P-Value Calculator: Find p from z or t, Two-Tailed

What does p = 0.049996 mean?

It means that if the null hypothesis were true, a test statistic at least as far from zero as 1.96 would show up about 5% of the time by luck alone. The p-value is the area in the tails of the null distribution beyond your statistic — for a z-test the normal curve, for a t-test the t curve with your degrees of freedom. On the defaults the p-value is 0.049996, just under the 0.05 line, so the result is called significant at 0.05. A p-value is not the probability that the null hypothesis is true, and it is not the size of the effect — it is only how surprising the data would be if nothing were going on.

Three things to check before you trust the number

One tail or two, decided before you look. Two-tailed tests both directions — a result far out either way counts — while one-tailed tests one direction only. z = 1.96 is p = 0.049996 two-tailed and p = 0.024998 one-tailed, exactly half. Picking one tail after seeing the data halves your p-value for free, which is why the choice has to come first.

z or t, and the degrees of freedom with it. A z-test uses the normal curve and is for large samples or a known standard deviation; a t-test uses the t curve with degrees of freedom and is for small samples. With 10 degrees of freedom, t = 2.0 gives p = 0.073388, while z = 2.0 gives p = 0.045500 — the t curve's fatter tails give a larger p-value, and the two agree only when the degrees of freedom are large.

0.05 is a line, not a law. A p-value of 0.049996 is significant at 0.05 and a p-value of 0.050012 is not, but nothing real changes between them. Report the p-value itself — 0.049996, unrounded 0.049995652 — not just the word significant, and a p-value just over the line is weak evidence, not proof of nothing.

Worked example

On the defaults: z = 1.96, two-tailed.

1. Find the normal CDF at 1.96: the area under the bell curve up to 1.96 is 0.975002.
2. Take the upper tail: 1 − 0.975002 = 0.024998.
3. Double it for two tails: 2 × 0.024998 = 0.049996.
4. Compare with 0.05: 0.049996 is below 0.05, so the result is significant at 0.05.
5. Table value: to 4 decimals the p-value is 0.0500 — the number the printed tables show.
6. Smallest z for p of at most 0.05: 1.96 — at 1.96 p is 0.049996 (unrounded 0.049995652), and at 1.95 it is 0.051176 (unrounded 0.051175981).
7. Smallest z for p of at most 0.01: 2.58 — at 2.58 p is 0.009880 (unrounded 0.009880092), and at 2.57 it is 0.010170 (unrounded 0.010169909).

Every branch, checked

• z = 1.96, one-tailed: z = 1.96, one-tailed: p = 0.024998 (0.0250); one-tailed z-test; Significant at 0.05. z = 1.96 one-tailed — p is 0.024998, half the two-tailed value, because one tail is half of two.
• z = 2.58, two-tailed: z = 2.58, two-tailed: p = 0.009880 (0.0099); two-tailed z-test; Significant at 0.01. z = 2.58 two-tailed — p is 0.009880, the 1% level, the table value 0.0099.
• z = 1.645, two-tailed: z = 1.645, two-tailed: p = 0.099970 (0.1000); two-tailed z-test; Not significant at 0.05. z = 1.645 two-tailed — p is 0.099970, just above 0.10, not significant, because 1.645 is the one-tailed 5% point.
• t = 2.0, df 10, two-tailed: t = 2.0, df 10, two-tailed: p = 0.073388 (0.0734); two-tailed t-test (df 10); Not significant at 0.05. t = 2.0, df 10, two-tailed — p is 0.073388, the table value 0.0734, fatter tails than the normal curve's 0.045500.
• t = 2.228, df 10, two-tailed: t = 2.228, df 10, two-tailed: p = 0.050012 (0.0500); two-tailed t-test (df 10); Not significant at 0.05. t = 2.228, df 10, two-tailed — p is 0.050012, the 5% point for 10 degrees of freedom, the table value 0.0500.
• z = 0, two-tailed: z = 0, two-tailed: p = 1.000000 (1.0000); two-tailed z-test; Not significant at 0.05. z = 0 two-tailed — p is 1.000000, a statistic of zero is exactly what the null hypothesis predicts.
• z = 3.0, two-tailed: z = 3.0, two-tailed: p = 0.002700 (0.0027); two-tailed z-test; Significant at 0.01. z = 3.0 two-tailed — p is 0.002700, three standard deviations out, significant at 0.01.
• z = 1.0, two-tailed: z = 1.0, two-tailed: p = 0.317311 (0.3173); two-tailed z-test; Not significant at 0.05. z = 1.0 two-tailed — p is 0.317311, one standard deviation out is nothing unusual.
• t = 1.812, df 10, two-tailed: t = 1.812, df 10, two-tailed: p = 0.100075 (0.1001); two-tailed t-test (df 10); Not significant at 0.05. t = 1.812, df 10, two-tailed — p is 0.100075, the 10% point for 10 degrees of freedom.
• z = 2.326, one-tailed: z = 2.326, one-tailed: p = 0.010009 (0.0100); one-tailed z-test; Significant at 0.05. z = 2.326 one-tailed — p is 0.010009, the 1% one-tailed point.

