P-Value Calculator – Turn a z Score or t Score Into a Clear Verdict
A test statistic on its own does not tell you whether a result is real or just luck. The p-value does: it is the probability of seeing a result at least this extreme if nothing were really going on. Type in your z or t statistic, pick your tail, and get the p-value with the significance verdict at your level.
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Result
The numbers behind it
Below 0.05 is the common rule of thumb for calling a result statistically significant. It is a guide, not a proof: a small p-value can still come from a fluke, and a large one does not prove there is no effect.
- A p-value is the probability of seeing a result at least this extreme if nothing were really going on — z = 1.96 two-tailed gives 0.0500, Significant.
- The verdict follows the unrounded value: z 1.96 is Significant while t 2.262 with df 9 is Not significant, although both display 0.0500.
- Pick your tail before you look — two-tailed is the honest default — and remember significance is not importance.
- Classic z cut-offs, at a glance
- How it works, in three steps
- How the calculation works
- Going deeper
- Reading a p-value without over-reading it
- Why two results can both display 0.0500 and get different verdicts
- When to use a t test instead of a z test
- A checklist for reporting any p-value from this page
- Significance is not importance, and other sentences to retire
- Classic cut-offs, and why they are landmarks, not walls
- What to do with a borderline p-value
- Frequently asked questions
Classic z cut-offs, at a glance
These are the reference values every statistics textbook leans on. A two-tailed z of 1.96 sits exactly on the 0.05 line, which is why 1.96 shows up in so many confidence intervals.
| z statistic | Area below (Φ) | Two-tailed p-value |
|---|---|---|
| 1.000 | 0.8413 | 0.3173 |
| 1.645 | 0.9500 | 0.1000 |
| 1.960 | 0.9750 | 0.0500 |
| 2.576 | 0.9950 | 0.0100 |
| 3.000 | 0.9987 | 0.0027 |
Related tools
How it works, in three steps
Pick the right test
Use z when the population standard deviation is known or the sample is large. Use t when the sample is small and the standard deviation is estimated from your own data, and type in the degrees of freedom.
Pick a tail before you look
Two-tailed is the honest default: it counts extreme results in both directions. A one-tailed test only counts one direction, so it should be chosen before seeing the data, not after.
Read the verdict, not just the number
Below your significance level, the result is statistically significant. It is still only a guide: a small p-value is not a proof, and a large one is not a proof of no effect.
How the calculation works
Two-tailed p = 2 × (1 − Φ(|z|)) · Right-tailed p = 1 − Φ(z) · Left-tailed p = Φ(z)
Worked example. z = 1.96, two-tailed. The normal curve area below 1.96 is Φ = 0.9750, so the area above is 1 − 0.9750 = 0.0250. Both tails together give p = 2 × 0.0250 = 0.0500, which sits exactly on the usual 0.05 line.
Second example. z = 2.58, two-tailed. Φ(2.58) = 0.9951, so the area above is 0.0049, and p = 2 × 0.0049 = 0.0099 — significant at 0.05 and at 0.01.
t test example. t = 2.262 with df = 9, two-tailed. The t distribution with 9 degrees of freedom gives p = 0.0500, the classic critical value for a 0.05 two-tailed t test at df 9.
One-tailed example. z = 1.645, right-tailed. Φ(1.645) = 0.9500, so p = 1 − 0.9500 = 0.0500. Notice that the same number 1.645 is the 0.10 cut-off two-tailed but the 0.05 cut-off one-tailed: counting one tail halves the p-value.
Going deeper
The 0.05 line means: if nothing were really going on, a result at least this extreme would still turn up about 1 time in 20 by luck alone. That is why 20 tests on pure noise will typically hand you one false significant result. Significance is also not the same as importance — a tiny, meaningless effect can be statistically significant with a huge sample, and a big, real effect can miss 0.05 with a tiny one. Always read the p-value next to the size of the effect.
A two-tailed test asks whether the result is unusual in either direction, and that is the safe default for most work. A one-tailed test only asks about one direction, so it gives a smaller p-value for the same statistic — z = 1.645 is 0.1000 two-tailed but 0.0500 right-tailed. That smaller number is only honest if the direction was fixed before the data were collected. Switching to one tail after seeing which way the result went is moving the goalposts, and it roughly doubles the false-positive rate.
Reading a p-value without over-reading it
A p-value is the probability of seeing a result at least as extreme as yours if nothing were really going on. That sentence is worth reading twice, because almost every misreading of a p-value comes from dropping one of its clauses. It is not the probability that nothing is going on. It is not the probability that your result is wrong. It is the probability of the data, or more extreme data, under the no-effect story. A small p-value says your result would be unusual under that story. A large one says it would be quite ordinary. Neither one, on its own, proves or disproves anything.
