Z-score to P-value Calculator

Using the Z-score to P-value calculator

A Z-score tells you how many standard deviations an observation sits above or below what the null hypothesis predicts. This calculator turns that distance into a probability: the chance of seeing a result at least as extreme if the null hypothesis were true.

1. Type your Z-score into the first box. It can be positive or negative and may have decimals, for example 1.65 or -2.33.
2. Pick One-tailed if you only care about one direction of difference, or Two-tailed if a difference in either direction would count as evidence against the null.
3. Press the button to read the p-value. If you need the reverse direction, use the P-value to Z-score calculator.

Because the standard normal curve is symmetric, the sign of the Z-score does not change the answer: -1.96 and 1.96 give the same p-value. The sign only tells you which tail the observation fell in.

The p-value is the exact normal tail area, worked out to about 15 significant digits and shown to ten decimals. A Z-score of 0 therefore gives exactly 0.50 one-tailed and 1 two-tailed. Anything smaller than 0.00000000005 is rounded away and prints as 0.

What is a p-value?

A p-value is the probability of getting data at least as extreme as yours, assuming the null hypothesis is true. It is a statement about how surprising the data would be in a world where nothing interesting is going on, not about how likely any hypothesis is to be correct.

For a Z-test, the test statistic follows the standard normal distribution under the null hypothesis, so the p-value is simply an area under that bell curve. The area beyond your Z-score in one direction is the one-tailed p-value; counting both ends of the curve gives the two-tailed value.

one-tailed p = 1 − Φ(|z|)Φ is the cumulative distribution function of the standard normal distribution.

two-tailed p = 2 × (1 − Φ(|z|))

Worked example. A machine is supposed to fill bags with a mean of 50 g and a standard deviation of 8 g. A sample of 64 bags averages 52.1 g. The standard error is 8 ÷ √64 = 1, so z = (52.1 − 50) ÷ 1 = 2.1. The area beyond 2.1 is about 0.0179, so the one-tailed p-value is 0.0179 and the two-tailed p-value is 0.0357. At a 5% significance level the two-tailed result would be called significant, and the same data would still be significant at the 4% level but not at 3%.

If you are starting from raw data instead of a Z-score, the Z-score calculator helps you get the statistic first.

How to interpret a low p-value

A small p-value means the data would be unusual if the null hypothesis were true. Researchers compare it with a threshold chosen in advance, the significance level α, often 0.05. When p is below α the result is called statistically significant and the null hypothesis is rejected. Otherwise you fail to reject it, which is not the same as proving it true.

A low p-value can arise for three reasons: the null hypothesis is false, a rare event happened by chance, or the model behind the test (independence, normality, the correct variance) is wrong. The p-value alone cannot tell these apart, which is why design and context matter. The P-value significance calculator shows how a p-value compares with common thresholds.

• p < 0.05 is a conventional cut-off, not a law of nature.
• Stricter fields use 0.01, 0.001 or even 5 sigma (about 3 in 10 million) before claiming a discovery.
• Always report the effect size and a confidence interval next to the p-value.

Common misinterpretations of p-values

• It is not the probability that the null hypothesis is true. It is the probability of the data (or more extreme data) given the null.
• It is not the probability that the result happened by chance alone. That phrase hides the assumption that the null is true.
• A large p-value is not evidence of no effect. A small sample or a noisy measurement can produce a large p-value even when a real effect exists.
• A tiny p-value does not mean a big effect. With enough data, trivial differences become significant.
• 0.05 is not a cliff. A result with p = 0.049 and one with p = 0.051 carry nearly the same evidence.

Z-score to p-value lookup table

These values come from the exact standard normal distribution, rounded to six decimals. They are the critical values you meet most often in practice.

The familiar rule of thumb follows directly: a two-tailed test at the 5% level rejects when |z| exceeds 1.96, and a one-tailed test at the same level rejects when z exceeds 1.645.

One-tailed versus two-tailed p-values

A one-tailed test asks whether the effect goes in a specific direction, such as whether a new process is faster than the old one. The whole rejection region sits in one tail, so for the same Z-score the p-value is half that of a two-tailed test.

A two-tailed test asks only whether there is a difference, in either direction. It splits the rejection region between both tails and is the safer default when you have no strong reason to expect one direction.

Choose the direction before you look at the data. Switching to a one-tailed test after seeing which way the result went halves the p-value without any new evidence and inflates the false-positive rate.

For other test statistics use the matching tool, for example the chi-square to p-value calculator, or plot the full curve with the normal distribution calculator.