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Calculator Answers

P Value Calculator

Source: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1 (Normal Distribution) · Source verified August 6, 2026

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

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.

See also: Z Score Calculator, Confidence Interval Calculator, and T-Test Calculator.

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

z = 1.96-4-204
Standard normal curve. Both tails beyond plus and minus 1.96 are shaded, together covering 0.049996 of the area.
p-value · two-tailed

Two-tailed test

0.049996

p = 2 × P(Z ≥ |1.96|) ≈ 2 × 0.024998 = 0.049996

z-score1.96
TestTwo-tailed
Area to the left0.9750
Area to the right0.0250
Alpha0.05
Against alphap is below alpha

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.
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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.

Working it out by hand

Take a z-score of 2.1 with a two-tailed test, which is what the calculator runs by default.

  1. Find the area in the upper tail beyond the statistic: P(Z ≥ 2.1) = 0.017864. That is the share of the standard normal curve lying to the right of 2.1.
  2. A two-tailed test counts departures in both directions, and the normal curve is symmetric, so double it: 2 × 0.017864 = 0.035729.
  3. The calculator displays 0.0357, and compares the unrounded value against the alpha you set.

At an alpha of 0.05 that result falls below the threshold. What that licenses is narrow: a statistic this far into the tail would be uncommon if the null hypothesis were true. It is not a statement about whether the null hypothesis holds, and it says nothing about how large any underlying difference is.

A one-tailed test at the same z-score skips the doubling and returns 0.017864 instead. That is the whole of the difference, and it is why the choice of tail has to be fixed before the data are seen rather than after.

How p-values are misread

  • Reading it backwards. The p-value is computed by assuming the null hypothesis and asking about the data. Reading it as a probability attached to the hypothesis reverses the conditional, which is a different quantity the calculation never produces.
  • Treating 0.05 as a law. It is a convention. The threshold is a choice made in advance, and a p-value of 0.049 and one of 0.051 describe almost identical data. The calculator shows extra digits near your alpha precisely so the pair are not mistaken for opposites.
  • Confusing a threshold with importance. The p-value carries no information about effect size. A negligible difference measured across a very large sample clears any threshold, and a substantial difference in a small sample often does not.
  • Reading a large p-value as support for the null. It is an absence of evidence against, which is weaker. Small samples routinely produce large p-values whatever is going on underneath.
  • Switching tails after the fact. Moving from two tails to one halves the p-value. Done after seeing the data, it is a way of clearing a threshold rather than a hypothesis about direction.

Your inputs stay in the page

The z-score and significance level you enter are processed by this page in your browser. They are not sent to a server, not stored after you close the tab, and not visible to anyone else. There is no account and nothing to sign up for.

Sources

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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.

No. It is a convention that hardened into a habit, and the threshold you use is a choice you make before seeing the data rather than a property of the world. Particle physics works at roughly five sigma, many clinical fields use 0.01, and some analyses report the p-value without applying a threshold at all. The calculator lets you set alpha to whatever your field uses because there is no single correct value to hard-code.

Alpha is the threshold you pick in advance; the p-value is what the data produce. Comparing them is a decision rule, not a measurement. The calculator keeps them separate for that reason: it computes the p-value from your z-score alone, then reports whether it falls below the alpha you set.

No. The calculation runs in the opposite direction: it assumes the null hypothesis is true and asks how unusual your statistic would be under that assumption. It never assigns a probability to any hypothesis. A small p-value says the data sit far into the tail of the distribution you assumed, which is a reason to look further, not a verdict on which hypothesis holds.

Because 1.96 is itself a rounded critical value. The two-tailed p-value for exactly 1.96 is 0.049996, and the value that would give exactly 0.05 is closer to 1.959964. The calculator shows the extra digits when a result sits this near your threshold, so a number that rounds to 0.0500 is never mistaken for one that is exactly at the line.

Use z when the population standard deviation is known or the sample is large enough that the t distribution has effectively converged on the normal. Use t when you estimated the standard deviation from a small sample. This calculator maps z-scores through the standard normal only, so a t statistic entered here returns a p-value that is too small. The t-test calculator handles degrees of freedom.

Report it as what it is: the data were not unusual enough, under the assumed distribution, to clear the threshold you chose. That is not a finding of no effect. Consider the effect size, the sample size and the width of the interval around your estimate, all of which carry information a single threshold comparison discards.

Yes. The alpha field accepts any level you enter, and the comparison uses that value rather than a hard-coded default. Set it before you look at the result: choosing the threshold after seeing the p-value is what turns a decision rule into a way of confirming whatever the data happened to show.

Yes. It is free to use with no account, no sign-up, and no limit on calculations. Everything runs inside this page in your browser.