P Value Calculator
Source: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1 (Normal Distribution) · Source verified August 6, 2026
A p-value calculator computes the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis is true. This calculator maps a z-score to its probability using the standard normal distribution, and supports one-tailed and two-tailed tests depending on the directional hypothesis. It does not accept t-scores, chi-square values, or F-statistics, because each of those requires its own distribution and degrees of freedom. Researchers, data scientists, and students use a p-value during hypothesis testing to judge the statistical significance of a result.
Enter a z-score, choose a tail, and set your significance level. The calculator returns the p-value from the standard normal distribution, shows the area on each side of your z-score, and draws the region the p-value measures.
Quick Answer
Find the p-value for your z-score. Enter a z-score, select a left-tailed, right-tailed or two-tailed test, and optionally set your significance level. This calculator uses the standard normal distribution and accepts z-scores only.
Negative values are fine. Plus or minus 1.96 is the 95% two-tailed cutoff. · e.g. 1.96
Tail
Two-tailed is the usual default. Choose one tail only if you fixed the direction of the hypothesis before seeing the data.
Used only to compare against your p-value. · e.g. 0.05
Two-tailed test
0.049996
p = 2 × P(Z ≥ |1.96|) ≈ 2 × 0.024998 = 0.049996
Extra digits are shown because the p-value is close to the selected alpha.
The p-value is below the selected alpha of 0.05. This result meets the selected statistical-significance threshold. A p-value is one piece of evidence. Read it alongside effect size, sample size, study design, how many tests were run, and what was already known.
How this p-value was reached
- Formula for the selected tail
- p = 2 × P(Z ≥ |z|)
- Substitution
- p = 2 × P(Z ≥ |1.96|) ≈ 2 × 0.024998 = 0.049996
- What the shading shows
- Both tails beyond the magnitude of your z-score. The two-tailed p depends only on |z|, so +1.96 and -1.96 give the same p-value and shade the same two regions. Only the dashed z marker moves, mirrored to the other side of zero.
- What it does not show
- The p-value is the probability of a statistic at least this extreme if the null hypothesis holds. It is not the probability that the null hypothesis holds, and it does not measure how large or how important an effect is.
Examples
z = 1.96, two-tailed
p = 0.049996
z = 2.58, two-tailed
p = 0.0099
z = 1.645, right-tailed
p = 0.04998
z = −1.96, left-tailed
p = 0.0250
How it works
Formula · Two-tailed: p = 2 × P(Z ≥ |z|) · Right-tailed: p = P(Z ≥ z) · Left-tailed: p = P(Z ≤ z)
The p-value is the area under the standard normal curve in the region your test treats as extreme. Which region that is depends on the tail you select, so the calculator shows the formula it actually used along with the substituted numbers.
Two-tailed · p = 2 × P(Z ≥ |z|)
Right-tailed · p = P(Z ≥ z)
Left-tailed · p = P(Z ≤ z)
Tail areas come from Hart's rational approximation, computed directly rather than as one minus the opposite tail, so very large z-scores keep their precision instead of rounding to zero. The drawn curve spans four standard deviations either side of the mean, which holds 99.994% of the area.
This calculator takes z-scores only. A t statistic, a chi-square statistic and an F statistic each follow a different distribution, and converting them through the standard normal gives the wrong answer. If you have a t statistic, use the t-test calculator. If you have a raw score rather than a standardised one, convert it first with the z-score calculator.
How to read the curve
The shaded region is the region whose area is your p-value, so the picture and the number always describe the same thing.
- A two-tailed test shades both tails beyond the magnitude of your z-score. Because it depends only on |z|, entering +1.96 and −1.96 produces the same p-value and shades the same two regions. The dashed z marker is the only thing that moves, mirrored to the other side of zero.
- A right-tailed test shades everything above your z-score. For a negative z-score that is most of the curve, not a tail, and the p-value is correspondingly large.
- A left-tailed test shades everything below your z-score, with the mirror-image behaviour.
What a p-value cannot tell you
A p-value answers one narrow question: how unusual is a statistic at least this extreme, if the null hypothesis is true. Several things follow from that, and the American Statistical Association's 2016 statement sets them out directly.