The default case on this page sits exactly on the line that most readers care about. A z statistic of 1.96 with a two-tailed test gives a p-value of 0.0500, with the normal curve area below 1.96 equal to 0.9750. The formula is two-tailed p equals 2 times (1 minus Phi of the absolute z), so 2 times (1 minus 0.9750) is 2 times 0.0250, which is 0.0500. The page calls this result Significant at the 0.05 level, because the unrounded p-value sits just below the line even though the rounded display reads 0.0500. That rounding detail matters, and it returns below in the section on the verdict line.
Move the statistic and the p-value moves fast. A z of 2.58, two-tailed, gives a p-value of 0.0099: Phi of 2.58 is 0.9951, the area above is 0.0049, and doubling it for two tails gives 0.0099. That result is significant at 0.05 and at 0.01, because 0.0099 is below both levels. The jump from 1.96 to 2.58 looks small on the statistic scale but cuts the p-value by a factor of five, because the normal curve’s tails thin out quickly. That is why reporting the statistic alongside the p-value matters: two results with similar p-values can come from very different statistics, and two similar statistics can give very different p-values once you move into the tails.
Tails change the answer for the same statistic, and the choice of tail must come before looking at the data. A two-tailed test counts extreme results in both directions, and it is the honest default. A right-tailed test counts only the upper direction, so the same z of 2.58 gives a p-value of 0.0049 right-tailed instead of 0.0099 two-tailed: one tail instead of two, so half the p-value. The base guide’s 1.645 example shows the same halving at the boundary: z of 1.645 is 0.1000 two-tailed but 0.0500 right-tailed, which is why 1.645 is the 0.10 cut-off two-tailed and the 0.05 cut-off one-tailed. Choosing one tail after seeing which way the result went is moving the goalposts, and it roughly doubles the false-positive rate, because results that would have failed a two-tailed test suddenly pass.
| Case | Statistic | Tail | Area below | p-value | Verdict at 0.05 |
|---|---|---|---|---|---|
| Default | z 1.96 | Two-tailed | 0.9750 | 0.0500 | Significant |
| Stronger evidence | z 2.58 | Two-tailed | 0.9951 | 0.0099 | Significant |
| Same statistic, one tail | z 2.58 | Right-tailed | 0.9951 | 0.0049 | Significant |
| t version of the default | t 1.96, df 9 | Two-tailed | 0.9592 | 0.0816 | Not significant |
| t critical value | t 2.262, df 9 | Two-tailed | 0.9750 | 0.0500 | Not significant |
Every figure in that table is a displayed result from this page’s own calculator. The last two rows are the ones to study, because they show why the t distribution exists at all: with only 9 degrees of freedom, the same statistic of 1.96 gives 0.0816 instead of 0.0500, and it takes a larger statistic, 2.262, to reach the 0.05 line. Small samples carry more uncertainty, and the t distribution charges for it with fatter tails.
Why two results can both display 0.0500 and get different verdicts
The most instructive quirk on this page is not a bug, and it is worth understanding before it confuses you. The z case at 1.96 displays a p-value of 0.0500 and a verdict of Significant. The t case at 2.262 with 9 degrees of freedom also displays a p-value of 0.0500, but its verdict is Not significant. Same displayed number, different verdict. The explanation is rounding: the display shows four decimal places, while the verdict follows the unrounded value.
For z equals 1.96, the exact two-tailed p-value sits just below 0.05, so the verdict is Significant even though the rounded display reads 0.0500. For t equals 2.262 with df 9, the exact two-tailed p-value sits just above 0.05, because the classic critical value at df 9 is 2.262 to three decimals but a touch higher in full precision, so the verdict is Not significant even though the rounded display again reads 0.0500. Neither verdict contradicts the other; both follow the same rule, applied to unrounded values that happen to round to the same four-decimal display.
The practical lesson is to treat a displayed p-value near your significance level as a borderline result, not a clean pass or fail. If the display reads 0.0500, look at the verdict, and if the verdict matters for a decision, recompute with more precision or report the result as borderline. A p-value of 0.0499 and a p-value of 0.0501 are nearly identical as evidence, but a hard 0.05 rule treats them as opposites. The page’s own guide makes the same point in general terms: 0.05 is a guide, not a proof, and a small p-value can still come from luck while a large one is not proof of no effect.
