P-Value To Z-Score Calculator

What the p-value to z-score calculator does

This calculator runs the standard normal curve in reverse. You give it a probability, and it finds the z-score that cuts off that much area in the tail. It is the inverse of turning a z-score into a p-value, and it answers the question of how far out a value has to be to be that unlikely.

Its main job is finding critical values, the z-score cut-offs that a test statistic has to beat to count as significant. Below is how it works and where it fits.

How to use it

1. Enter the p-value, a probability between 0 and 1.
2. Choose one-tailed or two-tailed to match your test.
3. Press Calculate for the z-score, or Reset to clear it.

How the z-score is worked out

The calculator uses the inverse of the normal distribution's area function. Where the forward direction takes a z-score and returns the tail area beyond it, this takes the tail area you want and returns the z-score that produces it. It is asking which point on the bell curve leaves exactly this much probability out in the tail.

For a one-tailed value, it finds the z-score with your chosen probability in a single tail. For a two-tailed value, it splits the probability across both tails and returns the matching cut-off, which is why the two-tailed z-score for a given p-value is the larger of the two.

One-tailed or two-tailed

The tail choice should match the test you are running. A one-tailed cut-off puts all of the probability in one tail, which suits a directional hypothesis where you are only interested in a change in one direction. A two-tailed cut-off divides the probability between the two tails, for a test that counts a difference in either direction.

This is why the same p-value gives two different z-scores. For a probability of 0.05, the one-tailed cut-off is about 1.645, while the two-tailed cut-off is about 1.96, because each tail only holds half the probability and so has to sit further out.

Critical values in testing

The z-score this calculator returns is a critical value: the line a test statistic must cross to be called significant at your chosen level. If your test's z-score is larger than the critical value, the result falls in the tail beyond the cut-off, and you reject the idea that only chance was at work.

This is the mirror image of the p-value approach. Instead of turning your test statistic into a p-value and comparing it to a threshold, you turn the threshold into a critical z-score and compare it to your statistic. Both roads lead to the same decision, and this calculator gives you the critical value for the second one.

A worked example

Suppose you are working at a significance level of 0.05.

Entering 0.05 as a one-tailed p-value returns a z-score of about 1.645, the cut-off for a directional test. Entering it as a two-tailed p-value returns about 1.96, since each tail then holds only 0.025. These two numbers, 1.645 and 1.96, are the critical values you meet most often in introductory statistics.

Entering the p-value

Enter a p-value strictly between 0 and 1, and pick the tail that matches your test. The z-score is shown to several decimal places with trailing zeros trimmed. If you want to go the other way, from a z-score to its p-value, our z-score to p-value calculator handles that direction.

Questions people ask

What does this calculator find?

The z-score that leaves a given probability in the tail of the standard normal curve. It is the reverse of converting a z-score into a p-value.

What is a critical value?

The z-score cut-off a test statistic must exceed to be significant at a chosen level. If your statistic is beyond it, the result is called significant.

Why do one-tailed and two-tailed give different z-scores?

Because a two-tailed test splits the probability between both tails, so each tail holds only half and its cut-off sits further out. For 0.05, that is 1.645 one-tailed versus 1.96 two-tailed.

Where do 1.645 and 1.96 come from?

They are the z-score cut-offs for a probability of 0.05, one-tailed and two-tailed. They appear constantly because 0.05 is the usual significance level.

References

A quick note on where the methods here come from. Critical values and the inverse of the normal distribution are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering critical values and hypothesis testing.

1. NIST/SEMATECH e-Handbook of Statistical Methods (percent points of the normal distribution). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (critical values and hypothesis testing). https://openstax.org/details/books/introductory-statistics-2e