Normal Distribution Z-Score Calculator
A z-score expresses how far a value sits from the mean in units of standard deviation, and the standard normal curve turns that into a probability. Enter a raw value, the mean, and the standard deviation to get the z-score and the left-tail, right-tail, and two-tail probabilities. This is the core step for percentiles, test scores, and process control.
Z-score and probability formula
z = (x - mean) / standard deviation
Left tail = CDF(z) = 0.5 * (1 + erf(z / sqrt(2)))
Right tail = 1 - CDF(z)
Two-tail = 2 * (1 - CDF(|z|))
CDF is the standard normal cumulative distribution function and erf is the error function. The two-tail probability is the chance of a value at least |z| standard deviations from the mean in either direction.
Worked example
For an IQ-style scale with mean 100 and standard deviation 15, a value of 130 gives z = (130 minus 100) divided by 15 = 2.00. The left-tail probability is CDF(2.00) = 0.9772, so about 97.72% of values fall at or below 130, and about 2.28% fall above it.
Z-score: frequently asked questions
What is a z-score?
A z-score measures how many standard deviations a value lies from the mean. It is found by subtracting the mean from the value and dividing by the standard deviation. A z-score of 0 sits at the mean; a z-score of 2 is two standard deviations above it.
How is the probability calculated?
The left-tail probability is the area under the standard normal curve to the left of the z-score, given by the cumulative distribution function. This calculator uses a high-accuracy approximation of the error function to evaluate it without lookup tables.
What do the tail probabilities mean?
The left tail is the chance of a value at or below your input. The right tail is the chance of a value at or above it, equal to one minus the left tail. The two-tail probability is the chance of a value at least as far from the mean in either direction.
What is the empirical rule?
For a normal distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. These correspond to z-scores of plus or minus 1, 2, and 3.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Normal distribution.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.