That last clause deserves emphasis, because it is the most consequential misreading in statistics. A p-value below 0.05 does not prove the effect is real, and a p-value above 0.05 does not prove there is no effect. It says only how unusual the data would be under the no-effect story. A tiny, meaningless effect can be statistically significant with a huge sample, because even a small deviation becomes unusual when the sample is large enough. A big, real effect can miss 0.05 with a tiny sample, because the data are too noisy to distinguish it from luck. Always read the p-value next to the size of the effect, as the base guide instructs, and ask whether the effect is large enough to matter, not only whether it clears a line.
When to use a t test instead of a z test
Use z when the population standard deviation is known or the sample is large. Use t when the sample is small and the standard deviation is estimated from your own data, and type in the degrees of freedom. For a one-sample t test, degrees of freedom are the sample size minus 1, so a sample of 10 gives df 9, which is the default on this page. The degrees of freedom must be greater than 0, because a t distribution with no degrees of freedom is not a distribution at all.
The cost of using t is visible in the table above: at df 9, the same statistic of 1.96 gives 0.0816 instead of 0.0500, and reaching the 0.05 line takes a statistic of 2.262 instead of 1.96. The t distribution has fatter tails than the normal distribution, which is its way of accounting for the extra uncertainty that comes from estimating the standard deviation rather than knowing it. As the degrees of freedom grow, the t distribution approaches the normal distribution, and the difference between the two tests shrinks. With a large sample, the choice barely matters. With a small sample, it matters a great deal, and using z on a small sample with an estimated standard deviation will flatter your p-value, making weak evidence look stronger than it is.
Multiple testing is the final trap, and the base guide states it plainly: the 0.05 line means that if nothing were really going on, a result at least this extreme would still turn up about 1 time in 20 by luck alone. Run 20 tests on pure noise and you will typically hand yourself one false significant result. Run 100 and you will typically hand yourself five. That does not make any single significant result worthless, but it does mean that a significant result found after trying many tests, many outcomes, or many subgroups is weaker evidence than the same result from a single pre-planned test. If you test many things, say so, and treat the significant ones as leads to confirm, not as conclusions.
A checklist for reporting any p-value from this page
First, name the test: z or t, with degrees of freedom if t. A p-value without its test is incomplete, because the same statistic gives different p-values under the two distributions. Second, name the tail, and confirm it was chosen before looking at the data. A one-tailed p-value chosen after seeing the direction is not the same evidence as the same number from a pre-planned one-tailed test. Third, report the statistic and the p-value together, not the p-value alone. Fourth, report the significance level you are judging against, and whether the result clears it on the unrounded value, not only on the rounded display. Fifth, report the size of the effect alongside the p-value, because significance is not importance. Sixth, say how many tests were run, if more than one, because a significant result among many tests is weaker than the same result standing alone.
Two questions to ask of any p-value you did not compute yourself will catch most abuses. Was the tail chosen before the data were seen? And was this the only test, or one of many? A one-tailed test chosen after the fact, or a significant result cherry-picked from twenty attempts, should be treated as a suggestion to run a confirmatory test, not as a finding. The calculator cannot ask those questions for you: it computes the number you ask for, from the inputs you give it, and the integrity of the result depends on the integrity of the choices around it.
Finally, remember the scope of this tool. It covers z tests and t tests with a statistic you supply. It does not compute the statistic from raw data, it does not check the assumptions behind the test, and it does not tell you whether the test you picked is the right one for your design. Normality, independence, and the other assumptions of the test are yours to check, and a p-value from a test whose assumptions fail is not rescued by being small. Use the calculator for the arithmetic, and bring the judgment yourself.
Significance is not importance, and other sentences to retire
Several sentences about p-values sound precise and mean nothing, or mean the opposite of what they seem to say. Retire them. “The p-value is 0.03, so there is a 97 percent chance the effect is real” is wrong: the p-value is not the probability that the effect is real, and one minus the p-value is not that probability either. “The result was not significant, so there is no effect” is wrong: a large p-value says the data would be ordinary under the no-effect story, not that the no-effect story is true. “The result was highly significant, so the effect must be large” is wrong: significance measures unusualness under the no-effect story, and a huge sample makes even a tiny effect unusual. “We proved the hypothesis with p less than 0.05” is wrong twice over: a p-value does not prove, and 0.05 is a convention, not a law of nature.
Replace those sentences with ones that say what the numbers actually support. “With z equals 2.58 two-tailed, p equals 0.0099, so a result at least this extreme would turn up about 1 time in 100 by luck alone under the no-effect story” is correct, and it carries its own qualification. “The estimated effect is small, and although it is statistically significant, its practical importance is doubtful” is correct, and it separates the two judgments that significance is often asked to collapse. “The study was small, the p-value is 0.0816, and the data are consistent with both no effect and a moderate effect” is correct, and it says more than “not significant” ever could, because it names the ambiguity instead of hiding it behind a line.
