Statistics
P-Value Calculator
Use this p-value calculator to turn a z-score or t statistic from a hypothesis test into a p-value.
Pick the test type, enter the statistic, choose the tail that matches your alternative hypothesis, and read the p-value alongside significance decisions at alpha = 0.05 and alpha = 0.01. Every answer shows the CDF lookup and tail arithmetic, so it doubles as a worked solution for statistics homework.
Use z when the population standard deviation is known or the sample is large; use t otherwise.
The z or t value from your test, for example 1.96 or -2.31.
Match the tail to your alternative hypothesis: not-equal is two-tailed, less-than is left, greater-than is right.
Try an example
Result
p-value
—
Probability of a test statistic at least this extreme if the null hypothesis is true.
- Decision at alpha = 0.05
- —
- Decision at alpha = 0.01
- —
- CDF value used
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the p-value calculator solves
Use this The chance of seeing results this extreme if nothing real were going on. calculator to turn a How many standard deviations a value sits above or below the mean. or t statistic from a hypothesis test into a p-value. Pick the test type, enter the statistic, choose the tail that matches your alternative hypothesis, and read the p-value alongside significance decisions at alpha = 0.05 and alpha = 0.01. Every answer shows the CDF lookup and tail arithmetic, so it doubles as a worked solution for statistics homework.
Formula
How to use it
- Choose z-test or t-test; for a t-test, enter the degrees of freedom (df = n - 1 for one A smaller group taken from a population, used to estimate facts about the whole group.).
- Enter the test statistic from your hypothesis test, keeping its sign.
- Pick the tail that matches your alternative hypothesis: two-tailed for not-equal, left for less-than, right for greater-than.
- Read the p-value, then compare it to your significance level using the alpha = 0.05 and alpha = 0.01 decisions.
How to read the answer
The p-value is the probability, computed as if the null hypothesis were true, of getting a test statistic at least as extreme as yours. Small p-values (at or below alpha) mean the data would be surprising under H0, so you reject H0; larger p-values mean you fail to reject H0. The p-value is not the probability that H0 is true, and failing to reject is not proof that H0 is correct.
Common mistakes and edge cases
- Forgetting to double the tail probability for a two-tailed test — a two-tailed p is twice the one-tailed p for the same statistic.
- Using the z distribution when the Every member of the group you actually care about, not just the ones measured. A number saying how far, on average, the data values sit from the mean. is unknown and the sample is small; that calls for the t distribution with df = n - 1.
- Reading the p-value as the probability that the null hypothesis is true — it is a probability about the data, not about H0.
- Mixing up tail direction: a left-tailed test with a positive statistic (or right-tailed with a negative one) gives p > 0.5, which usually signals the wrong tail was selected.
- Reporting p = 0 for a huge test statistic — the true p-value is tiny but never exactly zero, so write p < 0.0001 instead.
Worked examples
Critical two-tailed z
z = 1.96 is the classic 5% two-tailed boundary.
p-value
0.049996
Right-tailed z test
z = 1.645 sits at the 5% right-tail boundary.
p-value
0.049985
Two-tailed t test
t = 2.045 with df = 29 lands almost exactly on the 5% cutoff.
p-value
0.050024
Negative statistic, left tail
A below-null result: z = -2.5 with a less-than alternative.
p-value
0.0062097
Zero statistic
z = 0 matches the null exactly, so the two-tailed p-value is 1.
p-value
1.0000
Extreme statistic
z = 10 is so far out that the p-value falls below the display floor.
p-value
< 1e-15
Frequently asked questions
What is a p-value?+
It is the probability, assuming the null hypothesis is true, of getting a test statistic at least as extreme as the one you observed. A p-value of 0.03 means data this extreme would show up about 3% of the time under H0.
Is p = 0.05 statistically significant?+
By the usual convention you reject H0 when p <= alpha, so p = 0.05 is right on the boundary at the 5% level. Most courses treat p <= 0.05 as significant at alpha = 0.05 and p <= 0.01 as significant at alpha = 0.01.
Should I use a one-tailed or two-tailed p-value?+
Match the alternative hypothesis. Use two-tailed for 'not equal to', left-tailed for 'less than', and right-tailed for 'greater than'. A two-tailed p-value is twice the one-tailed value for the same statistic, so the choice changes the conclusion near the boundary.
When do I use a t-test instead of a z-test?+
Use t when the population standard deviation is unknown and you estimated it from the sample, especially with small samples. With large df the t distribution is nearly identical to the standard normal, so the two p-values converge.
What degrees of freedom should I enter?+
For a one-sample or paired t-test, df = n - 1. For a two-sample pooled t-test, df = n1 + n2 - 2. Your test output or textbook formula tells you the df; this calculator accepts 1 through 1000.
Can a p-value ever be exactly 0?+
No. For any finite test statistic the tail area is positive, just possibly astronomically small. When the value drops below this calculator's 1e-15 display floor, it shows an inequality — report such results as p < 0.0001.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026