Mathematics · Statistics
Z-Score & Normal Distribution Calculator — Percentile and Probability
Turn a value into a z-score with the mean and standard deviation, and get the probability below it, above it and its percentile. Find the probability between two values, or go backwards from a percentile (or probability) to the value and z-score — the normal-distribution table, without the table.
Your inputs
Quick start — try a typical case
Use 0 and 1 to work directly with z-scores.
Your result Live
Z-score
2
130 is 2 standard deviations above the mean
- Percentile
- 97.725th — higher than 97.72% of values
- P(X < x)
- 0.97725 (97.725%)
- P(X > x)
- 0.0227501 (2.275%)
- P(within ±|z| of the mean)
- 0.9545 (95.45%)
How this is calculated
z = (x − μ) ÷ σ · P(X < x) = Φ(z) · x = μ + z·σ for a percentile
- Φ
- Standard normal cumulative distribution
A z-score says how many standard deviations a value is from the mean. On a normal (bell) curve, the area to the left of a z-score is the share of values below it — its percentile. About 68 % of values lie within 1 standard deviation, 95 % within 1.96, and 99.7 % within 3.
Assumptions & sources
- The data follow a normal distribution.
- 1NIST/SEMATECH e-Handbook of Statistical Methods — US National Institute of Standards and Technology
- 2Mathematics, Classes 11–12 — Probability and Statistics — NCERT
Calculator ID MATH-ZSCORE-001 · version 1.0.0
Common questions
What does a z-score of 2 mean?
The value is 2 standard deviations above the mean — higher than about 97.7 % of values.
An IQ of 130 is what percentile?
With mean 100 and SD 15, z = 2, so about the 97.7th percentile.
What z-score is the 95th percentile?
1.645 (one-sided); 1.96 cuts off the top and bottom 2.5 % each.