from the mean. A percentile is a 0–100% rank — the share of the distribution below it. They map one-to-one: Z = 0 ↔ 50%, Z = 1 ↔ ~84%, Z = 1.96 ↔ 97.5%. This calculator shows both side-by-side.
When should I use a one- vs. two-sided p-value?
Use a two-sided p-value when either tail counts as a deviation worth flagging — e.g. quality control where both too-heavy and too-light are problems. Use one-sided only when the direction of interest is pre-specified (e.g. "this drug can only raise the marker, not lower it"). In practice, two-sided is the conservative default for most research.
Why is an IQ of 130 at the 97.7th percentile rather than the 95th?
IQ is standardised at mean 100, SD 15. An IQ of 130 gives Z = (130 − 100) / 15 = 2, and Φ(2) ≈ 0.9772 — the 97.7th percentile. Mensa's "top 2 %" cutoff is Z ≈ 2.05 (IQ ≈ 131). The 95 % figure people sometimes cite is shorthand for the two-sided Z = 1.645 boundary, not the actual percentile of an IQ-130 score.
Can the mean or standard deviation be negative?
The mean can be any real number, including negative — fully supported. The standard deviation, however, must be strictly positive: σ = 0 would mean every observation is identical (no variation at all) and σ < 0 makes no physical sense. The tool flags zero or negative SD as an input error.
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