Statistics
Statistics
p-Value Calculator
Compute p-values from t-, z-, and chi-square statistics, with significance interpretation in plain language.
Input
Test details
The t, z, or χ² value reported by your statistical software.
For an independent-samples t-test: df = n₁ + n₂ − 2.
Output
p-value
Two-tailed t-test
- p
- 0.0266
- p < .05, significant
LaTeX:
$p = 0.027$p-values are computed using standard analytic CDFs (Abramowitz & Stegun for normal, regularized incomplete beta for t and chi-square). Accurate to ~6 decimals for typical research ranges.
What is a p-value?
A p-value is the probability of observing a test statistic at least as extreme as the one you got, assuming the null hypothesis is true. It answers one narrow question: "if there were really no effect, how surprising would this data be?" A small p-value means your data would be rare under the null hypothesis, which is taken as evidence against it.
Three things a p-value is not: it is not the probability that the null hypothesis is true, it is not the probability that your result is due to chance, and it is not a measure of effect size. A tiny p-value with a huge sample can describe a difference too small to matter, which is why journals increasingly ask for effect sizes and confidence intervals alongside p-values.
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The p-value formula
There is no single closed-form "p-value formula" — the p-value is an area under the sampling distribution of your test statistic. For a two-tailed z-test it is the area in both tails beyond your observed |z|:
These integrals have no elementary antiderivative, which is why p-values come from software or tables rather than hand calculation. The calculator above evaluates the exact CDFs.
How to find a p-value from a test statistic
Whatever the test, the recipe is the same four steps:
- Identify the distribution. A z-statistic follows the standard normal; a t-statistic follows Student's t with your degrees of freedom; a χ² statistic follows chi-square.
- Decide one- or two-tailed. Two-tailed unless you pre-registered a directional hypothesis (see below).
- Take the tail area beyond your statistic. That area is the p-value — double it for a two-tailed test.
- Compare to your significance level. p ≤ α means statistically significant at that level.
Worked example. You run an independent-samples t-test with n₁ = n₂ = 15, so df = 28, and get t = 2.34. The two-tailed tail area of t(28) beyond ±2.34 is p ≈ .027 — the calculator's default inputs show exactly this. Since .027 < .05, the difference is significant at the 5% level, and you would report it as t(28) = 2.34, p = .027. From a z-score the logic is identical: z = 1.96 gives the textbook two-tailed p = .050.
p-value table: critical values at a glance
Before software, researchers compared their statistic against critical values. If your |z| or |t| exceeds the critical value for a significance level, then p is below that level. The tables are still the fastest sanity check on a computed result.
| Significance level (α) | Critical z, two-tailed | Critical z, one-tailed |
|---|---|---|
| 0.10 | 1.645 | 1.282 |
| 0.05 | 1.960 | 1.645 |
| 0.01 | 2.576 | 2.326 |
| 0.001 | 3.291 | 3.090 |
| Degrees of freedom | Critical t (α = .05, two-tailed) | Critical t (α = .01, two-tailed) |
|---|---|---|
| 10 | 2.228 | 3.169 |
| 20 | 2.086 | 2.845 |
| 30 | 2.042 | 2.750 |
| 60 | 2.000 | 2.660 |
| 120 | 1.980 | 2.617 |
| ∞ (= z) | 1.960 | 2.576 |
Example: with df = 30 and t = 2.20, your statistic clears the .05 critical value (2.042) but not the .01 value (2.750), so .01 < p < .05. The calculator gives the exact figure.
Interpreting p-values: what does p < 0.05 mean?
The 0.05 threshold is a convention, not a law of nature — it says you accept a 1-in-20 false-positive rate. Interpretation at a glance:
- p < .001 — very strong evidence against the null hypothesis; report as p < .001.
- p < .01 — strong evidence; significant even under stricter conventions.
- p < .05 — statistically significant by the default convention; the null hypothesis is rejected.
- .05 ≤ p < .10 — sometimes described as "marginal"; treat as inconclusive, not as a weaker kind of significance.
- p ≥ .10 — the data are compatible with the null hypothesis. This is not proof of no effect.
Two cautions. First, significance is binary only by convention — p = .049 and p = .051 are nearly identical evidence. Second, with large samples everything becomes significant, so pair the p-value with an effect size before claiming a finding matters.
How to find a p-value in Excel or Google Sheets
The same functions work in both. Replace t, z, x2, and df with your values or cell references:
| Test | Formula |
|---|---|
| Two-tailed t-test | =T.DIST.2T(ABS(t), df) |
| One-tailed t-test (upper) | =T.DIST.RT(t, df) |
| Two-tailed z-test | =2*(1-NORM.S.DIST(ABS(z), TRUE)) |
| One-tailed z-test (upper) | =1-NORM.S.DIST(z, TRUE) |
| Chi-square (upper tail) | =CHISQ.DIST.RT(x2, df) |
If you have raw data rather than a statistic, Excel's T.TEST(range1, range2, tails, type) returns the p-value directly. The calculator above is the quicker check when a paper or output table already gives you the statistic.
One- vs two-tailed, and which you should use
A two-tailed test asks "is there a difference?" (either direction). A one-tailed test asks "is the value larger / smaller?" (specified direction). Two-tailed is the default in psychology, education, and most social sciences; one- tailed is justifiable only when your hypothesis truly is directional and pre-registered. The convention exists because one-tailed tests halve the p-value, making it easier to find "significance", which is why journals are suspicious of them when used post hoc.
Frequently asked questions
What does p < 0.05 mean?
It means that if the null hypothesis were true, you would see a test statistic this extreme less than 5% of the time. By the most common convention, the result is then called statistically significant and the null hypothesis is rejected. It does not mean there is a 95% chance your hypothesis is true, and it says nothing about how large or practically important the effect is.
What is a good p-value?
There is no universally “good” p-value — there are conventions. Most fields use α = 0.05 as the significance cutoff; stricter work uses 0.01 or 0.001. A p-value just below your cutoff is weak evidence; the further below the cutoff, the stronger the evidence against the null hypothesis. Always report the exact value rather than only “p < .05”.
Can a p-value be zero?
No. A p-value is a tail probability from a continuous distribution, so it can be arbitrarily small but never exactly zero. When software prints 0.000, the value is smaller than the displayed precision — report it as p < .001 in APA style, not as p = 0.
What is the difference between the p-value and alpha?
Alpha (α) is the threshold you choose before running the test — the false-positive rate you are willing to accept, usually 0.05. The p-value is what the data actually produced. You compare the two: if p ≤ α, the result is significant at that level.
How do I report a p-value in APA style?
Report the test statistic, degrees of freedom, and exact p-value to two or three decimals: t(28) = 2.34, p = .027. Drop the leading zero for values that cannot exceed 1, and use p < .001 for anything smaller than .001. This calculator prints the APA and LaTeX forms for every result.
Does a non-significant p-value prove there is no effect?
No. A large p-value means the data are compatible with the null hypothesis, not that the null is true. The study may simply be underpowered. Check the confidence interval: if it is wide and includes both meaningful effects and zero, the test was inconclusive rather than negative.