The habit behind those replacements is simple: report the statistic, the test, the tail, the p-value, and the effect size, and let the reader judge. A single number, however precisely computed, cannot carry the weight of a conclusion, and this calculator, which computes that number from a statistic you supply, is honest about its own limits. It does not check assumptions, it does not supply the effect size, and it does not count how many tests you ran. Those responsibilities stay with you, and a report that meets them will survive scrutiny that a bare p-value will not.
Classic cut-offs, and why they are landmarks, not walls
The page’s own cut-off landmarks are worth memorising, because they recur across textbooks and papers. For a two-tailed z test, 1.96 marks the 0.05 line and 2.58 marks the 0.01 line, approximately: the displayed p-values are 0.0500 and 0.0099, with the rounding caveat from the verdict section applying to the first. For a right-tailed z test, 1.645 marks the 0.05 line, with a p-value of 0.0500, while the same 1.645 two-tailed gives 0.1000. Those three statistics, 1.645, 1.96 and 2.58, are the landmarks most readers will recognise, and recognising them lets you sanity-check a result at a glance: a z statistic below 1.645 two-tailed cannot be significant at 0.05, whatever the display says, and a z statistic above 2.58 two-tailed is significant at 0.01 as well as at 0.05.
Landmarks are not walls. A result just inside a landmark and a result just outside it are nearly identical as evidence, and treating them as opposites is the sharpest form of the 0.05 fallacy. The t distribution adds its own landmarks, which depend on the degrees of freedom: at df 9, the 0.05 two-tailed landmark is 2.262, not 1.96, and the gap between the two landmarks is the price of a small sample. As the degrees of freedom grow, the t landmarks converge on the z landmarks, and with a large sample the distinction fades. Knowing both sets, and knowing which one applies to your test, is the difference between reading a p-value and merely repeating one.
What to do with a borderline p-value
A borderline p-value, one that sits near your significance level, deserves a different response from a clear one. Do not rerun the test with a different tail, a different test, or a different exclusion rule until it passes: that is exactly the goalpost-moving the base guide warns against, and it manufactures significance from noise. Do not discard the result either, as if a p-value of 0.06 meant nothing: the data still carry information, and the effect size still matters. The honest responses are to report the result as borderline with its statistic and effect size, to collect more data if the question matters enough to justify it, and to treat the finding as a lead for a pre-planned confirmatory test rather than as a conclusion. A borderline result is an invitation to learn more, not a verdict to engineer. If the question matters, the next study should be designed before it is run, with its test, tail, significance level, and sample size fixed in advance, so that its p-value can be read without the suspicion that attaches to a result found by searching. Pre-planning does not make a p-value more powerful, but it makes it more believable, and believability is the currency a borderline result lacks. A pre-planned confirmatory test that returns a clear result settles the question in a way that no re-analysis of the original data can, because its choices were made before the data could influence them.
The Bayesian alternative in one sketch
Bayesian analysis answers the question p-values do not: how likely is the hypothesis, given the data? It does so by combining a prior probability, what was believed before the data, with the likelihood, how probable the data are under each hypothesis, to give a posterior probability. A small p-value with a very implausible prior still gives a low posterior, which is why extraordinary claims need extraordinary evidence: the prior matters, and frequentist p-values never ask about it.
Bayesian methods require stating a prior, which critics call subjective and supporters call honest, because every interpretation smuggles in prior beliefs anyway. The practical point for a p-value user is narrower: a p-value is not a posterior, and no rewording turns it into one.
Power: the chance of detecting a real effect
Statistical power is the probability that a study detects an effect of a given size if it is real, typically targeted at 80%. An underpowered study, with too few participants for the effect size, will often return non-significant results even when the effect is real, and its significant results, when they occur, will exaggerate the effect, because only large random fluctuations cross the threshold in a small sample.
Power analysis before data collection, choosing sample size for the smallest effect worth detecting, is the defence. A study designed for an effect of a certain size that returns a non-significant result has genuinely ruled out effects of that size or larger, while an underpowered study has ruled out almost nothing.
Replication: the real significance test
A single significant result, however small its p-value, is weaker evidence than the same result replicated independently. Replication guards against the failures p-values cannot see: bias, measurement error, selective reporting, and chance fluctuations that happened to cross a threshold. The replication crisis in several fields was driven less by misunderstood arithmetic than by publication practices that favoured significant results and hid the rest.
