Z-Score to Probability Calculator
A z-score tells you where a value sits relative to the mean of a normal distribution, measured in standard deviations. This calculator turns that z-score into a probability: the area under the standard normal bell curve to the left of your value, which is the chance of seeing a result less than or equal to it. That cumulative probability is what statistical tables of the standard normal distribution provide, and it is the foundation of confidence intervals, hypothesis tests and percentile rankings. Because the normal cumulative distribution function has no simple algebraic form, it is evaluated through the error function, a closely related special function with high-accuracy numerical approximations. This tool uses one of those approximations, so the figure it reports is computed deterministically rather than estimated by eye from a printed table, and the worked example reconciles exactly with the value on screen. Enter any z-score, positive or negative, and read off the left-tail probability; the right-tail and two-tailed probabilities follow by simple subtraction, as explained below. The classic value of 1.96, which returns about 0.9750, is the boundary behind the 95 percent confidence level. Use the tool to convert results into percentiles or to check a critical value.
This tool returns the standard normal cumulative probability, the area to the left of a z-score: P = 0.5 x (1 + erf(z / sqrt(2))). For a z-score of 1.96, the cumulative probability is 0.9750, so about 97.50 percent of values fall below it.
Normal CDF formula
P(Z < z) = 0.5 x ( 1 + erf( z / sqrt(2) ) )
z = the z-score (standard deviations from the mean)
erf = the error function
sqrt(2) = the square root of 2 (about 1.414214)
The cumulative probability is the area under the standard normal curve up to z. It is computed from the error function, which is built into scientific software, because the normal cumulative distribution function cannot be written in elementary terms.
Worked example
Find the cumulative probability for a z-score of 1.96.
- z / sqrt(2) = 1.96 / 1.414214 = 1.385929
- erf(1.385929) = 0.950004
- P = 0.5 x (1 + 0.950004) = 0.975002
- Rounded: P(Z < 1.96) = 0.9750
About 97.50 percent of a normal distribution lies below a z-score of 1.96, and 2.50 percent lies above it. This is the calculator's default input, so the result above matches the widget exactly.
Common z-scores and cumulative probability
| Z-score | Left-tail P | Right-tail P |
|---|---|---|
| 0.00 | 0.5000 | 0.5000 |
| 1.00 | 0.8413 | 0.1587 |
| 1.645 | 0.9500 | 0.0500 |
| 1.96 | 0.9750 | 0.0250 |
| 2.576 | 0.9950 | 0.0050 |
Reference: US National Institute of Standards and Technology (NIST).
Z-score calculator: frequently asked questions
What is a z-score?
A z-score measures how many standard deviations a value sits above or below the mean of a normal distribution. A z-score of zero is exactly at the mean, a positive z-score is above it, and a negative z-score is below it. Standardizing values into z-scores lets you compare measurements from different distributions on one common scale.
What probability does this calculator return?
It returns the cumulative probability, the area under the standard normal curve to the left of your z-score. This is the chance of observing a value less than or equal to that z-score. For a z-score of 1.96 the cumulative probability is about 0.9750, meaning roughly 97.50 percent of the distribution lies below it.
How is the normal CDF computed?
The cumulative distribution function of the standard normal has no simple closed form, so it is evaluated using the error function: the probability equals one half times one plus the error function of z divided by the square root of two. This calculator uses a high-accuracy numerical approximation of the error function, the same approach used in statistical software.
How do I get the right-tail or two-tail probability?
The right-tail probability (the area above your z-score) is one minus the cumulative probability shown here. A two-tailed probability for a symmetric interval is two times the smaller tail. For z equal to 1.96, the right tail is about 0.0250 and the two-tailed area outside is about 0.0500, which is why 1.96 marks the 95 percent confidence boundary.
Why is the standard normal distribution so common?
By the central limit theorem, sums and averages of many independent effects tend toward a normal shape regardless of the original distribution. Converting to z-scores then lets a single standard normal table or function answer probability questions for a wide range of real measurements, from test scores to manufacturing tolerances.
Official sources
- Statistics and the normal distribution reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.