Convert a standard-normal Z-score into one-sided or two-sided probability areas, or work backward from any one area to Z. Enter exactly one field at a time; all other normal-curve probabilities are calculated together.
How to use this calculator
This tool uses the standard normal distribution only. Enter a Z-score and leave all probability fields blank. To work backward, clear the default Z-score and enter exactly one probability as a decimal strictly between 0 and 1: enter 0.05 for 5%, not 5. The zero-to-Z center area must also be below 0.5. The calculator returns all the listed areas.
Use the left-tail result for a lower-tail Z test, the right-tail result for an upper-tail Z test, and the two-tail result for a two-sided Z test. Center and between areas are normal-curve areas, not p-values for those tests. Choose the test direction before examining the result. There are no degrees-of-freedom inputs: a t, chi-square, or F statistic needs its own distribution rather than being entered as Z.
Examples
Z-score of 2
For Z = 2, P(x < Z) is 0.97725, the right-tail probability is
0.022750, and the combined two-tail probability is 0.045500.
The same magnitude with a negative sign
For Z = −2, the left-tail probability is about 0.022750 and the right-tail probability is 0.97725. The two-tailed probability remains 0.045500 because it uses the magnitude of Z. A one-sided result depends on which direction the alternative hypothesis specifies.
Formula
If Φ is the standard normal cumulative distribution function:
left tail = Φ(Z)
right tail = 1 − Φ(Z)
two tails = 2 × (1 − Φ(|Z|))
between = 2 × Φ(|Z|) − 1
When a probability is supplied, the calculator uses the inverse normal CDF to find the corresponding Z-score. A symmetric between-area or two-tail input determines the magnitude of Z; it cannot identify its original sign, so the inverse result uses the nonnegative value.
P-value interpretation
A p-value is calculated under the null model and is not the probability that the null hypothesis is true. It does not measure effect size, practical importance, study quality, or the probability that a result will replicate. Tail choice and test statistic must be specified before looking at the result; multiple testing needs an appropriate analysis plan. The American Statistical Association statement explains these principles.