Z-Score To P-Value Calculator

What the z-score to p-value calculator does

A p-value is the probability of seeing a result at least as extreme as yours, if nothing but chance were at work. This calculator turns a z-score into that probability, using the standard normal curve, and lets you choose a one-tailed or two-tailed answer.

It is the step that takes you from a test statistic to a decision about significance. Below is how it works and the choice of tail that changes the answer.

How to use it

1. Enter the z-score from your test.
2. Choose one-tailed or two-tailed to match your hypothesis.
3. Press Calculate for the p-value, or Reset to clear it.

Enter the z-score as a positive value, its size, since the p-value depends on how far the score is from zero rather than on its direction.

How the p-value is worked out

The p-value is an area under the standard normal curve, the bell curve with a mean of zero and a standard deviation of one. The calculator finds where your z-score sits on that curve and measures the area in the tail beyond it, which is the probability of landing that far out or further by chance.

For a one-tailed test it measures the area in a single tail. For a two-tailed test it measures both tails, since a result that extreme could have fallen either side, so the two-tailed p-value is simply twice the one-tailed one.

One-tailed or two-tailed

The choice comes down to the question you asked before you saw the data. Use a one-tailed test when your hypothesis has a direction, when you specifically expected an increase, or specifically a decrease. All the probability sits in that one tail.

Use a two-tailed test when you only expected a difference, without saying which way. Then a surprising result in either direction counts, so the probability is split across both tails and the p-value is doubled. Two-tailed is the more cautious and more common default, because it does not assume you knew the direction in advance. Picking the tail after seeing which way the data went is a well-known way to fool yourself, so decide it up front.

What the p-value means

A small p-value means your result would be unlikely if only chance were operating, which is taken as evidence against the idea that nothing is going on. The common threshold is 0.05, below which a result is often called statistically significant, though that line is a convention, not a law of nature.

It is worth being clear on what the p-value is not. It is not the probability that your hypothesis is true, and a small p-value does not measure how large or important an effect is, only how surprising the result would be under pure chance. A tiny effect can be significant with enough data, and a big one can miss significance with too little.

A worked example

Suppose your test gives a z-score of 1.96.

The area in one tail beyond 1.96 is about 0.025, so the one-tailed p-value is 0.025. Doubling it for both tails gives a two-tailed p-value of about 0.05. That is the famous pairing: a z-score of 1.96 sits right at the 0.05 two-tailed threshold, which is why 1.96 turns up so often in statistics.

Entering the z-score

Enter the z-score as a positive number representing its magnitude, and choose the tail that matches your hypothesis. The p-value is shown to several decimal places with trailing zeros trimmed. If you have a p-value already and want the z-score behind it, our p-value to z-score calculator runs the conversion the other way.

Questions people ask

What is a p-value?

The probability of getting a result at least as extreme as yours if only chance were at work. A small p-value suggests the result is unlikely to be chance alone.

What is the difference between one-tailed and two-tailed?

A one-tailed test puts all the probability in a single tail and suits a directional hypothesis. A two-tailed test uses both tails for a difference in either direction, giving a p-value twice as large.

What p-value is significant?

By common convention, below 0.05, though this is a threshold people agree on rather than a hard rule. The right cut-off depends on the field and the stakes.

Does a small p-value mean a big effect?

No. It measures how surprising the result would be under chance, not how large the effect is. With enough data even a tiny effect can produce a small p-value.

References

A quick note on where the methods here come from. The link between a z-score and a tail probability of the normal distribution is 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 p-values and hypothesis testing.

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