P-Value Calculator

1. Pick the distribution. Choose z for a z test. Choose t for a t test, then enter the degrees of freedom.
2. Pick the tail that matches your alternative hypothesis. Not equal is two-tailed, less than is left-tailed, and greater than is right-tailed.
3. Type your test statistic. You can also change alpha, which is 0.05 unless you pick another value.
4. Read the p-value, the decision, and the critical values. The bell curve shades the area that makes up the p-value.

For example, a t test with 11 observations and t = 2.228 uses df = 11 - 1 = 10. Choose t, two-tailed, and enter 2.228 and 10.

Every hypothesis test starts with a null hypothesis, H0. It usually says there is no effect or no difference. The p-value measures how well your data fit with H0.

The key word is if. The p-value is worked out by pretending H0 is true. Then it asks how often chance alone would give a result as far out as yours, or farther.

Here is an example. Suppose a z test gives z = 2.576. If H0 were true, a z value this far from 0 in either direction happens only about 1% of the time. So p is about 0.01, and the data look unusual under H0.

The tail tells the calculator which results count as extreme. It must match the alternative hypothesis H1, and you should choose it before you look at the data.

A two-tailed p-value is twice the smaller tail. That is why z = 1.645 gives p = 0.049985 right-tailed but p = 0.099970 two-tailed. Switching to one tail after seeing the data, just to get a smaller p, is not allowed.

Both statistics compare a sample mean with a claimed mean. The difference is the standard deviation in the bottom of the fraction.

Use z when the population standard deviation, sigma, is known. Use t when you estimate it with the sample standard deviation, s. That is the usual case in practice.

The t curve has heavier tails than the normal curve, so the same statistic gives a larger p-value. The gap is biggest for small samples. For t = 2.0 with df = 10, two-tailed p = 0.073388. For z = 2.0, p = 0.045500.

As df grows, the t values move toward the z values. Our z-score calculator helps you find a z statistic first.

Alpha is the cutoff you set before the test. It is also the chance of rejecting H0 when H0 is actually true, called a Type I error.

You can reach the same decision with critical values. Reject H0 when your statistic lands past the critical value, in the shaded tail. For alpha = 0.05 two-tailed with z, the critical values are -1.959964 and 1.959964.

Fail to reject is not the same as accept. It only means the data did not give strong enough evidence against H0. A small sample can miss a real effect.

Watch results near the cutoff. For t = 2.228 with df = 10, p = 0.050012. That is just above 0.05, so you fail to reject H0, even though a rounded table value looks like a tie.

1. The P-Value Is Not the Chance That H0 Is True

The p-value assumes H0 is true. So p = 0.03 does not mean a 3% chance that H0 is right. The ASA statement on p-values makes this point directly.

2. Significant Does Not Mean Important

A tiny effect can give a small p-value when the sample is huge. Always look at the size of the effect, such as the difference in means, and its confidence interval.

3. A Large P-Value Does Not Prove H0

p = 0.40 only says the data fit fine with H0. Other hypotheses may fit just as well.

4. Do Not Test Until Something Works

Running many tests, or trying other tails and subsets, will turn up small p-values by chance. Report every test you ran.

A p-value tells you how surprising your data would be if the null hypothesis were true. It is the chance of getting a test statistic at least as extreme as the one you got, assuming H0 is true. A small p-value means your result would be rare under H0.

It means that if H0 were true, a result this extreme would happen less than 5% of the time. With a significance level of 0.05, you would reject H0. It does not prove the alternative is true, and it says nothing about how large or important the effect is.

No. The p-value is calculated by assuming H0 is true, so it cannot also be the chance that H0 is true. A p-value of 0.03 means data this extreme would show up 3% of the time under H0. It is not a 3% chance that H0 is correct.

Use a two-tailed test unless you decided before looking at the data that only one direction matters. A one-tailed test puts all of alpha in one tail, so it is easier to reach significance in that direction but it cannot detect an effect in the other direction.

Use t when you estimate the population standard deviation from your sample, which is the usual case. This matters most for small samples. Use z when the population standard deviation is truly known, or for a large-sample test of a proportion.

For a one-sample or paired t test, df = n - 1, where n is the number of observations or pairs. For a pooled two-sample t test, df = n1 + n2 - 2. The Welch test can give a decimal df, and this calculator accepts decimal values of 1 or more.

Very small p-values round to 0.000000 at 6 decimals, which could look like zero. A p-value is never exactly 0, so the calculator shows it as less than 0.000001 and also gives the value in scientific notation, such as 1.973e-9 for z = 6 two-tailed.

• NIST/SEMATECH e-Handbook of Statistical Methods, section 7.1.3.1: Critical values and p values (how critical values and p-values lead to a decision).
• NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.2: Are the data consistent with the assumed process mean? (one-sided and two-sided tests; t with N - 1 df when sigma is estimated).
• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7.2: Critical Values of the Student's t Distribution (t table used to check critical values).
• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1: Normal Distribution (normal density and cumulative distribution).
• Wasserstein, R. L., and Lazar, N. A. (2016). The ASA Statement on p-Values: Context, Process, and Purpose. The American Statistician, 70(2), 129-133 (what p-values do and do not mean).
• Illowsky, B., and Dean, S. Introductory Statistics 2e. OpenStax, including section 8.2 on the Student t distribution and chapter 9 on hypothesis testing.