When reading a striking result, ask whether it has replicated, whether the replication was preregistered, and whether the effect size held up. A result that shrinks on replication was probably exaggerated in the first place, which is the expected pattern for underpowered original studies.
Frequently asked questions
What is a p-value in simple words?
A p-value is the probability of seeing a result at least as extreme as yours if nothing were really going on. A small p-value says your result would be unusual under that no-effect story; a large one says it would be quite ordinary.
How do you calculate a p-value from a z score?
For a two-tailed test, take the normal curve area below the absolute z value, subtract it from 1, and double the result: p = 2 times (1 minus Phi of the absolute z). For example, z = 1.96 gives an area of 0.9750, so p = 2 times 0.0250 = 0.0500.
What is the p-value for z = 1.96?
Two-tailed, the p-value for z = 1.96 is 0.0500, because the normal curve area below 1.96 is 0.9750 and 2 times 0.0250 is 0.0500. It sits exactly on the usual 0.05 significance line.
What is the difference between a one-tailed and a two-tailed p-value?
A two-tailed p-value counts extreme results in both directions, while a one-tailed p-value counts only one direction. The same statistic therefore gives a smaller one-tailed p-value: z = 1.645 is 0.1000 two-tailed but 0.0500 right-tailed.
Does a p-value below 0.05 prove the effect is real?
No. A p-value below 0.05 means the result would be unusual if nothing were going on, but unusual things still happen by luck about 1 time in 20 at that level. It is a guide for calling a result statistically significant, not a proof, and it says nothing about how large or important the effect is.
When should I use a t test instead of a z test?
Use a t test when the sample is small and the population standard deviation is estimated from your own data. The t distribution has heavier tails than the normal curve, so it gives larger p-values for the same statistic, and it needs the degrees of freedom: for a one-sample t test, df is the sample size minus 1.
Why do z 1.96 and t 2.262 both show 0.0500 but get different verdicts?
Because the display rounds to four decimal places while the verdict follows the unrounded value. The exact two-tailed p-value for z 1.96 sits just below 0.05, so the verdict is Significant, while the exact value for t 2.262 with df 9 sits just above 0.05, so the verdict is Not significant. Both displays round to 0.0500. Treat any displayed p-value near your significance level as borderline, and judge it by the verdict and the unrounded value, not by the rounded display alone.
Why does the same statistic give a larger p-value under a t test?
Because the t distribution has fatter tails than the normal distribution, to account for the extra uncertainty of estimating the standard deviation from a small sample. At df 9, a statistic of 1.96 gives 0.0816 under a t test but 0.0500 under a z test, and reaching the 0.05 line takes 2.262 under t but only 1.96 under z. As the degrees of freedom grow, the t distribution approaches the normal distribution and the difference shrinks.
Does a p-value of 0.0099 mean there is only a 1 percent chance the result is luck?
No. A p-value of 0.0099, as for z 2.58 two-tailed, means that if nothing were really going on, a result at least this extreme would turn up about 1 time in 100 by luck alone. It is not the probability that nothing is going on, and it is not the probability that your result is wrong. It is the probability of data this extreme under the no-effect story, and it must be read next to the size of the effect and the number of tests run.
Can I switch from two-tailed to one-tailed after seeing the result?
No, not honestly. A one-tailed test gives a smaller p-value for the same statistic: z 2.58 is 0.0099 two-tailed but 0.0049 right-tailed, and z 1.645 is 0.1000 two-tailed but 0.0500 right-tailed. That smaller number is only honest if the direction was fixed before the data were collected. Switching after seeing which way the result went roughly doubles the false-positive rate, because results that would have failed a two-tailed test suddenly pass.
What is the Bayesian alternative to p-values?
Combining a prior probability with the data likelihood to give a posterior probability for the hypothesis. It answers the question p-values do not, at the cost of stating a prior.
What is statistical power?
The probability a study detects an effect of a given size if it is real, usually targeted at 80%. Underpowered studies miss real effects and exaggerate the ones they happen to find.
Why does replication matter more than one p-value?
Because replication guards against bias, measurement error, selective reporting, and chance fluctuations that a single p-value cannot see. A result that shrinks on replication was probably exaggerated.
More calculators
This calculator is an educational tool for working out p-values from z and t statistics. It does not give statistical advice, and its results should be checked before being used in any formal report or decision. A significance verdict is a guide, not a proof that an effect is real or absent.
P-Values Explained
Correct definition, the 0.05 convention, and the misreadings to retire.