P-Value Calculator
Converts a z or t test statistic into a p-value, one-tailed or two-tailed.
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Your details
Result
- p-value
- 0.04999579
- At α = 0.05
- Reject H₀
- At α = 0.01
- Fail to reject H₀
- At α = 0.001
- Fail to reject H₀
- Critical z at 5%
- 1.95996
4.9996% — two-tailed
Statistically significant
z = 1.96…
p = 2 × P(X > |stat|)
Cumulative probability = 0.9750021
2 × (1 − 0.9750021)
p = 0.04999579
Formula — written step by step
A z-score of 1.96, two-tailed…
p = 2 × P(Z > |z|)
Φ(1.96) = 0.975002
1 − 0.975002 = 0.024998
2 × 0.024998
p = 0.049996
The p-value is the probability of observing a result at least this extreme if the null hypothesis were true. A z of 1.96 lands almost exactly on the conventional 0.05 threshold, which is where that critical value comes from.
A worked example
The same statistic, one tail or two
- Test statistic
- z = 1.75
- Two-tailed test
- no direction predicted
- One-tailed test
- direction predicted in advance
- Φ(1.75) = 0.959941
- One-tailed: p = 1 − 0.959941 = 0.040059
- Two-tailed: p = 2 × 0.040059 = 0.080118
- One is significant at 5%, the other is not
p = 0.040 one-tailed, 0.080 two-tailed
The identical data crosses the threshold or fails to, depending only on which test was chosen. This is why the direction must be declared before seeing the data. Switching to a one-tailed test afterwards, because it produces significance, is a well-known way of fooling yourself.
How to use this calculator
- Choose whether your statistic is a z-score or a t-statistic; for t, enter the degrees of freedom.
- Enter the test statistic your analysis produced.
- Pick one-tailed or two-tailed — and pick it before you see the result, not after.
- Read the p-value and compare it against the significance level you set in advance.
What a p-value is
A p-value is the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true. A small p means the data would be surprising if nothing were going on, which counts as evidence that something is.
It answers a narrow question. It does not give the probability that the null hypothesis is true, nor the probability that your finding is real — both are common misreadings. The p-value is computed under the assumption that the null holds; it cannot then tell you how likely that assumption is.
One tail or two
A two-tailed test asks whether there is a difference in either direction. A one-tailed test asks whether there is a difference in a specific direction, and gives exactly half the p-value for the same statistic.
That halving is why the choice must be made before the data is seen. Choosing a one-tailed test after observing which way the result went, because it brings p under 0.05, doubles the false-positive rate while appearing rigorous. Two-tailed is the safer default, and is what most fields expect unless a directional prediction was registered in advance.
Significance is not importance
Statistical significance says an effect is probably not zero. It says nothing about whether the effect is large enough to matter. With a large enough sample, a trivial difference will reach significance, because the p-value depends on both effect size and sample size and cannot separate them.
This is why effect sizes and confidence intervals are increasingly reported alongside p-values, and why some journals have dropped the p-value threshold entirely. A confidence interval shows both whether zero is plausible and how large the effect might be — which is usually the question that was actually being asked.
Formula & methodology
What this calculator assumes
- A p-value is the probability of data at least this extreme if the null hypothesis were true. It is not the probability that the null is true, nor that the finding is real.
- The tail must be chosen before seeing the data. Switching to one-tailed afterwards halves the p-value and doubles the false-positive rate.
- Statistical significance is not practical importance. With a large enough sample, a trivial effect reaches any threshold.
- The 0.05 line is a convention adopted from Fisher, not a property of nature.
Sources
Conventional thresholds
| p-value | Often described as | What it means |
|---|---|---|
| p ≥ 0.10 | Not significant | Weak or no evidence against the null |
| 0.05 ≤ p < 0.10 | Marginal | Suggestive, conventionally not significant |
| 0.01 ≤ p < 0.05 | Significant | Below the usual threshold |
| 0.001 ≤ p < 0.01 | Highly significant | Strong evidence against the null |
| p < 0.001 | Very highly significant | Very strong evidence |
The 0.05 line is a convention, adopted from Fisher, not a property of nature. A p of 0.049 and one of 0.051 describe almost identical evidence, and treating them as categorically different is a mistake the threshold encourages.
Frequently asked questions
Does p < 0.05 mean my result is 95% likely to be true?
No. It means data this extreme would occur under 5% of the time if the null hypothesis were true. The probability that a hypothesis is true is a different quantity and requires a Bayesian approach.
When should I use t instead of z?
When the population standard deviation is estimated from a sample, which is the usual case. With more than about 30 degrees of freedom the two give nearly identical p-values.
What does a p-value of exactly 1 mean?
That the observed statistic is exactly what the null hypothesis predicts. It is not evidence the null is true — only that this data gives no evidence against it.
Can I use a one-tailed test to reach significance?
Only if the direction was predicted before collecting the data. Switching afterwards to cross the threshold doubles your false-positive rate.