Enter a raw score, mean, and standard deviation to compute the Z-score and its cumulative probability P(Z < z) — or enter a Z-score directly to find the corresponding raw score.
How to use this calculator
Enter the mean and standard deviation of your distribution.
In 'Raw score → Z-score' mode: enter the raw score x to get z and its percentile.
In 'Z-score → Raw score' mode: enter z to find the corresponding raw value.
Formula & method
Z-score formula and normal CDF
z = (x − μ) / σ · x = μ + z × σ · P(Z < z) computed from normal CDF
CDF computed via polynomial approximation (Abramowitz & Stegun). Accurate to ±5 × 10⁻⁴.
Z = 0 is exactly at the mean; Z = ±1 covers the middle 68.3% of a normal distribution.
Z = ±2 covers 95.4%; Z = ±3 covers 99.7% (the '68-95-99.7 rule').
Z-scores enable comparison across different scales: a test score and a height can be placed on the same standard scale.
The empirical rule: Z > 1.96 places you in the top 2.5% for a two-tailed test at α = 0.05.
How to interpret your result
Percentile
P(Z < z) × 100 gives the percentile: z = 1.5 means your value exceeds 93.3% of the distribution.
Standardisation
Z-scores remove units and scale differences, allowing direct comparison across datasets with different means and SDs.
Frequently asked questions
What is a Z-score?
The number of standard deviations a value is from the mean: z = (x − μ) / σ. A Z-score of 2 means 2 SDs above the mean.
What is a good Z-score?
Context-dependent. In a hypothesis test, |z| > 1.96 is statistically significant at α = 0.05. In grading, z = 1.5 puts you in the top ~7%.
What is the 68-95-99.7 rule?
In a normal distribution: 68.3% of values fall within ±1 SD, 95.4% within ±2 SD, 99.7% within ±3 SD of the mean.
Related calculators
Standard Deviation Calculator — Calculate sample and population standard deviation, variance, standard error, and coefficient of variation for any dataset.
Confidence Interval Calculator — Calculate confidence intervals for a sample mean or proportion at 80%, 90%, 95%, 98%, or 99% confidence levels.
P-value Calculator — Calculate the P-value from a Z-statistic for one-tailed or two-tailed hypothesis tests and determine statistical significance.