- It is not the probability that the null hypothesis is correct, nor the probability that your data arose by chance alone.
- It does not measure the size of an effect or how much it matters. A trivial difference in a very large sample can clear any threshold.
- A result on one side of a threshold and a result just on the other are not qualitatively different findings.
- A p-value above your threshold is not evidence that the null hypothesis is correct. It is an absence of evidence against it, which is a weaker statement.
Interpret any p-value alongside the effect size, the sample size, the study design, how many comparisons were made, and the prior evidence. Where several tests were run, the chance of at least one small p-value rises with the number of tests, and the threshold needs adjusting.
Assumptions and limits
- The test statistic is assumed to follow the standard normal distribution, with mean 0 and standard deviation 1.
- Probabilities are displayed to four decimals, to five decimals between 0.0001 and 0.001, and in scientific notation below 0.0001 so a deep-tail value never reads as zero. Additional digits appear when that rounding would make the p-value look as though it landed on, or crossed, your significance level. The comparison itself always uses the unrounded p-value, and every place the page states the p-value shows the same figure.
- The approximation's worst relative error against SciPy over |z| up to 37 is within 9e-09, measured at z = 7.78. Below |z| = 5, where every conventional threshold sits, it is within 5e-11. Both are far below the displayed precision.
- The drawn curve is truncated at four standard deviations, so the shaded area is short of the true p-value by less than 6.4e-05. The number shown is never the drawing; it is computed over the full distribution. A z-score at or beyond the edge of that range leaves nothing visible to shade, and the caption says so rather than describing a band you cannot see.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution for the distribution and its tail areas.
- Wasserstein & Lazar (2016), The ASA Statement on p-Values: Context, Process, and Purpose, The American Statistician 70(2), 129 to 133 for the interpretation limits described above. The American Statistical Association's announcement of that statement summarises its six principles and links the paper above.
Related statistics calculators
- Z-score calculator turns a raw score into the z-score this page expects.
- T-test calculator for a t statistic, which needs its own distribution and degrees of freedom.
- Confidence interval calculator builds the interval that pairs with a two-tailed p-value.
- Normal distribution calculator for tail and between-value probabilities directly.
- Margin of error calculator for the half-width on a sample proportion.
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Frequently asked questions
A p-value is the probability of observing a test statistic at least as extreme as yours, assuming the null hypothesis is true. A small p-value means your data would be unusual if the null hypothesis held. It is not the probability that the null hypothesis holds, and it is not the probability that your result happened by chance.
No. This calculator converts a z-score through the standard normal distribution only. A t statistic follows Student's t distribution, whose shape depends on the degrees of freedom, so putting a t value in here returns a p-value that is too small. The error is large enough to change the conclusion: t = 3.0 with 3 degrees of freedom has a true two-tailed p of about 0.058, while the standard normal gives about 0.0027. Use the t-test calculator instead.
Two-tailed tests detect a difference in either direction and are the usual default. A one-tailed test is appropriate only when the direction was fixed as part of the hypothesis before the data were seen. Choosing one tail after looking at the data halves the p-value without any statistical justification.
There is no universal threshold. The 0.05 convention is a convention, not a property of nature, and many fields use 0.01 or stricter. The American Statistical Association's 2016 statement warns against treating any single threshold as a verdict, and against reporting a result as meaningful on the basis of a p-value alone.
That your data would be unusual if the null hypothesis were true, at the threshold you chose. It does not establish that the null hypothesis is wrong, that the effect is large, or that it matters in practice. A tiny effect measured in a very large sample can produce a small p-value, and a real effect measured in a small sample can fail to.
Only that your data are not unusual enough to clear the threshold you picked. It is not evidence that the null hypothesis is correct, and it is not evidence that there is nothing to find. Absence of a significant result is not the same as a demonstrated absence of an effect, particularly when the sample is small.
The calculator uses Hart's rational approximation of the standard normal distribution, computing each tail directly rather than by subtracting the other from 1. That matters for large z-scores: deriving a small tail from a value near 1 quantises it, and beyond about z = 8.3 it collapses to exactly zero. Computing the tail directly keeps a meaningful value out to z = 37 and beyond.
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