The reverses, worked backwards

• Smallest z for the two-tailed p-value to be no more than 0.05: 1.96 — at 1.96 the figures are p = 0.049996 (0.0500); two-tailed z-test; Significant at 0.05 (unrounded 0.049995652); at the lower neighbor the figures are p = 0.051176 (0.0512); two-tailed z-test; Not significant at 0.05 (unrounded 0.051175981); at the upper neighbor the figures are p = 0.048838 (0.0488); two-tailed z-test; Significant at 0.05 (unrounded 0.048838232) — verdict: STEP.
• Smallest z for the two-tailed p-value to be no more than 0.01: 2.58 — at 2.58 the figures are p = 0.009880 (0.0099); two-tailed z-test; Significant at 0.01 (unrounded 0.009880092); at the lower neighbor the figures are p = 0.010170 (0.0102); two-tailed z-test; Significant at 0.05 (unrounded 0.010169909); at the upper neighbor the figures are p = 0.009598 (0.0096); two-tailed z-test; Significant at 0.01 (unrounded 0.009597658) — verdict: STEP.
• Largest z for the one-tailed p-value to stay at or above 0.10: 1.28 — at 1.28 the figures are p = 0.100273 (0.1003); one-tailed z-test; Not significant at 0.05 (unrounded 0.100272637); at the upper neighbor the figures are p = 0.098525 (0.0985); one-tailed z-test; Not significant at 0.05 (unrounded 0.098525397); at the lower neighbor the figures are p = 0.102042 (0.1020); one-tailed z-test; Not significant at 0.05 (unrounded 0.102042384) — verdict: STEP.
• Smallest t for the two-tailed p-value to be no more than 0.05: 2.23 — at 2.23 the figures are p = 0.049842 (0.0498); two-tailed t-test (df 10); Significant at 0.05 (unrounded 0.049842471); at the lower neighbor the figures are p = 0.050695 (0.0507); two-tailed t-test (df 10); Not significant at 0.05 (unrounded 0.050694547); at the upper neighbor the figures are p = 0.049004 (0.0490); two-tailed t-test (df 10); Significant at 0.05 (unrounded 0.049004240) — verdict: STEP.
• Largest t for the two-tailed p-value to stay at or above 0.10: 1.81 — at 1.81 the figures are p = 0.100402 (0.1004); two-tailed t-test (df 10); Not significant at 0.05 (unrounded 0.100402295); at the upper neighbor the figures are p = 0.098777 (0.0988); two-tailed t-test (df 10); Not significant at 0.05 (unrounded 0.098776906); at the lower neighbor the figures are p = 0.102052 (0.1021); two-tailed t-test (df 10); Not significant at 0.05 (unrounded 0.102052243) — verdict: STEP.
• Smallest z for the one-tailed p-value to be no more than 0.025: 1.96 — at 1.96 the figures are p = 0.024998 (0.0250); one-tailed z-test; Significant at 0.05 (unrounded 0.024997826); at the lower neighbor the figures are p = 0.025588 (0.0256); one-tailed z-test; Significant at 0.05 (unrounded 0.025587991); at the upper neighbor the figures are p = 0.024419 (0.0244); one-tailed z-test; Significant at 0.05 (unrounded 0.024419116) — verdict: STEP.

What the defaults assume

The defaults are z = 1.96, two-tailed — the 5% point of the normal curve, the most quoted number in statistics — and the p-value is 0.049996, the table value 0.0500, and the result is significant at 0.05. They are a starting point, not a benchmark: your statistic may be a t with 10 degrees of freedom, or a one-tailed z, and the right answer for you is the one your test describes. Change the statistic, the test, the degrees of freedom and the tail to your own and the calculator recomputes instantly. Take the area in the tails, compare it with 0.05, and report the p-value itself.

These notes repeat in every explainer section because this page is built to be read start to finish: the defaults are one statistic, not a standard; one tail or two, decided before you look; z or t, and the degrees of freedom with it; and 0.05 is a line, not a law.

Output · formula · meaning

Who uses this

• A student checking a statistics assignment — z = 1.96 two-tailed gives p = 0.049996, significant at 0.05, by the area in the tails, not by guessing.
• A researcher reading a paper's p-value — p = 0.049996 is just under the 0.05 line, so the result is reported as significant, and the unrounded 0.049995652 shows how close to the line it sits.
• A data analyst testing whether a new website version beats the old one — a t-test with 10 degrees of freedom and t = 2.0 gives p = 0.073388, not significant, so the test runs longer.
• A quality-control engineer checking whether a batch differs from the standard — z = 3.0 gives p = 0.002700, strong evidence of a real difference.

Accuracy & limitations

The calculator uses the Abramowitz & Stegun 7.1.26 approximation for the normal curve, good to 1.5e-7, and the Numerical Recipes continued fraction for the t curve, good to about 3e-14 — every p-value on this page rounds to the same 6 decimals as the exact value, and every figure is checked against an independent model. What it cannot do is decide your significance level or your hypothesis: it finds the p-value for the statistic you give it, one-tailed or two-tailed, z-test or t-test, so if your test is a chi-square or an F-test, this is the wrong calculator. And a p-value is not an effect size — p = 0.049996 on a huge sample can be a trivial difference, and p = 0.073388 on a small one can hide a large one. The page prints the unrounded p-value so the rounding does not mislead you — 0.049996, unrounded 0.049995652.

Cite this calculator

A p-value is the area in the tails of the null distribution beyond the test statistic. On the defaults — z = 1.96, two-tailed — the p-value is 0.049996, the table value 0.0500, and the result is significant at 0.05. Smallest z for the two-tailed p-value to be no more than 0.05: 1.96. Smallest z for the two-tailed p-value to be no more than 0.01: 2.58. Largest z for the one-tailed p-value to stay at or above 0.10: 1.28. Smallest t for the two-tailed p-value to be no more than 0.05: 2.23. Largest t for the two-tailed p-value to stay at or above 0.10: 1.81. Smallest z for the one-tailed p-value to be no more than 0.025: 1.96. Source: Bright Side Kit, brightoolbox.com.

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