P-Value Calculator — Step-by-Step Solutions

Q: What does a p-value of 0.05 mean?

It means that if the null hypothesis were true, you would see a result at least this extreme about 5 times in 100 repeated experiments. It does not mean there is a 95% chance your hypothesis is right. Most fields use 0.05 as the conventional cut-off for calling a result statistically significant, but it is a convention, not a law.

Q: How do I find the p-value from a z-score?

Look up the area under the standard normal curve beyond your z-score. For a right-tailed test use the area to the right, for a left-tailed test the area to the left, and for a two-tailed test double the smaller of the two. This calculator does the lookup for you and shows all three values.

Q: Should I use a one-tailed or two-tailed p-value?

Use two-tailed unless you decided before collecting data that only one direction matters. A two-tailed p-value is twice the one-tailed value for a symmetric distribution, so switching to one-tailed after seeing the data to push p under 0.05 is not valid.

Q: Why does the t p-value depend on degrees of freedom?

The t distribution has heavier tails than the normal curve when the sample is small. The same t-statistic of 2.5 gives a two-tailed p of 0.0314 with 10 degrees of freedom, but a smaller value with 100. Degrees of freedom are usually n - 1 for a one-sample test.

Q: Can a p-value be exactly 0?

No. A p-value can be extremely small, which the calculator shows in scientific notation, but a continuous distribution always leaves some area in the tails. Report very small values as p < 0.001 rather than p = 0.

Q: Does a small p-value mean the effect is large?

No. With a large sample, tiny and unimportant differences can produce very small p-values. The p-value tells you about evidence against the null hypothesis, not the size of the effect. Check the effect size and a confidence interval as well.

The P-Value Calculator helps you determine statistical significance from test statistics. Compute p-values from z-scores (one-tailed or two-tailed), t-statistics with degrees of freedom, or chi-square statistics. The calculator provides step-by-step working and a visual shaded region showing the p-value area under the distribution curve.

Enter your test statistic, choose the distribution type and tail direction, and get instant results with significance interpretation at any alpha level.

Z-Score

Tail Type

Your calculations will appear here

How to Use the P-Value Calculator

Pick the mode that matches the statistic you already have: From Z-Score, From T-Statistic, From Chi-Square, or Significance Test.
For a z-score or t-statistic, type the value, choose the tail type (two-tailed, left-tailed or right-tailed), and for a t-statistic also enter the degrees of freedom. For chi-square, enter the statistic and its degrees of freedom. Chi-square is always right-tailed, so there is no tail option.
Read the p-value in the result panel. The z and t modes also list the left-tail, right-tail and two-tailed values side by side, so you can check which one your hypothesis needs. The curve below the result shades the area that the p-value measures.
To turn a p-value into a decision, switch to Significance Test, enter the p-value and your alpha (0.05 is the default) and the calculator tells you whether the result is significant and how strong the evidence is.

Formula and Theory

A p-value is the probability of seeing a test statistic at least as extreme as the one you observed, assuming the null hypothesis is true. It is not the probability that the null hypothesis is true, and it is not the probability that your result happened by chance alone. A small p-value says the data would be unusual if the null hypothesis held, which is evidence against it.

Which tail you use depends on the question asked before looking at the data. A two-tailed test asks whether the value differs in either direction and doubles the smaller tail area. A right-tailed test asks whether the value is larger than the null predicts; a left-tailed test asks whether it is smaller. The same statistic gives different p-values under different tails, so pick the tail from the hypothesis, not from the result.

For a z-score the area comes from the standard normal curve. For a t-statistic it comes from the t distribution, whose tails are heavier when the degrees of freedom are small; as the degrees of freedom grow it approaches the normal curve. For chi-square the p-value is the upper tail area of the chi-square distribution.

You reject the null hypothesis when p is smaller than the significance level alpha you chose in advance. Alpha = 0.05 means you accept a 5% risk of rejecting a true null hypothesis.

Worked Examples

Example 1: Two-tailed p-value from a z-score

Problem: A z-test gives z = 1.96. What is the two-tailed p-value?

Solution: The area to the left of z = 1.96 on the standard normal curve is 0.97500, so the right tail is 1 - 0.97500 = 0.02500. A two-tailed test counts both ends: 2 x 0.02500 = 0.05000.

Answer: p = 0.0500 (right tail 0.0250)

Example 2: Two-tailed p-value from a t-statistic

Problem: A t-test gives t = 2.5 with 10 degrees of freedom. Find the two-tailed and right-tailed p-values.

Solution: The right tail of the t distribution beyond 2.5 with 10 degrees of freedom is 0.01572. Two-tailed: 2 x 0.01572 = 0.03145. At alpha = 0.05 the two-tailed result is significant because 0.0314 < 0.05.

Answer: Two-tailed p = 0.0314; right-tailed p = 0.0157

Example 3: Right-tailed p-value from a z-score

Problem: A one-sided test of whether a new process raises output gives z = 2.33. What is the p-value?

Solution: The hypothesis points in one direction (output is higher), so use the right tail only. The area to the right of z = 2.33 is 1 - 0.99010 = 0.00990.

Answer: p = 0.0099, significant at alpha = 0.01

Example 4: P-value from a chi-square statistic

Problem: A chi-square test gives a statistic of 7.88 with 3 degrees of freedom. Find the p-value.

Solution: The chi-square p-value is the area to the right of 7.88 under the chi-square curve with 3 degrees of freedom. That area is 0.04856. The 5% critical value for 3 degrees of freedom is 7.815, and 7.88 is just above it, so p is just below 0.05.

Answer: p = 0.0486

Frequently Asked Questions

What does a p-value of 0.05 mean?

It means that if the null hypothesis were true, you would see a result at least this extreme about 5 times in 100 repeated experiments. It does not mean there is a 95% chance your hypothesis is right. Most fields use 0.05 as the conventional cut-off for calling a result statistically significant, but it is a convention, not a law.

How do I find the p-value from a z-score?

Look up the area under the standard normal curve beyond your z-score. For a right-tailed test use the area to the right, for a left-tailed test the area to the left, and for a two-tailed test double the smaller of the two. This calculator does the lookup for you and shows all three values.

Should I use a one-tailed or two-tailed p-value?

Use two-tailed unless you decided before collecting data that only one direction matters. A two-tailed p-value is twice the one-tailed value for a symmetric distribution, so switching to one-tailed after seeing the data to push p under 0.05 is not valid.

Why does the t p-value depend on degrees of freedom?

The t distribution has heavier tails than the normal curve when the sample is small. The same t-statistic of 2.5 gives a two-tailed p of 0.0314 with 10 degrees of freedom, but a smaller value with 100. Degrees of freedom are usually n - 1 for a one-sample test.

Can a p-value be exactly 0?

No. A p-value can be extremely small, which the calculator shows in scientific notation, but a continuous distribution always leaves some area in the tails. Report very small values as p < 0.001 rather than p = 0.

Does a small p-value mean the effect is large?

No. With a large sample, tiny and unimportant differences can produce very small p-values. The p-value tells you about evidence against the null hypothesis, not the size of the effect. Check the effect size and a confidence interval as well.